Finite Group Databasebeta

← #33093829

Subgroups — #33093829

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619719C211yesmaximal, characteristic
2#22618868C211yesmaximal, characteristic, complemented
3#22618359C211yesmaximal, characteristic
4#22610874C211yesmaximal, characteristic, complemented
5#22610704C211yesmaximal, characteristic, complemented
6#22610528C211yesmaximal, characteristic, complemented
7#22610528C211yesmaximal, characteristic, complemented
8#22610204C211yesmaximal, characteristic
9#22610204C211yesmaximal, characteristic
10#22609814C211yesmaximal, characteristic, complemented
11#22609809C211yesmaximal, characteristic, complemented
12#22609806C211yesmaximal, characteristic, complemented
13#22609776C211yesmaximal, characteristic
14#22609769C211yesmaximal, characteristic
15C73×#1806C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17C2×#14556598C2211yescharacteristic
18#17989144C2211yescharacteristic
19#17988894C411yescharacteristic, complemented
20#17988894C411yescharacteristic, complemented
21#17988887C411yescharacteristic, complemented
22#17988887C411yescharacteristic, complemented
23#17988882C411yescharacteristic, complemented
24#17988877C411yescharacteristic, complemented
25#17988872C411yescharacteristic, complemented
26#17988867C411yescharacteristic, complemented
27#17988716—21no—
28#17988716—21no—
29#17988716—21no—
30#17988716—21no—
31C2×#14556526C2211yescharacteristic
32C2×#14556526C2211yescharacteristic
33#17988623C2211yescharacteristic
34#17988623C2211yescharacteristic
35#17988615C2211yescharacteristic
36#17988615C2211yescharacteristic
37C2×C4⋊#10596831C2211yescharacteristic
38#17988583C2211yescharacteristic
39#17988583C2211yescharacteristic
40#17988545C2211yescharacteristic
41#17988545C2211yescharacteristic
42#17988543—21no—
43#17988543—21no—
44#17988543C2211yescharacteristic
45#17988543C2211yescharacteristic
46C4×#11924012—21no—
47C4×#11924012C2211yescharacteristic
48C2×#14556502—21no—
49C2×#14556502C2211yescharacteristic
50C2×#14556502C2211yescharacteristic
51C2×C4×#10596831C2211yescharacteristic
52#17988351C2211yescharacteristic
53#17988351C2211yescharacteristic
54#17988347C2211yescharacteristic
55#17988347C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988347C2211yescharacteristic
58#17988342C2211yescharacteristic
59#17988342C2211yescharacteristic
60C4×#41×C73C2211yescharacteristic
61#423×C73C2211yescharacteristic
62C2×#104×C73C2211yescharacteristic
63C2×#104×C73C2211yescharacteristic
64C4×C4⋊C4×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#339×C73C2211yescharacteristic
67#14556598—22no—
68#14556598—22no—
69#14556598—22no—
70#14556598Q814yescomplemented
71C2×#11924012—21no—
72C2×#11924012—21no—
73C2×#11924012C2311yescharacteristic
74C2×#11924012C2311yescharacteristic
75C22×#10596831—21no—
76C22×#10596831—21no—
77C22×#10596831C2311yescharacteristic
78C22×#10596831C2311yescharacteristic
79C22×#10596831C2311yescharacteristic
80C22×#10596831C2311yescharacteristic
81#14556591C2×C411yescharacteristic
82#14556591C2×C411yescharacteristic
83#14556583C2×C411yescharacteristic
84#14556583C2×C411yescharacteristic
85#14556570—21no—
86#14556570—21no—
87C4×Dic146—21no—
88C4×Dic146—21no—
89#14556550C2311yescharacteristic
90#14556550C2311yescharacteristic
91#14556526—22no—
92#14556526—22no—
93#14556524C2×C411yescharacteristic
94#14556524C2×C411yescharacteristic
95#14556524C2×C411yescharacteristic
96#14556524C2×C411yescharacteristic
97#14556522C2×C411yescharacteristic
98#14556522C2×C411yescharacteristic
99#14556522C2×C411yescharacteristic
100#14556522C2×C411yescharacteristic
101C4⋊#10596831C2×C411yescharacteristic
102C4⋊#10596831C2×C411yescharacteristic
