Finite Group Databasebeta

← #33242949

Subgroups — #33242949

Order 18688.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621618C211yesmaximal, characteristic, complemented
2#22612170C211yesmaximal, characteristic, complemented
3#22612128C211yesmaximal, characteristic, complemented
4#22612116C211yesmaximal, characteristic, complemented
5#22612075C211yesmaximal, characteristic, complemented
6#22612032C211yesmaximal, characteristic, complemented
7#22612025C211yesmaximal, characteristic, complemented
8#22611986C211yesmaximal, characteristic, complemented
9#22611852C211yesmaximal, characteristic, complemented
10#22611842C211yesmaximal, characteristic, complemented
11#22611805C211yesmaximal, characteristic, complemented
12#22611757C211yesmaximal, characteristic, complemented
13#22611641C211yesmaximal, characteristic, complemented
14C2×#17988387C211yesmaximal, characteristic, complemented
15C73×#3334C211yesmaximal, characteristic, complemented
16C2×#14556613C2211yescharacteristic
17C22×D584C2211yescharacteristic
18#17989233—21no—
19C2×#14556535C2211yescharacteristic, complemented
20C2×#14556532C2211yescharacteristic
21C2×#14556532C2211yescharacteristic
22C2×#14556531C2211yescharacteristic, complemented
23C22×#11924003C2211yescharacteristic
24#17988971C2211yescharacteristic
25#17988828C2211yescharacteristic
26#17988823C2211yescharacteristic
27#17988803—21no—
28#17988740—21no—
29#17988734—21no—
30C4×D584C2211yescharacteristic
31#17988414C2211yescharacteristic, complemented
32#17988414C2211yescharacteristic, complemented
33#17988414C2211yescharacteristic, complemented
34C4⋊#11924003C2211yescharacteristic, complemented
35#17988390C2211yescharacteristic, complemented
36#17988390C2211yescharacteristic, complemented
37#17988390C2211yescharacteristic, complemented
38#17988388C2211yescharacteristic, complemented
39#17988387C2211yescharacteristic
40#17988387C2211yescharacteristic
41#17988387C2211yescharacteristic
42#17988387C2211yescharacteristic
43#17988387C2211yescharacteristic
44#17988387C2211yescharacteristic
45C22⋊#11924003C2211yescharacteristic
46C22⋊#11924003C2211yescharacteristic, complemented
47#17988363C2211yescharacteristic, complemented
48#547×C73—21no—
49#500×C73C2211yescharacteristic, complemented
50#480×C73—21no—
51#480×C73—21no—
52#474×C73—21no—
53#466×C73C2211yescharacteristic
54#466×C73C2211yescharacteristic, complemented
55C4×D8×C73C2211yescharacteristic, complemented
56C8×D4×C73C2211yescharacteristic
57C2×#107×C73C2211yescharacteristic
58C2×#107×C73C2211yescharacteristic, complemented
59C2×#11924028—21no—
60C22×D292—21no—
61#14556613—22no—
62#14556613—41no—
63C2×Dic292C2311yescharacteristic
64C2×#11924017—21no—
65C2×#11924017—21no—
66C2×D584—22no—
67C2×D584—21no—
68C2×D584—41no—
69C2×D584C2311yescharacteristic
70C2×D584C2311yescharacteristic
71#14556579—21no—
72#14556576—41no—
73#14556570—21no—
74#14556565—21no—
75#14556564—41no—
76#14556559—41no—
77C4×D292—21no—
78C22⋊C4×D73—41no—
79#14556535—41no—
80#14556532—22no—
81#14556532—41no—
82#14556531—41no—
83C2×#11924003—41no—
84C2×#11924003C2311yescharacteristic
85C2×#11924003C2311yescharacteristic
86C2×#11924003C2311yescharacteristic
87C2×#11924003C2311yescharacteristic
88#14556525C2311yescharacteristic
89#14556516—21no—
90#14556513—21no—
91#14556513—21no—
92C2×#41×C73—41no—
