Finite Group Databasebeta

← #33104492

Subgroups — #33104492

Order 18688.

850 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618815C211yesmaximal, characteristic
2#22618807C211yesmaximal, characteristic
3#22618531C211yesmaximal, characteristic, complemented
4#22618526C211yesmaximal, characteristic, complemented
5#22610717C211yesmaximal, characteristic, complemented
6#22610712C211yesmaximal, characteristic, complemented
7#22610166C211yesmaximal, characteristic, complemented
8#22610166C211yesmaximal, characteristic, complemented
9#22610159C211yesmaximal, characteristic, complemented
10#22610159C211yesmaximal, characteristic, complemented
11#22610099C211yesmaximal, characteristic, complemented
12#22609995C211yesmaximal, characteristic, complemented
13#22609990C211yesmaximal, characteristic, complemented
14#22609815C211yesmaximal, characteristic, complemented
15C2×#444×C73C211yesmaximal, characteristic
16#17989070C2211yescharacteristic
17#17989070C2211yescharacteristic
18#17989070C2211yescharacteristic
19#17989070C2211yescharacteristic
20#17988699—21no—
21#17988694—21no—
22#17988688—21no—
23#17988688—21no—
24Q16×Dic73—21no—
25SD16×Dic73—21no—
26SD16×Dic73—21no—
27D8×Dic73—21no—
28C2×#14556527C2211yescharacteristic
29#17988629—21no—
30#17988629—21no—
31#17988629—21no—
32#17988629—21no—
33#17988629C2211yescharacteristic
34#17988629C2211yescharacteristic
35#17988629C2211yescharacteristic
36#17988629C2211yescharacteristic
37#17988625—21no—
38#17988625—21no—
39#17988625C2211yescharacteristic
40C2×#14556525—21no—
41C2×#14556525C2211yescharacteristic
42C2×#14556525C2211yescharacteristic
43C2×#14556523—21no—
44C2×#14556523C2211yescharacteristic
45C2×#14556523C2211yescharacteristic
46#17988588C2211yescharacteristic
47#17988588C2211yescharacteristic
48#17988588C2211yescharacteristic
49#17988588C2211yescharacteristic
50#17988542C2211yescharacteristic
51#17988542C2211yescharacteristic
52#17988457C2211yescharacteristic
53#17988453C2211yescharacteristic
54#17988355C2211yescharacteristic
55#17988355C2211yescharacteristic
56#17988355C2211yescharacteristic
57#17988355C2211yescharacteristic
58#17988347C2211yescharacteristic
59#17988347C2211yescharacteristic
60C2×#135×C73C2211yescharacteristic
61C2×#131×C73C2211yescharacteristic
62C2×#131×C73C2211yescharacteristic
63#444×C73C418yescomplemented
64C2×#108×C73C2211yescharacteristic
65C2×#104×C73C2211yescharacteristic
66C2×#104×C73C2211yescharacteristic
67C2×#97×C73C2211yescharacteristic
68C2×#11924012C2311yescharacteristic
69C2×#11924012C2311yescharacteristic
70Q8×Dic73—21no—
71Q8×Dic73—21no—
72Q8×Dic73—21no—
73Q8×Dic73—21no—
74D4×Dic73—21no—
75D4×Dic73—21no—
76D4×Dic73—21no—
77D4×Dic73—21no—
78D4×Dic73—41no—
79D4×Dic73—41no—
80D4×Dic73—41no—
81D4×Dic73—41no—
82#14556550—21no—
83#14556550—21no—
84#14556550—21no—
85#14556550—21no—
86#14556550C2311yescharacteristic
87#14556550C2311yescharacteristic
88C2×C4⋊Dic73—21no—
89C2×C4⋊Dic73—21no—
90C2×C4⋊Dic73—21no—
91C2×C4⋊Dic73—21no—
92C2×C4⋊Dic73—41no—
93C2×C4⋊Dic73C2311yescharacteristic
94C2×C4⋊Dic73C2311yescharacteristic
95C2×C4⋊Dic73C2311yescharacteristic
96C2×C4⋊Dic73C2311yescharacteristic
97#14556527—22no—
98#14556525—21no—
