Finite Group Databasebeta

← #33104494

Subgroups — #33104494

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
1C2×#17988709C211yesmaximal, characteristic
2#22618815C211yesmaximal, characteristic
3#22618529C211yesmaximal, characteristic, complemented
4#22618524C211yesmaximal, characteristic, complemented
5#22610714C211yesmaximal, characteristic, complemented
6#22610711C211yesmaximal, characteristic, complemented
7#22610164C211yesmaximal, characteristic, complemented
8#22610162C211yesmaximal, characteristic, complemented
9#22610157C211yesmaximal, characteristic, complemented
10#22610155C211yesmaximal, characteristic, complemented
11#22610098C211yesmaximal, characteristic, complemented
12#22609993C211yesmaximal, characteristic, complemented
13#22609988C211yesmaximal, characteristic, complemented
14#22609816C211yesmaximal, characteristic, complemented
15C73×#2951C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17C2×#14556598C2211yescharacteristic
18#17989070C2211yescharacteristic
19#17989070C2211yescharacteristic
20#17988709C418yescomplemented
21#17988699—21no—
22#17988694—21no—
23SD16×Dic73—21no—
24SD16×Dic73—21no—
25C2×#14556527C2211yescharacteristic
26C2×#14556526C2211yescharacteristic
27#17988630—21no—
28#17988630—21no—
29#17988630C2211yescharacteristic
30#17988630C2211yescharacteristic
31#17988627—21no—
32#17988627—21no—
33#17988627C2211yescharacteristic
34#17988627C2211yescharacteristic
35#17988623—21no—
36#17988623—21no—
37#17988623C2211yescharacteristic
38C2×#14556524—21no—
39C2×#14556524C2211yescharacteristic
40C2×#14556523C2211yescharacteristic
41C2×#14556522—21no—
42C2×#14556522C2211yescharacteristic
43C2×#14556518C2211yescharacteristic
44C2×#14556518C2211yescharacteristic
45#17988588C2211yescharacteristic
46#17988588C2211yescharacteristic
47#17988542C2211yescharacteristic
48C2×#14556501C2211yescharacteristic
49#17988456C2211yescharacteristic
50#17988452C2211yescharacteristic
51#17988356C2211yescharacteristic
52#17988356C2211yescharacteristic
53#17988355C2211yescharacteristic
54#17988355C2211yescharacteristic
55#17988348C2211yescharacteristic
56#17988346C2211yescharacteristic
57C2×#135×C73C2211yescharacteristic
58#520×C73C2211yescharacteristic
59#520×C73C2211yescharacteristic
60#440×C73—21no—
61#440×C73—21no—
62C4×Q16×C73—21no—
63C4×D8×C73—21no—
64C2×#107×C73C2211yescharacteristic
65#417×C73C2211yescharacteristic
66#417×C73C2211yescharacteristic
67#405×C73C2211yescharacteristic
68#14556598—22no—
69#14556598—22no—
70#14556598C2×C414yescomplemented
71#14556598C2×C414yescomplemented
72C2×#11924012—21no—
73C2×#11924012—21no—
74C2×#11924012C2311yescharacteristic
75C2×#11924012C2311yescharacteristic
76C2×#11924012C2311yescharacteristic
77C22×#10596831—21no—
78C22×#10596831—21no—
79C22×#10596831C2311yescharacteristic
80Q8×Dic73—21no—
81Q8×Dic73—21no—
82D4×Dic73—21no—
83D4×Dic73—21no—
84D4×Dic73—41no—
85D4×Dic73—41no—
86#14556550—21no—
87#14556550—21no—
88#14556550C2311yescharacteristic
89C2×C4⋊Dic73—21no—
90C2×C4⋊Dic73—21no—
91C2×C4⋊Dic73C2311yescharacteristic
92C2×C4⋊Dic73C2311yescharacteristic
93C2×#11924007—41no—
94C2×C4×Dic73C2311yescharacteristic
95#14556527C2×C414yescomplemented
96#14556526—22no—
97#14556524—21no—
98#14556524—21no—
99#14556524—21no—
100#14556524—21no—
101#14556524—21no—
102#14556524—21no—
103#14556524D411yescharacteristic
104#14556524D411yescharacteristic
105#14556523—21no—
106#14556523—21no—
107#14556522—21no—
108#14556522—21no—
109#14556522—21no—
110#14556522—21no—
111#14556522—21no—
112#14556522—21no—
113#14556522D411yescharacteristic
114#14556522D411yescharacteristic
115#14556518C2×C414yescomplemented
116#14556518C2×C414yescomplemented
117#14556515—21no—
118#14556515—21no—
119#14556512—21no—
120#14556512—21no—
121#14556501—21no—
122#14556501C2×C414yescomplemented
123C8×Dic73—21no—
124#14556498—41no—
125C2×#41×C73C2311yescharacteristic
126C2×#41×C73C2311yescharacteristic
127#135×C73C2×C414yes—
128#135×C73Q814yes—
129C2×Q16×C73D411yescharacteristic
130C2×SD16×C73D411yescharacteristic
131C2×SD16×C73D411yescharacteristic
132C2×D8×C73D411yescharacteristic
133C2×M16×C73C2311yescharacteristic
134C22×C8×C73C2311yescharacteristic
135C4×Q8×C73—21no—
136C4×Q8×C73—21no—
137C4×D4×C73—21no—
138C4×D4×C73—21no—
139C4×D4×C73—41no—
140C4×D4×C73—41no—
141#117×C73—21no—
142#117×C73—21no—
143#117×C73C2311yescharacteristic
