Finite Group Databasebeta

← #10796775

Subgroups — #10796775

Order 768.

1694 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
1C2×#5724C211yesmaximal, characteristic
2D3×#519C211yesmaximal, characteristic, complemented
3#78956C212yesmaximal
4C2×D3×#101C211yesmaximal, characteristic, complemented
5C3×#1886C211yesmaximal, characteristic, complemented
6#71784C211yesmaximal, characteristic, complemented
7#71661C211yesmaximal, characteristic, complemented
8#70345C212yesmaximal, complemented
9#70104C211yesmaximal, characteristic, complemented
10#68766C212yesmaximal, complemented
11#68753C212yesmaximal, complemented
12#20954—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C22×D3×Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C3×#519C2211yescharacteristic
17C2×#947C2211yescharacteristic
18#5812C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×D3×M16C2212yes—
21#5724—24no—
22#5724C418yescomplemented
23#5572—24no—
24C4×D3×Q8—24no—
25D3×#117—22no—
26D3×#117C2212yes—
27#5516—22no—
28#5516C2212yes—
29C2×#896—22no—
30C2×#894—22no—
31C2×#875C2211yescharacteristic
32C2×#872C2211yescharacteristic
33C3×#419C2212yes—
34C2×C3×#101C2211yescharacteristic
35C2×#836C2211yescharacteristic
36#5225C2212yes—
37C2×#824C2212yes—
38#5131C2212yes—
39#5001C2212yes—
40#4804—24no—
41#4803—24no—
42D3×#101—24no—
43C3×#339C2212yes—
44#4555C2212yes—
45#4532C2212yes—
46#4523C2212yes—
47C2×#536—31no—
48C2×#519—31no—
49#2896—32no—
50C22×#101—31no—
51#1888—31no—
52#1888—32no—
53#1888—31no—
54#1886—32no—
55#1886—31no—
56#1886D311yescharacteristic, complemented
57#1733—32no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C2×#239—22no—
61C2×#239—22no—
62C2×#239—22no—
63C2×#239—24no—
64C2×#239C2×C414yes—
65C2×D3×Q8—22no—
66C2×D3×Q8—22no—
67C2×D3×Q8—24no—
68C2×D3×Q8C2×C414yes—
69C2×D3×Q8D414yes—
70C22×D12—21no—
71C22×D12C2311yescharacteristic
72C22×C4×D3—21no—
73C22×C4×D3D412yes—
74C22×C4×D3C2311yescharacteristic
75C22×Dic6D412yes—
76C22×Dic6C2311yescharacteristic
77C2×C3×M16C2312yes—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8C2×C414yescomplemented
80C3×C4×Q8—24no—
81C3×#117—22no—
82C3×#117C2312yes—
83C2×C3×C4⋊C4—22no—
84C2×C3×C42C2311yescharacteristic
85#947—22no—
86#947C2×C414yescomplemented
87Q8×Dic3—24no—
88#944—22no—
89#944C2×C414yescomplemented
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93#926—22no—
94#926C2312yes—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—22no—
97C2×C4×Dic3C2311yescharacteristic
98C2×#208C2312yes—
99D3×M16—24no—
100D3×M16—24no—
101C2×ΓL(1,𝔽25)—22no—
102C2×D3×C8—22no—
103#896—216no—
104#894—216no—
105#892—24no—
106#892D414yes—
107D3×C4⋊C4—24no—
108D3×C4⋊C4D414yes—
109#887—28no—
110#887—24no—
111D3×C22⋊C4—22no—
112#877—22no—
113#877—24no—
114#875—22no—
115#875C2×C414yescomplemented
116#872—24no—
117#872—22no—
118#872—22no—
119#872C2×C414yes—
120C42×D3—24no—
121C42×D3—22no—
122C4×Dic6—28no—
123C4×Dic6—24no—