99#14556525—21no—
100#14556525—21no—
101#14556525—21no—
102#14556525—21no—
103#14556525—21no—
104#14556525—21no—
105#14556525—21no—
106#14556525D411yescharacteristic
107#14556525D411yescharacteristic
108#14556523—21no—
109#14556523—21no—
110#14556523—21no—
111#14556523—21no—
112#14556523—21no—
113#14556523—21no—
114#14556523—21no—
115#14556523—21no—
116#14556523D411yescharacteristic
117#14556523D411yescharacteristic
118#14556515—21no—
119#14556515—21no—
120#14556515—21no—
121#14556515—21no—
122#14556512—21no—
123#14556512—21no—
124#14556512—21no—
125#14556512—21no—
126#14556501—21no—
127#14556501—21no—
128C8×Dic73—21no—
129C8×Dic73—21no—
130#14556498—41no—
131C2×#41×C73C2311yescharacteristic
132C2×#41×C73C2311yescharacteristic
133#135×C73C2×C414yes—
134#135×C73Q814yes—
135C2×Q16×C73D411yescharacteristic
136C2×SD16×C73D411yescharacteristic
137C2×SD16×C73D411yescharacteristic
138C2×D8×C73D411yescharacteristic
139#131×C73—22no—
140#131×C73—22no—
141#131×C73C2×C414yescomplemented
142#131×C73C2×C414yescomplemented
143C2×M16×C73—21no—
144C2×M16×C73—21no—
145C2×M16×C73C2311yescharacteristic
146C2×M16×C73C2311yescharacteristic
147C22×C8×C73—21no—
148C22×C8×C73—21no—
149C22×C8×C73C2311yescharacteristic
150C22×C8×C73C2311yescharacteristic
151C2×C42×C73C2311yescharacteristic
152#108×C73C2×C414yescomplemented
153#104×C73C2×C414yescomplemented
154#104×C73C2×C414yescomplemented
155#97×C73C2×C414yescomplemented
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73—21no—
160C22×Dic73—41no—
161C22×Dic73—41no—
162C22×Dic73—41no—
163C22×Dic73—41no—
164#11924012—21no—
165#11924012—21no—
166#11924012—21no—
167#11924012—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73—21no—
178C4⋊Dic73—41no—
179C4⋊Dic73—41no—
180C4⋊Dic73—41no—
181C4⋊Dic73—41no—
182C4⋊Dic73—82no—
183C4⋊Dic73C2×D411yescharacteristic
184C4⋊Dic73C2×D411yescharacteristic
185C4⋊Dic73C2×D411yescharacteristic
186C4⋊Dic73C2×D411yescharacteristic
187C22⋊Dic73—21no—
188C22⋊Dic73—21no—
189C22⋊Dic73—41no—
190C22⋊Dic73—41no—
191C22⋊Dic73—41no—
192C22⋊Dic73—41no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—41no—
196C4×Dic73—41no—
197C4×Dic73—41no—
198C4×Dic73—41no—
199#41×C73—21no—
200#41×C73—21no—
201#41×C73—21no—
202#41×C73—21no—
203#41×C73C22×C412yes—
204#41×C73C22×C412yes—
205#41×C73C2×Q812yes—
206#41×C73C2×Q812yes—
207C2×Q8×C73C2×D411yescharacteristic
208C2×Q8×C73C2×D411yescharacteristic
209C2×D4×C73—21no—
210C2×D4×C73—21no—
211C2×D4×C73C2×D411yescharacteristic
212C2×D4×C73C2×D411yescharacteristic
213C22×C4×C73—21no—
214C22×C4×C73—21no—
215C22×C4×C73—21no—
216C22×C4×C73C2411yescharacteristic
217Q16×C73C4⋊C414yescomplemented
218SD16×C73C4⋊C414yescomplemented
219SD16×C73C4⋊C414yescomplemented
220D8×C73C4⋊C414yescomplemented
221M16×C73—21no—
222M16×C73—21no—
223M16×C73—24no—
224M16×C73—24no—
225M16×C73C22×C412yes—
226M16×C73C22×C412yes—
227C2×C8×C73—21no—
228C2×C8×C73—21no—
229C2×C8×C73—24no—
230C2×C8×C73—24no—
231C2×C8×C73C22×C412yes—
232C2×C8×C73C22×C412yes—
233C2×C8×C73C2×D411yescharacteristic
234C2×C8×C73C2×D411yescharacteristic
235C2×C8×C73C2×D411yescharacteristic
236C2×C8×C73C2×D411yescharacteristic
237C2×C8×C73C2×Q812yes—
238C2×C8×C73#4112yes—
239C42×C73—21no—
