Finite Group Databasebeta

← #10778178

Subgroups — #10778178

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79302C211yesmaximal, characteristic
2#79025C211yesmaximal, characteristic
3#78540C211yesmaximal, characteristic, complemented
4#78524C211yesmaximal, characteristic
5C3×#1565C211yesmaximal, characteristic, complemented
6#70781C211yesmaximal, characteristic, complemented
7#70439C211yesmaximal, characteristic, complemented
8#68595C211yesmaximal, characteristic, complemented
9#68593C211yesmaximal, characteristic, complemented
10#68581C211yesmaximal, characteristic, complemented
11#68577C211yesmaximal, characteristic, complemented
12#68555C211yesmaximal, characteristic, complemented
13#68551C211yesmaximal, characteristic, complemented
14#68539C211yesmaximal, characteristic, complemented
15#68537C211yesmaximal, characteristic, complemented
16#19181—31nomaximal, Sylow
17C4×#235C2211yescharacteristic
18#5470C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C3×C4×M16C2211yescharacteristic
23#5137—21no—
24#5137—21no—
25#5127—21no—
26#5127—21no—
27#5113—21no—
28#5106—21no—
29#4926C2211yescharacteristic
30#4851C2211yescharacteristic
31#4851C2211yescharacteristic
32#4850C2211yescharacteristic
33#4849C2211yescharacteristic
34#4847—21no—
35#4846C2211yescharacteristic
36#4846C2211yescharacteristic
37#4842C2211yescharacteristic
38#4841C2211yescharacteristic
39#4829—21no—
40#4733—21no—
41#4723—21no—
42#4716—21no—
43#4716—21no—
44#4705—21no—
45D3×#97—21no—
46#4699C2211yescharacteristic
47C4×Dic12—21no—
48C4×#204—21no—
49C8×D12C2211yescharacteristic
50C8×D12C2211yescharacteristic
51#4681C2211yescharacteristic
52#4681C2211yescharacteristic
53#4680C2211yescharacteristic
54#4679C2211yescharacteristic
55#4679C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4486C2211yescharacteristic
64#4482C2211yescharacteristic
65#4482C2211yescharacteristic
66#4481C2211yescharacteristic
67#4481C2211yescharacteristic
68#3185—31no—
69#3107—31no—
70#3076—31no—
71#2970—31no—
72#1704—31no—
73#1700—31no—
74#1692—31no—
75#1690—31no—
76#1683—31no—
77#1679—31no—
78#1671—31no—
79#1669—31no—
80#1565D311yescharacteristic, complemented
81#1554—31no—
82#1552—31no—
83C2×#235D411yescharacteristic
84C2×C3×M16D411yescharacteristic
85C2×C3×M16D411yescharacteristic
86C22×C3×C8—21no—
87C2×C3×C42C2311yescharacteristic
88C4×#67—21no—
89C4×#67—21no—
90#926—21no—
91#926—21no—
92#926D411yescharacteristic
93#924—21no—
94#924—21no—
95#924—21no—
96#924—21no—
97C2×Dic12—21no—
98C2×Dic12—21no—
99C2×Dic12—21no—
100C2×#204—21no—
101C2×D3×C8—21no—
102C2×D3×C8—21no—
103C4×D12D411yescharacteristic
104C4×D12C2311yescharacteristic
105#872—21no—
106C42×D3—21no—
107#870—21no—
108#870—21no—
109#869C2311yescharacteristic
110#869C2311yescharacteristic
111#869C2311yescharacteristic
112#869C2311yescharacteristic
113C4×Dic6—21no—
114C4×Dic6—21no—
115C4×Dic6—21no—
116C4×Dic6—21no—
117C4×Dic6D411yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×C4⋊C8—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C22⋊C8—21no—
125C3×#97D411yescharacteristic
126C3×#97D411yescharacteristic
127C3×C4×C8—21no—
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#830—21no—
131#821—21no—
132#821—21no—
133#821—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136#818—21no—
137#818—21no—
138#818—21no—
139#817—21no—
140#817—21no—
141#817—21no—
142#817—21no—
143#817—21no—
144#816—21no—
145#816—21no—
146#816—21no—
147#816—21no—
148#816—21no—
149#816—21no—
150#816—21no—
151#816—21no—
152#816—21no—
153#815—21no—
154#815—21no—
155Dic3⋊C8—21no—
156Dic3⋊C8—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160#533—31no—
161#533—31no—
162C4×#41—31no—
163#498—31no—
164#494—31no—
165#487—31no—
166#487—31no—
167#485—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#476—31no—
172#475—31no—
173#475—31no—
174#474—31no—
175#474—31no—
176#470—61no—
177#470—61no—
178#462—31no—
179#461—31no—
180#452—61no—
181#446—31no—
182#446—31no—
183#441—61no—
184#441—61no—
185C4×Q16—61no—
186C4×SD16—61no—
187#435—61no—
188#435—61no—
189C8×D4—31no—
190C8×D4—31no—
191#432—31no—
192#432—61no—
193#431C2×D311yescharacteristic
194#427—61no—
195C2×C4⋊C8C2×D311yescharacteristic
196#416—61no—
197C4×M16C2×D311yescharacteristic
198C2×#97—61no—
199#335C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#334C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4111yescharacteristic
214C2×#67—21no—
215#235—22no—
216#235—41no—
217C2×D12C2×D411yescharacteristic
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C2×D411yescharacteristic
226C3×M16—22no—
227C3×M16—41no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—42no—
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42#4111yescharacteristic
240C3×C42C2411yescharacteristic
241#217—21no—
242#217—41no—
243#217—41no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3C2×D411yescharacteristic
255C4⋊Dic3C2×D411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261Dic3⋊C4—41no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—41no—
