Finite Group Databasebeta

← #10786708

Subgroups — #10786708

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3331C211yesmaximal, characteristic, complemented
2#83628C211yesmaximal, characteristic, complemented
3#83516C211yesmaximal, characteristic
4#79686C211yesmaximal, characteristic, complemented
5#71926C211yesmaximal, characteristic, complemented
6#71911C211yesmaximal, characteristic, complemented
7#71235C211yesmaximal, characteristic, complemented
8#71214C211yesmaximal, characteristic, complemented
9#71146C211yesmaximal, characteristic, complemented
10#71133C211yesmaximal, characteristic, complemented
11#71062C211yesmaximal, characteristic, complemented
12#71053C211yesmaximal, characteristic, complemented
13SD16×C3⋊C8C211yesmaximal, characteristic, complemented
14#70876C211yesmaximal, characteristic, complemented
15#70859C211yesmaximal, characteristic, complemented
16#19903—31nomaximal, Sylow
17C3×#554C2211yescharacteristic
18C3×#547C2211yescharacteristic
19#5579C2211yescharacteristic
20#5555C2211yescharacteristic
21C3×#487C2211yescharacteristic
22C3×#483—21no—
23C3×#476C2211yescharacteristic
24C3×#470—21no—
25C3×#462C2211yescharacteristic
26C3×#452—21no—
27C3×C4×SD16C2211yescharacteristic
28C3×#436C2211yescharacteristic
29C3×#417—21no—
30#5173—21no—
31#5093C2211yescharacteristic
32#5087C2211yescharacteristic
33#5084C2211yescharacteristic
34#5079C2211yescharacteristic
35#5078C2211yescharacteristic
36#5078C2211yescharacteristic
37#5066—21no—
38#5065—21no—
39#5038—21no—
40#5033—21no—
41#5031C2211yescharacteristic
42Q8×C3⋊C8C2211yescharacteristic
43C4×#213—21no—
44C4×#213C2211yescharacteristic
45D4×C3⋊C8C2211yescharacteristic
46#5009—21no—
47#5008C2211yescharacteristic
48#5008C2211yescharacteristic
49#5007C2211yescharacteristic
50#4967—21no—
51C2×#807—21no—
52#4862—21no—
53#4816—21no—
54#4771—21no—
55#4688C2211yescharacteristic
56#4681C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4535C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61D4⋊(C3⋊C8)C2211yescharacteristic
62#4483C2211yescharacteristic
63#4482C2211yescharacteristic
64#4481C2211yescharacteristic
65#4479C2211yescharacteristic
66#4457C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3380—31no—
69#3334—31no—
70#3331—31no—
71#3331D311yescharacteristic, complemented
72#1710—31no—
73#1703—31no—
74#1703—31no—
75#1675—31no—
76#1674—31no—
77#1644—31no—
78#1644—31no—
79#1635—31no—
80#1635—31no—
81#1589—31no—
82C8×SD16—31no—
83C2×C3×#41—21no—
84C2×#237—21no—
85C2×C3×Q16—21no—
86C2×C3×SD16D411yescharacteristic
87C2×C3×M16—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×#125—21no—
91C3×#123—41no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—21no—
94C3×C22⋊Q8—41no—
95C3×#121—21no—
96C3×C4×Q8C2311yescharacteristic
97C3×C4×D4—21no—
98C3×C4×D4C2311yescharacteristic
99C3×C4×D4C2311yescharacteristic
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102#946—21no—
103C2×#216—21no—
104C2×#213D411yescharacteristic
105#930—21no—
106C2×C4⋊Dic3—21no—
107C22×C3⋊C8—21no—
108C2×ΓL(1,𝔽25)—21no—
109#897—21no—
110#896—41no—
111D3×C4⋊C4—21no—
112#886—41no—
113C4×D12—21no—
114C4×D12C2311yescharacteristic
115#870—21no—
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C3×#106—21no—
119C3×#106D411yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×#103—21no—
122C3×#103—21no—
123C3×#103—21no—
124C3×#103D411yescharacteristic
125C3×#102—21no—
126C3×#102D411yescharacteristic
127C3×C22⋊C8—21no—
128C3×C4×C8C2311yescharacteristic
129#835—41no—
130#832—21no—
131#832—21no—
132#832—21no—
133#832D411yescharacteristic
134#830—21no—
135(C2×D3)⋊C8—21no—
136#816—21no—
137#816—21no—
138#810—21no—
139#810—41no—
140#809—21no—
141#809—21no—
142#809D411yescharacteristic
143#808—21no—
144#808—21no—
145#808—21no—
146#807—21no—
147#807—21no—
148#807—21no—
