Finite Group Databasebeta

← #34019258

Subgroups — #34019258

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626878C211yesmaximal, characteristic
2#22626831C211yesmaximal, characteristic, complemented
3#22625570C211yesmaximal, characteristic, complemented
4#22625568C211yesmaximal, characteristic, complemented
5#22625561C211yesmaximal, characteristic, complemented
6#22625555C211yesmaximal, characteristic, complemented
7#22625555C211yesmaximal, characteristic, complemented
8#22625546C211yesmaximal, characteristic, complemented
9#22625519C211yesmaximal, characteristic, complemented
10#22625514C211yesmaximal, characteristic, complemented
11#22625506C211yesmaximal, characteristic, complemented
12#22625282C211yesmaximal, characteristic
13C2×#17989437C211yesmaximal, characteristic, complemented
14#22625136C211yesmaximal, characteristic, complemented
15#22620008C211yesmaximal, characteristic
16#17989639—21no—
17#17989639—21no—
18#17989637C2211yescharacteristic
19#17989637C2211yescharacteristic
20#17989637C2211yescharacteristic
21#17989636C2211yescharacteristic
22#17989636C2211yescharacteristic
23#17989636C2211yescharacteristic
24#17989622C2211yescharacteristic
25#17989618—21no—
26D4×#10596833—21no—
27D4×#10596833C2211yescharacteristic
28#17989611—21no—
29#17989608C2211yescharacteristic
30#17989607—21no—
31#17989604C2211yescharacteristic
32#17989603—21no—
33#17989593—21no—
34C2×C4⋊#10596833C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989450C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989449C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42#17989437C2211yescharacteristic
43#17989437C2211yescharacteristic
44#17989437C2211yescharacteristic
45#17989437C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50C8×#10596833C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989118C2211yescharacteristic
53C2×#14556556C2211yescharacteristic
54#17988917C411yescharacteristic
55#17988893C411yescharacteristic
56#17988893C411yescharacteristic
57#17988808C411yescharacteristic
58#17988808C411yescharacteristic
59#17988745C411yescharacteristic
60#17988725C411yescharacteristic
61#17988681C2211yescharacteristic
62#17988681C2211yescharacteristic
63#17988678C2211yescharacteristic
64#17988678C2211yescharacteristic
65#17988637C2211yescharacteristic
66#480×C73C411yescharacteristic
67C2×#11924037—21no—
68C22×#10596833—21no—
69C22×#10596833—21no—
70C22×#10596833—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658D411yescharacteristic
75#14556658D411yescharacteristic
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833D411yescharacteristic
88C22⋊#10596833D411yescharacteristic
89#14556646—21no—
90#14556646—21no—
91#14556646—21no—
92#14556646—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C22×Dic146C2×C411yescharacteristic
102C2×Dic292C2×C411yescharacteristic
103C2×Dic292C2×C411yescharacteristic
104C2×#11924016C2×C411yescharacteristic
105C2×#11924016C2×C411yescharacteristic
106#14556584—21no—
107#14556582C2×C411yescharacteristic
108#14556578C2×C411yescharacteristic
109#14556572—21no—
110#14556570C2×C411yescharacteristic
111#14556556—21no—
112#14556556—21no—
113#14556556C2×C411yescharacteristic
114#14556556C2×C411yescharacteristic
115#14556556C2311yescharacteristic
116#14556556C2311yescharacteristic
117#14556556C2311yescharacteristic
118D4×Dic73—21no—
119D4×Dic73C2311yescharacteristic
120#14556547—21no—
121C2×#11924007—21no—
122C2×C4×Dic73—21no—
123C2×C4×Dic73C2311yescharacteristic
124#14556522C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556514C2×C411yescharacteristic
131#14556514C2×C411yescharacteristic
132#14556514C2×C411yescharacteristic
133#14556514C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137#121×C73C2×C411yescharacteristic
138C2×C4⋊C4×C73C2×C411yescharacteristic
139#107×C73C2×C411yescharacteristic
140#107×C73C2×C411yescharacteristic
141#102×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924037—81no—
145C2×#10596833—24no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—81no—
