Finite Group Databasebeta

← #10595437

Subgroups — #10595437

Order 576.

533 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64155C211yesmaximal, characteristic, complemented
2C3×#947C211yesmaximal, characteristic, complemented
3C3×#946C211yesmaximal, characteristic, complemented
4#63922C211yesmaximal, characteristic
5#63921C211yesmaximal, characteristic, complemented
6D3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7#63891C211yesmaximal, characteristic, complemented
8#63884C211yesmaximal, characteristic, complemented
9#63881C211yesmaximal, characteristic, complemented
10#63879C211yesmaximal, characteristic, complemented
11#63877C211yesmaximal, characteristic, complemented
12#63861C211yesmaximal, characteristic, complemented
13#63858C211yesmaximal, characteristic, complemented
14#63854C211yesmaximal, characteristic, complemented
15#63836C211yesmaximal, characteristic, complemented
16#5679—31nomaximal
17#5637—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C32×Q8C2211yescharacteristic, complemented
20C2×C4×C3⋊D3C2211yescharacteristic
21C2×C3×D12C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×#626C2211yescharacteristic
24C2×#625C2211yescharacteristic
25C2×#624C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic, complemented
28#3752C2211yescharacteristic
29#3752C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
31#3750C2211yescharacteristic, complemented
32#3750C2211yescharacteristic, complemented
33C3×(C2×D3)⋊C4C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
38C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
39C3×C4⋊Dic3C2211yescharacteristic, complemented
40C3×Dic3⋊C4C2211yescharacteristic, complemented
41C3×Dic3⋊C4C2211yescharacteristic, complemented
42C3×C4×Dic3C2211yescharacteristic, complemented
43#3724C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic
46(C2×D3)⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic, complemented
51Dic32C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#124—31no—
54C3×C22⋊Q8—31no—
55#947D311yescharacteristic, complemented
56#946—31no—
57#946—31no—
58#946—61no—
59#946D311yescharacteristic, complemented
60#937—31no—
61#935—31no—
62C4×#67—31no—
63C2×(C2×D3)⋊C4—31no—
64#898—31no—
65#898—31no—
66#894—31no—
67#894—31no—
68#893—31no—
69#892—31no—
70#886—31no—
71#883—31no—
72#883—31no—
73#882—31no—
74#67⋊C4—31no—
75#67⋊C4—31no—
76D3×C22⋊C4—31no—
77D3×C22⋊C4—31no—
78D3×C22⋊C4—31no—
79#879—31no—
80#879—31no—
81#877—31no—
82#872—31no—
83#872—31no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×D32—22no—
89C2×D32—22no—
90C2×D32—41no—
91C32×Q8—41no—
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C32×C4C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C4×C3⋊D3—41no—
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×D12—41no—
105C3×C4×D3—41no—
106#626—22no—
107#625—22no—
108#624—22no—
109D3×Dic3—22no—
110D3×Dic3—41no—
111#551—91noSylow
112C23×D3—31no—
113C23×D3—31no—
114C2×C3×Q8—21no—
115C2×C3×Q8C2×D311yescharacteristic, complemented
116C2×C3×Q8C2×D311yescharacteristic, complemented
117C2×C3×D4—31no—
118C22×C3×C4—31no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C22×Dic3—31no—
123C22×Dic3—31no—
124C2×D12—31no—
125C2×D12C2×D311yescharacteristic
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3—31no—
132C2×C4×D3—61no—
133C2×C4×D3C2×D311yescharacteristic
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—31no—
