Finite Group Databasebeta

← #34019206

Subgroups — #34019206

Order 18688.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626762C211yesmaximal, characteristic
2#22626732C211yesmaximal, characteristic, complemented
3#22625572C211yesmaximal, characteristic, complemented
4#22625570C211yesmaximal, characteristic, complemented
5#22625561C211yesmaximal, characteristic, complemented
6#22625555C211yesmaximal, characteristic, complemented
7#22625543C211yesmaximal, characteristic, complemented
8#22625534C211yesmaximal, characteristic, complemented
9#22625526C211yesmaximal, characteristic, complemented
10#22625514C211yesmaximal, characteristic, complemented
11#22625507C211yesmaximal, characteristic, complemented
12#22625286C211yesmaximal, characteristic
13#22625235C211yesmaximal, characteristic, complemented
14#22625148C211yesmaximal, characteristic, complemented
15#22620008C211yesmaximal, characteristic
16#17989639—21no—
17#17989639—21no—
18#17989637C2211yescharacteristic
19#17989637C2211yescharacteristic
20#17989637C2211yescharacteristic
21#17989636C2211yescharacteristic
22#17989636C2211yescharacteristic
23#17989636C2211yescharacteristic
24#17989622—21no—
25#17989622C2211yescharacteristic
26#17989618—21no—
27D4×#10596833C2211yescharacteristic
28#17989611—21no—
29#17989603C2211yescharacteristic
30#17989603C2211yescharacteristic
31#17989590—21no—
32#17989590C2211yescharacteristic
33#17989551—21no—
34C4×#11924037—21no—
35C4×#11924037C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42#17989437C2211yescharacteristic
43#17989437C2211yescharacteristic
44#17989437C2211yescharacteristic
45#17989437C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
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—
68C2×#11924037—21no—
69C2×#11924037D411yescharacteristic
70C2×#11924037D411yescharacteristic
71C22×#10596833—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658—21no—
75#14556658—21no—
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⋊#10596833—21no—
88#14556646—21no—
89#14556646—21no—
90#14556646—21no—
91#14556646D411yescharacteristic
92#14556646D411yescharacteristic
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—22no—
145#11924037—41no—
146#11924037—42no—
147#11924037—42no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—42no—
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159C2×Dic146—22no—
160C2×Dic146—21no—
161C2×Dic146—41no—
162C2×Dic146C22×C411yescharacteristic
163C2×Dic146C22×C411yescharacteristic
164C2×Dic146C22×C411yescharacteristic
165C22×Dic73—21no—
166C22×Dic73—21no—
167C22×Dic73C2×D411yescharacteristic
168C22×Dic73#4111yescharacteristic
169Dic292—22no—
170Dic292—41no—
171#11924016—22no—
172#11924016—41no—
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C4⋊Dic73—21no—
176C4⋊Dic73C22×C411yescharacteristic
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186C22⋊Dic73—21no—
187C22⋊Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73C22×C411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C4×Dic73C2411yescharacteristic
197C2×D4×C73—21no—
198C2×D4×C73C22×C411yescharacteristic
199C22×C4×C73—21no—
200C22×C4×C73C22×C411yescharacteristic
201C2×C8×C73C22×C411yescharacteristic
202C2×C8×C73C22×C411yescharacteristic
203C4⋊C4×C73—21no—
204C4⋊C4×C73C22×C411yescharacteristic
205C4⋊C4×C73C22×C411yescharacteristic
206C4⋊C4×C73C22×C411yescharacteristic
207C22⋊C4×C73—21no—
208C23×C73—21no—
209C23×C73C4×D411yescharacteristic
