Finite Group Databasebeta

← #10595002

Subgroups — #10595002

Order 576.

707 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D3C211yesmaximal, characteristic, complemented
2C2×#3798C211yesmaximal, characteristic, complemented
3C4×#638C211yesmaximal, characteristic, complemented
4C3×C4×#67C211yesmaximal, characteristic, complemented
5#63978C211yesmaximal, characteristic, complemented
6#63977C211yesmaximal, characteristic, complemented
7#63976C211yesmaximal, characteristic, complemented
8#63964C211yesmaximal, characteristic, complemented
9#63906C211yesmaximal, characteristic, complemented
10#63905C211yesmaximal, characteristic, complemented
11C4×#626C211yesmaximal, characteristic, complemented
12C4×#625C211yesmaximal, characteristic, complemented
13#63855C211yesmaximal, characteristic, complemented
14#63851C211yesmaximal, characteristic, complemented
15#63850C211yesmaximal, characteristic, complemented
16#5791—31nomaximal
17#5547—31nomaximal
18C23×C3×D3C2211yescharacteristic
19C22×C32×C4C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic, complemented
22C2×C3×#67C2211yescharacteristic
23C22×C3×Dic3C2211yescharacteristic
24C2×C3×C4×D3—24no—
25C2×C3×C4×D3C2211yescharacteristic
26C2×C3×C4×D3C2211yescharacteristic
27C2×C3×C4×D3C2211yescharacteristic
28C2×C3×C4×D3C2211yescharacteristic
29C2×C3×C4×D3C2211yescharacteristic, complemented
30C2×#627C2211yescharacteristic, complemented
31C2×#626C2211yescharacteristic
32C2×#626C2211yescharacteristic
33C2×#626C2211yescharacteristic, complemented
34C2×#625C2211yescharacteristic
35C2×#625C2211yescharacteristic
36#3798—24no—
37#3757C2211yescharacteristic
38#3752C2211yescharacteristic, complemented
39#3750C2211yescharacteristic, complemented
40C4×C3⋊Dic3C2211yescharacteristic, complemented
41C3×#217C2211yescharacteristic
42C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C3×Dic3⋊C4C2211yescharacteristic, complemented
44C3×C4×Dic3C2211yescharacteristic, complemented
45#3724C2211yescharacteristic, complemented
46Dic3⋊Dic3C2211yescharacteristic, complemented
47Dic3⋊Dic3C2211yescharacteristic, complemented
48Dic3⋊Dic3C2211yescharacteristic, complemented
49#3722C2211yescharacteristic, complemented
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52(C2×D3)⋊Dic3C2211yescharacteristic
53(C2×D3)⋊Dic3C2211yescharacteristic
54(C2×D3)⋊Dic3C2211yescharacteristic, complemented
55C23×C3×C4—31no—
56C2×#235—31no—
57C2×#235—31no—
58C22×C4×D3D311yescharacteristic, complemented
59C3×C4×D4—31no—
60#953—31no—
61#953—31no—
62#939—31no—
63#937—31no—
64#930—31no—
65#930—31no—
66#929—31no—
67#929—31no—
68C4×#67—31no—
69C4×#67—31no—
70C4×#67—31no—
71C4×#67—31no—
72C4×#67—31no—
73C4×#67—61no—
74C4×#67D311yescharacteristic, complemented
75#926—31no—
76#924—31no—
77#924—31no—
78#896—31no—
79#894—31no—
80#884—31no—
81#883—31no—
82#882—31no—
83#878—31no—
84C4×D12—31no—
85#872—31no—
86C23×C32C2311yescharacteristic
87C22×C3⋊D3C2311yescharacteristic
88C22×C3×D3—22no—
89C22×C3×D3—22no—
90C22×C3×D3—22no—
91C22×C3×D3C2311yescharacteristic
92C22×C3×D3C2311yescharacteristic
93C22×C3×D3C2311yescharacteristic