124C3×#103—24no—
125C3×#101—22no—
126#836—22no—
127#835—24no—
128#824—24no—
129#821—24no—
130#809—24no—
131C22×#41—31no—
132C23×Q8—31no—
133C22×M16—32no—
134#536—38no—
135#536—64no—
136C2×C4⋊Q8—31no—
137C2×C4⋊Q8C2×D311yescharacteristic
138C2×#124—31no—
139C2×#124—31no—
140C2×#123—62no—
141C2×C22⋊Q8—62no—
142#519—31no—
143#519—32no—
144#519—62no—
145#519—64no—
146#519C2×D311yescharacteristic
147C2×C4×Q8—64no—
148C2×#117—31no—
149C2×#117—32no—
150C2×#117—62no—
151#419—32no—
152#419C2×D312yes—
153#418—32no—
154#418—32no—
155#417—64no—
156#416—64no—
157C2×#101—31no—
158C2×#101—32no—
159C2×#101—64no—
160C2×#101C2×D311yescharacteristic
161#339—32no—
162#339—32no—
163#339—32no—
164#339C2×D312yes—
165C23×D3—21no—
166C23×D3C2×D411yescharacteristic
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8C22×C412yes—
173C2×C3×Q8C22×C412yes—
174C2×C3×Q8C2×D412yes—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C22×Dic3—21no—
180C22×Dic3C2×D412yes—
181C22×Dic3C2×D411yescharacteristic
182#239—28no—
183#239—28no—
184#239—44no—
185#239—44no—
186D3×Q8—28no—
187D3×Q8—28no—
188D3×Q8—28no—
189D3×Q8—28no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—24no—
193C2×D12—21no—
194C2×D12—41no—
195C2×D12—41no—
196C2×D12—42no—
197C2×D12C22×C412yes—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—24no—
201C2×C4×D3—24no—
202C2×C4×D3—24no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3C22⋊C418yes—
206C2×C4×D3C22×C412yes—
207C2×C4×D3C2×D412yes—
208C2×C4×D3C2×D412yes—
209C2×C4×D3C2×D412yes—
210C2×C4×D3#4114yes—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—24no—
214C2×Dic6—24no—
215C2×Dic6C22⋊C418yes—
216C2×Dic6C22×C412yes—
217C2×Dic6C2×D412yes—
218C3×M16—22no—
219C3×M16—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—28no—
223C3×C4⋊C4C2×D414yes—
224C3×C22⋊C4—22no—
225C3×C42—24no—
226C3×C42—22no—
227C3×C42—21no—
228C3×C42C22×C412yes—
229#217—22no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—28no—
233(C2×D3)⋊C4—28no—
234C4⋊Dic3—24no—
235C4⋊Dic3C2×D414yes—
236Dic3⋊C4—28no—
237Dic3⋊C4—24no—
238Dic3⋊C4—28no—
239Dic3⋊C4—24no—
240C4×Dic3—28no—
241C4×Dic3—24no—
242C4×Dic3—22no—
243C4×Dic3—21no—
244C4×Dic3—24no—
245C4×Dic3C22×C412yes—
246#208—22no—
247#208—22no—
248C2×C3⋊C8—22no—
249ΓL(1,𝔽25)—44no—
250D3×C8—44no—
251C2×#41—34no—
252C2×#41—62no—
253C2×#41—62no—
254C2×#41—62no—
255C2×#41—64no—
256C22×Q8—34no—
257C22×Q8—34no—
258C22×Q8—32no—
259C22×Q8—31no—
260C22×Q8—62no—
261C22×Q8—62no—
262C22×Q8—64no—
263C22×Q8C22×D311yescharacteristic
264C22×Q8C22×D311yescharacteristic
265C22×D4—31no—
266C22×D4—61no—
267C23×C4—31no—
268C23×C4—32no—
269C23×C4—61no—
270C2×M16—32no—
271C2×M16—62no—
272C2×M16—64no—
273C2×M16—64no—
274C2×M16C22×D312yes—
275C22×C8—62no—
276C4⋊Q8—22no—
277C4⋊Q8—34no—
278C4⋊Q8—62no—
279C4⋊Q8C4×D314yescomplemented
280#124—34no—
281#124—34no—
282#124—62no—
283#124—62no—
284#123—616no—
285C22⋊Q8—616no—
286C4×Q8—24no—
287C4×Q8—64no—
288C4×Q8—68no—