140C3×C22⋊C4—31no—
141C3×C22⋊C4—31no—
142C3×C22⋊C4—31no—
143C3×C42—31no—
144#217—31no—
145#217—31no—
146#217—31no—
147#217—31no—
148#217—31no—
149(C2×D3)⋊C4—31no—
150(C2×D3)⋊C4—31no—
151(C2×D3)⋊C4—31no—
152(C2×D3)⋊C4—31no—
153(C2×D3)⋊C4—31no—
154(C2×D3)⋊C4—31no—
155(C2×D3)⋊C4—31no—
156(C2×D3)⋊C4—31no—
157(C2×D3)⋊C4—31no—
158(C2×D3)⋊C4—31no—
159(C2×D3)⋊C4—31no—
160(C2×D3)⋊C4—61no—
161(C2×D3)⋊C4—61no—
162(C2×D3)⋊C4C2×D311yescharacteristic
163(C2×D3)⋊C4C2×D311yescharacteristic
164(C2×D3)⋊C4C2×D311yescharacteristic
165(C2×D3)⋊C4C2×D311yescharacteristic
166(C2×D3)⋊C4C2×D311yescharacteristic, complemented
167(C2×D3)⋊C4C2×D311yescharacteristic, complemented
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—61no—
172C4⋊Dic3C2×D311yescharacteristic, complemented
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—31no—
180Dic3⋊C4—61no—
181Dic3⋊C4—61no—
182Dic3⋊C4C2×D311yescharacteristic, complemented
183Dic3⋊C4C2×D311yescharacteristic, complemented
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3—31no—
188C4×Dic3C2×D311yescharacteristic, complemented
189C22×C32C2411yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3#4112yes—
193C2×C3×D3—21no—
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3#4112yes—
202D32—44no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—21no—
208C3⋊Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—21no—
213C3×Dic3—21no—
214#126—91no—
215#126—91no—
216#124—31no—
217#123—91no—
218#123—91no—
219#123—91no—
220C22⋊Q8—31no—
221C22⋊Q8—91no—
222#120—91no—
223C4×D4—91no—
224C4×D4—91no—
225#117—91no—
226#117—91no—
227C2×C22⋊C4—91no—
228C2×C22⋊C4—91no—
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—31no—
232C22×D3—31no—
233C22×D3—31no—
234C22×D3—61no—
235C22×D3—62no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—121no—
240C22×D3—121no—
241C22×D3C22×D311yescharacteristic
242C22×D3C22×D311yescharacteristic
243C22×D3C22×D311yescharacteristic
244C3×Q8—41no—
245C3×Q8—41no—
246C3×Q8—42no—
247C3×D4—121no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—121no—
257C2×C3×C4C22×D311yescharacteristic
258C2×C3×C4C22×D311yescharacteristic
259C2×C3×C4C22×D311yescharacteristic
260C2×C3×C4C22×D311yescharacteristic
261C2×C3×C4C22×D311yescharacteristic
262C2×C3×C4C22×D311yescharacteristic
263#67—62no—
264#67—62no—
265#67—62no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—31no—
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—62no—
276C2×Dic3—121no—
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282D12—41no—
283D12—62no—
284C4×D3—41no—
285C4×D3—62no—
286C4×D3—62no—
287C4×D3—62no—
288C4×D3—121no—
289C4×D3—121no—
290C4×D3—121no—
291C4×D3—122no—
292C2×C32C2×#4111yescharacteristic
293C2×C32C2×#4111yescharacteristic
294C2×C32H(5,𝔽2)11yescharacteristic
295C3⋊D3—22no—
296C3×D3—22no—
297C3×D3—41no—
298C3×D3—41no—
299C24—91no—
300C2×Q8D3211yescharacteristic, complemented
301C2×D4—31no—
302C2×D4—91no—
303C2×D4—91no—
304C22×C4—31no—
305C22×C4—91no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C4⋊C4—31no—
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—91no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C22⋊C4—31no—
318C22⋊C4—31no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—31no—
322C22⋊C4—31no—
323C22⋊C4—91no—
324C22⋊C4—91no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C22⋊C4—91no—
331C42—31no—
332C42—91no—
333C42—91no—
334C22×C3—21no—