94C2×C32×C4—22no—
95C2×C32×C4C2311yescharacteristic
96C2×C32×C4C2311yescharacteristic
97#638—22no—
98C2×C3⋊Dic3C2311yescharacteristic
99C2×C3⋊Dic3C2311yescharacteristic
100C2×C3⋊Dic3C2311yescharacteristic
101C4×C3⋊D3—22no—
102C3×#67—22no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C3×C4×D3—22no—
110C3×C4×D3—22no—
111C3×C4×D3—22no—
112C3×C4×D3—22no—
113C3×C4×D3D414yescomplemented
114#627—22no—
115#626—22no—
116#626—22no—
117#626—22no—
118#625—22no—
119#625—22no—
120#525—91noSylow
121C24×C3—31no—
122C23×D3C2×D311yescharacteristic
123C2×C3×D4—31no—
124C22×C3×C4—21no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—64no—
132C22×C3×C4C2×D311yescharacteristic
133C22×C3×C4C2×D311yescharacteristic
134C2×#67—31no—
135C2×#67—31no—
136C2×#67—31no—
137C2×#67—31no—
138C2×#67—31no—
139C2×#67—31no—
140C2×#67—31no—
141C2×#67—31no—
142C2×#67—31no—
143C2×#67—61no—
144C2×#67C2×D311yescharacteristic
145C22×Dic3C2×D311yescharacteristic
146#235—64no—
147#235—64no—
148C2×D12—31no—
149C2×D12—31no—
150C2×D12—31no—
151C2×C4×D3—24no—
152C2×C4×D3—31no—
153C2×C4×D3—31no—
154C2×C4×D3—61no—
155C2×C4×D3C2×D311yescharacteristic
156C2×C4×D3C2×D311yescharacteristic
157C2×C4×D3C2×D311yescharacteristic
158C2×C4×D3C2×D311yescharacteristic
159C2×C4×D3C2×D311yescharacteristic, complemented
160C2×Dic6—31no—
161C2×Dic6—31no—
162C3×C4⋊C4—31no—
163C3×C22⋊C4—31no—
164C3×C22⋊C4—31no—
165C3×C42—31no—
166#217—31no—
167#217—31no—
168#217—31no—
169#217—31no—
170#217—31no—
171#217—31no—
172#217—31no—
173#217—61no—
174#217C2×D311yescharacteristic
175(C2×D3)⋊C4—31no—
176(C2×D3)⋊C4—31no—
177(C2×D3)⋊C4—31no—
178(C2×D3)⋊C4—31no—
179(C2×D3)⋊C4—31no—
180(C2×D3)⋊C4—31no—
181(C2×D3)⋊C4—31no—
182(C2×D3)⋊C4—31no—
183(C2×D3)⋊C4—31no—
184(C2×D3)⋊C4—61no—
185(C2×D3)⋊C4C2×D311yescharacteristic, complemented
186C4⋊Dic3—31no—
187C4⋊Dic3—31no—
188C4⋊Dic3—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—31no—
191Dic3⋊C4—31no—
192Dic3⋊C4—31no—
193Dic3⋊C4—31no—
194Dic3⋊C4—31no—
195Dic3⋊C4—31no—
196Dic3⋊C4—61no—
197Dic3⋊C4C2×D311yescharacteristic, complemented
198C4×Dic3—31no—
199C4×Dic3—31no—
200C4×Dic3—61no—
201C4×Dic3C2×D311yescharacteristic, complemented
202C22×C32—21no—
203C22×C32—21no—
204C22×C32#4112yescomplemented
205C22×C32C2411yescharacteristic
206C2×C3⋊D3—21no—
207C2×C3⋊D3—21no—
208C2×C3⋊D3—21no—
209C2×C3×D3—28no—
210C2×C3×D3—21no—
211C2×C3×D3—21no—
212C2×C3×D3—21no—
213C2×C3×D3—21no—
214C2×C3×D3—21no—
215C2×C3×D3—21no—
216C2×C3×D3—21no—
217C2×C3×D3C2×D412yes—
218C2×C3×D3#4112yescomplemented
219C32×C4—21no—
220C32×C4C2×D412yes—
221C3⋊Dic3—21no—
222C3⋊Dic3—21no—
223C3⋊Dic3—21no—
224C3×Dic3—21no—
225C3×Dic3—21no—
226C3×Dic3—21no—
227C3×Dic3—21no—
228C3×Dic3C2×D412yes—
229C2×#41—91no—
230C23×C4—31no—
231#123—91no—
232#123—91no—
233C22⋊Q8—91no—
234C22⋊Q8—91no—
235#121—91no—
236#121—91no—
237#120—91no—
238#120—91no—
239C4×D4—31no—
240C4×D4—91no—
241C4×D4—91no—
242C4×D4—91no—
243#117—91no—
244C23×C3—21no—
245C23×C3—31no—
246C23×C3—31no—
247C23×C3—31no—
248C23×C3—62no—
249C23×C3—62no—
250C23×C3—62no—
251C23×C3C22×D311yescharacteristic
252C23×C3C22×D311yescharacteristic
