Finite Group Databasebeta

← #10594660

Subgroups — #10594660

Order 576.

537 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64105C211yesmaximal, characteristic, complemented
2C3×#898C211yesmaximal, characteristic, complemented
3C3×#895C211yesmaximal, characteristic, complemented
4#63922C211yesmaximal, characteristic, complemented
5#63922C211yesmaximal, characteristic, complemented
6#63921C211yesmaximal, characteristic, complemented
7#63916C211yesmaximal, characteristic, complemented
8#63913C211yesmaximal, characteristic, complemented
9#63905C211yesmaximal, characteristic, complemented
10#63904C211yesmaximal, characteristic, complemented
11#63885C211yesmaximal, characteristic, complemented
12#63884C211yesmaximal, characteristic, complemented
13#63880C211yesmaximal, characteristic, complemented
14#63860C211yesmaximal, characteristic, complemented
15#63858C211yesmaximal, characteristic, complemented
16#5704—31nomaximal
17#5613—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×#634C2211yescharacteristic, complemented
21C2×C3×D12C2211yescharacteristic
22C2×C3×D12C2211yescharacteristic
23C2×C3×D12C2211yescharacteristic, complemented
24C2×C3×C4×D3C2211yescharacteristic, complemented
25C2×#626C2211yescharacteristic
26C2×#626C2211yescharacteristic
27C2×#625C2211yescharacteristic
28C2×#625C2211yescharacteristic
29C2×#624C2211yescharacteristic
30C2×D3×Dic3C2211yescharacteristic, complemented
31C32×C4⋊C4C2211yescharacteristic, complemented
32#3752C2211yescharacteristic
33#3752C2211yescharacteristic
34#3750C2211yescharacteristic, complemented
35#3750C2211yescharacteristic, complemented
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C3×C4⋊Dic3C2211yescharacteristic, complemented
42C3×Dic3⋊C4C2211yescharacteristic, complemented
43C3×C4×Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45(C2×D3)⋊Dic3C2211yescharacteristic
46(C2×D3)⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic, complemented
50(C2×D3)⋊Dic3C2211yescharacteristic, complemented
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×#121—31no—
55#939—31no—
56#938—31no—
57#937—31no—
58#937—31no—
59#936—31no—
60#935—31no—
61#929—31no—
62C4×#67—31no—
63#898—31no—
64#898D311yescharacteristic, complemented
65#896—31no—
66#896—31no—
67#896—61no—
68#895—31no—
69#895D311yescharacteristic, complemented
70#894—31no—
71#893—31no—
72#885—31no—
73#885—31no—
74#884—31no—
75#883—31no—
76#882—31no—
77#882—31no—
78#67⋊C4—31no—
79D3×C22⋊C4—31no—
80D3×C22⋊C4—31no—
81#879—31no—
82#875—31no—
83C4×D12—31no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C22×C3×D3C2311yescharacteristic
89C2×D32—22no—
90C2×D32—41no—
91C2×D32—41no—
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C32×C4C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C4×C3⋊D3—41no—
99#634—41no—
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×D12—41no—
105C3×D12—41no—
106C3×D12—41no—
107C3×C4×D3—41no—
108#626—22no—
109#626—22no—
110#625—41no—
111#625—41no—
112#624—41no—
113D3×Dic3—41no—
114#538—91noSylow
115C23×D3—31no—
116C23×D3—31no—
117C2×C3×D4—31no—
118C2×C3×D4—31no—
119C2×C3×D4—31no—
120C22×C3×C4—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×#67—31no—
124C2×#67—31no—
125C2×#67—31no—
126C2×#67—31no—
127C22×Dic3—31no—
128C2×D12—31no—
129C2×D12—31no—
130C2×D12C2×D311yescharacteristic
131C2×D12C2×D311yescharacteristic
132C2×D12C2×D311yescharacteristic, complemented
133C2×C4×D3—31no—
134C2×C4×D3—31no—
135C2×C4×D3—31no—
136C2×C4×D3—31no—
137C2×C4×D3—31no—
138C2×C4×D3—61no—
139C2×C4×D3C2×D311yescharacteristic, complemented
140C2×Dic6—31no—
141C2×Dic6—31no—
142C2×Dic6—61no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4C2×D311yescharacteristic, complemented