93C2×D8×C73C2311yescharacteristic
94C22×C8×C73C2311yescharacteristic
95C22×C8×C73C2311yescharacteristic
96C4⋊Q8×C73—21no—
97#123×C73—41no—
98#123×C73—41no—
99C22⋊Q8×C73—41no—
100#121×C73—21no—
101#121×C73—21no—
102C4×D4×C73—21no—
103C4×D4×C73D411yescharacteristic
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106#107×C73—21no—
107#107×C73—21no—
108#107×C73—21no—
109#107×C73C2311yescharacteristic
110#107×C73C2311yescharacteristic
111#107×C73C2311yescharacteristic
112C4⋊C8×C73D411yescharacteristic
113#102×C73—21no—
114#102×C73—21no—
115#102×C73—21no—
116C22⋊C8×C73D411yescharacteristic
117C22⋊C8×C73D411yescharacteristic
118C4×C8×C73C2311yescharacteristic
119C23×D73—41no—
120#11924028—42no—
121#11924028—81no—
122C2×Dic146—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292—42no—
126C2×D292—41no—
127C2×D292—81no—
128C2×C22⋊D73—41no—
129C2×C22⋊D73—41no—
130C2×C4×D73—41no—
131C2×C4×D73—41no—
132Dic292—21no—
133Dic292—21no—
134#11924017—42no—
135#11924017—81no—
136D584—21no—
137D584—21no—
138D584—21no—
139D584—42no—
140D584#4112yescomplemented
141C4⋊Dic73—21no—
142#11924007—41no—
143C22⋊Dic73—41no—
144#11924005—41no—
145#11924005—41no—
146#11924005—41no—
147#11924005—41no—
148#11924003—21no—
149#11924003—21no—
150#11924003—21no—
151#11924003—21no—
152#41×C73—84no—
153C2×Q8×C73—41no—
154C2×D4×C73—21no—
155C2×D4×C73—41no—
156C2×D4×C73—41no—
157C2×D4×C73D811yescharacteristic
158C22×C4×C73—41no—
159C22×C4×C73—41no—
160C22×C4×C73—41no—
161C22×C4×C73C2×D411yescharacteristic
162C22×C4×C73C2×D411yescharacteristic
163D8×C73—21no—
164D8×C73—21no—
165C2×C8×C73—21no—
166C2×C8×C73—21no—
167C2×C8×C73—21no—
168C2×C8×C73—21no—
169C2×C8×C73C2×D411yescharacteristic
170C2×C8×C73C2×D411yescharacteristic
171C2×C8×C73C2×D411yescharacteristic
172C2×C8×C73C2411yescharacteristic
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—41no—
177C4⋊C4×C73—41no—
178C4⋊C4×C73—41no—
179C4⋊C4×C73D811yescharacteristic
180C22⋊C4×C73—41no—
181C22⋊C4×C73—41no—
182C22⋊C4×C73—41no—
183C22⋊C4×C73D811yescharacteristic
184C22⋊C4×C73D811yescharacteristic
185C42×C73C2×D411yescharacteristic
186C23×C73—41no—
187C23×C73C2×D811yescharacteristic
188C23×C73C2×D811yescharacteristic
189C22×D73—41no—
190C22×D73—41no—
191C22×D73—82no—
192C22×D73—82no—
193C22×D73—161no—
194Q8×C73—82no—
195D4×C73—22no—
196D4×C73—41no—
197D4×C73—41no—
198D4×C73—82no—
199D4×C73—82no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—82no—
208C2×C4×C73—82no—
209C2×C4×C73—82no—
210C2×C4×C73C2×D811yescharacteristic
211C2×C4×C73C2×D811yescharacteristic
212C2×C4×C73C2×D811yescharacteristic
213C2×C4×C73C2×D811yescharacteristic
214C2×C4×C73C22×D411yescharacteristic
215C22⋊D73—82no—
216C22⋊D73—161no—
217C2×Dic73—41no—
218C2×Dic73—41no—
219D292—22no—
220D292—41no—
221D292—41no—
222D292—41no—
223D292—82no—
224C4×D73—82no—
225C4×D73—161no—
226Dic146—41no—
227Dic146—41no—
228C8×C73—21no—
229C8×C73—21no—
230C8×C73—21no—
231C8×C73C2×#4111yescharacteristic
232C8×C73H(5,𝔽2)11yescharacteristic
233C22×C73—21no—
234C22×C73—21no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73—21no—
238C22×C73—81no—
239C22×C73—81no—
240C22×C73—81no—
241C22×C73#50912yes—
242C22×C73C22×D811yescharacteristic
243C2×D73—42no—
244C2×D73—81no—
245C2×D73—81no—
246C2×D73—81no—
247C2×D73—81no—
248C2×D73—81no—
249C2×D73—164no—
250C4×C73—21no—
