Finite Group Databasebeta

← #10897640

Subgroups — #10897640

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78980C211yesmaximal, characteristic, complemented
2#78517C211yesmaximal, characteristic, complemented
3C3×#1932C211yesmaximal, characteristic, complemented
4#70735C211yesmaximal, characteristic, complemented
5#70729C211yesmaximal, characteristic, complemented
6#70406C211yesmaximal, characteristic
7#70398C211yesmaximal, characteristic, complemented
8#70382C211yesmaximal, characteristic, complemented
9#70144C211yesmaximal, characteristic, complemented
10#70136C211yesmaximal, characteristic, complemented
11#69874C211yesmaximal, characteristic, complemented
12#69036C211yesmaximal, characteristic, complemented
13#69024C211yesmaximal, characteristic, complemented
14#68756C211yesmaximal, characteristic, complemented
15#68749C211yesmaximal, characteristic, complemented
16#29384—31nomaximal, Sylow
17#5828C2211yescharacteristic
18#5822C2211yescharacteristic
19#5819C2211yescharacteristic
20#5745C2211yescharacteristic
21#5524C2211yescharacteristic
22C2×C42×D3C2211yescharacteristic
23C3×#430C2211yescharacteristic
24C3×#421C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#413C2211yescharacteristic
27C3×C4×M16C2211yescharacteristic
28C2×#837C2211yescharacteristic
29#5139C2211yescharacteristic
30C2×#825C2211yescharacteristic
31#5132C2211yescharacteristic
32#5126C2211yescharacteristic
33#5126C2211yescharacteristic
34#5126C2211yescharacteristic
35#5126C2211yescharacteristic
36#5124C2211yescharacteristic
37#5124C2211yescharacteristic
38#5118C2211yescharacteristic
39#5118C2211yescharacteristic
40C2×#805C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4824—22no—
43#4823—22no—
44#4705—22no—
45#4685—22no—
46C3×#344C2211yescharacteristic
47C3×#339C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4559C2211yescharacteristic
50#4559C2211yescharacteristic
51#4555C2211yescharacteristic
52#4533C2211yescharacteristic
53#4532C2211yescharacteristic
54#4531C2211yescharacteristic
55#4523C2211yescharacteristic
56#2956—31no—
57#2920—31no—
58#2102—31no—
59#2091—31no—
60#2050—31no—
61#2010—31no—
62#1932—31no—
63#1932D311yescharacteristic, complemented
64#1896—31no—
65#1896—31no—
66#1891—31no—
67#1891—31no—
68#1861—31no—
69#1861—31no—
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C2×C3×M16C2311yescharacteristic
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C3×#117C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81#951—21no—
82#950—22no—
83#949—21no—
84#948—22no—
85#948—41no—
86#946—21no—
87#942—21no—
88#938—21no—
89#937—21no—
90#935—21no—
91#932—21no—
92C4×#67—41no—
93#926C2311yescharacteristic
94C2×C4×Dic3—21no—
95C2×C4×Dic3C2311yescharacteristic
96C2×#208—41no—
97C2×#208C2311yescharacteristic
98C2×#208C2311yescharacteristic
99C2×#208C2311yescharacteristic
100C2×#208C2311yescharacteristic
101#907—22no—
102D3×M16—41no—
103#901—41no—
104#897—21no—
105#895—21no—
106#886—21no—
107#882—21no—
108C4×D12—41no—
109C42×D3—21no—
110C42×D3—21no—
111C42×D3—21no—
112C42×D3—21no—
113C3×#108—41no—
114C3×#104—21no—
115C3×#104—21no—
116C3×#104—41no—
117C3×#101—21no—
118C3×#100—21no—
119C3×#97—21no—
120C3×C4×C8—21no—
121#837—21no—
122#837—21no—
123#825—21no—
124#825—21no—
125#825—41no—
126#824—21no—
127#823—21no—
128#822—22no—
129#822—41no—
130(C2×D3)⋊C8—22no—
131(C2×D3)⋊C8—22no—
132(C2×D3)⋊C8—41no—
133(C2×D3)⋊C8—41no—
134Dic3⋊C8—22no—
135Dic3⋊C8—22no—
136#805—21no—
137#805—21no—
138#803D412yescomplemented
139C4×C3⋊C8D412yescomplemented
140#575—31no—
141#568—31no—
142#568—31no—
143#525—31no—
144#525—31no—
145C22×C42—31no—
146#444—62no—
147#443—62no—
148#432—62no—
149#431—62no—
150#430—31no—
151#430—31no—
152#430C2×D311yescharacteristic
153#423—31no—
154#423—31no—
155#421—31no—
156#421C2×D311yescharacteristic