144C2×C4⋊C4×C73—21no—
145C2×C4⋊C4×C73—21no—
146C2×C4⋊C4×C73C2311yescharacteristic
147C2×C4⋊C4×C73C2311yescharacteristic
148#107×C73—21no—
149#107×C73—21no—
150#103×C73—21no—
151#103×C73—21no—
152#102×C73—21no—
153#102×C73—21no—
154#97×C73—21no—
155C4×C8×C73—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73—41no—
160C22×Dic73—41no—
161#11924012—21no—
162#11924012—21no—
163#11924012—24no—
164#11924012—24no—
165#11924012—21no—
166#11924012—21no—
167#11924012C22×C412yes—
168#11924012C22×C412yes—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—24no—
172C2×#10596831—24no—
173C2×#10596831—21no—
174C2×#10596831C22×C412yes—
175C2×#10596831C2×D411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C2×#10596831#4112yes—
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73—21no—
182C4⋊Dic73—41no—
183C4⋊Dic73—41no—
184C4⋊Dic73C2×D411yescharacteristic
185C4⋊Dic73C2×D411yescharacteristic
186#11924007—82no—
187C22⋊Dic73—21no—
188C22⋊Dic73—41no—
189C22⋊Dic73—41no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—41no—
193C4×Dic73—41no—
194C4×Dic73C22×C412yes—
195#41×C73—21no—
196#41×C73—21no—
197#41×C73—21no—
198#41×C73—21no—
199#41×C73C22×C412yes—
200#41×C73C22×C412yes—
201#41×C73C2×Q812yes—
202#41×C73C2×Q812yes—
203C2×Q8×C73C2×D411yescharacteristic
204C2×Q8×C73C2×D411yescharacteristic
205C2×D4×C73—21no—
206C2×D4×C73—21no—
207C2×D4×C73C2×D411yescharacteristic
208C2×D4×C73C2×D411yescharacteristic
209C22×C4×C73—21no—
210C22×C4×C73—21no—
211C22×C4×C73—21no—
212C22×C4×C73—21no—
213C22×C4×C73—41no—
214C22×C4×C73—41no—
215C22×C4×C73C2411yescharacteristic
216Q16×C73C4⋊C414yescomplemented
217SD16×C73C4⋊C414yescomplemented
218SD16×C73C4⋊C414yescomplemented
219D8×C73C4⋊C414yescomplemented
220M16×C73—21no—
221M16×C73—21no—
222C2×C8×C73—21no—
223C2×C8×C73C22×C412yes—
224C2×C8×C73C2×D411yescharacteristic
225C2×C8×C73C2×D411yescharacteristic
226C2×C8×C73C2×Q812yes—
227C4⋊C4×C73—21no—
228C4⋊C4×C73—21no—
229C4⋊C4×C73—21no—
230C4⋊C4×C73—21no—
231C4⋊C4×C73—41no—
232C4⋊C4×C73—41no—
233C4⋊C4×C73C2×D411yescharacteristic
234C4⋊C4×C73C2×D411yescharacteristic
235C22⋊C4×C73—21no—
236C22⋊C4×C73—41no—
237C22⋊C4×C73—41no—
238C42×C73—21no—
239C42×C73—41no—
240C42×C73—41no—
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—21no—
263C2×C4×C73—21no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73—81no—
267C2×C4×C73—81no—
268C2×C4×C73—161no—
269C2×C4×C73—161no—
270C2×C4×C73C23×C411yescharacteristic
271C2×C4×C73C22×D411yescharacteristic
272C2×C4×C73C22×D411yescharacteristic
273C2×C4×C73C22×Q811yescharacteristic
274C2×C4×C73C2×#4111yescharacteristic
275C2×C4×C732−1+411yescharacteristic
276C2×Dic73—22no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—81no—
284C2×Dic73—81no—
285C2×Dic73—161no—
286C2×Dic73—161no—
287C8×C73—41no—
288C8×C73C2×C4⋊C412yes—
289C8×C73C2×C4⋊C412yes—
290#10596831—22no—
291#10596831—22no—
292#10596831—41no—
293#10596831C4×D412yes—
294#10596831C4×D412yes—
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—41no—
309C4×C73—41no—
310C4×C73—81no—
311C4×C73—81no—
312C4×C73C22×C4⋊C411yescharacteristic
313C4×C73C2×C4×D411yescharacteristic
314C4×C73#54711yescharacteristic
315C4×C73D4×Q811yescharacteristic
316Dic73—42no—
317Dic73—41no—
318Dic73—41no—
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×#444—731no—
330#2952—731no—
331#2951D7311yescharacteristic
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×#131—731no—
347C2×#131—731no—
348#520—731no—
349#520—731no—
350#520C2×D7311yescharacteristic
351#520C2×D7311yescharacteristic
352#444—738no—
353#442—1461no—
354#441—1461no—
355#440—21no—
356#440—21no—
357C4×Q16—21no—
358C4×SD16—1461no—
359C4×SD16—1461no—
360C4×D8—21no—
361C2×#108—731no—
362C2×#108—731no—
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—1461no—
376C2×#107C2×D7311yescharacteristic
377C2×#106—731no—
378C2×#106—731no—
379C2×#106—1461no—
380C2×#104—731no—
381C2×#104—731no—
382#417—731no—
383#417—731no—
384#417C2×D7311yescharacteristic
385#417C2×D7311yescharacteristic
386#405—731no—
387#405C2×D7311yescharacteristic
388C2×#97—731no—
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#135C4×D7314yes—
401C2×Q16C22⋊D7311yescharacteristic
402C2×SD16C22⋊D7311yescharacteristic
403C2×SD16C22⋊D7311yescharacteristic
404C2×D8C22⋊D7311yescharacteristic
405#131—734no—
406#131—734no—
407#131—1462no—
408#131—1462no—