335C22×C3—32no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3C23×D311yescharacteristic
345C22×C3C23×D311yescharacteristic
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—32no—
355C2×D3—32no—
356C2×D3—62no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—124no—
364C2×D3—124no—
365C2×D3#23912yes—
366C3×C4—22no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—61no—
379C3×C4—61no—
380Dic3—21no—
381Dic3—21no—
382Dic3—21no—
383Dic3—21no—
384Dic3—21no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—61no—
390Dic3—61no—
391Dic3—121no—
392Dic3—121no—
393Dic3—121no—
394C32#55111yescharacteristic, complemented, Sylow
395C23—31no—
396C23—31no—
397C23—31no—
398C23—91no—
399C23—182no—
400C23—182no—
401C23—361no—
402Q8—41no—
403D4—121no—
404D4—182no—
405D4—182no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—121no—
415C2×C4—182no—
416C2×C4—182no—
417C2×C4—361no—
418C2×C4—361no—
419C2×C4C2×D3211yescharacteristic
420C2×C4C2×D3211yescharacteristic
421C2×C4C2×D3211yescharacteristic
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—62no—
426C2×C3—121no—
427C2×C3—121no—
428C2×C3#100411yescharacteristic
429C2×C3#100411yescharacteristic
430C2×C3C2×#23911yescharacteristic
431C2×C3D3×#4111yescharacteristic
432C2×C3D3×#4111yescharacteristic
433C2×C3D3×#4111yescharacteristic
434D3—22no—
435D3—41no—
436D3—41no—
437D3—62no—
438D3—62no—
439D3—122no—
440C22—32no—
441C22—61no—
442C22—61no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22—92no—
450C22—181no—
451C22—181no—
452C22—364no—
453C22C22×D3211yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—61no—
458C4—61no—
459C4—61no—
460C4—61no—
461C4—61no—
462C4—181no—
463C4—181no—
464C4—181no—
465C3—21no—
466C3#563711yescharacteristic, complemented
467C3#567911yescharacteristic, complemented
468C2—62no—
469C2—121no—
470C2—121no—
471C2—182no—
472C2D3×#23911yescharacteristic
473C2#6431911yescharacteristic
474C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radical#63922C2
hypercenterC2×Q8D32
hypocenterC32#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #64155C2semidirect
2. C3×#947C2semidirect
3. C3×#946C2semidirect
5. #63921C2semidirect
6. D3×(C2×D3)⋊C4C2semidirect
7. #63891C2semidirect
8. #63884C2semidirect
9. #63881C2semidirect
10. #63879C2semidirect
11. #63877C2semidirect
12. #63861C2semidirect
13. #63858C2semidirect
14. #63854C2semidirect
15. #63836C2semidirect
19. C2×C32×Q8C22semidirect
27. C2×D3×Dic3C22semidirect
30. C4⋊(C3⋊Dic3)C22semidirect
31. #3750C22semidirect
32. #3750C22semidirect
37. C3×(C2×D3)⋊C4C22semidirect
38. C3×(C2×D3)⋊C4C22semidirect
39. C3×C4⋊Dic3C22semidirect
40. C3×Dic3⋊C4C22semidirect
41. C3×Dic3⋊C4C22semidirect
42. C3×C4×Dic3C22semidirect
43. #3724C22semidirect
44. Dic3⋊Dic3C22semidirect
50. (C2×D3)⋊Dic3C22semidirect
51. Dic32C22semidirect
52. Dic32C22semidirect
55. #947D3semidirect
59. #946D3semidirect
115. C2×C3×Q8C2×D3semidirect
116. C2×C3×Q8C2×D3semidirect
166. (C2×D3)⋊C4C2×D3semidirect
167. (C2×D3)⋊C4C2×D3semidirect
172. C4⋊Dic3C2×D3semidirect
182. Dic3⋊C4C2×D3semidirect
183. Dic3⋊C4C2×D3semidirect
188. C4×Dic3C2×D3semidirect
300. C2×Q8D32semidirect
394. C32#551semidirect
466. C3#5637semidirect
467. C3#5679semidirect

Maximal subgroups

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

subgroupindex
#641552
C3×#9472
C3×#9462
#639222
#639212
D3×(C2×D3)⋊C42
#638912
#638842
#638812
#638792
#638772
#638612
#638582
#638542
#638362
#56793
#56373