251C4×C73—21no—
252C4×C73—21no—
253C4×C73—21no—
254C4×C73—81no—
255C4×C73—81no—
256C4×C73—81no—
257C4×C73#54611yescharacteristic
258C4×C73#57711yescharacteristic
259Dic73—81no—
260Dic73—81no—
261#33873—731nomaximal, Sylow
262C2×C73—22no—
263C2×C73—41no—
264C2×C73—161no—
265C2×C73#329411yescharacteristic
266C2×C73C2×#50911yescharacteristic
267C2×C73#341611yescharacteristic
268D73—82no—
269D73—161no—
270#3334D7311yescharacteristic, complemented
271#3280—731no—
272#2243—731no—
273#2235—731no—
274#2233—731no—
275#2226—731no—
276#2218—731no—
277#2217—731no—
278#2210—731no—
279#2186—731no—
280#2184—731no—
281#2177—731no—
282#2168—731no—
283#2139—731no—
284C2×#357—731no—
285C73#3387311yescharacteristic, complemented, Sylow
286C2×#135—731no—
287C22×D8—731no—
288#547—21no—
289#546—1461no—
290C2×SD32—731no—
291C2×D16—731no—
292C2×D16—731no—
293C2×M32—731no—
294C22×C16—731no—
295#500C2×D7311yescharacteristic, complemented
296#495—731no—
297#480—21no—
298#480—21no—
299#474—21no—
300#467—731no—
301#466—731no—
302#466C2×D7311yescharacteristic
303#466C2×D7311yescharacteristic, complemented
304#463—1461no—
305#450—1461no—
306#449—1461no—
307C4×D8—731no—
308C4×D8C2×D7311yescharacteristic, complemented
309C8×D4C2×D7311yescharacteristic
310C2×#107C2×D7311yescharacteristic
311C2×#107C2×D7311yescharacteristic, complemented
312#367—731no—
313#367—731no—
314#367—731no—
315C4⋊C16—731no—
316#358—731no—
317#358—731no—
318#358—731no—
319#357—731no—
320#357—731no—
321#357—731no—
322#357—731no—
323#357—731no—
324#357—731no—
325#357—731no—
326C22⋊C16—731no—
327C22⋊C16—731no—
328#346—731no—
329C2×#41—41no—
330C2×#41—1461no—
331C22×D4—1461no—
332#135—2921no—
333#135—1462no—
334C2×Q16—731no—
335C2×SD16—1461no—
336C2×SD16—1461no—
337C2×D8—2921no—
338C2×D8—731no—
339C2×D8—731no—
340C2×D8—1462no—
341C2×D8—1461no—
342C2×D8C22×D7311yescharacteristic
343C22×C8C22×D7311yescharacteristic
344C22×C8C22×D7311yescharacteristic
345C4⋊Q8—21no—
346#124—1461no—
347#123—41no—
348#123—41no—
349#123—2921no—
350C22⋊Q8—41no—
351C22⋊Q8—1461no—
352#121—21no—
353#121—21no—
354#121—2921no—
355#121—1461no—
356#120—2921no—
357C4×D4—21no—
358C4×D4—1461no—
359C4×D4C22⋊D7311yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C22⋊C4—2921no—
363SD32—2921no—
364D16—2921no—
365D16—1462no—
366M32—2921no—
367C2×C16—2921no—
368C2×C16—731no—
369C2×C16—731no—
370C2×C16—731no—
371C2×C16—731no—
372#107—21no—
373#107—21no—
374#107—21no—
375#107—731no—
376#107C22×D7311yescharacteristic
377#107C22×D7311yescharacteristic
378#107C22×D7311yescharacteristic
379C4⋊C8C22⋊D7311yescharacteristic
380#103—1461no—
381#102—21no—
382#102—21no—
383#102—21no—
384#102—1461no—
385#102—1461no—
386C22⋊C8C22⋊D7311yescharacteristic
387C22⋊C8C22⋊D7311yescharacteristic
388C4×C8C22×D7311yescharacteristic
389C24—2921no—
390#41—84no—
391#41—2922no—
392#41—5841no—
393C2×Q8—41no—
394C2×Q8—1461no—
395C2×D4—21no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—2922no—
399C2×D4—2921no—
400C2×D4—2921no—
401C2×D4—2921no—
402C2×D4—5841no—
403C2×D4—1461no—
404C2×D4—1461no—
405C2×D4#1192401311yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—2921no—
410C22×C4—2921no—
411C22×C4C2×C22⋊D7311yescharacteristic
412C22×C4C2×C22⋊D7311yescharacteristic
413Q16—1461no—
414Q16—1461no—
415SD16—2922no—
416SD16—5841no—
417D8—21no—
418D8—21no—
