Finite Group Databasebeta

← #10780313

Subgroups — #10780313

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
1C3×#3141C211yesmaximal, characteristic, complemented
2#83312C211yesmaximal, characteristic
3#83217C211yesmaximal, characteristic, complemented
4#78511C211yesmaximal, characteristic, complemented
5#71808C211yesmaximal, characteristic, complemented
6#71774C211yesmaximal, characteristic, complemented
7#71243C211yesmaximal, characteristic, complemented
8#71241C211yesmaximal, characteristic, complemented
9#71235C211yesmaximal, characteristic, complemented
10#71233C211yesmaximal, characteristic, complemented
11#71210C211yesmaximal, characteristic, complemented
12#71208C211yesmaximal, characteristic, complemented
13#71202C211yesmaximal, characteristic, complemented
14#71200C211yesmaximal, characteristic, complemented
15#69851C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19#5479C2211yescharacteristic
20C3×#489—21no—
21C3×#489—21no—
22C3×#487C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#486C2211yescharacteristic
26C3×#417—21no—
27C3×#417—21no—
28C3×C4×M16C2211yescharacteristic
29#5128—21no—
30#5127—21no—
31#5094C2211yescharacteristic
32#5091C2211yescharacteristic
33#5091C2211yescharacteristic
34#5089—21no—
35#5087C2211yescharacteristic
36#5084C2211yescharacteristic
37#5084C2211yescharacteristic
38#5081—21no—
39#5075C2211yescharacteristic
40#5072C2211yescharacteristic
41#5042—21no—
42#5039—21no—
43#5016—21no—
44#5009—21no—
45#4971—21no—
46#4967—21no—
47#4926C2211yescharacteristic
48#4925C2211yescharacteristic
49#4924C2211yescharacteristic
50#4709—21no—
51#4708—21no—
52#4688C2211yescharacteristic
53C8×D12C2211yescharacteristic
54#4679C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4546C2211yescharacteristic
60#4546C2211yescharacteristic
61#4546C2211yescharacteristic
62#4544C2211yescharacteristic
63#4544C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3251—31no—
69#3238—31no—
70#3141D311yescharacteristic, complemented
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#1548—31no—
83C2×C3×#41—21no—
84C22×C3×Q8—21no—
85C2×C3×M16D411yescharacteristic
86C2×C3×M16D411yescharacteristic
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—21no—
89C3×C4⋊Q8—21no—
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C3×C4⋊Q8C2311yescharacteristic
93C3×#127C2311yescharacteristic
94C3×#124—21no—
95C3×#121—21no—
96C3×#121—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C42C2311yescharacteristic
102C2×#214—21no—
103C2×#213—21no—
104#930D411yescharacteristic
105#929—21no—
106C4×#67—21no—
107#924D411yescharacteristic
108C2×#208—21no—
109C2×ΓL(1,𝔽25)—21no—
110C2×D3×C8—21no—
111#876—21no—
112#875D411yescharacteristic
113C4×D12C2311yescharacteristic
114#872—21no—
115#870D411yescharacteristic
116C4×Dic6C2311yescharacteristic
117C3×#103—21no—
118C3×#103—21no—
119C3×#103—21no—
120C3×#103—21no—
121C3×#102—21no—
122C3×#102—21no—
123C3×#102—21no—
124C3×#102—21no—
125C3×#97D411yescharacteristic
126C3×#97D411yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129#832—21no—
130#832—21no—
131#830—21no—
132#830—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135#815—21no—
136Dic3⋊C8—21no—
137Dic3⋊C8—21no—
138C8×Dic3—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#809—21no—
143#809—21no—
144#809—21no—
145#808—21no—
146#808—21no—
147#808—21no—
148#807—21no—
149#807—21no—
150#807—21no—
151C4⋊(C3⋊C8)—21no—
152C4⋊(C3⋊C8)C2311yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158#803—21no—
159C4×C3⋊C8—21no—
160#532C2×D311yescharacteristic
161C2×C4⋊Q8C2×D311yescharacteristic
162#529—31no—
163#501—61no—
164#489—21no—
165#489—21no—
166#489—61no—
167#487C2×D311yescharacteristic
168#487C2×D311yescharacteristic
169#486C2×D311yescharacteristic
170#486C2×D311yescharacteristic
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#431—31no—
194#423—31no—
195#417—21no—
196#417—21no—
197#405—61no—
198C4×M16C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C3×#41—42no—
212C3×#41—42no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—41no—
217C2×C3×Q8—82no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—41no—
222C22×C3×C4—41no—
223C22×C3×C4—41no—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4#4111yescharacteristic
227C2×#67—21no—
228C2×D12C2×D411yescharacteristic
229C2×C4×D3—21no—
230C2×Dic6C2×D411yescharacteristic
231C3×M16—41no—
232C3×M16—41no—
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C22⋊C4—41no—
246C3×C22⋊C4—41no—
247C3×C42C2×D411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42#4111yescharacteristic
250C3×C42C2411yescharacteristic
251#217—21no—
252#214—81no—
253#213—81no—
254(C2×D3)⋊C4—21no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—21no—
258C4⋊Dic3C2×D411yescharacteristic
259C4⋊Dic3C2×D411yescharacteristic
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263Dic3⋊C4—21no—
264C4×Dic3—21no—