240C42×C73C22×C412yes—
241C23×C73—21no—
242C23×C73—21no—
243C23×C73H(5,𝔽2)11yescharacteristic
244Q8×C73—21no—
245Q8×C73—21no—
246Q8×C73C2×C4⋊C412yes—
247Q8×C73C2×C4⋊C412yes—
248D4×C73—22no—
249D4×C73—22no—
250D4×C73—22no—
251D4×C73—22no—
252D4×C73—21no—
253D4×C73—21no—
254D4×C73C2×C4⋊C412yes—
255D4×C73C2×C4⋊C412yes—
256C2×C4×C73—21no—
257C2×C4×C73—21no—
258C2×C4×C73—22no—
259C2×C4×C73—22no—
260C2×C4×C73—22no—
261C2×C4×C73—22no—
262C2×C4×C73—22no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73C23×C411yescharacteristic
266C2×C4×C73C22×D411yescharacteristic
267C2×C4×C73C22×D411yescharacteristic
268C2×C4×C73C22×Q811yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C732−1+411yescharacteristic
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—21no—
274C2×Dic73—21no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—81no—
280C2×Dic73—81no—
281C2×Dic73—81no—
282C2×Dic73—81no—
283C2×Dic73—161no—
284C2×Dic73—161no—
285C2×Dic73—161no—
286C2×Dic73—161no—
287C8×C73—22no—
288C8×C73—22no—
289C8×C73C2×C4⋊C412yes—
290C8×C73C2×C4⋊C412yes—
291C8×C73C4×D412yes—
292C8×C73C4×D412yes—
293#10596831—41no—
294#10596831—41no—
295C22×C73—21no—
296C22×C73—21no—
297C22×C73—22no—
298C22×C73—22no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73—41no—
303C22×C73#52011yescharacteristic
304C22×C73D4211yescharacteristic
305C22×C73#55411yescharacteristic
306C4×C73—22no—
307C4×C73—22no—
308C4×C73—42no—
309C4×C73C22×C4⋊C411yescharacteristic
310C4×C73C2×C4×D411yescharacteristic
311C4×C73#54711yescharacteristic
312C4×C73D4×Q811yescharacteristic
313Dic73—41no—
314Dic73—41no—
315Dic73—41no—
316Dic73—41no—
317Dic73—81no—
318Dic73—81no—
319Dic73—81no—
320Dic73—81no—
321#22693—731nomaximal, Sylow
322C2×C73—21no—
323C2×C73—21no—
324C2×C73—42no—
325C2×C73—42no—
326C2×C73D4×C4⋊C411yescharacteristic
327C2×C73#328811yescharacteristic
328C2×C73#328911yescharacteristic
329C2×#444D7311yescharacteristic
330#2952—731no—
331#2951—731no—
332#2913—731no—
333#2912—731no—
334#1952—731no—
335#1951—731no—
336#1860—731no—
337#1860—731no—
338#1859—731no—
339#1859—731no—
340#1847—731no—
341#1816—731no—
342#1815—731no—
343#1773—731no—
344C73#2269311yescharacteristic, complemented, Sylow
345C2×#135C2×D7311yescharacteristic
346C2×#131C2×D7311yescharacteristic
347C2×#131C2×D7311yescharacteristic
348#520—731no—
349#520—731no—
350#520—731no—
351#520—731no—
352#444Dic7318yescomplemented
353#442—1461no—
354#441—1461no—
355#440—1461no—
356#440—1461no—
357C4×Q16—1461no—
358C4×SD16—1461no—
359C4×SD16—1461no—
360C4×D8—1461no—
361C2×#108—731no—
362C2×#108C2×D7311yescharacteristic
363#428—731no—
364#428—731no—
365#428—731no—
366#428—731no—
367#428—1461no—
368#428—1461no—
369#428—1461no—
370#428—1461no—
371#427—731no—
372#427—1461no—
373#427—1461no—
374C2×#107—731no—
375C2×#107—731no—
376C2×#107—1461no—
377C2×#106—731no—
378C2×#106—731no—
379C2×#106—1461no—
380C2×#104C2×D7311yescharacteristic
381C2×#104C2×D7311yescharacteristic
382#417—731no—
383#417—731no—
384#417—731no—
385#417—731no—
386#405—731no—
387#405—731no—
388C2×#97C2×D7311yescharacteristic
389#384—731no—
390#382—731no—
391#343—731no—
392#343—731no—
393#343—731no—
394#343—731no—
395#339—731no—
