Finite Group Databasebeta

← #10804402

Subgroups — #10804402

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85847C211yesmaximal, characteristic
2#78715C211yesmaximal, characteristic, complemented
3#78689C211yesmaximal, characteristic, complemented
4C3×#1827C211yesmaximal, characteristic, complemented
5#71644C211yesmaximal, characteristic, complemented
6#70139C211yesmaximal, characteristic, complemented
7#70048C211yesmaximal, characteristic, complemented
8#68777C212yesmaximal, complemented
9#68766C212yesmaximal, complemented
10#68713C212yesmaximal, complemented
11#68708C212yesmaximal, complemented
12#21785—31nomaximal, Sylow
13C2×#1010C2211yescharacteristic
14C2×#1009C2211yescharacteristic
15C3×C4×#41C2211yescharacteristic
16#5745C2212yescomplemented
17#5576—22no—
18#5554—24no—
19#5554—22no—
20#5526C412yescomplemented
21#5526C2211yescharacteristic
22#5525C412yescomplemented
23#5525C2211yescharacteristic
24#5516C2212yescomplemented
25#5477C412yescomplemented
26#5477C2211yescharacteristic
27C2×#875C412yescomplemented
28C2×#875C2211yescharacteristic
29C3×#413C2211yescharacteristic
30C3×#410C2211yescharacteristic
31#5240C2211yescharacteristic
32#5237C2211yescharacteristic
33#5139C2212yes—
34C2×#824C2212yescomplemented
35#5131C2212yes—
36#5006C2212yescomplemented
37#5005C2212yes—
38#5001C2212yes—
39D3×#101—22no—
40#4745—22no—
41#4740—24no—
42C3×#339C2212yescomplemented
43C3×H(3,ℤ/4)C2212yescomplemented
44#4531C2212yescomplemented
45#4523C2212yescomplemented
46#3500—31no—
47#2889—31no—
48#2884—31no—
49#1912—32no—
50#1910—32no—
51#1888—32no—
52#1884—32no—
53#1827—31no—
54#1827D311yescharacteristic, complemented
55#1795—31no—
56#1795—31no—
57C2×C3×#41C2311yescharacteristic
58#1010—42no—
59#1010—42no—
60#1009—42no—
61#1009—42no—
62C2×#239—21no—
63C2×D3×Q8D412yes—
64C2×#237D414yes—
65C2×#237D412yes—
66C2×D3×D4—22no—
67C2×D3×D4—21no—
68C2×#235C2×C412yes—
69C2×#235C2×C412yes—
70C2×#235C2311yescharacteristic
71C2×#235C2311yescharacteristic
72C22×D12C2×C412yes—
73C22×D12C2311yescharacteristic
74C22×Dic6C2×C412yes—
75C22×Dic6C2311yescharacteristic
76C2×C3×M16C2312yes—
77C3×C4×Q8—21no—
78C3×C4×D4—22no—
79C3×C4×D4—21no—
80C3×#117C2×C412yes—
81C3×#117C2312yes—
82C3×#117C2311yescharacteristic
83C2×C3×C42C2×C412yes—
84C2×C3×C42C2311yescharacteristic
85#930—24no—
86#930—22no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—22no—
89#926C2312yes—
90C2×#208C2312yes—
91#907—42no—
92D3×M16—24no—
93#901—42no—
94#897—24no—
95#897—22no—
96#895—24no—
97#895—22no—
98#887—42no—
99#886—24no—
100#886—22no—
101#882—24no—
102#882—22no—
103#877—24no—
104#875—24no—
105#875—22no—
106#875C2×C411yescharacteristic
107#875C2×C412yes—
108#875C2×C412yes—
109#875C2×C412yes—
110#875C2×C412yes—
111#875C2×C411yescharacteristic
112#874C2×C412yes—
113#874C2×C411yescharacteristic
114C4×D12—24no—
115C4×D12—22no—
116#869C2×C412yes—
117#869C2×C411yescharacteristic
118C4×Dic6—42no—
119C3×#101—21no—
120C3×#100—21no—
121C3×#99—22no—
122#836—21no—
123#834—22no—
124#833—21no—
125#825—42no—
