Finite Group Databasebeta

← #10897638

Subgroups — #10897638

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#82049C211yesmaximal, characteristic, complemented
2#78980C211yesmaximal, characteristic, complemented
3C3×#1896C211yesmaximal, characteristic, complemented
4#71715C211yesmaximal, characteristic
5#71714C211yesmaximal, characteristic, complemented
6#71532C211yesmaximal, characteristic, complemented
7#70379C211yesmaximal, characteristic, complemented
8#70378C211yesmaximal, characteristic, complemented
9#70375C211yesmaximal, characteristic, complemented
10#69992C211yesmaximal, characteristic, complemented
11#69987C211yesmaximal, characteristic, complemented
12#68773C211yesmaximal, characteristic, complemented
13#68751C211yesmaximal, characteristic, complemented
14#68750C211yesmaximal, characteristic, complemented
15#68749C211yesmaximal, characteristic, complemented
16#29384—31nomaximal, Sylow
17C3×#525C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19#5822C2211yescharacteristic
20#5819C2211yescharacteristic
21#5791C2211yescharacteristic
22#5745C2211yescharacteristic
23C2×C3×#104C2211yescharacteristic
24C2×C3×#104C2211yescharacteristic
25C3×#407C2211yescharacteristic
26#5237C2211yescharacteristic
27#5236C2211yescharacteristic
28#5207C2211yescharacteristic
29#5207C2211yescharacteristic
30#5207C2211yescharacteristic
31#5132C2211yescharacteristic
32#5125C2211yescharacteristic
33#5124C2211yescharacteristic
34#5124C2211yescharacteristic
35#5005C2211yescharacteristic
36C2×#805C2211yescharacteristic
37C2×#805C2211yescharacteristic
38#4926—22no—
39#4925—22no—
40#4924C2211yescharacteristic
41#4924C2211yescharacteristic
42C4×#208C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4824—22no—
45#4823—22no—
46C3×#339C2211yescharacteristic
47C3×#339C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4559C2211yescharacteristic
50#4556C2211yescharacteristic
51#4533C2211yescharacteristic
52#4533C2211yescharacteristic
53#4531C2211yescharacteristic
54#4523C2211yescharacteristic
55#4523C2211yescharacteristic
56#2956—31no—
57#2920—31no—
58#2102—31no—
59#2091—31no—
60#2050—31no—
61#2010—31no—
62#1932—31no—
63#1932—31no—
64#1896—31no—
65#1896D311yescharacteristic, complemented
66#1891—31no—
67#1891—31no—
68#1861—31no—
69#1861—31no—
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C2×#235C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C2×C3×M16C2311yescharacteristic
77C3×#123—21no—
78C3×C22⋊Q8—21no—
79C3×#121—21no—
80C3×#120—21no—
81C3×C4×D4—41no—
82C3×#117C2311yescharacteristic
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42—21no—
86C2×C3×C42—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90#953—21no—
91#951—21no—
92#950—22no—
93#949—21no—
94#948—22no—
95#948—41no—
96#942—21no—
97#932—21no—
98#930—21no—
99#929—21no—
100C4×#67—41no—
101#926C2311yescharacteristic
102#924—21no—
103C2×#208—41no—
104C2×#208C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C2×#208C2311yescharacteristic
110#907—22no—
111D3×M16—41no—
112#901—41no—
113C3×#104—21no—
114C3×#104—21no—
115C3×#104—21no—
116C3×#104—21no—
117C3×C22⋊C8—41no—
118#837—41no—
119#836—21no—
120#833—21no—
121#830—22no—
122#830—22no—
123#830—41no—
124#824—21no—
125#823—21no—
126#822—22no—
127#822—41no—
128#819—41no—
129#805—21no—
130#805—21no—
131#805—21no—
132#805—21no—
133#805—41no—
134C4⋊(C3⋊C8)—22no—
135C4⋊(C3⋊C8)—22no—
136#803—21no—
137#803D412yescomplemented
138C4×C3⋊C8—21no—
139C4×C3⋊C8D412yescomplemented
140#575—31no—
141#568—31no—
142#568—31no—
143#525—31no—
144#525C2×D311yescharacteristic
145C22×C42C2×D311yescharacteristic
146#444—62no—
147#443—62no—
148#432—62no—
149#431—62no—
150#430—31no—
151#430—31no—
152#430—31no—
153#423—31no—
154#423—31no—
155#421—31no—
156#421—31no—
157C2×#104—31no—
158C2×#104—31no—
159C2×#104C2×D311yescharacteristic
160C2×#104C2×D311yescharacteristic
161#413—31no—
162#413—31no—
163#407—31no—
164#407—31no—
165#407—31no—
166#407C2×D311yescharacteristic
167C4×M16—31no—
168C4×M16—31no—
169#344—31no—
170#344—31no—
171#344—31no—
172#344—31no—
173#339—31no—
174#339—31no—
175#339C2×D311yescharacteristic
176#339C2×D311yescharacteristic
177#338—31no—
178H(3,ℤ/4)—31no—
179C24×C3#4111yescharacteristic
180C3×#41—21no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C2×C3×Q8#4111yescharacteristic
185C2×C3×D4—21no—
186C2×C3×D4—41no—
187C2×C3×D4#4111yescharacteristic
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—44no—
202C22×C3×C4#4111yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C2×#67—21no—
205C2×#67—41no—
206#235—21no—
207#235—21no—
208#235—21no—
209#235—21no—
210C2×D12#4111yescharacteristic
211C2×C4×D3—21no—
212C2×Dic6#4111yescharacteristic
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—21no—
216C3×M16—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—41no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—41no—
223C3×C22⋊C4—41no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42#4111yescharacteristic
234C3×C42#4111yescharacteristic
235#217—21no—
236#217—41no—
237#216—22no—
238#215—22no—
239#214—22no—
240#213—22no—
241(C2×D3)⋊C4—41no—
242C4⋊Dic3—21no—
243Dic3⋊C4—41no—
244C4×Dic3—21no—
245#208—21no—
246#208—21no—