396#339—731no—
397C2×#41C22×D7311yescharacteristic
398C2×#41C22×D7311yescharacteristic
399#135Dic14614yes—
400#135C2×Dic7314yes—
401C2×Q16D29211yescharacteristic
402C2×SD16D29211yescharacteristic
403C2×SD16D29211yescharacteristic
404C2×D8D29211yescharacteristic
405#131—22no—
406#131—22no—
407#131C2×Dic7314yescomplemented
408#131C2×Dic7314yescomplemented
409C2×M16—21no—
410C2×M16—21no—
411C2×M16—731no—
412C2×M16—731no—
413C2×M16C22×D7311yescharacteristic
414C2×M16C22×D7311yescharacteristic
415C22×C8—21no—
416C22×C8—21no—
417C22×C8C22×D7311yescharacteristic
418C22×C8C22×D7311yescharacteristic
419C4×Q8—1461no—
420C4×Q8—1461no—
421C4×Q8—1461no—
422C4×Q8—1461no—
423C4×D4—2921no—
424C4×D4—2921no—
425C4×D4—2921no—
426C4×D4—2921no—
427C4×D4—1461no—
428C4×D4—1461no—
429C4×D4—1461no—
430C4×D4—1461no—
431#117—731no—
432#117—731no—
433#117—1461no—
434#117—1461no—
435#117—1461no—
436#117—1461no—
437C2×C4⋊C4—2921no—
438C2×C4⋊C4—731no—
439C2×C4⋊C4—731no—
440C2×C4⋊C4—731no—
441C2×C4⋊C4—731no—
442C2×C4⋊C4—1461no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4—1461no—
445C2×C4⋊C4—1461no—
446C2×C42C22×D7311yescharacteristic
447#108—1462no—
448#108C2×Dic7314yescomplemented
449#107—731no—
450#107—731no—
451#107—1461no—
452#107—1461no—
453#107—1461no—
454#107—1461no—
455#107—1461no—
456#107—1461no—
457#107—1461no—
458#107—1461no—
459#106—731no—
460#106—731no—
461#106—1461no—
462#106—1461no—
463#106—1461no—
464#106—1461no—
465#106—1461no—
466#106—1461no—
467#106—1461no—
468#106—1461no—
469#104C2×Dic7314yescomplemented
470#104C2×Dic7314yescomplemented
471#103—1461no—
472#103—1461no—
473#103—1461no—
474#103—1461no—
475#102—1461no—
476#102—1461no—
477#102—1461no—
478#102—1461no—
479#97—1461no—
480#97—1461no—
481#97C2×Dic7314yescomplemented
482C4×C8—1461no—
483C4×C8—1461no—
484#95—2921no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—21no—
489#41C22×Dic7312yes—
490#41C22×Dic7312yes—
491#41C2×Dic14612yes—
492#41C2×Dic14612yes—
493C2×Q8C2×D29211yescharacteristic
494C2×Q8C2×D29211yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4C2×D29211yescharacteristic
498C2×D4C2×D29211yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—2921no—
503C22×C4—2921no—
504C22×C4—2921no—
505C22×C4—2921no—
506C22×C4—1461no—
507C22×C4—1461no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4C23×D7311yescharacteristic
511Q16C4⋊Dic7314yescomplemented
512SD16C4⋊Dic7314yescomplemented
513SD16C4⋊Dic7314yescomplemented
514D8C4⋊Dic7314yescomplemented
515M16—21no—
516M16—21no—
517M16—24no—
518M16—24no—
519M16—1461no—
520M16—1461no—
521M16—1461no—
522M16—1461no—
523M16C22×Dic7312yes—
524M16C22×Dic7312yes—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—1461no—
530C2×C8—1461no—
531C2×C8C22×Dic7312yes—
532C2×C8C22×Dic7312yes—
533C2×C8D4×D7311yescharacteristic
534C2×C8D4×D7311yescharacteristic
535C2×C8C2×D29211yescharacteristic
536C2×C8C2×D29211yescharacteristic
537C2×C8C2×Dic14612yes—
538C2×C8#1192402712yes—
539C4⋊C4—2921no—
540C4⋊C4—2921no—
541C4⋊C4—2921no—
542C4⋊C4—2921no—
543C4⋊C4—5842no—
544C4⋊C4—731no—
545C4⋊C4—731no—
546C4⋊C4—731no—
547C4⋊C4—731no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4—1461no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C4⋊C4—1461no—