126#824—24no—
127#821—42no—
128#809—42no—
129#806—24no—
130#805—42no—
131C2×2−1+4—31no—
132C2×H(5,𝔽2)—31no—
133#568—32no—
134#548—62no—
135#546—64no—
136#546—62no—
137#536—32no—
138#536—31no—
139#535—32no—
140#535—31no—
141#532—32no—
142#532—31no—
143C2×#124—32no—
144C2×#124—31no—
145#519—32no—
146C4×#41C2×D311yescharacteristic
147#421—32no—
148#421—32no—
149#418—32no—
150#418—32no—
151#413—31no—
152#413—62no—
153#413C2×D311yescharacteristic
154C2×#101—32no—
155C2×#101—62no—
156#410—31no—
157#410—32no—
158#410—64no—
159#410C2×D311yescharacteristic
160#339—32no—
161#339C2×D312yescomplemented
162H(3,ℤ/4)—32no—
163H(3,ℤ/4)C2×D312yescomplemented
164C23×D3—22no—
165C23×D3—21no—
166C3×#41—22no—
167C3×#41—22no—
168C2×C3×Q8C2×D411yescharacteristic
169C2×C3×D4C2×D412yes—
170C2×C3×D4C2×D411yescharacteristic
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4C22×C412yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—22no—
176C2×#67—21no—
177C2×#67C22⋊C414yes—
178C2×#67C2×D412yes—
179C22×Dic3C22⋊C414yes—
180C22×Dic3C2×D412yes—
181#239—44no—
182D3×Q8—44no—
183#237—44no—
184#237—48no—
185D3×D4—48no—
186D3×D4—44no—
187#235—22no—
188#235—22no—
189#235—22no—
190#235—22no—
191#235—42no—
192#235—42no—
193#235—42no—
194#235—42no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—21no—
199C2×D12—24no—
200C2×D12—22no—
201C2×D12—42no—
202C2×D12—42no—
203C2×D12C22×C411yescharacteristic
204C2×D12C22×C412yes—
205C2×D12C22×C411yescharacteristic
206C2×C4×D3—22no—
207C2×C4×D3—21no—
208C2×C4×D3C22⋊C414yes—
209C2×C4×D3C2×D412yes—
210C2×Dic6—22no—
211C2×Dic6—22no—
212C2×Dic6—42no—
213C2×Dic6—42no—
214C2×Dic6C22⋊C414yes—
215C2×Dic6C22×C411yescharacteristic
216C2×Dic6C22×C411yescharacteristic
217C2×Dic6C22×C412yes—
218C2×Dic6C2×D412yes—
219C3×M16—22no—
220C3×M16—22no—
221C2×C3×C8—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—22no—
228C3×C42—22no—
229C3×C42—21no—
230C3×C42—22no—
231C3×C42C22×C412yes—
232C3×C42C22×C411yescharacteristic
233C3×C42C22×C411yescharacteristic
234#217—22no—
235(C2×D3)⋊C4—22no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—24no—
241C4⋊Dic3—22no—
242C4⋊Dic3—24no—
243C4⋊Dic3—22no—
244Dic3⋊C4—44no—
245C4×Dic3—22no—
246C4×Dic3—44no—
247#208—22no—
248#208—22no—
249C2×C3⋊C8—22no—
250ΓL(1,𝔽25)—44no—
251D3×C8—44no—
2522−1+4—122no—
2532−1+4—122no—
254H(5,𝔽2)—122no—
255H(5,𝔽2)—122no—
256C2×#41—34no—
257C2×#41—32no—
258C2×#41—32no—
259C2×#41—31no—
260C2×#41—32no—
261C2×#41—31no—
262C2×#41—61no—
263C2×#41C22×D311yescharacteristic
264C22×Q8—32no—
265C22×Q8—32no—
266C22×Q8—31no—
267C22×D4—32no—
268C22×D4—31no—
269C22×D4—62no—
270C22×D4—61no—
271#131—122no—
272#131—122no—
273C2×M16—32no—
274C2×M16—64no—
275C2×M16C22×D312yes—
276C4⋊Q8—32no—
277C4⋊Q8—31no—
278#127—32no—
279#127—31no—
280#124—31no—
281#124—32no—
282#124—32no—
283#124—32no—
284#124—32no—
285#124—31no—
286#124—64no—
287#124—62no—
288#123—64no—
289#123—62no—
290C22⋊Q8—64no—
291C22⋊Q8—62no—
292#121—64no—
293#121—62no—
294#121—64no—
295#121—62no—
296#120—64no—
297#120—62no—
