Finite Group Databasebeta

← #10780325

Subgroups — #10780325

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#83217C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic, complemented
3#79280C211yesmaximal, characteristic, complemented
4#79062C211yesmaximal, characteristic
5C3×#1548C211yesmaximal, characteristic, complemented
6#70775C211yesmaximal, characteristic, complemented
7#70492C211yesmaximal, characteristic, complemented
8#68515C211yesmaximal, characteristic, complemented
9#68514C211yesmaximal, characteristic, complemented
10#68511C211yesmaximal, characteristic, complemented
11#68510C211yesmaximal, characteristic, complemented
12#68503C211yesmaximal, characteristic, complemented
13#68502C211yesmaximal, characteristic, complemented
14#68499C211yesmaximal, characteristic, complemented
15#68498C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18#5479C2211yescharacteristic
19#5477C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×#423C2211yescharacteristic
22#5135—21no—
23#5134—21no—
24#5093C2211yescharacteristic
25#5093C2211yescharacteristic
26#5091C2211yescharacteristic
27#5091C2211yescharacteristic
28#5089—21no—
29#5089—21no—
30#5089—21no—
31#5088C2211yescharacteristic
32#5088C2211yescharacteristic
33#5084C2211yescharacteristic
34#5084C2211yescharacteristic
35#5081—21no—
36#4971—21no—
37#4967—21no—
38#4966—21no—
39#4966—21no—
40#4926C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4851C2211yescharacteristic
43#4848C2211yescharacteristic
44#4847—21no—
45#4846C2211yescharacteristic
46#4844—21no—
47#4843C2211yescharacteristic
48#4838C2211yescharacteristic
49#4834C2211yescharacteristic
50#4831C2211yescharacteristic
51#4830C2211yescharacteristic
52#4729—21no—
53#4728—21no—
54#4690—21no—
55#4689—21no—
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#333C2211yescharacteristic
59C3×#333C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
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#1569—31no—
81#1553—31no—
82#1548D311yescharacteristic, complemented
83C22×C3×Q8—21no—
84C2×#235—21no—
85C2×C3×M16—21no—
86C3×C4⋊Q8—21no—
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—21no—
89C3×C4⋊Q8C2311yescharacteristic
90C3×C4⋊Q8C2311yescharacteristic
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C42C2311yescharacteristic
94#930—21no—
95#930—21no—
96#930D411yescharacteristic
97#929—21no—
98C4×#67—21no—
99#924D411yescharacteristic
100C2×#208D411yescharacteristic
101C2×#208D411yescharacteristic
102C2×D24—21no—
103C2×#204—21no—
104C2×ΓL(1,𝔽25)—21no—
105C2×D3×C8—21no—
106#876—21no—
107#875—21no—
108#875D411yescharacteristic
109#874C2311yescharacteristic
110C4×D12—21no—
111C4×D12—21no—
112C4×D12C2311yescharacteristic
113#872—21no—
114#870D411yescharacteristic
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×C4×C8—21no—
124#830—21no—
125#821—21no—
126#821—21no—
127#821—21no—
128(C2×D3)⋊C8—21no—
129(C2×D3)⋊C8—21no—
130#818—21no—
131#817—21no—
132#816—21no—
133#815—21no—
134Dic3⋊C8—21no—
135Dic3⋊C8—21no—
136C8×Dic3—21no—
137#810—21no—
138#810—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#810—21no—
143#809—21no—
144#809—21no—
145#809—21no—
146#809—21no—
147#809—21no—
148#809—21no—
149#808—21no—
150#808—21no—
151#807—21no—
152#807—21no—
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155#803—21no—
156#803D411yescharacteristic
157#803D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8C2×D311yescharacteristic
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#432—31no—
193#431C2×D311yescharacteristic
194#423C2×D311yescharacteristic
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333C2×D311yescharacteristic
204#333C2×D311yescharacteristic
205#333C2×D311yescharacteristic
206#333C2×D311yescharacteristic
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—41no—
214C2×C3×Q8—82no—
215C22×C3×C4—41no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4#4111yescharacteristic
219C2×#67—21no—
220C2×#67—41no—
221#235—42no—
222#235—42no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12C2×D411yescharacteristic
227C2×C4×D3—21no—
228C2×C4×D3—41no—
229C2×Dic6—21no—
230C2×Dic6C2×D411yescharacteristic
231C3×M16—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—41no—
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⋊Dic3C2×D411yescharacteristic
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261C4×Dic3—21no—