266C4×Dic3—41no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269Dic12—42no—
270Dic12—42no—
271Dic12—42no—
272#204—42no—
273D3×C8—42no—
274D3×C8—42no—
275C2×#41—31no—
276C2×Q16—61no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×SD16—61no—
280C2×M16D1211yescharacteristic
281C2×M16D1211yescharacteristic
282C22×C8—21no—
283C22×C8—61no—
284C22×C8—61no—
285C4⋊Q8—31no—
286C4⋊Q8—31no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289#125—61no—
290#125—61no—
291C22⋊Q8—61no—
292C22⋊Q8—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305#117—31no—
306#117—61no—
307#117—61no—
308#117—61no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311#107—61no—
312#107—61no—
313#107—61no—
314#106—61no—
315#106—61no—
316#106—61no—
317#106—61no—
318#106—61no—
319C4⋊C8—21no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340C22⋊C8—21no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—61no—
345#97—61no—
346#97—61no—
347#97D1211yescharacteristic
348#97D1211yescharacteristic
349C4×C8—21no—
350C4×C8C22×D311yescharacteristic
351C4×C8C22×D311yescharacteristic
352C23×C3#12111yescharacteristic
353C22×D3—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4#12111yescharacteristic
360C2×C3×C4#12111yescharacteristic
361C2×C3×C4#12111yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365#67—42no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—21no—
374D12#13712yes—
375C4×D3—42no—
376C4×D3—41no—
377C4×D3—41no—
378C4×D3—81no—
379Dic6—21no—
380Dic6—21no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6—81no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—21no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—41no—
395C3×C8#13512yes—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—62no—
399#41—121no—
400C2×Q8—31no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×D4—31no—
406C2×D4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4C2×D1211yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4#23911yescharacteristic
413Q16—122no—
414Q16—122no—
415Q16—122no—
416SD16—122no—
417M16—22no—
418M16—41no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—42no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—122no—
427C2×C8—122no—
428C2×C8C2×D1211yescharacteristic
429C2×C8C2×D1211yescharacteristic
430C2×C8C2×D1211yescharacteristic
431C2×C8C2×D1211yescharacteristic
432C4⋊C4—31no—
433C4⋊C4—31no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C4⋊C4—121no—
447C4⋊C4—121no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—121no—
452C22⋊C4—121no—
453C42—61no—
454C42—61no—
455C42—61no—
456C42—121no—
457C42—121no—
458C42C2×D1211yescharacteristic
459C42D3×D411yescharacteristic
460C42#23911yescharacteristic
461C42C23×D311yescharacteristic
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3C2×#12111yescharacteristic
466C2×D3—22no—
467C2×D3—41no—
468C2×D3—41no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4C2×#13511yescharacteristic
474C3×C4C2×#13711yescharacteristic
475C3×C4#57511yescharacteristic
476C3×C4#57511yescharacteristic
477Dic3—41no—
478Dic3—41no—
479Dic3—41no—
480Dic3—81no—
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484C23—61no—
485C23#89511yescharacteristic
486Q8—61no—
487Q8—61no—
488Q8—121no—
489Q8—121no—
490Q8—121no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8—241no—
495D4—32no—
496D4—61no—
497D4—122no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—241no—
514C2×C4#89511yescharacteristic
515C2×C4#89511yescharacteristic
516C2×C4#89511yescharacteristic
517C2×C4C22×D1211yescharacteristic
518C2×C4C2×D3×D411yescharacteristic
519C2×C4C2×#23911yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—121no—
528C8—121no—
529C8#90412yes—
530C2×C3—41no—
531C2×C3#303511yescharacteristic
532C2×C3#305611yescharacteristic
533C2×C3#306911yescharacteristic
534D3—42no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—62no—
539C22—121no—
540C22—121no—
541C22C2×#89511yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—121no—
547C4—121no—
548C4—121no—
549C4—241no—
550C4—241no—
551C4—241no—
552C4—241no—
553C4C2×#90411yescharacteristic
554C4#574811yescharacteristic
555C4#577311yescharacteristic
556C4D3×#13711yescharacteristic
557C3#1918111yescharacteristic, complemented, Sylow
558C2—41no—
559C2—122no—
560C2#7902111yescharacteristic
561C2#7921211yescharacteristic
562C2#7926211yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1565C2
socleC22×C3C2×#121
prime radicalC2×#235D4
hypercenter#1565D3
hypocenterC3#19181
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
3. #78540C2semidirect
5. C3×#1565C2semidirect
6. #70781C2semidirect
7. #70439C2semidirect
8. #68595C2semidirect
9. #68593C2semidirect
10. #68581C2semidirect
11. #68577C2semidirect
12. #68555C2semidirect
13. #68551C2semidirect
14. #68539C2semidirect
15. #68537C2semidirect
80. #1565D3semidirect
557. C3#19181semidirect

Maximal subgroups

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

subgroupindex
#793022
#790252
#785402
#785242
C3×#15652
#707812
#704392
#685952
#685932
#685812
#685772
#685552
#685512
#685392
#685372
#191813