157C2×#104—31no—
158C2×#104—31no—
159C2×#104—31no—
160C2×#104C2×D311yescharacteristic
161#413—31no—
162#413C2×D311yescharacteristic
163#407—31no—
164#407—31no—
165#407—31no—
166#407—31no—
167C4×M16—31no—
168C4×M16C2×D311yescharacteristic
169#344—31no—
170#344—31no—
171#344—31no—
172#344C2×D311yescharacteristic
173#339—31no—
174#339—31no—
175#339—31no—
176#339C2×D311yescharacteristic
177#338—31no—
178H(3,ℤ/4)—31no—
179C23×D3#4111yescharacteristic
180C3×#41—21no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C2×C3×Q8#4111yescharacteristic
185C2×C3×D4—21no—
186C2×C3×D4#4111yescharacteristic
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2411yescharacteristic
190C2×#67—21no—
191C2×#67—41no—
192C22×Dic3—21no—
193C22×Dic3#4111yescharacteristic
194#235—21no—
195#235—21no—
196#235—21no—
197#235—21no—
198C2×D12—41no—
199C2×D12#4111yescharacteristic
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—44no—
212C2×Dic6#4111yescharacteristic
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—21no—
216C3×M16—21no—
217C3×M16—21no—
218C3×M16—21no—
219C3×M16—21no—
220C3×M16—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C3×C4⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231#217—21no—
232#217—41no—
233#216—22no—
234#215—22no—
235#214—22no—
236#213—22no—
237(C2×D3)⋊C4—41no—
238(C2×D3)⋊C4—41no—
239(C2×D3)⋊C4—41no—
240C4⋊Dic3—21no—
241C4⋊Dic3—41no—
242Dic3⋊C4—41no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3#4111yescharacteristic
250C4×Dic3#4111yescharacteristic
251#208—21no—
252#208—21no—
253#208—21no—
254#208—21no—
255#208—21no—
256#208—21no—
257#208—21no—
258#208—21no—
259#208—42no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—42no—
263C2×C3⋊C8C2×D412yes—
264C2×C3⋊C8C2×D412yes—
265ΓL(1,𝔽25)—42no—
266D3×C8—42no—
267C2×#41—31no—
268C2×#41C22×D311yescharacteristic
269C23×C4—31no—
270C23×C4—61no—
271#137—61no—
272#137—61no—
273AGL(1,ℤ/8)—61no—
274AGL(1,ℤ/8)—61no—
275#135—62no—
276#131—62no—
277#131—62no—
278#131—121no—
279#131—121no—
280C2×M16—31no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—121no—
285C2×M16—121no—
286C2×M16C22×D311yescharacteristic
287C2×M16C22×D311yescharacteristic
288C2×M16C22×D311yescharacteristic
289C2×M16C22×D311yescharacteristic
290#123—61no—
291#123—61no—
292C22⋊Q8—61no—
293C22⋊Q8—61no—
294#121—61no—
295#121—61no—
296#120—61no—
297#120—61no—
298C4×D4—121no—
299C4×D4—121no—
300#117—31no—
301#117C22×D311yescharacteristic
302C2×C42—31no—
303C2×C42—61no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#108—41no—
310#108—62no—
311#108—121no—
312C4⋊C8—62no—
313C4⋊C8—62no—
314#104—21no—
315#104—21no—
316#104—41no—
317#104—61no—
318#104—61no—
319#104—61no—
320#104—61no—
321#104—61no—
322#104—61no—
323#104—121no—
324#101—21no—
325#101—61no—
326#100—21no—
327#100—61no—
328C22⋊C8—62no—
329C22⋊C8—62no—
330C22⋊C8—121no—
331C22⋊C8—121no—
332#97—21no—
333#97—32no—
334C4×C8—21no—
335C4×C8—32no—
336C23×C3—21no—
337C23×C3C2×#4111yescharacteristic
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—42no—
343C3×Q8—21no—
344C3×Q8—21no—
345C3×D4—21no—
346C3×D4—21no—
347C3×D4—41no—
348C3×D4—41no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362#67—41no—
363#67—41no—
364#67—82no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373D12—21no—
374D12—21no—
375D12—82no—
376C4×D3—24no—
377C4×D3—42no—
378C4×D3—42no—
379C4×D3—42no—
380C4×D3—41no—
381C4×D3—41no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—41no—
386Dic6—21no—
387Dic6—21no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—22no—
393C3⋊C8—22no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C24—31no—
397#41—21no—
398#41—21no—
399#41—21no—
400#41—21no—
401#41—61no—
402#41—61no—
403#41—61no—
404#41—61no—
405C2×Q8—31no—
406C2×Q8#23511yescharacteristic
407C2×D4—21no—
408C2×D4—31no—
409C2×D4—61no—