554C4⋊C4—1461no—
555C4⋊C4—1461no—
556C22⋊C4—2921no—
557C22⋊C4—2921no—
558C22⋊C4—2921no—
559C22⋊C4—2921no—
560C22⋊C4—1461no—
561C22⋊C4—1461no—
562C42—21no—
563C42—2921no—
564C42—2921no—
565C42—2921no—
566C42—2921no—
567C42—1461no—
568C42—1461no—
569C42C22×Dic7312yes—
570C23—21no—
571C23—21no—
572C23#1455663711yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8C2×C4⋊Dic7312yes—
576Q8C2×C4⋊Dic7312yes—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—21no—
582D4—21no—
583D4C2×C4⋊Dic7312yes—
584D4C2×C4⋊Dic7312yes—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—5841no—
599C2×C4—5841no—
600C2×C4—5841no—
601C2×C4—5841no—
602C2×C4—11681no—
603C2×C4—11681no—
604C2×C4—11681no—
605C2×C4—11681no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4C23×Dic7311yescharacteristic
611C2×C4C2×D4×D7311yescharacteristic
612C2×C4C22×D29211yescharacteristic
613C2×C4C22×Dic14611yescharacteristic
614C2×C4C2×#1192402711yescharacteristic
615C2×C4#1455663811yescharacteristic
616C8—22no—
617C8—22no—
618C8—2921no—
619C8—2921no—
620C8C2×C4⋊Dic7312yes—
621C8C2×C4⋊Dic7312yes—
622C8D4×Dic7312yes—
623C8D4×Dic7312yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22#1798907011yescharacteristic
633C22D4×D29211yescharacteristic
634C22#1798929411yescharacteristic
635C4—22no—
636C4—22no—
637C4—42no—
638C4—2921no—
639C4—2921no—
640C4—2921no—
641C4—2921no—
642C4—5841no—
643C4—5841no—
644C4—5841no—
645C4—5841no—
646C4C22×C4⋊Dic7311yescharacteristic
647C4C2×D4×Dic7311yescharacteristic
648C4#1798924111yescharacteristic
649C4D4×Dic14611yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—42no—
654C2D4×C4⋊Dic7311yescharacteristic
655C2#2262169811yescharacteristic
656C2#2262170711yescharacteristic
subgroupG/H
centerC22D4×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#444×C73C2
socleC22×C73D42
prime radicalC2×#135×C73C22
cosocleC22×C4×C73C24
hypercenterC2×#444D73
hypocenterC73#22693
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#444

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22618531C2semidirect
4. #22618526C2semidirect
5. #22610717C2semidirect
6. #22610712C2semidirect
7. #22610166C2semidirect
8. #22610166C2semidirect
9. #22610159C2semidirect
10. #22610159C2semidirect
11. #22610099C2semidirect
12. #22609995C2semidirect
13. #22609990C2semidirect
14. #22609815C2semidirect
63. #444×C73C4semidirect
141. #131×C73C2×C4semidirect
142. #131×C73C2×C4semidirect
152. #108×C73C2×C4semidirect
153. #104×C73C2×C4semidirect
154. #104×C73C2×C4semidirect
155. #97×C73C2×C4semidirect
217. Q16×C73C4⋊C4semidirect
218. SD16×C73C4⋊C4semidirect
219. SD16×C73C4⋊C4semidirect
220. D8×C73C4⋊C4semidirect
344. C73#22693semidirect
352. #444Dic73semidirect
407. #131C2×Dic73semidirect
408. #131C2×Dic73semidirect
448. #108C2×Dic73semidirect
469. #104C2×Dic73semidirect
470. #104C2×Dic73semidirect
481. #97C2×Dic73semidirect
511. Q16C4⋊Dic73semidirect
512. SD16C4⋊Dic73semidirect
513. SD16C4⋊Dic73semidirect
514. D8C4⋊Dic73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226188152
#226188072
#226185312
#226185262
#226107172
#226107122
#226101662
#226101592
#226100992
#226099952
#226099902
#226098152
C2×#444×C732
#2269373