253C22×D3—22no—
254C22×D3—22no—
255C22×D3—22no—
256C22×D3—31no—
257C22×D3—31no—
258C22×D3—61no—
259C22×D3C22×D311yescharacteristic
260C22×D3C22×D311yescharacteristic
261C22×D3C22×D311yescharacteristic
262C3×D4—62no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—31no—
268C2×C3×C4—34no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—42no—
274C2×C3×C4—62no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—62no—
279C2×C3×C4C22×D311yescharacteristic
280C2×C3×C4C22×D311yescharacteristic
281C2×C3×C4C22×D311yescharacteristic
282C2×C3×C4C22×D311yescharacteristic
283#67—22no—
284#67—62no—
285#67—62no—
286#67—62no—
287#67—62no—
288#67—64no—
289#67—62no—
290#67—62no—
291#67—62no—
292#67—62no—
293#67—62no—
294C2×Dic3—22no—
295C2×Dic3—31no—
296C2×Dic3—31no—
297C2×Dic3—31no—
298C2×Dic3—31no—
299C2×Dic3—31no—
300C2×Dic3—31no—
301C2×Dic3—61no—
302C2×Dic3—61no—
303C2×Dic3—61no—
304C2×Dic3C22×D311yescharacteristic
305C2×Dic3C22×D311yescharacteristic
306C2×Dic3C22×D311yescharacteristic
307C2×Dic3C22×D311yescharacteristic
308C2×Dic3C22×D311yescharacteristic
309D12—62no—
310D12—62no—
311D12—62no—
312C4×D3—22no—
313C4×D3—22no—
314C4×D3—22no—
315C4×D3—22no—
316C4×D3—62no—
317C4×D3—62no—
318C4×D3—122no—
319C4×D3#6714yescomplemented
320Dic6—62no—
321Dic6—62no—
322C2×C32—22no—
323C2×C32C22×D411yescharacteristic
324C2×C32C2×#4111yescharacteristic
325C2×C32C2×#4111yescharacteristic
326C3⋊D3—41no—
327C3×D3—22no—
328C3×D3—22no—
329C3×D3—41no—
330C24—31no—
331#41—184no—
332C2×Q8—91no—
333C2×D4—31no—
334C2×D4—91no—
335C2×D4—91no—
336C2×D4—91no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—91no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—64no—
347C22×C4—91no—
348C22×C4D3211yescharacteristic
349C4⋊C4—31no—
350C4⋊C4—91no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C22⋊C4—31no—
356C22⋊C4—31no—
357C22⋊C4—91no—
358C22⋊C4—91no—
359C22⋊C4—91no—
360C22⋊C4—91no—
361C22⋊C4—91no—
362C22⋊C4—91no—
363C22⋊C4—91no—
364C22⋊C4—91no—
365C42—31no—
366C42—91no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—32no—
374C22×C3—32no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—68no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—61no—
384C22×C3—61no—
385C22×C3#23512yescomplemented
386C22×C3#23512yescomplemented
387C22×C3C23×D311yescharacteristic
388C22×C3C23×D311yescharacteristic
389C2×D3—28no—
390C2×D3—21no—
391C2×D3—21no—
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—21no—
396C2×D3—21no—
397C2×D3—62no—
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—61no—
402C2×D3—61no—
403C2×D3—61no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3#23512yescomplemented
407C2×D3C2×#6712yes—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—32no—
412C3×C4—41no—
413C3×C4—61no—
414C3×C4—61no—
415C3×C4—61no—
416C3×C4—61no—
417C3×C4D3×D412yes—
418C3×C4C2×#6712yes—
419Dic3—21no—
420Dic3—21no—
421Dic3—21no—
422Dic3—21no—
423Dic3—62no—
424Dic3—61no—
425Dic3—61no—
426Dic3—61no—
427Dic3—61no—
428Dic3—61no—
429Dic3—61no—
430Dic3—121no—
431Dic3—121no—
432Dic3C2×#6712yes—
433C32#52511yescharacteristic, complemented, Sylow