103C4⋊#10596831C2×C411yescharacteristic
104C4⋊#10596831C2×C411yescharacteristic
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556515C2×C411yescharacteristic
110#14556515C2×C411yescharacteristic
111#14556515C2×C411yescharacteristic
112#14556515C2×C411yescharacteristic
113#14556512C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115#14556512C2×C411yescharacteristic
116#14556512C2×C411yescharacteristic
117#14556502—21no—
118#14556502—21no—
119#14556502—22no—
120#14556502—21no—
121#14556502D411yescharacteristic
122#14556502D411yescharacteristic
123#14556502D411yescharacteristic
124#14556502D411yescharacteristic
125#14556501—21no—
126#14556501—21no—
127C4×#10596831—21no—
128C4×#10596831—21no—
129C4×#10596831—22no—
130C4×#10596831—21no—
131C8×Dic73—21no—
132C8×Dic73—21no—
133C2×#41×C73C2311yescharacteristic
134C2×M16×C73C2311yescharacteristic
135C2×M16×C73C2311yescharacteristic
136C4×Q8×C73C2×C411yescharacteristic
137C4×Q8×C73C2×C411yescharacteristic
138C4×D4×C73—21no—
139C4×D4×C73—21no—
140C4×D4×C73C2×C411yescharacteristic
141C4×D4×C73C2×C411yescharacteristic
142#117×C73—21no—
143#117×C73C2311yescharacteristic
144C2×C4⋊C4×C73—21no—
145C2×C42×C73—21no—
146C2×C42×C73C2311yescharacteristic
147C2×C42×C73C2311yescharacteristic
148C2×C42×C73C2311yescharacteristic
149C4⋊C8×C73—21no—
150C4⋊C8×C73—21no—
151#104×C73—22no—
152#104×C73—22no—
153#95×C73—21no—
154C2×Dic146—21no—
155C2×Dic146—21no—
156#11924012—24no—
157#11924012—21no—
158#11924012—21no—
159#11924012—24no—
160#11924012C2×Q812yes—
161#11924012#4112yes—
162C2×#10596831—24no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—24no—
172C2×#10596831C22×C411yescharacteristic
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C2×#10596831C22×C411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C2×#10596831C2×D411yescharacteristic
178C2×#10596831C2×D411yescharacteristic
179C2×#10596831C2×D411yescharacteristic
180C2×#10596831C2×Q812yes—
181C2×#10596831#4112yes—
182C4⋊Dic73—21no—
183C4⋊Dic73—21no—
184C4⋊Dic73C22×C411yescharacteristic
185C4⋊Dic73C22×C411yescharacteristic
186C4⋊Dic73C22×C411yescharacteristic
187C4⋊Dic73C22×C411yescharacteristic
188#11924007—21no—
189#11924007—21no—
190#11924007—21no—
191#11924007—21no—
192C22⋊Dic73—21no—
193C22⋊Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196#41×C73—21no—
197#41×C73—21no—
198#41×C73C2×Q812yes—
199#41×C73#4112yes—
200C2×Q8×C73C22×C411yescharacteristic
201C2×D4×C73—21no—
202C2×D4×C73C22×C411yescharacteristic
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207C22×C4×C73—21no—
208C22×C4×C73#4111yescharacteristic
209C22×C4×C73#4111yescharacteristic
210C22×C4×C73C2411yescharacteristic
211M16×C73—21no—
212M16×C73—21no—
213M16×C73—21no—
214M16×C73—21no—
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C4⋊C4×C73—21no—
218C4⋊C4×C73—21no—
219C4⋊C4×C73—42no—
220C4⋊C4×C73C22×C411yescharacteristic
221C4⋊C4×C73C22×C411yescharacteristic
222C22⋊C4×C73—21no—
223C22⋊C4×C73—21no—
224C22⋊C4×C73—21no—
225C42×C73—21no—
226C42×C73—21no—
227C42×C73—21no—
228C42×C73—21no—
229C42×C73—21no—
230C42×C73—21no—
231C42×C73—21no—
232C42×C73C22×C411yescharacteristic
233C42×C73C22×C411yescharacteristic
234C42×C73C2×D411yescharacteristic
235C42×C73C2×D411yescharacteristic
236C23×C73—21no—
237C23×C73C2×#4111yescharacteristic
238Q8×C73—21no—
239Q8×C73C4×Q812yes—
240D4×C73—22no—
241D4×C73—22no—
242D4×C73—21no—
243D4×C73C4×Q812yes—
244C2×C4×C73—21no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—81no—
258C2×C4×C73—81no—
259C2×C4×C73C4×D411yescharacteristic
260C2×C4×C73C4×D411yescharacteristic
261C2×C4×C73C4×D411yescharacteristic
262C2×C4×C73C4×D411yescharacteristic
263C2×C4×C73C23×C411yescharacteristic