289C4×Q8—68no—
290C4×Q8—64no—
291C4×Q8—64no—
292#117—22no—
293#117—34no—
294#117—32no—
295#117—34no—
296#117—62no—
297#117—62no—
298#117—64no—
299#117—62no—
300#117—64no—
301#117—64no—
302#117—62no—
303#117C22×D312yes—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—34no—
306C2×C4⋊C4—64no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—62no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—62no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—62no—
313C2×C42—31no—
314C2×C42—62no—
315C2×C42—64no—
316C2×C42C22×D311yescharacteristic
317#103—24no—
318#103—64no—
319#102—64no—
320#102—64no—
321#101—22no—
322#101—64no—
323#101—62no—
324C23×C3C22×D411yescharacteristic
325C22×D3—22no—
326C22×D3—42no—
327C22×D3—44no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3C2×C22⋊C412yes—
331C22×D3#12011yescharacteristic
332C22×D3#12011yescharacteristic
333C22×D3#12312yes—
334C3×Q8—24no—
335C3×Q8—24no—
336C3×Q8—24no—
337C3×Q8—42no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4C2×C22⋊C414yes—
347C2×C3×C4C23×C411yescharacteristic
348C2×C3×C4C22×D412yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4112yes—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—24no—
355C2×Dic3—42no—
356C2×Dic3C2×C22⋊C414yes—
357C2×Dic3C2×C22⋊C412yes—
358C2×Dic3#12012yes—
359C2×Dic3#12312yes—
360D12—24no—
361D12—44no—
362D12—44no—
363D12—42no—
364C4×D3—28no—
365C4×D3—28no—
366C4×D3—48no—
367C4×D3—48no—
368C4×D3C4×D414yes—
369C4×D3C4×D414yes—
370C4×D3#12114yes—
371C4×D3#12114yes—
372Dic6—24no—
373Dic6—24no—
374Dic6—216no—
375C3×C8—42no—
376C3⋊C8—42no—
377C24—31no—
378C24—61no—
379#41—68no—
380#41—68no—
381#41—124no—
382#41—124no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—38no—
389C2×Q8—32no—
390C2×Q8—32no—
391C2×Q8—64no—
392C2×Q8—64no—
393C2×Q8—68no—
394C2×Q8—68no—
395C2×Q8—68no—
396C2×Q8—68no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×Q8C2×C4×D312yes—
400C2×Q8C2×C4×D312yes—
401C2×Q8D3×D412yes—
402C2×D4—32no—
403C2×D4—64no—
404C2×D4—62no—
405C2×D4—61no—
406C2×D4—62no—
407C2×D4—122no—
408C2×D4—121no—
409C2×D4—121no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—31no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—32no—
417C22×C4—34no—
418C22×C4—32no—
419C22×C4—38no—
420C22×C4—64no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—61no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4D3×D412yes—
429C22×C4C23×D311yescharacteristic
430M16—22no—
431M16—22no—
432M16—62no—
433M16—62no—
434M16—124no—
435C2×C8—22no—
436C2×C8—62no—
437C2×C8—124no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—34no—
441C4⋊C4—64no—
442C4⋊C4—64no—
443C4⋊C4—68no—
444C4⋊C4—68no—
445C4⋊C4—64no—
446C4⋊C4D3×D414yes—
447C22⋊C4—22no—
448C22⋊C4—68no—
449C22⋊C4—64no—
450C22⋊C4—64no—
451C22⋊C4—62no—
452C22⋊C4—68no—
453C42—24no—
454C42—22no—
455C42—21no—
456C42—32no—
457C42—64no—
458C42—64no—
459C42—68no—
460C42—61no—
461C42—62no—
462C42C2×C4×D312yes—