298C4×Q8—21no—
299C4×Q8—122no—
300C4×Q8—122no—
301C4×D4—22no—
302C4×D4—21no—
303C4×D4—64no—
304C4×D4—62no—
305#117—32no—
306#117—64no—
307#117C4×D312yes—
308#117C22×D312yes—
309#117C22×D311yescharacteristic
310C2×C22⋊C4—64no—
311C2×C22⋊C4—62no—
312C2×C42C4×D312yes—
313C2×C42C22×D311yescharacteristic
314#104—122no—
315#104—122no—
316#102—122no—
317#102—122no—
318#101—21no—
319#101—64no—
320#101—61no—
321#100—21no—
322#100—61no—
323#99—22no—
324#99—64no—
325#99—62no—
326C23×C3C2×C22⋊C412yes—
327C23×C3C22×D411yescharacteristic
328C22×D3—22no—
329C22×D3—21no—
330C22×D3—44no—
331C22×D3—42no—
332C22×D3—82no—
333C22×D3—84no—
334C22×D3C2×C22⋊C412yes—
335C3×Q8—22no—
336C3×D4—24no—
337C3×D4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—42no—
344C2×C3×C4—81no—
345C2×C3×C4—82no—
346C2×C3×C4C2×C22⋊C412yes—
347C2×C3×C4C23×C411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4H(5,𝔽2)12yes—
350C2×C3×C4H(5,𝔽2)12yes—
351#67—24no—
352#67—44no—
353#67—44no—
354#67—42no—
355C2×Dic3—24no—
356C2×Dic3—21no—
357C2×Dic3—44no—
358C2×Dic3—42no—
359C2×Dic3C2×C22⋊C412yes—
360C2×Dic3C2×C22⋊C414yes—
361D12—21no—
362D12—21no—
363D12—22no—
364D12—22no—
365D12—21no—
366D12—21no—
367D12—44no—
368D12—42no—
369D12—84no—
370D12—82no—
371C4×D3—24no—
372C4×D3—44no—
373C4×D3—44no—
374C4×D3—42no—
375Dic6—22no—
376Dic6—22no—
377Dic6—21no—
378Dic6—21no—
379Dic6—24no—
380Dic6—21no—
381Dic6—21no—
382Dic6—44no—
383C3×C8—42no—
384C3⋊C8—42no—
385C24—62no—
386C24—61no—
387#41—22no—
388#41—22no—
389#41—62no—
390#41—62no—
391#41—62no—
392#41—62no—
393#41—122no—
394#41—122no—
395#41—122no—
396#41—122no—
397#41—128no—
398#41—124no—
399#41—124no—
400C2×Q8—34no—
401C2×Q8—32no—
402C2×Q8—31no—
403C2×Q8—32no—
404C2×Q8—31no—
405C2×Q8—62no—
406C2×Q8—62no—
407C2×Q8—122no—
408C2×Q8—122no—
409C2×Q8—124no—
410C2×Q8D3×D411yescharacteristic
411C2×D4—32no—
412C2×D4—34no—
413C2×D4—31no—
414C2×D4—32no—
415C2×D4—31no—
416C2×D4—64no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—61no—
420C2×D4—62no—
421C2×D4—61no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4—128no—
427C2×D4—124no—
428C2×D4D3×D412yes—
429C2×D4D3×D411yescharacteristic
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—32no—
433C22×C4—34no—
434C22×C4—32no—
435C22×C4—34no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4C2×C4×D312yes—
439C22×C4C23×D311yescharacteristic
440M16—22no—
441M16—22no—
442M16—62no—
443M16—62no—
444M16—124no—
445C2×C8—22no—
446C2×C8—62no—
447C2×C8—124no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—64no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—124no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C22⋊C4—62no—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—62no—
464C22⋊C4—62no—
465C42—22no—
466C42—21no—
467C42—22no—
468C42—62no—
469C42—124no—
470C42C2×C4×D311yescharacteristic
471C42C2×C4×D312yes—
472C42C2×C4×D311yescharacteristic
473C22×C3—22no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3C22×C22⋊C411yescharacteristic
477C22×C3#53412yes—
478C2×D3—22no—
479C2×D3—42no—
