Finite Group Databasebeta

← #10897649

Subgroups — #10897649

Order 768.

662 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82209C211yesmaximal, characteristic, complemented
2#78980C211yesmaximal, characteristic, complemented
3C3×#1897C211yesmaximal, characteristic, complemented
4#71715C211yesmaximal, characteristic
5#71714C211yesmaximal, characteristic, complemented
6#71591C211yesmaximal, 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#29385—31nomaximal, Sylow
17C3×#525C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
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
36#5000—22no—
37#5000—22no—
38C2×#805C2211yescharacteristic
39C2×#805C2211yescharacteristic
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#2921—31no—
58#2102—31no—
59#2091—31no—
60#2050—31no—
61#2010—31no—
62#1932—31no—
63#1932—31no—
64#1897—31no—
65#1897D311yescharacteristic, complemented
66#1891—31no—
67#1891—31no—
68#1861—31no—
69#1861—31no—
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C2×#235C2311yescharacteristic
74C2×C3×M16C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C3×#123—21no—
77C3×C22⋊Q8—21no—
78C3×#121—21no—
79C3×#120—21no—
80C3×C4×D4—41no—
81C3×#117—21no—
82C3×#117—21no—
83C3×#117—22no—
84C3×#117C2311yescharacteristic
85C2×C3×C4⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#953—21no—
90#951—21no—
91#950—22no—
92#949—21no—
93#948—22no—
94#948—41no—
95#942—21no—
96#932—21no—
97#930—21no—
98#929—21no—
99C4×#67—41no—
100#926C2311yescharacteristic
101#924—21no—
102C2×#208—41no—
103C2×#208C2311yescharacteristic
104C2×#208C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109#907—22no—
110D3×M16—41no—
111#901—41no—
112C3×#104—21no—
113C3×#104—21no—
114C3×#104—22no—
115C3×C22⋊C8—41no—
116#837—41no—
117#836—21no—
118#833—21no—
119#830—22no—
120#830—22no—
121#830—41no—
122#824—21no—
123#823—21no—
124#822—22no—
125#822—41no—
126#819—41no—
127#805—21no—
128#805—21no—
129#805—22no—
130#805—41no—
131C4⋊(C3⋊C8)—22no—
132C4⋊(C3⋊C8)—22no—
133#803—21no—
134#803D412yescomplemented
135C4×C3⋊C8—21no—
136C4×C3⋊C8D412yescomplemented
137#575—31no—
138#568—31no—
139#568—31no—
140#525—31no—
141#525C2×D311yescharacteristic
142C2×#117C2×D311yescharacteristic
143#444—62no—
144#443—62no—
145#433—62no—
146#433—62no—
147#430—31no—
148#430—31no—
149#430—31no—
150#423—31no—
151#423—31no—
152#421—31no—
153#421—31no—
154C2×#104—31no—
155C2×#104—31no—
156C2×#104C2×D311yescharacteristic
157C2×#104C2×D311yescharacteristic
158#413—31no—
159#413—31no—
160#407—31no—
161#407—31no—
162#407—31no—
163#407C2×D311yescharacteristic
164C4×M16—31no—
165C4×M16—31no—
166#344—31no—
167#344—31no—
168#344—31no—
169#344—31no—
170#339—31no—
171#339—31no—
172#339C2×D311yescharacteristic
173#339C2×D311yescharacteristic
174#338—31no—
175H(3,ℤ/4)—31no—
176C24×C3#4111yescharacteristic
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C3×#41—21no—
181C2×C3×Q8#4111yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4—41no—
184C2×C3×D4#4111yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
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#4111yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C2×#67—21no—
197C2×#67—41no—
198#235—21no—
199#235—21no—
200#235—21no—
201#235—21no—
202C2×D12#4111yescharacteristic
203C2×C4×D3—21no—
204C2×Dic6#4111yescharacteristic
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—21no—
208C3×M16—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—41no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—41no—
219C3×C22⋊C4—41no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42C2×D411yescharacteristic
224C3×C42C2×D411yescharacteristic
225C3×C42#4112yes—
226#217—21no—
227#217—41no—
228#216—22no—
229#215—22no—
230#214—22no—
231#213—22no—
232(C2×D3)⋊C4—41no—
233C4⋊Dic3—21no—
234Dic3⋊C4—41no—
235C4×Dic3—21no—
236#208—21no—
237#208—21no—
238#208—21no—
239#208—21no—
240#208—21no—
241#208—21no—
242#208—21no—
243#208—21no—
244#208—21no—
245#208—21no—
246#208—21no—
247#208—21no—
248#208—42no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—42no—
254C2×C3⋊C8C2×D412yes—
255C2×C3⋊C8C2×D412yes—
256ΓL(1,𝔽25)—42no—
257D3×C8—42no—
258C2×#41—31no—
259C2×#41C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261#137—61no—
262#137—61no—
263AGL(1,ℤ/8)—61no—
264AGL(1,ℤ/8)—61no—
265#135—62no—
266#131—62no—
267#131—62no—
268#131—121no—
269#131—121no—
270C2×M16—31no—
271C2×M16—31no—
272C2×M16—31no—
273C2×M16—31no—
274C2×M16—31no—
275C2×M16—31no—
276C2×M16—121no—
277C2×M16—121no—
278C2×M16C22×D311yescharacteristic
279C2×M16C22×D311yescharacteristic
280#123—21no—
281#123—61no—
282C22⋊Q8—21no—
283C22⋊Q8—61no—
284#121—21no—
285#121—61no—
286#120—21no—
287#120—61no—
288C4×D4—41no—
289C4×D4—121no—
290#117—21no—
291#117—21no—
292#117—22no—
293#117—31no—
294#117C22×D311yescharacteristic
295C2×C4⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42C22×D311yescharacteristic
298C2×C42C22×D311yescharacteristic
299#108—62no—