262#208—41no—
263#208—41no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C2×D411yescharacteristic
267C2×C3⋊C8C2×D411yescharacteristic
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270D24—81no—
271#204—81no—
272ΓL(1,𝔽25)—42no—
273D3×C8—42no—
274C2×#41—61no—
275C22×Q8—21no—
276C2×SD16—61no—
277C2×D8—61no—
278C2×M16—21no—
279C2×M16—31no—
280C2×M16—31no—
281C2×M16—61no—
282C22×C8—61no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—31no—
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
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—21no—
306C2×C4⋊C4—21no—
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—31no—
344#97—31no—
345#97—61no—
346#97—61no—
347C4×C8—21no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—61no—
351C23×C3#12111yescharacteristic
352C22×D3—21no—
353C22×D3—41no—
354C3×Q8—81no—
355C3×Q8—81no—
356C3×Q8—82no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—82no—
365C2×C3×C4#12111yescharacteristic
366C2×C3×C4#12111yescharacteristic
367C2×C3×C4#12111yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371#67—81no—
372#67—82no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377D12—21no—
378D12—21no—
379D12—41no—
380D12—41no—
381D12—41no—
382D12—41no—
383D12—81no—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—82no—
387Dic6—21no—
388Dic6—21no—
389Dic6—41no—
390Dic6—41no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—41no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—122no—
402#41—122no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—31no—
406C2×Q8—41no—
407C2×Q8—61no—
408C2×Q8—82no—
409C2×D4—31no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—121no—
415C22×C4—41no—
416C22×C4—61no—
417C22×C4—121no—
418C22×C4#23511yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421SD16—241no—
422D8—241no—
423M16—81no—
424M16—122no—
425M16—121no—
426M16—121no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—31no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—122no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—31no—
443C4⋊C4—31no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C22⋊C4—61no—
453C22⋊C4—61no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C42—61no—
460C42#23511yescharacteristic
461C42D3×D411yescharacteristic
462C42D3×D411yescharacteristic
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—81no—
479C3×C4—81no—
480C3×C4#57511yescharacteristic
481C3×C4#57511yescharacteristic
482C3×C4#57711yescharacteristic
483C3×C4#57811yescharacteristic
484Dic3—41no—
485Dic3—41no—
486Dic3—41no—
487Dic3—81no—
488C23—61no—
489C23—121no—
490C23#88411yescharacteristic
491Q8—61no—
492Q8—61no—
493Q8—81no—
494Q8—81no—
495Q8—82no—
496Q8—121no—
497Q8—121no—
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—41no—
513C2×C4—41no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—82no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—242no—
522C2×C4#88411yescharacteristic
523C2×C4#88411yescharacteristic
524C2×C4#88411yescharacteristic
525C2×C4C2×#23511yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4C2×D3×D411yescharacteristic
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—61no—
533C8—61no—
534C8—61no—
535C8—61no—
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×#88411yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—81no—
559C4—81no—
560C4—121no—
561C4—121no—
562C4—121no—
563C4—241no—
564C4#576211yescharacteristic
565C4#576611yescharacteristic
566C4#577711yescharacteristic
567C4#577711yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—41no—
570C2—241no—
571C2—481no—
572C2#7870311yescharacteristic
573C2#7880011yescharacteristic
574C2#7897511yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1548C2
socleC22×C3C2×#121
prime radical#79062C2
hypercenter#1548D3
hypocenterC3#19382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #83217C2semidirect
2. #83199C2semidirect
3. #79280C2semidirect
5. C3×#1548C2semidirect
6. #70775C2semidirect
7. #70492C2semidirect
8. #68515C2semidirect
9. #68514C2semidirect
10. #68511C2semidirect
11. #68510C2semidirect
12. #68503C2semidirect
13. #68502C2semidirect
14. #68499C2semidirect
15. #68498C2semidirect
82. #1548D3semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
#832172
#831992
#792802
#790622
C3×#15482
#707752
#704922
#685152
#685142
#685112
#685102
#685032
#685022
#684992
#684982
#193823