264C2×C4×C73C22×D411yescharacteristic
265C2×C4×C73C22×Q811yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—21no—
273Dic146—81no—
274Dic146—81no—
275C8×C73—41no—
276C8×C73—41no—
277#10596831—21no—
278#10596831—21no—
279#10596831—21no—
280#10596831—21no—
281#10596831—22no—
282#10596831—21no—
283#10596831—22no—
284#10596831—21no—
285#10596831C22⋊Q812yes—
286#10596831C22⋊Q812yes—
287C22×C73—21no—
288C22×C73—21no—
289C22×C73—22no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73C2×C4×D411yescharacteristic
293C22×C73#52911yescharacteristic
294C22×C73#53311yescharacteristic
295C4×C73—21no—
296C4×C73—21no—
297C4×C73—21no—
298C4×C73—21no—
299C4×C73—22no—
300C4×C73—42no—
301C4×C73—41no—
302C4×C73—41no—
303C4×C73—81no—
304C4×C73C2×C4×Q811yescharacteristic
305C4×C73C4×#4111yescharacteristic
306C4×C73C2×C22⋊Q811yescharacteristic
307C4×C73#52511yescharacteristic
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311Dic73—41no—
312#21629—731nomaximal, Sylow
313C2×C73—21no—
314C2×C73—21no—
315C2×C73—42no—
316C2×C73C4×C22⋊Q811yescharacteristic
317C2×C73#295611yescharacteristic
318C2×C73#295711yescharacteristic
319#3076—731no—
320#2961—731no—
321#2875—731no—
322#1986—731no—
323#1950—731no—
324#1922—731no—
325#1922—731no—
326#1866—731no—
327#1866—731no—
328#1806D7311yescharacteristic
329#1773—731no—
330#1772—731no—
331#1771—731no—
332#1765—731no—
333#1764—731no—
334C73#2162911yescharacteristic, complemented, Sylow
335C2×#131—731no—
336C2×#131—731no—
337#533—731no—
338C4×#41C2×D7311yescharacteristic
339#479—731no—
340#479—731no—
341#478—731no—
342#478—731no—
343#477—731no—
344#476—731no—
345#475—731no—
346#474—731no—
347#446—1461no—
348#446—1461no—
349#446—1461no—
350#446—1461no—
351C2×#108—731no—
352C2×#108—731no—
353#427—731no—
354#427—731no—
355#424—731no—
356#424—731no—
357#423C2×D7311yescharacteristic
358C2×C4⋊C8—731no—
359C2×#104C2×D7311yescharacteristic
360C2×#104C2×D7311yescharacteristic
361#416—731no—
362#416—731no—
363#405—731no—
364#405—731no—
365#405—731no—
366#405—731no—
367#405—1461no—
368#405—1461no—
369C4×M16—731no—
370C4×M16—1461no—
371C2×#97—731no—
372C2×#97—731no—
373C2×#97—1461no—
374C2×C4×C8—731no—
375C4×C4⋊C4C2×D7311yescharacteristic
376#341—731no—
377#341—731no—
378#339—731no—
379#339—731no—
380#339—731no—
381#339—731no—
382#339C2×D7311yescharacteristic
383#339C2×D7311yescharacteristic
384#336—731no—
385#336—731no—
386C2×#41C22×D7311yescharacteristic
387#131—734no—
388#131—1462no—
389#131—1462no—
390#131—1462no—
391C2×M16—731no—
392C2×M16—731no—
393C2×M16—1461no—
394C2×M16—1461no—
395C2×M16C22×D7311yescharacteristic
396C2×M16C22×D7311yescharacteristic
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—731no—
401C22×C8—1461no—
402C22×C8—1461no—
403C4⋊Q8—731no—
404C4⋊Q8—731no—
405#125—731no—
406#125—731no—
407C22⋊Q8—1461no—
408C22⋊Q8—1461no—
409C4×Q8—1461no—
410C4×Q8—1461no—
411C4×Q8C4×D7311yescharacteristic
412C4×Q8C4×D7311yescharacteristic
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C4×D7311yescharacteristic
416C4×D4C4×D7311yescharacteristic
417#117—21no—
418#117—731no—
419#117—731no—
420#117C22×D7311yescharacteristic
421C2×C4⋊C4—21no—
422C2×C42—21no—
423C2×C42C22×D7311yescharacteristic
424C2×C42C22×D7311yescharacteristic
425C2×C42C22×D7311yescharacteristic
426#108—1462no—
427#108—1462no—
428#107—731no—
429#107—731no—
430#107—731no—
431#107—731no—
432#106—731no—
433#106—731no—
434#106—731no—
435#106—731no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—731no—
439C4⋊C8—731no—
440C4⋊C8—731no—
441C4⋊C8—731no—
442C4⋊C8—1461no—
443C4⋊C8—1461no—
444C4⋊C8—1461no—
445C4⋊C8—1461no—
446#104—22no—
447#104—22no—
448#103—731no—
449#103—731no—
450#103—731no—
451#103—731no—
452#102—731no—
453#102—731no—