419D8—2922no—
420D8—732no—
421D8—1461no—
422D8—1461no—
423D8—1461no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×C22⋊D7311yescharacteristic
429C2×C8C2×C22⋊D7311yescharacteristic
430C2×C8C2×C22⋊D7311yescharacteristic
431C2×C8C23×D7311yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—2921no—
439C4⋊C4—1461no—
440C4⋊C4#1192401311yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—2921no—
445C22⋊C4—2921no—
446C22⋊C4—2921no—
447C22⋊C4—2921no—
448C22⋊C4—2921no—
449C22⋊C4#1192401311yescharacteristic
450C22⋊C4#1192401311yescharacteristic
451C42C2×C22⋊D7311yescharacteristic
452C16—1461no—
453C16—1461no—
454C16—1461no—
455C16—1461no—
456C23—41no—
457C23—2921no—
458C23—2921no—
459C23—5842no—
460C23—5842no—
461C23—11681no—
462C23C2×#1192401311yescharacteristic
463C23C2×#1192401311yescharacteristic
464Q8—82no—
465Q8—2921no—
466Q8—2921no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—82no—
471D4—82no—
472D4—2921no—
473D4—2921no—
474D4—2921no—
475D4—5842no—
476D4—5842no—
477D4—11681no—
478D4—1462no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—2921no—
490C2×C4—2921no—
491C2×C4—5842no—
492C2×C4—11681no—
493C2×C4C2×#1192401311yescharacteristic
494C2×C4C2×#1192401311yescharacteristic
495C2×C4C2×#1192401311yescharacteristic
496C2×C4C2×#1192401311yescharacteristic
497C2×C4C22×C22⋊D7311yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8#41×D7311yescharacteristic
502C8#1455663711yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—2922no—
512C22—5841no—
513C22—5841no—
514C22—5841no—
515C22—5841no—
516C22—5841no—
517C22—11684no—
518C22#1798902312yes—
519C22C22×#1192401311yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—5841no—
528C4—5841no—
529C4#1798922511yescharacteristic
530C4#1798939511yescharacteristic
531C2—22no—
532C2—41no—
533C2—161no—
534C2—5842no—
535C2—11681no—
536C2#2262173811yescharacteristic
537C2C2×#1798902311yescharacteristic
538C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3334C2
socleC22×C73C22×D8
hypercenter#3334D73
hypocenterC73#33873
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #22621618C2semidirect
2. #22612170C2semidirect
3. #22612128C2semidirect
4. #22612116C2semidirect
5. #22612075C2semidirect
6. #22612032C2semidirect
7. #22612025C2semidirect
8. #22611986C2semidirect
9. #22611852C2semidirect
10. #22611842C2semidirect
11. #22611805C2semidirect
12. #22611757C2semidirect
13. #22611641C2semidirect
14. C2×#17988387C2semidirect
15. C73×#3334C2semidirect
19. C2×#14556535C22semidirect
22. C2×#14556531C22semidirect
31. #17988414C22semidirect
32. #17988414C22semidirect
33. #17988414C22semidirect
34. C4⋊#11924003C22semidirect
35. #17988390C22semidirect
36. #17988390C22semidirect
37. #17988390C22semidirect
38. #17988388C22semidirect
46. C22⋊#11924003C22semidirect
47. #17988363C22semidirect
49. #500×C73C22semidirect
54. #466×C73C22semidirect
55. C4×D8×C73C22semidirect
58. C2×#107×C73C22semidirect
140. D584#41semidirect
270. #3334D73semidirect
285. C73#33873semidirect
295. #500C2×D73semidirect
303. #466C2×D73semidirect
308. C4×D8C2×D73semidirect
311. C2×#107C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226216182
#226121702
#226121282
#226121162
#226120752
#226120322
#226120252
#226119862
#226118522
#226118422
#226118052
#226117572
#226116412
C2×#179883872
C73×#33342
#3387373