149#807—21no—
150#807—21no—
151#807—21no—
152#807D411yescharacteristic
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#554—31no—
161#554C2×D311yescharacteristic
162#547—31no—
163#547C2×D311yescharacteristic
164#500—31no—
165#500—31no—
166#499—61no—
167#487—31no—
168#487—31no—
169#487—31no—
170#487C2×D311yescharacteristic
171#483—21no—
172#483—61no—
173#480—61no—
174#480—61no—
175#478—61no—
176#478—61no—
177#476—31no—
178#476—31no—
179#476—31no—
180#476C2×D311yescharacteristic
181#474—31no—
182#474—31no—
183#470—21no—
184#470—61no—
185#466—61no—
186#462—31no—
187#462C2×D311yescharacteristic
188#452—21no—
189#452—61no—
190C8×Q8—31no—
191C4×SD16C2×D311yescharacteristic
192C4×D8—31no—
193C4×D8—61no—
194#436—31no—
195#436C2×D311yescharacteristic
196C8×D4—31no—
197C2×#107—61no—
198#417—21no—
199#417—61no—
200#335—31no—
201#334—31no—
202#333—31no—
203#333—31no—
204#332—31no—
205#332—31no—
206Q8⋊C8—31no—
207D4⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210C82—31no—
211C3×#41—42no—
212C3×#41—81no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—41no—
217C2×C3×D4C2×D411yescharacteristic
218C2×C3×D4C2×D411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—41no—
222C22×C3×C4—41no—
223C2×#67—41no—
224C22×Dic3—41no—
225#237—42no—
226#237—81no—
227C2×D12C2×D411yescharacteristic
228C2×C4×D3—21no—
229C2×C4×D3—41no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C3×Q16—81no—
233C3×SD16—22no—
234C3×M16—81no—
235C2×C3×C8—21no—
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
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×C4⋊C4—41no—
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C4⋊C4C2×D411yescharacteristic
248C3×C4⋊C4C2×D411yescharacteristic
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—41no—
252C3×C22⋊C4—41no—
253C3×C42—21no—
254C3×C42C2411yescharacteristic
255#216—81no—
256#213—22no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—41no—
259(C2×D3)⋊C4—41no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—41no—
265C4⋊Dic3C2×D411yescharacteristic
266Dic3⋊C4—41no—
267Dic3⋊C4—41no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—42no—
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275ΓL(1,𝔽25)—81no—
276C2×#41—21no—
277C2×#41—61no—
278C2×Q16—21no—
279C2×Q16—61no—
280C2×SD16#6711yescharacteristic
281C2×D8—31no—
282C2×M16—21no—
283C2×M16—61no—
284C22×C8—61no—
285C4⋊Q8—31no—
286C4⋊Q8—31no—
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
289#125—21no—
290#125—61no—
291#123—41no—
292#123—121no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—41no—
296C22⋊Q8—61no—
297C22⋊Q8—61no—
298C22⋊Q8—121no—
299#121—21no—
300#121—61no—
301C4×Q8C22×D311yescharacteristic
302C4×D4—21no—
303C4×D4—31no—
304C4×D4—61no—
305C4×D4C22×D311yescharacteristic
306C4×D4C22×D311yescharacteristic
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311#107—31no—
312#107—61no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#107—61no—
318#106—21no—
319#106—61no—
320#106—61no—
321#106—61no—
322#106#6711yescharacteristic
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8—61no—
327C4⋊C8C22×D311yescharacteristic
328#103—21no—
329#103—21no—
330#103—21no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—121no—
335#103—121no—
336#103#6711yescharacteristic
337#102—21no—
338#102—31no—
339#102—31no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344#102—61no—
345#102#6711yescharacteristic
346C22⋊C8—21no—
347C22⋊C8—61no—
348C22⋊C8—61no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C23×C3—21no—
354C23×C3—21no—
355C22×D3—21no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—81no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—82no—
365C3×D4#13512yes—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—82no—
377C2×C3×C4—161no—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381#67—82no—