409C2×M16—731no—
410C2×M16—731no—
411C2×M16—731no—
412C2×M16—1461no—
413C2×M16—1461no—
414C2×M16C22×D7311yescharacteristic
415C22×C8—731no—
416C22×C8—1461no—
417C22×C8—1461no—
418C22×C8C22×D7311yescharacteristic
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—1461no—
422C4×Q8—1461no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—41no—
426C4×D4—41no—
427C4×D4—2921no—
428C4×D4—2921no—
429C4×D4—1461no—
430C4×D4—1461no—
431#117—21no—
432#117—21no—
433#117—731no—
434#117—1461no—
435#117—1461no—
436#117C22×D7311yescharacteristic
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—2921no—
440C2×C4⋊C4—731no—
441C2×C4⋊C4—731no—
442C2×C4⋊C4—1461no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4C22×D7311yescharacteristic
445C2×C4⋊C4C22×D7311yescharacteristic
446C2×C42—731no—
447#108—734no—
448#108—1462no—
449#107—21no—
450#107—21no—
451#107—731no—
452#107—731no—
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#104—734no—
470#104—734no—
471#103—21no—
472#103—21no—
473#103—1461no—
474#103—1461no—
475#102—21no—
476#102—21no—
477#102—1461no—
478#102—1461no—
479#97—21no—
480#97—734no—
481#97—1461no—
482C4×C8—21no—
483C4×C8—1461no—
484#95—2921no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—21no—
489#41C2×C4×D7312yes—
490#41C2×C4×D7312yes—
491#41C2×Dic14612yes—
492#41C2×Dic14612yes—
493C2×Q8C2×C22⋊D7311yescharacteristic
494C2×Q8C2×C22⋊D7311yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4C2×C22⋊D7311yescharacteristic
498C2×D4C2×C22⋊D7311yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—2921no—
506C22×C4—2921no—
507C22×C4—1461no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4C23×D7311yescharacteristic
511Q16#1192400714yescomplemented
512SD16#1192400714yescomplemented
513SD16#1192400714yescomplemented
514D8#1192400714yescomplemented
515M16—21no—
516M16—21no—
517M16—732no—
518M16—732no—
519M16—1461no—
520M16—1461no—
521M16—1464no—
522M16—1464no—
523M16—1461no—
524M16—1461no—
525C2×C8—21no—
526C2×C8—732no—
527C2×C8—731no—
528C2×C8—731no—
529C2×C8—732no—
530C2×C8—1461no—
531C2×C8—1464no—
532C2×C8—1464no—
533C2×C8—1461no—
534C2×C8—1461no—
535C2×C8C2×C4×D7312yes—
536C2×C8C2×C22⋊D7311yescharacteristic
537C2×C8C2×C22⋊D7311yescharacteristic
538C2×C8C2×Dic14612yes—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—2921no—
546C4⋊C4—2921no—
547C4⋊C4—5842no—
548C4⋊C4—731no—
549C4⋊C4—731no—
550C4⋊C4—1461no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C4⋊C4—1461no—
554C4⋊C4D4×D7311yescharacteristic
555C4⋊C4D4×D7311yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—2921no—
560C22⋊C4—2921no—
561C22⋊C4—1461no—
562C42—21no—
563C42—41no—
564C42—41no—
565C42—2921no—
566C42—2921no—
567C42—732no—
568C42—1461no—
569C42—1461no—
570C23—21no—
571C23—21no—
572C23#1455663611yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8C2×#1192400712yes—
576Q8C2×#1192400712yes—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—21no—
582D4—21no—
583D4C2×#1192400712yes—
584D4C2×#1192400712yes—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—161no—
598C2×C4—161no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—5841no—
604C2×C4—5841no—
605C2×C4—11681no—
606C2×C4—11681no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1462no—
610C2×C4C22×C4×D7311yescharacteristic
611C2×C4C2×D4×D7311yescharacteristic
612C2×C4C22×C22⋊D7311yescharacteristic
613C2×C4C22×Dic14611yescharacteristic
614C2×C4#41×D7311yescharacteristic
615C2×C4#1455663811yescharacteristic
616C8—41no—
617C8—2921no—
618C8—732no—
619C8—732no—
620C8—1462no—
621C8—1462no—
622C8C2×#1192400712yes—
623C8C2×#1192400712yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22#1798906611yescharacteristic
633C22D4×C22⋊D7311yescharacteristic
634C22#1798929611yescharacteristic
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—2922no—
642C4—2921no—
643C4—2921no—
644C4—5841no—
645C4—5841no—
646C4C22×#1192400711yescharacteristic
647C4C4×D4×D7311yescharacteristic
648C4#1798923611yescharacteristic
649C4D4×Dic14611yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—42no—
654C2D4×#1192400711yescharacteristic
655C2#2262169511yescharacteristic
656C2#2262170211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2951C2
socleC22×C73D42
prime radicalC2×#135×C73C22
cosocleC22×C4×C73C24
hypercenter#2951D73
hypocenterC73#22693
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2951