463C22×C3—22no—
464C22×C3C22×C22⋊C411yescharacteristic
465C22×C3C2×#12011yescharacteristic
466C22×C3C2×#12311yescharacteristic
467C2×D3—22no—
468C2×D3—22no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—44no—
472C2×D3—42no—
473C2×D3—82no—
474C2×D3—82no—
475C2×D3#38612yes—
476C2×D3#38612yes—
477C3×C4—24no—
478C3×C4—42no—
479C3×C4—44no—
480C3×C4—42no—
481C3×C4C2×C4×D412yes—
482C3×C4C2×#12112yes—
483Dic3—24no—
484Dic3—42no—
485Dic3—44no—
486Dic3#38614yes—
487C23—31no—
488C23—31no—
489C23—32no—
490C23—32no—
491C23—62no—
492C23—122no—
493C23—124no—
494C23—121no—
495C23—121no—
496C23C2×D3×D411yescharacteristic
497Q8—24no—
498Q8—24no—
499Q8—24no—
500Q8—42no—
501Q8—616no—
502Q8—64no—
503Q8—64no—
504D4—64no—
505D4—124no—
506D4—124no—
507D4—122no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—32no—
514C2×C4—32no—
515C2×C4—32no—
516C2×C4—34no—
517C2×C4—34no—
518C2×C4—34no—
519C2×C4—34no—
520C2×C4—34no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—68no—
525C2×C4—68no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—64no—
530C2×C4—128no—
531C2×C4—128no—
532C2×C4—122no—
533C2×C4D3×C22⋊C414yes—
534C2×C4C22×C4×D311yescharacteristic
535C2×C4C2×D3×D412yes—
536C2×C4C2×D3×D411yescharacteristic
537C2×C4D3×#4112yes—
538C8—42no—
539C8—122no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3C2×#38611yescharacteristic
543C2×C3#302012yes—
544D3—22no—
545D3—22no—
546D3—82no—
547C22—22no—
548C22—32no—
549C22—32no—
550C22—62no—
551C22—62no—
552C22—122no—
553C22—122no—
554C22—124no—
555C22—122no—
556C22—242no—
557C22—242no—
558C22C2×D3×C22⋊C411yescharacteristic
559C22D3×#12011yescharacteristic
560C22D3×#12311yescharacteristic
561C4—24no—
562C4—34no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—64no—
567C4—122no—
568C4—124no—
569C4C4×D3×D412yes—
570C4D3×#12112yes—
571C3#2095411yescharacteristic, complemented, Sylow
572C2—21no—
573C2—21no—
574C2—62no—
575C2—62no—
576C2—242no—
577C2D3×#38611yescharacteristic
578C2#8029912yes—
subgroupG/H
centerC22D3×#120
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1886C2
socleC22×C3C2×#120
prime radicalC22×#239C22
cosocleC22×C3×C4C24
hypercenter#1886D3
hypocenterC3#20954
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1886

Complements

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

NH ≅ G/Ntype
2. D3×#519C2semidirect
4. C2×D3×#101C2semidirect
5. C3×#1886C2semidirect
6. #71784C2semidirect
7. #71661C2semidirect
8. #70345C2semidirect
9. #70104C2semidirect
10. #68766C2semidirect
11. #68753C2semidirect
22. #5724C4semidirect
56. #1886D3semidirect
79. C3×C4⋊Q8C2×C4semidirect
86. #947C2×C4semidirect
89. #944C2×C4semidirect
115. #875C2×C4semidirect
279. C4⋊Q8C4×D3semidirect
571. C3#20954semidirect

Maximal subgroups

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

subgroupindex
C2×#57242
D3×#5192
#789562
C2×D3×#1012
C3×#18862
#717842
#716612
#703452
#701042
#687662
#687532
#209543