265#208—81no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272ΓL(1,𝔽25)—42no—
273D3×C8—42no—
274C2×#41—21no—
275C22×Q8—21no—
276C2×SD16—61no—
277C2×D8—61no—
278C2×M16—61no—
279C2×M16—61no—
280C2×M16#6711yescharacteristic
281C2×M16#6711yescharacteristic
282C22×C8—61no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8C22×D311yescharacteristic
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
289#127C22×D311yescharacteristic
290#126—61no—
291#125—31no—
292#124—21no—
293#124—31no—
294#123—61no—
295C22⋊Q8—31no—
296#121—21no—
297#121—21no—
298#121—31no—
299C4×Q8—31no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—31no—
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—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323#103—21no—
324#103—21no—
325#103—21no—
326#103—21no—
327#103—61no—
328#103—61no—
329#103—61no—
330#102—21no—
331#102—21no—
332#102—21no—
333#102—21no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—61no—
344#97—61no—
345#97#6711yescharacteristic
346#97#6711yescharacteristic
347C4×C8—61no—
348C4×C8—61no—
349C4×C8C22×D311yescharacteristic
350C4×C8C22×D311yescharacteristic
351C23×C3—41no—
352C23×C3#12111yescharacteristic
353C22×D3—21no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—81no—
357C3×Q8—81no—
358C3×Q8—82no—
359C3×D4—41no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—81no—
364C3×D4—82no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—82no—
374C2×C3×C4—82no—
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4#12111yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381#67—81no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385D12—21no—
386D12—21no—
387C4×D3—41no—
388C4×D3—41no—
389Dic6—21no—
390Dic6—21no—
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—42no—
402#41—42no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—31no—
407C2×Q8—41no—
408C2×Q8—82no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—31no—
413C2×D4—41no—
414C2×D4—61no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—61no—
418C22×C4D3×D411yescharacteristic
419C22×C4#23711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421SD16—241no—
422D8—241no—
423M16—41no—
424M16—41no—
425M16—122no—
426M16—241no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—122no—
434C2×C8C2×#6711yescharacteristic
435C2×C8C2×#6711yescharacteristic
436C2×C8C2×#6711yescharacteristic
437C2×C8C2×#6711yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—31no—
445C4⋊C4—31no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C42—61no—
460C42D3×D411yescharacteristic
461C42#23711yescharacteristic
462C42C2×#6711yescharacteristic
463C42C23×D311yescharacteristic
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3C2×#12111yescharacteristic
471C2×D3—41no—
472C2×D3—41no—
473C2×D3—41no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4#57511yescharacteristic
482C3×C4#57511yescharacteristic
483C3×C4#57711yescharacteristic
484C3×C4#57811yescharacteristic
485Dic3—41no—
486Dic3—41no—
487Dic3—41no—
488C23—41no—
489C23—61no—
490C23#93811yescharacteristic
491Q8—41no—
492Q8—41no—
493Q8—61no—
494Q8—61no—
495Q8—81no—
496Q8—81no—
497Q8—82no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—61no—
503D4—61no—
504D4—81no—
505D4—82no—
506D4—241no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4#93811yescharacteristic
523C2×C4#93811yescharacteristic
524C2×C4#93811yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×#23711yescharacteristic
527C2×C4C22×#6711yescharacteristic
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—121no—
533C8—121no—
534C8—121no—
535C8—121no—
536C8—121no—
537C8—121no—
538C2×C3—41no—
539C2×C3—161no—
540C2×C3#304011yescharacteristic
541C2×C3#306311yescharacteristic
542C2×C3#306811yescharacteristic
543D3—81no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—121no—
551C22—121no—
552C22—121no—
553C22C2×#93811yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—121no—
562C4—121no—
563C4—121no—
564C4#577811yescharacteristic
565C4#577911yescharacteristic
566C4#582211yescharacteristic
567C4#582211yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—41no—
570C2—161no—
571C2—241no—
572C2#8294511yescharacteristic
573C2#8396311yescharacteristic
574C2#8398611yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3141C2
socleC22×C3C2×#121
prime radical#83312C2
hypercenter#3141D3
hypocenterC3#19382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
1. C3×#3141C2semidirect
3. #83217C2semidirect
4. #78511C2semidirect
5. #71808C2semidirect
6. #71774C2semidirect
7. #71243C2semidirect
8. #71241C2semidirect
9. #71235C2semidirect
10. #71233C2semidirect
11. #71210C2semidirect
12. #71208C2semidirect
13. #71202C2semidirect
14. #71200C2semidirect
15. #69851C2semidirect
70. #3141D3semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
C3×#31412
#833122
#832172
#785112
#718082
#717742
#712432
#712412
#712352
#712332
#712102
#712082
#712022
#712002
#698512
#193823