210D4×C73—21no—
211D4×C73—21no—
212D4×C73—81no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—81no—
219C2×C4×C73C4×D411yescharacteristic
220C2×C4×C73C23×C411yescharacteristic
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—41no—
228C2×Dic73—42no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73C4×D411yescharacteristic
232C2×Dic73C4×D411yescharacteristic
233C2×Dic73C22×D411yescharacteristic
234C2×Dic73C2×#4111yescharacteristic
235Dic146—21no—
236Dic146—21no—
237Dic146—21no—
238Dic146—21no—
239Dic146—21no—
240Dic146—42no—
241Dic146#13112yes—
242#10596833—21no—
243#10596833—21no—
244#10596833—21no—
245#10596833—41no—
246#10596833—41no—
247#10596833—41no—
248#10596833—41no—
249#10596833—41no—
250#10596833—41no—
251#10596833#13712yes—
252C8×C73—21no—
253C8×C73—21no—
254#10596831—21no—
255#10596831—21no—
256C22×C73—21no—
257C22×C73—21no—
258C22×C73—21no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73—41no—
262C22×C73C2×C4×D411yescharacteristic
263C4×C73—21no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73—41no—
267C4×C73C2×#13111yescharacteristic
268C4×C73#56811yescharacteristic
269Dic73—22no—
270Dic73—21no—
271Dic73—21no—
272Dic73—21no—
273Dic73—41no—
274Dic73—41no—
275Dic73—41no—
276Dic73C2×#13711yescharacteristic
277Dic73#57511yescharacteristic
278#18010—731nomaximal, Sylow
279C2×C73—41no—
280C2×C73—81no—
281C2×C73#293311yescharacteristic
282C2×C73C4×#13711yescharacteristic
283C2×C73#295711yescharacteristic
284#3108—731no—
285#2969—731no—
286Q8×M16—731no—
287#1592—731no—
288#1591—731no—
289#1588—731no—
290#1587—731no—
291#1584—731no—
292#1582—731no—
293#1579—731no—
294C8×SD16—731no—
295#1574—731no—
296#1495—731no—
297#1479—731no—
298#1453—731no—
299C73#1801011yescharacteristic, complemented, Sylow
300#529—731no—
301C2×C4×Q8—731no—
302#483—731no—
303#480#6440811yescharacteristic
304#479—731no—
305#479—731no—
306#464—731no—
307#464—731no—
308#451—731no—
309#448—731no—
310#446—731no—
311#446—731no—
312#446—731no—
313#446—1461no—
314#446—1461no—
315C8×Q8—731no—
316C8×Q8—731no—
317C8×Q8—731no—
318C4×Q16—731no—
319C4×Q16—731no—
320C4×SD16—731no—
321C4×SD16—731no—
322#436—731no—
323#436—1461no—
324#435—1461no—
325C8×D4—731no—
326#433—1461no—
327#431—731no—
328#431—731no—
329#431—731no—
330#423—731no—
331#423—1461no—
332#405—1461no—
333C4×M16—731no—
334C4×M16—1461no—
335#335—731no—
336#335—731no—
337#334—731no—
338#334—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343Q8⋊C8—731no—
344Q8⋊C8—731no—
345D4⋊C8—731no—
346D4⋊C8—731no—
347#322—731no—
348#322—731no—
349#322—731no—
350C82—731no—
351C22×Q8—731no—
352C2×Q16—731no—
353C2×Q16—731no—
354C2×SD16—731no—
355C2×SD16—731no—
356C2×M16—731no—
357C2×M16—731no—
358C2×M16—1461no—
359C2×M16—1461no—
360C22×C8—1461no—
361#126—1461no—
362#125—731no—
363#124—731no—
364#123—1461no—
365C22⋊Q8—731no—
366#121C2×#6440811yescharacteristic
367C4×Q8—731no—
368C4×Q8—731no—
369C4×Q8—731no—
370C4×Q8—731no—
371C4×Q8—731no—
372C4×Q8—1461no—
373C4×Q8—1461no—
374C4×D4—731no—
375C4×D4—1461no—
376#117—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4C2×#6440811yescharacteristic
379C2×C42—731no—
380C2×C42—1461no—
381#107C2×#6440811yescharacteristic
382#107C2×#6440811yescharacteristic
383#106—731no—
384#106—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—
398C4⋊C8—1461no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103—731no—
403#103—731no—
404#103—731no—