247#208—21no—
248#208—21no—
249#208—21no—
250#208—21no—
251#208—21no—
252#208—21no—
253#208—21no—
254#208—21no—
255#208—21no—
256#208—21no—
257#208—42no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
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—21no—
270C23×C4C22×D311yescharacteristic
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—31no—
285C2×M16—31no—
286C2×M16—121no—
287C2×M16—121no—
288C2×M16C22×D311yescharacteristic
289C2×M16C22×D311yescharacteristic
290#123—21no—
291#123—61no—
292C22⋊Q8—21no—
293C22⋊Q8—61no—
294#121—21no—
295#121—61no—
296#120—21no—
297#120—61no—
298C4×D4—41no—
299C4×D4—121no—
300#117—31no—
301#117C22×D311yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42C22×D311yescharacteristic
308C2×C42C22×D311yescharacteristic
309#108—62no—
310#108—121no—
311#108—121no—
312C4⋊C8—62no—
313C4⋊C8—62no—
314#104—21no—
315#104—21no—
316#104—21no—
317#104—21no—
318#104—61no—
319#104—61no—
320#104—61no—
321#104—61no—
322#104—121no—
323#104—121no—
324#101—61no—
325#101—61no—
326#100—61no—
327#100—61no—
328C22⋊C8—41no—
329C22⋊C8—62no—
330C22⋊C8—62no—
331C22⋊C8—121no—
332#97—32no—
333#97—61no—
334C4×C8—32no—
335C4×C8—61no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—42no—
341C23×C3C2×#4111yescharacteristic
342C22×D3—21no—
343C3×Q8—21no—
344C3×Q8—21no—
345C3×D4—21no—
346C3×D4—21no—
347C3×D4—41no—
348C3×D4—41no—
349C3×D4—82no—
350C2×C3×C4—24no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377#67—41no—
378#67—41no—
379#67—82no—
380C2×Dic3—21no—
381C2×Dic3—41no—
382D12—21no—
383D12—21no—
384C4×D3—41no—
385C4×D3—41no—
386Dic6—21no—
387Dic6—21no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—22no—
391C3⋊C8—22no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C24#23711yescharacteristic
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—41no—
410C2×D4—61no—
411C2×D4—121no—
412C2×D4#23511yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—44no—
427C22×C4—61no—
428C22×C4#23711yescharacteristic
429C22×C4C23×D311yescharacteristic
430Q16—62no—
431SD16—62no—
432SD16—62no—
433D8—62no—
434M16—21no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—61no—
439M16—61no—
440M16—61no—
441M16—61no—
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—32no—
455C2×C8—32no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—122no—
461C2×C8—122no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—61no—
465C4⋊C4—121no—
466C22⋊C4—21no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—61no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—61no—
480C42D3×D411yescharacteristic
481C42D3×D411yescharacteristic
482C42#23711yescharacteristic
483C42#23711yescharacteristic
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—81no—
494C22×C3#52511yescharacteristic
495C22×C3#55111yescharacteristic
496C22×C3#55111yescharacteristic
497C2×D3—41no—
498C2×D3—81no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—81no—
506C3×C4#52511yescharacteristic
507C3×C4#53511yescharacteristic
508C3×C4#55311yescharacteristic
509C3×C4#55311yescharacteristic
510Dic3—41no—
511Dic3—81no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—42no—
517C23—61no—
518C23C2×#23711yescharacteristic
519Q8—21no—
520Q8—21no—
521Q8—61no—
522Q8—61no—
523D4—21no—
524D4—21no—
525D4—41no—
526D4—41no—
527D4—61no—
528D4—61no—
529D4—82no—
530D4—121no—
531D4—121no—
532D4—242no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—61no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4C2×#23511yescharacteristic
559C2×C4C2×D3×D411yescharacteristic
560C2×C4C2×#23711yescharacteristic
561C2×C4#100411yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C2×C4#100911yescharacteristic
564C8—41no—
565C8—41no—
566C8—62no—
567C8—62no—
568C8—121no—
569C8—121no—
570C8—121no—
571C8—121no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—42no—
576C2×C3—81no—
577C2×C3#259811yescharacteristic
578D3—81no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—121no—
590C22—241no—
591C22#549511yescharacteristic
592C22#558611yescharacteristic
593C22#565811yescharacteristic
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—81no—
601C4—121no—
602C4—241no—
603C4#554711yescharacteristic
604C4#556511yescharacteristic
605C4#560711yescharacteristic
606C4#561411yescharacteristic
607C3#2938411yescharacteristic, complemented, Sylow
608C2—21no—
609C2—21no—
610C2—21no—
611C2—42no—
612C2—81no—
613C2—241no—
614C2#7477511yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1896C2
socleC2×C3#2598
prime radical#71715C2
hypercenter#1896D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1896

Complements

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

NH ≅ G/Ntype
1. #82049C2semidirect
2. #78980C2semidirect
3. C3×#1896C2semidirect
5. #71714C2semidirect
6. #71532C2semidirect
7. #70379C2semidirect
8. #70378C2semidirect
9. #70375C2semidirect
10. #69992C2semidirect
11. #69987C2semidirect
12. #68773C2semidirect
13. #68751C2semidirect
14. #68750C2semidirect
15. #68749C2semidirect
65. #1896D3semidirect
137. #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
#820492
#789802
C3×#18962
#717152
#717142
#715322
#703792
#703782
#703752
#699922
#699872
#687732
#687512
#687502
#687492
#293843