147C3×C4⋊C4C2×D311yescharacteristic, complemented
148C3×C22⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C22⋊C4—31no—
152C3×C22⋊C4—31no—
153C3×C42—31no—
154#217—31no—
155#217—31no—
156#217—31no—
157#217—31no—
158#217—31no—
159#217—31no—
160#217—31no—
161(C2×D3)⋊C4—31no—
162(C2×D3)⋊C4—31no—
163(C2×D3)⋊C4—31no—
164(C2×D3)⋊C4—31no—
165(C2×D3)⋊C4—31no—
166(C2×D3)⋊C4—31no—
167(C2×D3)⋊C4—31no—
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—31no—
170(C2×D3)⋊C4—31no—
171(C2×D3)⋊C4—31no—
172(C2×D3)⋊C4—61no—
173(C2×D3)⋊C4—61no—
174(C2×D3)⋊C4C2×D311yescharacteristic
175(C2×D3)⋊C4C2×D311yescharacteristic
176(C2×D3)⋊C4C2×D311yescharacteristic
177(C2×D3)⋊C4C2×D311yescharacteristic
178(C2×D3)⋊C4C2×D311yescharacteristic, complemented
179C4⋊Dic3—31no—
180C4⋊Dic3C2×D311yescharacteristic, complemented
181Dic3⋊C4—31no—
182Dic3⋊C4—31no—
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4—61no—
187Dic3⋊C4—61no—
188Dic3⋊C4C2×D311yescharacteristic, complemented
189C4×Dic3—31no—
190C4×Dic3—31no—
191C4×Dic3C2×D311yescharacteristic, complemented
192C22×C32C2411yescharacteristic
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3⋊D3#4112yes—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C2×C3×D3—21no—
203C2×C3×D3—21no—
204C2×C3×D3—21no—
205C2×C3×D3—21no—
206C2×C3×D3—21no—
207C2×C3×D3—21no—
208D32—44no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C3⋊Dic3—21no—
213C3⋊Dic3—21no—
214C3⋊Dic3—21no—
215C3×Dic3—21no—
216C3×Dic3—21no—
217C3×Dic3—21no—
218C3×Dic3—21no—
219#126—31no—
220#126—91no—
221#124—91no—
222#124—91no—
223#123—91no—
224#123—91no—
225C22⋊Q8—91no—
226#121—31no—
227#121—91no—
228#121—91no—
229#120—91no—
230#120—91no—
231C4×D4—91no—
232C4×D4—91no—
233C2×C22⋊C4—91no—
234C23×C3—31no—
235C23×C3—31no—
236C23×C3—31no—
237C23×C3—31no—
238C22×D3—31no—
239C22×D3—31no—
240C22×D3—61no—
241C22×D3—62no—
242C22×D3—62no—
243C22×D3—121no—
244C22×D3—121no—
245C22×D3—121no—
246C22×D3—121no—
247C22×D3C22×D311yescharacteristic
248C22×D3C22×D311yescharacteristic
249C22×D3C22×D311yescharacteristic
250C22×D3C22×D311yescharacteristic
251C3×D4—121no—
252C3×D4—121no—
253C3×D4—121no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—121no—
262C2×C3×C4C22×D311yescharacteristic
263C2×C3×C4C22×D311yescharacteristic
264C2×C3×C4C22×D311yescharacteristic
265C2×C3×C4C22×D311yescharacteristic
266C2×C3×C4C22×D311yescharacteristic
267C2×C3×C4C22×D311yescharacteristic
268#67—62no—
269#67—62no—
270#67—121no—
271#67—121no—
272#67—121no—
273#67—121no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—31no—
279C2×Dic3—31no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—121no—
284C2×Dic3C22×D311yescharacteristic
285C2×Dic3C22×D311yescharacteristic
286C2×Dic3C22×D311yescharacteristic
287C2×Dic3C22×D311yescharacteristic
288D12—41no—
289D12—41no—
290D12—41no—
291D12—62no—
292D12—62no—
293C4×D3—41no—
294C4×D3—121no—
295C4×D3—121no—
296C4×D3—121no—
297C4×D3—121no—
298C4×D3—122no—
299C4×D3—121no—
300Dic6—121no—
301Dic6—121no—
302Dic6—122no—
303C2×C32C2×#4111yescharacteristic
304C2×C32H(5,𝔽2)11yescharacteristic
305C2×C32H(5,𝔽2)11yescharacteristic
306C3⋊D3—22no—
307C3×D3—41no—
308C3×D3—41no—
309C3×D3—41no—
310C3×D3—41no—
311C24—91no—
312C2×Q8—91no—
313C2×D4—31no—
314C2×D4—31no—
315C2×D4—31no—
316C2×D4—91no—
317C2×D4—91no—
318C2×D4—91no—
319C2×D4—91no—
320C22×C4—31no—
321C22×C4—91no—
322C22×C4—91no—
323C22×C4—91no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—91no—
327C4⋊C4—91no—
328C4⋊C4—91no—
329C4⋊C4D3211yescharacteristic, complemented
330C22⋊C4—31no—
331C22⋊C4—31no—
332C22⋊C4—31no—
333C22⋊C4—31no—
334C22⋊C4—31no—
335C22⋊C4—91no—
336C22⋊C4—91no—
337C22⋊C4—91no—
338C22⋊C4—91no—
339C22⋊C4—91no—
340C22⋊C4—91no—