480C2×D3—44no—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—41no—
484C2×D3—162no—
485C2×D3—164no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—82no—
491C3×C4#51811yescharacteristic
492C3×C4#51811yescharacteristic
493C3×C4#53412yes—
494Dic3—24no—
495Dic3—22no—
496Dic3—41no—
497Dic3—82no—
498C23—32no—
499C23—62no—
500C23—61no—
501C23—124no—
502C23—122no—
503C23—242no—
504C23—244no—
505C23D3×C22⋊C412yes—
506C23C2×D3×D411yescharacteristic
507Q8—22no—
508Q8—64no—
509Q8—62no—
510Q8—62no—
511Q8—61no—
512Q8—61no—
513Q8—61no—
514Q8—61no—
515Q8—124no—
516D4—24no—
517D4—22no—
518D4—64no—
519D4—61no—
520D4—61no—
521D4—62no—
522D4—62no—
523D4—61no—
524D4—61no—
525D4—124no—
526D4—124no—
527D4—124no—
528D4—122no—
529D4—122no—
530D4—244no—
531D4—242no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—32no—
538C2×C4—34no—
539C2×C4—42no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—61no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—124no—
546C2×C4—124no—
547C2×C4—124no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4D3×C22⋊C412yes—
551C2×C4C22×C4×D311yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4#100912yes—
554C2×C4#100912yes—
555C8—42no—
556C8—122no—
557C2×C3—22no—
558C2×C3—21no—
559C2×C3—42no—
560C2×C3#230811yescharacteristic
561D3—42no—
562D3—82no—
563D3—81no—
564C22—22no—
565C22—41no—
566C22—42no—
567C22—62no—
568C22—122no—
569C22—124no—
570C22—122no—
571C22—122no—
572C22—121no—
573C22—482no—
574C22—484no—
575C22C2×D3×C22⋊C411yescharacteristic
576C22#565312yes—
577C4—22no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—62no—
582C4—64no—
583C4—82no—
584C4—121no—
585C4—242no—
586C4#552111yescharacteristic
587C4#552111yescharacteristic
588C4#565312yes—
589C3#2178511yescharacteristic, complemented, Sylow
590C2—21no—
591C2—22no—
592C2—42no—
593C2—122no—
594C2—242no—
595C2—241no—
596C2#7411911yescharacteristic
subgroupG/H
centerC2#74119
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1827C2
socleC2×C3#2308
prime radical#85847C2
cosocleC22×C3×C4C24
hypercenter#1827D3
hypocenterC3#21785
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1827

Complements

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

NH ≅ G/Ntype
2. #78715C2semidirect
3. #78689C2semidirect
4. C3×#1827C2semidirect
5. #71644C2semidirect
6. #70139C2semidirect
7. #70048C2semidirect
8. #68777C2semidirect
9. #68766C2semidirect
10. #68713C2semidirect
11. #68708C2semidirect
16. #5745C22semidirect
20. #5526C4semidirect
22. #5525C4semidirect
24. #5516C22semidirect
25. #5477C4semidirect
27. C2×#875C4semidirect
34. C2×#824C22semidirect
36. #5006C22semidirect
42. C3×#339C22semidirect
43. C3×H(3,ℤ/4)C22semidirect
44. #4531C22semidirect
45. #4523C22semidirect
54. #1827D3semidirect
161. #339C2×D3semidirect
163. H(3,ℤ/4)C2×D3semidirect
589. C3#21785semidirect

Maximal subgroups

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

subgroupindex
#858472
#787152
#786892
C3×#18272
#716442
#701392
#700482
#687772
#687662
#687132
#687082
#217853