434C23—31no—
435C23—31no—
436C23—31no—
437C23—62no—
438C23—62no—
439C23—62no—
440C23—91no—
441C23C2×D3211yescharacteristic
442Q8—182no—
443D4—62no—
444D4—182no—
445D4—182no—
446D4—182no—
447D4—182no—
448D4—182no—
449D4—182no—
450C2×C4—22no—
451C2×C4—31no—
452C2×C4—34no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—91no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—182no—
466C2×C4C2×D3211yescharacteristic
467C2×C4C2×D3211yescharacteristic
468C2×C3—22no—
469C2×C3—22no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—42no—
474C2×C3—62no—
475C2×C3—62no—
476C2×C3—121no—
477C2×C3C2×#23511yescharacteristic
478C2×C3C2×#23511yescharacteristic
479C2×C3C2×#23511yescharacteristic
480C2×C3C2×D3×D411yescharacteristic
481C2×C3D3×#4111yescharacteristic
482C2×C3C22×#6711yescharacteristic
483D3—22no—
484D3—22no—
485D3—41no—
486D3—122no—
487D3—121no—
488D3—121no—
489C22—21no—
490C22—21no—
491C22—32no—
492C22—32no—
493C22—68no—
494C22—61no—
495C22—61no—
496C22—61no—
497C22—61no—
498C22—61no—
499C22—61no—
500C22—61no—
501C22—181no—
502C22—181no—
503C22—181no—
504C22#379812yescomplemented
505C22C22×D3211yescharacteristic
506C4—21no—
507C4—32no—
508C4—61no—
509C4—61no—
510C4—61no—
511C4—61no—
512C4—181no—
513C4—181no—
514C4—181no—
515C4D3×#6712yes—
516C3—21no—
517C3#554711yescharacteristic, complemented
518C3#579111yescharacteristic, complemented
519C2—22no—
520C2—62no—
521C2—62no—
522C2—121no—
523C2—361no—
524C2C2×#379811yescharacteristic
525C2D3×#23511yescharacteristic
526C2C2×D3×#6711yescharacteristic
subgroupG/H
centerC2×C4C2×D32
commutator, socle, cosocleC22×C32C24
FrattiniC22C22×D32
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×D3C2semidirect
2. C2×#3798C2semidirect
3. C4×#638C2semidirect
4. C3×C4×#67C2semidirect
5. #63978C2semidirect
6. #63977C2semidirect
7. #63976C2semidirect
8. #63964C2semidirect
9. #63906C2semidirect
10. #63905C2semidirect
11. C4×#626C2semidirect
12. C4×#625C2semidirect
13. #63855C2semidirect
14. #63851C2semidirect
15. #63850C2semidirect
21. C2×C4×C3⋊D3C22semidirect
29. C2×C3×C4×D3C22semidirect
30. C2×#627C22semidirect
33. C2×#626C22semidirect
38. #3752C22semidirect
39. #3750C22semidirect
40. C4×C3⋊Dic3C22semidirect
42. C3×(C2×D3)⋊C4C22semidirect
43. C3×Dic3⋊C4C22semidirect
44. C3×C4×Dic3C22semidirect
45. #3724C22semidirect
46. Dic3⋊Dic3C22semidirect
47. Dic3⋊Dic3C22semidirect
48. Dic3⋊Dic3C22semidirect
49. #3722C22semidirect
54. (C2×D3)⋊Dic3C22semidirect
58. C22×C4×D3D3semidirect
74. C4×#67D3semidirect
113. C3×C4×D3D4semidirect
159. C2×C4×D3C2×D3semidirect
185. (C2×D3)⋊C4C2×D3semidirect
197. Dic3⋊C4C2×D3semidirect
201. C4×Dic3C2×D3semidirect
204. C22×C32#41semidirect
218. C2×C3×D3#41semidirect
319. C4×D3#67semidirect
385. C22×C3#235semidirect
386. C22×C3#235semidirect
406. C2×D3#235semidirect
433. C32#525semidirect
504. C22#3798semidirect
517. C3#5547semidirect
518. C3#5791semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D32
C2×#37982
C4×#6382
C3×C4×#672
#639782
#639772
#639762
#639642
#639062
#639052
C4×#6262
C4×#6252
#638552
#638512
#638502
#57913
#55473