Complements

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

NH ≅ G/Ntype
3. #22618529C2semidirect
4. #22618524C2semidirect
5. #22610714C2semidirect
6. #22610711C2semidirect
7. #22610164C2semidirect
8. #22610162C2semidirect
9. #22610157C2semidirect
10. #22610155C2semidirect
11. #22610098C2semidirect
12. #22609993C2semidirect
13. #22609988C2semidirect
14. #22609816C2semidirect
20. #17988709C4semidirect
70. #14556598C2×C4semidirect
71. #14556598C2×C4semidirect
95. #14556527C2×C4semidirect
115. #14556518C2×C4semidirect
116. #14556518C2×C4semidirect
122. #14556501C2×C4semidirect
216. Q16×C73C4⋊C4semidirect
217. SD16×C73C4⋊C4semidirect
218. SD16×C73C4⋊C4semidirect
219. D8×C73C4⋊C4semidirect
344. C73#22693semidirect
511. Q16#11924007semidirect
512. SD16#11924007semidirect
513. SD16#11924007semidirect
514. D8#11924007semidirect

Maximal subgroups

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

subgroupindex
C2×#179887092
#226188152
#226185292
#226185242
#226107142
#226107112
#226101642
#226101622
#226101572
#226101552
#226100982
#226099932
#226099882
#226098162
C73×#29512
#2269373