Finite Group Databasebeta

← #10780321

Subgroups — #10780321

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79322C211yesmaximal, characteristic, complemented
2#79062C211yesmaximal, characteristic
3#78528C211yesmaximal, characteristic, complemented
4#78511C211yesmaximal, characteristic, complemented
5C3×#1569C211yesmaximal, characteristic, complemented
6#70817C211yesmaximal, characteristic, complemented
7#70507C211yesmaximal, characteristic, complemented
8#68594C211yesmaximal, characteristic, complemented
9#68591C211yesmaximal, characteristic, complemented
10#68584C211yesmaximal, characteristic, complemented
11#68581C211yesmaximal, characteristic, complemented
12#68555C211yesmaximal, characteristic, complemented
13#68548C211yesmaximal, characteristic, complemented
14#68545C211yesmaximal, characteristic, complemented
15#68538C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17#5479C2211yescharacteristic
18#5477C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C3×C4×M16C2211yescharacteristic
23#5135—21no—
24#5134—21no—
25#5133—21no—
26#5133—21no—
27#5128—21no—
28#5127—21no—
29#4925C2211yescharacteristic
30#4851C2211yescharacteristic
31#4850C2211yescharacteristic
32#4850C2211yescharacteristic
33#4848C2211yescharacteristic
34#4847—21no—
35#4846C2211yescharacteristic
36#4844—21no—
37#4843C2211yescharacteristic
38#4842C2211yescharacteristic
39#4842C2211yescharacteristic
40#4736—21no—
41#4721—21no—
42#4714—21no—
43#4714—21no—
44#4714—21no—
45#4709—21no—
46#4708—21no—
47#4702—21no—
48#4694C2211yescharacteristic
49#4694C2211yescharacteristic
50#4688C2211yescharacteristic
51C8×D12C2211yescharacteristic
52#4681C2211yescharacteristic
53#4681C2211yescharacteristic
54#4679C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4481C2211yescharacteristic
65#4481C2211yescharacteristic
66#4481C2211yescharacteristic
67#4481C2211yescharacteristic
68#3251—31no—
69#3238—31no—
70#3141—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1698—31no—
75#1697—31no—
76#1684—31no—
77#1683—31no—
78#1678—31no—
79#1677—31no—
80#1569D311yescharacteristic, complemented
81#1553—31no—
82#1548—31no—
83C2×#235—21no—
84C22×Dic6—21no—
85C2×C3×M16—21no—
86C2×C3×M16D411yescharacteristic
87C2×C3×M16D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#930—21no—
90#930—21no—
91#930D411yescharacteristic
92#929—21no—
93C4×#67—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96#924D411yescharacteristic
97C2×D24—21no—
98C2×#204—21no—
99C2×ΓL(1,𝔽25)—21no—
100C2×D3×C8—21no—
101#876—21no—
102#875—21no—
103#875D411yescharacteristic
104#874C2311yescharacteristic
105C4×D12—21no—
106C4×D12—21no—
107C4×D12C2311yescharacteristic
108#872—21no—
109#870D411yescharacteristic
110#869—21no—
111#869—21no—
112#869—21no—
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6C2311yescharacteristic
117C3×C4⋊C8—21no—
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C22⋊C8—21no—
123C3×#97—21no—
124C3×#97D411yescharacteristic
125C3×#97D411yescharacteristic
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128#830—21no—
129#821—21no—
130#821—21no—
131#821—21no—
132#821—21no—
133#821—21no—
134#821—21no—
135#821—21no—
136#821—21no—
137#821—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140#818—21no—
141#818—21no—
142#818—21no—
143#817—21no—
144#817—21no—
145#817—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—
154Dic3⋊C8—21no—
155Dic3⋊C8—21no—
156C8×Dic3—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8—21no—
160#532—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#489—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#483—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477—31no—
176#476—31no—
177#476—31no—
178#475—31no—
179#474—31no—
180#469—61no—
181#468—61no—
182#463—61no—
183#460—31no—
184#459—31no—
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432C2×D311yescharacteristic
193#431—31no—
194#423C2×D311yescharacteristic
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16C2×D311yescharacteristic
199#335C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#334C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4111yescharacteristic
214C2×#67—21no—
215C2×#67—41no—
216C22×Dic3—41no—
217#235—42no—
218#235—42no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—41no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—41no—
229C2×Dic6—82no—
230C2×Dic6C2×D411yescharacteristic
231C3×M16—41no—
232C3×M16—41no—
233C3×M16—81no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2×D411yescharacteristic
244C3×C42#4111yescharacteristic
245C3×C42C2411yescharacteristic
246#217—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—41no—
252(C2×D3)⋊C4—41no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—41no—
260C4⋊Dic3—41no—
261C4⋊Dic3C2×D411yescharacteristic
262C4⋊Dic3C2×D411yescharacteristic
263Dic3⋊C4—21no—
264Dic3⋊C4—21no—
265Dic3⋊C4—21no—
266Dic3⋊C4—21no—
267C4×Dic3—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270D24—81no—
271#204—81no—
272ΓL(1,𝔽25)—42no—
273D3×C8—42no—
274C2×#41—61no—
275C22×Q8—61no—
276C2×SD16—61no—
277C2×D8—61no—
278C2×M16—21no—
279C2×M16—61no—
280C2×M16D1211yescharacteristic