382C2×Dic3—21no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—82no—
386D12—21no—
387D12—21no—
388C4×D3—41no—
389C4×D3—41no—
390C4×D3—161no—
391Dic6—41no—
392Dic6—41no—
393Dic6—81no—
394C3×C8—21no—
395C3×C8—21no—
396C3×C8—41no—
397C3⋊C8—21no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8#13712yes—
403#41—42no—
404#41—81no—
405#41—122no—
406#41—241no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8C2×#6711yescharacteristic
412C2×D4—31no—
413C2×D4—41no—
414C2×D4—121no—
415C2×D4D3×D411yescharacteristic
416C2×D4C2×#6711yescharacteristic
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—61no—
422C22×C4—121no—
423C22×C4—121no—
424Q16—81no—
425Q16—241no—
426SD16—22no—
427D8—62no—
428M16—81no—
429M16—241no—
430C2×C8—21no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—122no—
438C2×C8C2×#6711yescharacteristic
439C2×C8C2×#6711yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—31no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—121no—
454C4⋊C4—121no—
455C4⋊C4—121no—
456C4⋊C4D3×D411yescharacteristic
457C4⋊C4C2×#6711yescharacteristic
458C4⋊C4C2×#6711yescharacteristic
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—61no—
464C22⋊C4—121no—
465C22⋊C4—121no—
466C42—21no—
467C42C23×D311yescharacteristic
468C22×C3—22no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3D4211yescharacteristic
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C3×C4—21no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4C2×#13511yescharacteristic
486C3×C4C2×#13711yescharacteristic
487C3×C4#57611yescharacteristic
488C3×C4#57811yescharacteristic
489Dic3—41no—
490Dic3—81no—
491Dic3—81no—
492C23—21no—
493C23—21no—
494C23—61no—
495Q8—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—81no—
500Q8—121no—
501Q8—121no—
502Q8—241no—
503D4—21no—
504D4—21no—
505D4—21no—
506D4—61no—
507D4—61no—
508D4—82no—
509D4—242no—
510D4#95012yes—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—61no—
522C2×C4—82no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—161no—
528C2×C4—242no—
529C2×C4—481no—
530C2×C4C2×D3×D411yescharacteristic
531C2×C4#100411yescharacteristic
532C2×C4C22×#6711yescharacteristic
533C8—21no—
534C8—21no—
535C8—32no—
536C8—41no—
537C8—61no—
538C8—121no—
539C8—121no—
540C8—121no—
541C8—121no—
542C2×C3—42no—
543C2×C3—81no—
544C2×C3#327611yescharacteristic
545C2×C3#328011yescharacteristic
546C2×C3#329011yescharacteristic
547D3—81no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22D4×#6711yescharacteristic
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—81no—
564C4—81no—
565C4—121no—
566C4—241no—
567C4—241no—
568C4D3×#13711yescharacteristic
569C4#577911yescharacteristic
570C4C2×#95011yescharacteristic
571C4#583511yescharacteristic
572C3#1990311yescharacteristic, complemented, Sylow
573C2—42no—
574C2—81no—
575C2—241no—
576C2#8350511yescharacteristic
577C2#8399611yescharacteristic
578C2#8401211yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3331C2
socleC22×C3D42
prime radical#83516C2
hypercenter#3331D3
hypocenterC3#19903
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #3331

Complements

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

NH ≅ G/Ntype
1. C3×#3331C2semidirect
2. #83628C2semidirect
4. #79686C2semidirect
5. #71926C2semidirect
6. #71911C2semidirect
7. #71235C2semidirect
8. #71214C2semidirect
9. #71146C2semidirect
10. #71133C2semidirect
11. #71062C2semidirect
12. #71053C2semidirect
13. SD16×C3⋊C8C2semidirect
14. #70876C2semidirect
15. #70859C2semidirect
71. #3331D3semidirect
572. C3#19903semidirect

Maximal subgroups

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

subgroupindex
C3×#33312
#836282
#835162
#796862
#719262
#719112
#712352
#712142
#711462
#711332
#710622
#710532
SD16×C3⋊C82
#708762
#708592
#199033