410C2×D4—121no—
411C2×D4—121no—
412C2×D4#23511yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—31no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—124no—
429C22×C4C23×D311yescharacteristic
430Q16—62no—
431SD16—62no—
432SD16—62no—
433D8—62no—
434M16—21no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—21no—
439M16—21no—
440M16—21no—
441M16—21no—
442M16—61no—
443M16—61no—
444M16—61no—
445M16—61no—
446M16—61no—
447M16—61no—
448M16—61no—
449M16—61no—
450M16—122no—
451M16—122no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—32no—
457C2×C8—32no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—122no—
461C2×C8—122no—
462C4⋊C4—21no—
463C4⋊C4—61no—
464C4⋊C4—121no—
465C4⋊C4—121no—
466C22⋊C4—21no—
467C22⋊C4—61no—
468C22⋊C4—121no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C42—21no—
473C42—21no—
474C42—31no—
475C42—31no—
476C42—61no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—61no—
482C42D3×D411yescharacteristic
483C42D3×D411yescharacteristic
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—81no—
487C22×C3#52511yescharacteristic
488C22×C3#55111yescharacteristic
489C22×C3#55111yescharacteristic
490C2×D3—22no—
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—81no—
499C3×C4—22no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—81no—
503C3×C4#52511yescharacteristic
504C3×C4#53511yescharacteristic
505C3×C4#55311yescharacteristic
506C3×C4#55311yescharacteristic
507Dic3—22no—
508Dic3—41no—
509Dic3—41no—
510Dic3—41no—
511Dic3—81no—
512C23—21no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—122no—
518C23D3×#4111yescharacteristic
519Q8—21no—
520Q8—21no—
521Q8—61no—
522Q8—61no—
523D4—21no—
524D4—21no—
525D4—41no—
526D4—41no—
527D4—61no—
528D4—61no—
529D4—121no—
530D4—121no—
531D4—242no—
532D4—242no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—64no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4C2×#23511yescharacteristic
559C2×C4C2×D3×D411yescharacteristic
560C2×C4#100411yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C2×C4#100911yescharacteristic
564C8—41no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8—62no—
569C8—62no—
570C8—121no—
571C8—121no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—81no—
576C2×C3#259811yescharacteristic
577D3—42no—
578D3—81no—
579C22—41no—
580C22—41no—
581C22—62no—
582C22—62no—
583C22—62no—
584C22—81no—
585C22—121no—
586C22—121no—
587C22—121no—
588C22—121no—
589C22—121no—
590C22—241no—
591C22#552711yescharacteristic
592C22#565611yescharacteristic
593C22#567911yescharacteristic
594C4—22no—
595C4—41no—
596C4—41no—
597C4—62no—
598C4—81no—
599C4—121no—
600C4—121no—
601C4—121no—
602C4—241no—
603C4#554011yescharacteristic
604C4#554711yescharacteristic
605C4#560711yescharacteristic
606C4#570911yescharacteristic
607C3#2938411yescharacteristic, complemented, Sylow
608C2—21no—
609C2—21no—
610C2—21no—
611C2—81no—
612C2—122no—
613C2—241no—
614C2#7589511yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1932C2
socleC2×C3#2598
prime radical#70406C2
hypercenter#1932D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1932

Complements

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

NH ≅ G/Ntype
1. #78980C2semidirect
2. #78517C2semidirect
3. C3×#1932C2semidirect
4. #70735C2semidirect
5. #70729C2semidirect
7. #70398C2semidirect
8. #70382C2semidirect
9. #70144C2semidirect
10. #70136C2semidirect
11. #69874C2semidirect
12. #69036C2semidirect
13. #69024C2semidirect
14. #68756C2semidirect
15. #68749C2semidirect
63. #1932D3semidirect
138. #803D4semidirect
139. C4×C3⋊C8D4semidirect
607. C3#29384semidirect

Maximal subgroups

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

subgroupindex
#789802
#785172
C3×#19322
#707352
#707292
#704062
#703982
#703822
#701442
#701362
#698742
#690362
#690242
#687562
#687492
#293843