300#108—121no—
301#108—121no—
302C4⋊C8—62no—
303C4⋊C8—62no—
304#104—21no—
305#104—21no—
306#104—22no—
307#104—61no—
308#104—61no—
309#104—62no—
310#104—121no—
311#104—121no—
312#101—61no—
313#101—61no—
314#100—61no—
315#100—61no—
316C22⋊C8—41no—
317C22⋊C8—62no—
318C22⋊C8—62no—
319C22⋊C8—121no—
320#97—32no—
321#97—61no—
322C4×C8—32no—
323C4×C8—61no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—81no—
329C23×C3C2×#4111yescharacteristic
330C22×D3—21no—
331C3×Q8—21no—
332C3×Q8—21no—
333C3×D4—21no—
334C3×D4—21no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—82no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—81no—
353C2×C3×C4—81no—
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360#67—41no—
361#67—41no—
362#67—82no—
363C2×Dic3—21no—
364C2×Dic3—41no—
365D12—21no—
366D12—21no—
367C4×D3—41no—
368C4×D3—41no—
369Dic6—21no—
370Dic6—21no—
371C3×C8—41no—
372C3×C8—41no—
373C3⋊C8—22no—
374C3⋊C8—22no—
375C3⋊C8—41no—
376C3⋊C8—41no—
377C3⋊C8—41no—
378C3⋊C8—41no—
379C24#23711yescharacteristic
380#41—21no—
381#41—21no—
382#41—21no—
383#41—21no—
384#41—61no—
385#41—61no—
386#41—61no—
387#41—61no—
388C2×Q8—31no—
389C2×Q8#23511yescharacteristic
390C2×D4—21no—
391C2×D4—31no—
392C2×D4—41no—
393C2×D4—61no—
394C2×D4—121no—
395C2×D4#23511yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—61no—
406C22×C4#23711yescharacteristic
407C22×C4C23×D311yescharacteristic
408Q16—62no—
409SD16—62no—
410SD16—62no—
411D8—62no—
412M16—21no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—61no—
417M16—61no—
418M16—61no—
419M16—61no—
420M16—61no—
421M16—61no—
422M16—61no—
423M16—61no—
424M16—61no—
425M16—61no—
426M16—61no—
427M16—61no—
428M16—122no—
429M16—122no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—32no—
433C2×C8—32no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—122no—
439C2×C8—122no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—41no—
444C4⋊C4—61no—
445C4⋊C4—121no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—21no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—61no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—61no—
458C42D3×D411yescharacteristic
459C42D3×D411yescharacteristic
460C42#23712yes—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—42no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—81no—
470C22×C3#52511yescharacteristic
471C22×C3#55111yescharacteristic
472C22×C3#55111yescharacteristic
473C2×D3—41no—
474C2×D3—81no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—42no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—81no—
481C3×C4#52511yescharacteristic
482C3×C4#53511yescharacteristic
483C3×C4#55311yescharacteristic
484C3×C4#55311yescharacteristic
485Dic3—41no—
486Dic3—81no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—61no—
492C23—81no—
493C23C2×#23711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—61no—
497Q8—61no—
498D4—21no—
499D4—21no—
500D4—41no—
501D4—41no—
502D4—61no—
503D4—61no—
504D4—82no—
505D4—121no—
506D4—121no—
507D4—242no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—61no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4C2×#23511yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23711yescharacteristic
531C2×C4#100411yescharacteristic
532C2×C4D3×#4111yescharacteristic
533C2×C4#100911yescharacteristic
534C8—41no—
535C8—41no—
536C8—62no—
537C8—62no—
538C8—121no—
539C8—121no—
540C8—121no—
541C8—121no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—81no—
546C2×C3—81no—
547C2×C3#259811yescharacteristic
548D3—81no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—81no—
558C22—121no—
559C22—241no—
560C22#549511yescharacteristic
561C22#558611yescharacteristic
562C22#565811yescharacteristic
563C4—41no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—81no—
569C4—121no—
570C4—241no—
571C4#554711yescharacteristic
572C4#556511yescharacteristic
573C4#560711yescharacteristic
574C4#561411yescharacteristic
575C3#2938511yescharacteristic, complemented, Sylow
576C2—21no—
577C2—21no—
578C2—21no—
579C2—81no—
580C2—81no—
581C2—241no—
582C2#7477511yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1897C2
socleC2×C3#2598
prime radical#71715C2
hypercenter#1897D3
hypocenterC3#29385
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1897

Complements

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

NH ≅ G/Ntype
1. #82209C2semidirect
2. #78980C2semidirect
3. C3×#1897C2semidirect
5. #71714C2semidirect
6. #71591C2semidirect
7. #70379C2semidirect
8. #70378C2semidirect
9. #70375C2semidirect
10. #69992C2semidirect
11. #69987C2semidirect
12. #68773C2semidirect
13. #68751C2semidirect
14. #68750C2semidirect
15. #68749C2semidirect
65. #1897D3semidirect
134. #803D4semidirect
136. C4×C3⋊C8D4semidirect
575. C3#29385semidirect

Maximal subgroups

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

subgroupindex
#822092
#789802
C3×#18972
#717152
#717142
#715912
#703792
#703782
#703752
#699922
#699872
#687732
#687512
#687502
#687492
#293853