281C2×M16D1211yescharacteristic
282C22×C8—61no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8—31no—
286C4⋊Q8—61no—
287C4⋊Q8—61no—
288C4⋊Q8—61no—
289#127—31no—
290#126—61no—
291#125—31no—
292#124—31no—
293#124—61no—
294#123—61no—
295C22⋊Q8—31no—
296#121—31no—
297#121—61no—
298#121—61no—
299C4×Q8—31no—
300C4×D4—31no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4—61no—
304#117—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C42C22×D311yescharacteristic
308#107—61no—
309#107—61no—
310#107—61no—
311#106—61no—
312#106—61no—
313#106—61no—
314C4⋊C8—21no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8C22×D311yescharacteristic
320C4⋊C8C22×D311yescharacteristic
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—61no—
324#103—61no—
325#103—61no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#102—61no—
331#102—61no—
332#102—61no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—21no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—21no—
344#97—61no—
345#97D1211yescharacteristic
346#97D1211yescharacteristic
347C4×C8—61no—
348C4×C8—61no—
349C4×C8C22×D311yescharacteristic
350C4×C8C22×D311yescharacteristic
351C23×C3#12111yescharacteristic
352C22×D3—21no—
353C22×D3—41no—
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—81no—
366#67—82no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—82no—
374D12—21no—
375D12—21no—
376D12—41no—
377D12—41no—
378D12—41no—
379D12—41no—
380D12—81no—
381C4×D3—41no—
382C4×D3—41no—
383C4×D3—82no—
384Dic6—21no—
385Dic6—21no—
386Dic6—41no—
387Dic6—41no—
388Dic6—82no—
389Dic6—81no—
390Dic6—81no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—122no—
402#41—122no—
403C2×Q8—31no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—121no—
408C2×Q8—242no—
409C2×D4—31no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—121no—
415C22×C4—61no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4C2×D1211yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4#23911yescharacteristic
421SD16—241no—
422D8—241no—
423M16—41no—
424M16—41no—
425M16—81no—
426M16—122no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—122no—
434C2×C8C2×D1211yescharacteristic
435C2×C8C2×D1211yescharacteristic
436C2×C8C2×D1211yescharacteristic
437C2×C8C2×D1211yescharacteristic
438C4⋊C4—31no—
439C4⋊C4—31no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—121no—
451C4⋊C4—121no—
452C22⋊C4—61no—
453C22⋊C4—61no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C42—61no—
460C42C2×D1211yescharacteristic
461C42D3×D411yescharacteristic
462C42#23911yescharacteristic
463C42C23×D311yescharacteristic
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3C2×#12111yescharacteristic
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—81no—
472C2×D3—81no—
473C2×D3—81no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4#57511yescharacteristic
479C3×C4#57511yescharacteristic
480C3×C4#57711yescharacteristic
481C3×C4#57811yescharacteristic
482Dic3—41no—
483Dic3—41no—
484Dic3—41no—
485Dic3—81no—
486Dic3—81no—
487Dic3—81no—
488C23—61no—
489C23—121no—
490C23#89511yescharacteristic
491Q8—61no—
492Q8—61no—
493Q8—121no—
494Q8—121no—
495Q8—241no—
496Q8—241no—
497Q8—242no—
498D4—61no—
499D4—61no—
500D4—121no—
501D4—121no—
502D4—121no—
503D4—121no—
504D4—241no—
505D4—241no—
506D4—242no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—242no—
521C2×C4—242no—
522C2×C4#89511yescharacteristic
523C2×C4#89511yescharacteristic
524C2×C4#89511yescharacteristic
525C2×C4C22×D1211yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4C2×#23911yescharacteristic
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—121no—
537C8—121no—
538C2×C3—41no—
539C2×C3#304011yescharacteristic
540C2×C3#306311yescharacteristic
541C2×C3#306811yescharacteristic
542D3—81no—
543D3—161no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—121no—
548C22—121no—
549C22—121no—
550C22—241no—
551C22—241no—
552C22—241no—
553C22C2×#89511yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—121no—
559C4—121no—
560C4—121no—
561C4—241no—
562C4—241no—
563C4—241no—
564C4#574811yescharacteristic
565C4#574811yescharacteristic
566C4#577511yescharacteristic
567C4#577911yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—41no—
570C2—241no—
571C2—481no—
572C2#7905811yescharacteristic
573C2#7921411yescharacteristic
574C2#7926111yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1569C2
socleC22×C3C2×#121
prime radical#79062C2
hypercenter#1569D3
hypocenterC3#19382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1569

Complements

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

NH ≅ G/Ntype
1. #79322C2semidirect
3. #78528C2semidirect
4. #78511C2semidirect
5. C3×#1569C2semidirect
6. #70817C2semidirect
7. #70507C2semidirect
8. #68594C2semidirect
9. #68591C2semidirect
10. #68584C2semidirect
11. #68581C2semidirect
12. #68555C2semidirect
13. #68548C2semidirect
14. #68545C2semidirect
15. #68538C2semidirect
80. #1569D3semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
#793222
#790622
#785282
#785112
C3×#15692
#708172
#705072
#685942
#685912
#685842
#685812
#685552
#685482
#685452
#685382
#193823