405#102C2×#6440811yescharacteristic
406#102C2×#6440811yescharacteristic
407C22⋊C8—731no—
408C22⋊C8—1461no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8—1461no—
412C22⋊C8C2×#6440811yescharacteristic
413#97—731no—
414#97—731no—
415#97—1461no—
416#97—1461no—
417#97—1461no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—731no—
423C4×C8—731no—
424C4×C8—1461no—
425C4×C8—1461no—
426C4×C8—1461no—
427C4×C8—1461no—
428C2×Q8—2921no—
429C2×Q8—731no—
430C2×Q8—731no—
431C2×Q8—731no—
432C2×Q8—1462no—
433C2×Q8—1461no—
434C2×D4—21no—
435C2×D4C22×#6440811yescharacteristic
436C22×C4—21no—
437C22×C4—731no—
438C22×C4—731no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4C22×#6440811yescharacteristic
442Q16—2921no—
443Q16—1462no—
444SD16—2921no—
445SD16—1462no—
446M16—2922no—
447M16—2922no—
448M16—2921no—
449M16—1462no—
450C2×C8—2922no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—731no—
454C2×C8—731no—
455C2×C8—731no—
456C2×C8—731no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8C22×#6440811yescharacteristic
464C2×C8C22×#6440811yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—731no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4C22×#6440811yescharacteristic
476C4⋊C4C22×#6440811yescharacteristic
477C4⋊C4C22×#6440811yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—1461no—
480C22⋊C4—1461no—
481C22⋊C4—1461no—
482C22⋊C4—1461no—
483C42—731no—
484C42—731no—
485C42—731no—
486C42—731no—
487C42—1461no—
488C42—1461no—
489C42—1461no—
490C42—1461no—
491C42—1461no—
492C23—21no—
493C23D4×#6440811yescharacteristic
494Q8—2922no—
495Q8—732no—
496Q8—1461no—
497Q8—1461no—
498Q8—1461no—
499Q8—1461no—
500Q8—1461no—
501D4—21no—
502D4—21no—
503D4—81no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—81no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2922no—
513C2×C4—2921no—
514C2×C4—731no—
515C2×C4—731no—
516C2×C4—731no—
517C2×C4—731no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4—1461no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4D4×#6440811yescharacteristic
525C2×C4C23×#6440811yescharacteristic
526C8—21no—
527C8—21no—
528C8—2921no—
529C8—2921no—
530C8—2921no—
531C8—2921no—
532C8—2921no—
533C8—2921no—
534C8—732no—
535C8—1461no—
536C8—1461no—
537C8—1461no—
538C8—1461no—
539C8—1461no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×D4×#6440811yescharacteristic
547C4—21no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—2921no—
552C4—2921no—
553C4—2921no—
554C4—731no—
555C4—731no—
556C4—1462no—
557C4—1461no—
558C4—1461no—
559C4—1461no—
560C4D73⋊#13111yescharacteristic
561C4#1798966611yescharacteristic
562C2—41no—
563C2—81no—
564C2#2262666511yescharacteristic
565C2#137×#6440811yescharacteristic
566C2#2262670711yescharacteristic
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#18010
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. #22626732C2semidirect
3. #22625572C2semidirect
4. #22625570C2semidirect
5. #22625561C2semidirect
6. #22625555C2semidirect
7. #22625543C2semidirect
8. #22625534C2semidirect
9. #22625526C2semidirect
10. #22625514C2semidirect
11. #22625507C2semidirect
13. #22625235C2semidirect
14. #22625148C2semidirect
299. C73#18010semidirect

Maximal subgroups

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

subgroupindex
#226267622
#226267322
#226255722
#226255702
#226255612
#226255552
#226255432
#226255342
#226255262
#226255142
#226255072
#226252862
#226252352
#226251482
#226200082
#1801073