341C22⋊C4—91no—
342C22⋊C4—91no—
343C22⋊C4—91no—
344C42—31no—
345C42—91no—
346C22×C3—21no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—61no—
359C22×C3C23×D311yescharacteristic
360C22×C3C23×D311yescharacteristic
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—21no—
369C2×D3—21no—
370C2×D3—21no—
371C2×D3—21no—
372C2×D3—21no—
373C2×D3—32no—
374C2×D3—32no—
375C2×D3—62no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—121no—
381C2×D3—121no—
382C2×D3—124no—
383C2×D3—124no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—41no—
393C3×C4—61no—
394C3×C4—61no—
395C3×C4—61no—
396C3×C4—61no—
397Dic3—21no—
398Dic3—21no—
399Dic3—21no—
400Dic3—21no—
401Dic3—62no—
402Dic3—61no—
403Dic3—61no—
404Dic3—61no—
405Dic3—61no—
406Dic3—61no—
407Dic3—61no—
408Dic3—121no—
409Dic3—121no—
410C32#53811yescharacteristic, complemented, Sylow
411C23—31no—
412C23—31no—
413C23—31no—
414C23—31no—
415C23—91no—
416C23—182no—
417C23—361no—
418C23—361no—
419Q8—361no—
420D4—121no—
421D4—121no—
422D4—121no—
423D4—182no—
424D4—182no—
425D4—361no—
426D4—361no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—91no—
432C2×C4—91no—
433C2×C4—91no—
434C2×C4—121no—
435C2×C4—361no—
436C2×C4—361no—
437C2×C4—361no—
438C2×C4C2×D3211yescharacteristic
439C2×C4C2×D3211yescharacteristic
440C2×C4C2×D3211yescharacteristic
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—121no—
445C2×C3—121no—
446C2×C3—121no—
447C2×C3—121no—
448C2×C3#100411yescharacteristic
449C2×C3#100411yescharacteristic
450C2×C3D3×#4111yescharacteristic
451C2×C3D3×#4111yescharacteristic
452C2×C3#100911yescharacteristic
453C2×C3#100911yescharacteristic
454D3—41no—
455D3—41no—
456D3—41no—
457D3—41no—
458D3—62no—
459D3—62no—
460D3—122no—
461C22—61no—
462C22—61no—
463C22—61no—
464C22—61no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—61no—
471C22—61no—
472C22—61no—
473C22—92no—
474C22—181no—
475C22—181no—
476C22—364no—
477C22C22×D3211yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—61no—
482C4—61no—
483C4—61no—
484C4—61no—
485C4—181no—
486C4—181no—
487C4—181no—
488C3—21no—
489C3#561311yescharacteristic, complemented
490C3#570411yescharacteristic, complemented
491C2—121no—
492C2—121no—
493C2—121no—
494C2—121no—
495C2—182no—
496C2#6430711yescharacteristic
497C2#6431411yescharacteristic
498C2#6431911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64105C2semidirect
2. C3×#898C2semidirect
3. C3×#895C2semidirect
4. #63922C2semidirect
5. #63922C2semidirect
6. #63921C2semidirect
7. #63916C2semidirect
8. #63913C2semidirect
9. #63905C2semidirect
10. #63904C2semidirect
11. #63885C2semidirect
12. #63884C2semidirect
13. #63880C2semidirect
14. #63860C2semidirect
15. #63858C2semidirect
20. C2×#634C22semidirect
23. C2×C3×D12C22semidirect
24. C2×C3×C4×D3C22semidirect
30. C2×D3×Dic3C22semidirect
31. C32×C4⋊C4C22semidirect
34. #3750C22semidirect
35. #3750C22semidirect
40. C3×(C2×D3)⋊C4C22semidirect
41. C3×C4⋊Dic3C22semidirect
42. C3×Dic3⋊C4C22semidirect
43. C3×C4×Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
49. (C2×D3)⋊Dic3C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. Dic32C22semidirect
64. #898D3semidirect
69. #895D3semidirect
132. C2×D12C2×D3semidirect
139. C2×C4×D3C2×D3semidirect
146. C3×C4⋊C4C2×D3semidirect
147. C3×C4⋊C4C2×D3semidirect
178. (C2×D3)⋊C4C2×D3semidirect
180. C4⋊Dic3C2×D3semidirect
188. Dic3⋊C4C2×D3semidirect
191. C4×Dic3C2×D3semidirect
329. C4⋊C4D32semidirect
410. C32#538semidirect
489. C3#5613semidirect
490. C3#5704semidirect

Maximal subgroups

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

subgroupindex
#641052
C3×#8982
C3×#8952
#639222
#639212
#639162
#639132
#639052
#639042
#638852
#638842
#638802
#638602
#638582
#57043
#56133