153C2×#10596833—81no—
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×Dic146—22no—
159C2×Dic146—21no—
160C2×Dic146—41no—
161C2×Dic146C22×C411yescharacteristic
162C2×Dic146C22×C411yescharacteristic
163C2×Dic146C22×C411yescharacteristic
164C22×Dic73—21no—
165C22×Dic73—21no—
166C22×Dic73C2×D411yescharacteristic
167C22×Dic73#4111yescharacteristic
168Dic292—22no—
169Dic292—41no—
170#11924016—22no—
171#11924016—41no—
172C2×#10596831C22×C411yescharacteristic
173C2×#10596831C22×C411yescharacteristic
174C4⋊Dic73—21no—
175C4⋊Dic73C22×C411yescharacteristic
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183C22⋊Dic73—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186C22⋊Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73C22×C411yescharacteristic
193C4×Dic73C2×D411yescharacteristic
194C4×Dic73#4111yescharacteristic
195C4×Dic73C2411yescharacteristic
196C2×D4×C73—21no—
197C2×D4×C73C22×C411yescharacteristic
198C22×C4×C73—21no—
199C22×C4×C73C22×C411yescharacteristic
200C2×C8×C73C22×C411yescharacteristic
201C2×C8×C73C22×C411yescharacteristic
202C4⋊C4×C73—21no—
203C4⋊C4×C73C22×C411yescharacteristic
204C4⋊C4×C73C22×C411yescharacteristic
205C4⋊C4×C73C22×C411yescharacteristic
206C22⋊C4×C73—21no—
207C23×C73—21no—
208C23×C73C4×D411yescharacteristic
209D4×C73—21no—
210D4×C73—21no—
211D4×C73—42no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—42no—
218C2×C4×C73C4×D411yescharacteristic
219C2×C4×C73C23×C411yescharacteristic
220C2×Dic73—21no—
221C2×Dic73—22no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—42no—
228C2×Dic73—41no—
229C2×Dic73—41no—
230C2×Dic73C4×D411yescharacteristic
231C2×Dic73C4×D411yescharacteristic
232C2×Dic73C22×D411yescharacteristic
233C2×Dic73C2×#4111yescharacteristic
234Dic146—21no—
235Dic146—21no—
236Dic146—21no—
237Dic146—21no—
238Dic146—21no—
239Dic146—42no—
240Dic146#13112yes—
241#10596833—22no—
242#10596833—22no—
243#10596833—21no—
244#10596833—21no—
245#10596833—21no—
246#10596833—21no—
247#10596833—41no—
248#10596833—41no—
249#10596833—41no—
250#10596833—41no—
251C8×C73—21no—
252C8×C73—21no—
253#10596831—21no—
254#10596831—21no—
255C22×C73—21no—
256C22×C73—21no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73#44312yes—
261C22×C73C2×C4×D411yescharacteristic
262C4×C73—21no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73C2×#13111yescharacteristic
267C4×C73#56811yescharacteristic
268Dic73—22no—
269Dic73—21no—
270Dic73—21no—
271Dic73—21no—
272Dic73—41no—
273Dic73—41no—
274Dic73—41no—
275Dic73#57711yescharacteristic
276Dic73#57811yescharacteristic
277#18017—731nomaximal, Sylow
278C2×C73—22no—
279C2×C73—81no—
280C2×C73#293311yescharacteristic
281C2×C73#294711yescharacteristic
282C2×C73C2×#44311yescharacteristic
283#3108—731no—
284#2994—731no—
285#2984—731no—
286#1591—731no—
287#1590—731no—
288#1588—731no—
289#1587—731no—
290#1587—731no—
291#1585—731no—
292C8×Q16—731no—
293C8×SD16—731no—
294#1573—731no—
295#1494—731no—
296C2×Q8⋊C8—731no—
297#1450—731no—
298C73#1801711yescharacteristic, complemented, Sylow
299#529—731no—
300C2×C4×Q8—731no—
301#483—731no—
302#480#6440811yescharacteristic
303#479—731no—
304#479—731no—
305#464—731no—
306#464—731no—
307#451—731no—
308#448—731no—
309#446—731no—
310#446—731no—
311#446—731no—
312#446—1461no—
313#446—1461no—
314C8×Q8—731no—
315C8×Q8—731no—
316C8×Q8—731no—
317C4×Q16—731no—
318C4×Q16—731no—
319C4×SD16—731no—
320C4×SD16—731no—
321#436—731no—
322#435—1461no—
323C8×D4—731no—
324C8×D4—1461no—
325#433—1461no—
326#432—731no—
327#432—1461no—
328#431—731no—
329#431—731no—
330#431—1461no—
331#424—1461no—
332C2×C4⋊C8—731no—
333C2×C4×C8—731no—
334#335—731no—
335#335—731no—
336#334—731no—
337#334—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343Q8⋊C8—731no—
344D4⋊C8—731no—
345D4⋊C8—731no—
346#322—731no—
347#322—731no—
348C82—731no—
349C82—731no—
350C22×Q8—731no—
351C2×Q16—731no—
352C2×Q16—731no—
353C2×SD16—731no—
354C2×SD16—731no—
355C2×M16—1461no—
356C22×C8—1461no—
357C22×C8—1461no—
358C22×C8—1461no—
359#126—1461no—
360#125—731no—