454#102—731no—
455#102—731no—
456#97—731no—
457#97—731no—
458#97—731no—
459#97—731no—
460#97—1462no—
461#97—1461no—
462#97—1461no—
463#97—1461no—
464#97—1461no—
465#97—1461no—
466C4×C8—1462no—
467C4×C8—1461no—
468C4×C8—1461no—
469C4×C8—1461no—
470C4×C8—1461no—
471C4×C8—1461no—
472#95—21no—
473#41—21no—
474#41—21no—
475#41Q8×D7312yes—
476#41#1192402712yes—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×Q8C2×C4×D7311yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D7311yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4#1192402711yescharacteristic
488C22×C4#1192402711yescharacteristic
489C22×C4C23×D7311yescharacteristic
490M16—21no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—732no—
495M16—732no—
496M16—1464no—
497M16—1464no—
498M16—1461no—
499M16—1461no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—731no—
503C2×C8—731no—
504C2×C8—732no—
505C2×C8—732no—
506C2×C8—731no—
507C2×C8—731no—
508C2×C8—731no—
509C2×C8—731no—
510C2×C8—731no—
511C2×C8—731no—
512C2×C8—1464no—
513C2×C8—1464no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—731no—
526C4⋊C4—731no—
527C4⋊C4—731no—
528C4⋊C4—731no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4C2×C4×D7311yescharacteristic
536C4⋊C4C2×C4×D7311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—1461no—
541C22⋊C4—1461no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—1461no—
550C42—1461no—
551C42C2×C4×D7311yescharacteristic
552C42C2×C4×D7311yescharacteristic
553C42D4×D7311yescharacteristic
554C42D4×D7311yescharacteristic
555C23—21no—
556C23C2×#1192402711yescharacteristic
557Q8—21no—
558Q8—5841no—
559Q8—5841no—
560Q8#1455655612yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4#1455655612yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4#1455655111yescharacteristic
585C2×C4#1455655111yescharacteristic
586C2×C4#1455655111yescharacteristic
587C2×C4#1455655111yescharacteristic
588C2×C4C22×C4×D7311yescharacteristic
589C2×C4C2×D4×D7311yescharacteristic
590C2×C4C2×Q8×D7311yescharacteristic
591C2×C4#41×D7311yescharacteristic
592C2×C4#41×D7311yescharacteristic
593C2×C4C2×#1192402711yescharacteristic
594C8—41no—
595C8—41no—
596C8—732no—
597C8—732no—
598C8—1461no—
599C8—1461no—
600C8—1461no—
601C8—1461no—
602C8—1462no—
603C8—1461no—
604C8—1461no—
605C8—1462no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×#1455655111yescharacteristic
612C22#1798911811yescharacteristic
613C22#1798913911yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—2921no—
624C4—2921no—
625C4—2921no—
626C4—2921no—
627C4C2×#1455655611yescharacteristic
628C4#1798905511yescharacteristic
629C4C22⋊Q8×D7311yescharacteristic
630C4#1798909711yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2#2261245911yescharacteristic
635C2#2261883711yescharacteristic
636C2#2261884411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1806C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#1806D73
hypocenterC73#21629
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #22618868C2semidirect
4. #22610874C2semidirect
5. #22610704C2semidirect
6. #22610528C2semidirect
7. #22610528C2semidirect
10. #22609814C2semidirect
11. #22609809C2semidirect
12. #22609806C2semidirect
19. #17988894C4semidirect
20. #17988894C4semidirect
21. #17988887C4semidirect
22. #17988887C4semidirect
23. #17988882C4semidirect
24. #17988877C4semidirect
25. #17988872C4semidirect
26. #17988867C4semidirect
70. #14556598Q8semidirect
334. C73#21629semidirect

Maximal subgroups

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

subgroupindex
#226197192
#226188682
#226183592
#226108742
#226107042
#226105282
#226102042
#226098142
#226098092
#226098062
#226097762
#226097692
C73×#18062
#2162973