361#124—731no—
362#123—1461no—
363C22⋊Q8—731no—
364#121C2×#6440811yescharacteristic
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—731no—
368C4×Q8—731no—
369C4×Q8—731no—
370C4×Q8—1461no—
371C4×Q8—1461no—
372C4×D4—731no—
373C4×D4—1461no—
374#117—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4C2×#6440811yescharacteristic
377C2×C42—731no—
378C2×C42—1461no—
379#107C2×#6440811yescharacteristic
380#107C2×#6440811yescharacteristic
381#106—731no—
382#106—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—731no—
390C4⋊C8—731no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8—1461no—
397C4⋊C8—1461no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103—731no—
403#103—731no—
404#102C2×#6440811yescharacteristic
405#102C2×#6440811yescharacteristic
406C22⋊C8—731no—
407C22⋊C8—731no—
408C22⋊C8—731no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8—1461no—
412C22⋊C8C2×#6440811yescharacteristic
413#97—1461no—
414#97—1461no—
415#97—1461no—
416#97—1461no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—731no—
423C4×C8—1461no—
424C4×C8—1461no—
425C4×C8—1461no—
426C4×C8—1461no—
427C2×Q8—2921no—
428C2×Q8—731no—
429C2×Q8—731no—
430C2×Q8—731no—
431C2×Q8—1462no—
432C2×Q8—1461no—
433C2×D4—21no—
434C2×D4C22×#6440811yescharacteristic
435C22×C4—21no—
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4C22×#6440811yescharacteristic
441Q16—2921no—
442Q16—1462no—
443SD16—2921no—
444SD16—1462no—
445M16—5841no—
446C2×C8—5841no—
447C2×C8—5841no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—731no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1464no—
461C2×C8C22×#6440811yescharacteristic
462C2×C8C22×#6440811yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—731no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4C22×#6440811yescharacteristic
474C4⋊C4C22×#6440811yescharacteristic
475C4⋊C4C22×#6440811yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C22⋊C4—1461no—
480C22⋊C4—1461no—
481C42—731no—
482C42—731no—
483C42—731no—
484C42—731no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C42—1461no—
489C42—1461no—
490C23—21no—
491C23D4×#6440811yescharacteristic
492Q8—2922no—
493Q8—732no—
494Q8—1461no—
495Q8—1461no—
496Q8—1461no—
497Q8—1461no—
498Q8—1461no—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—42no—
508C2×C4—2921no—
509C2×C4—2921no—
510C2×C4—2922no—
511C2×C4—731no—
512C2×C4—731no—
513C2×C4—731no—
514C2×C4—731no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4—1461no—
521C2×C4—1462no—
522C2×C4D4×#6440811yescharacteristic
523C2×C4C23×#6440811yescharacteristic
524C8—21no—
525C8—21no—
526C8—2921no—
527C8—2921no—
528C8—2921no—
529C8—2921no—
530C8—1461no—
531C8—1461no—
532C8—1461no—
533C8—1461no—
534C8—1462no—
535C8—1462no—
536C8—1461no—
537C8—1461no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1798962912yes—
544C22C2×D4×#6440811yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—2921no—
550C4—2921no—
551C4—2921no—
552C4—731no—
553C4—731no—
554C4—1462no—
555C4—1461no—
556C4—1461no—
557C4—1461no—
558C4D73⋊#13111yescharacteristic
559C4#1798966611yescharacteristic
560C2—22no—
561C2—81no—
562C2#2262666511yescharacteristic
563C2#2262668911yescharacteristic
564C2C2×#1798962911yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#480×C73C4
socleC22×C73C2×C4×D4
prime radical#121×C73C2×C4
cosocleC4×Dic73C24
hypercenter#480#64408
hypocenterC73#18017
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #22626831C2semidirect
3. #22625570C2semidirect
4. #22625568C2semidirect
5. #22625561C2semidirect
6. #22625555C2semidirect
7. #22625555C2semidirect
8. #22625546C2semidirect
9. #22625519C2semidirect
10. #22625514C2semidirect
11. #22625506C2semidirect
13. C2×#17989437C2semidirect
14. #22625136C2semidirect
298. C73#18017semidirect

Maximal subgroups

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

subgroupindex
#226268782
#226268312
#226255702
#226255682
#226255612
#226255552
#226255462
#226255192
#226255142
#226255062
#226252822
C2×#179894372
#226251362
#226200082
#1801773