Finite Group Databasebeta

← #10594495

Subgroups — #10594495

Order 576.

420 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64098C211yesmaximal, characteristic, complemented
2C3×#894C211yesmaximal, characteristic
3C3×#888C211yesmaximal, characteristic, complemented
4#63899C211yesmaximal, characteristic, complemented
5#63887C211yesmaximal, characteristic, complemented
6#63884C211yesmaximal, characteristic
7#63870C211yesmaximal, characteristic, complemented
8#63863C211yesmaximal, characteristic
9#63861C211yesmaximal, characteristic, complemented
10#63859C211yesmaximal, characteristic, complemented
11#63859C211yesmaximal, characteristic, complemented
12#63858C211yesmaximal, characteristic, complemented
13#63856C211yesmaximal, characteristic, complemented
14#63851C211yesmaximal, characteristic, complemented
15#63839C211yesmaximal, characteristic, complemented
16#5735—31nomaximal
17#5651—31nomaximal
18C2×C3×D12C2211yescharacteristic
19C2×C3×C4×D3C2211yescharacteristic
20C2×C3×Dic6C2211yescharacteristic
21C2×C3×Dic6C2211yescharacteristic
22C2×#627C2211yescharacteristic
23C2×D3×Dic3C2211yescharacteristic
24C32×C4⋊C4C2211yescharacteristic
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26#3750C2211yescharacteristic
27#3750C2211yescharacteristic
28#3750C2211yescharacteristic
29#3750C2211yescharacteristic
30C4×C3⋊Dic3C2211yescharacteristic
31C3×(C2×D3)⋊C4C2211yescharacteristic
32C3×(C2×D3)⋊C4C2211yescharacteristic
33C3×(C2×D3)⋊C4C2211yescharacteristic
34C3×C4⋊Dic3C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×C4×Dic3C2211yescharacteristic
39#3724C2211yescharacteristic
40#3724C2211yescharacteristic
41#3724C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45(C2×D3)⋊Dic3C2211yescharacteristic
46(C2×D3)⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×C4⋊Q8—31no—
54C3×#123—31no—
55#947—31no—
56#946—31no—
57#946—31no—
58#944—31no—
59#936—31no—
60#935—31no—
61#924—31no—
62#924—31no—
63#898—31no—
64#898—31no—
65#896—31no—
66#896—31no—
67#894—31no—
68#894D311yescharacteristic
69#889—31no—
70#889—31no—
71#889—31no—
72#889—31no—
73#889—61no—
74#888—31no—
75#888D311yescharacteristic, complemented
76#879—31no—
77#879—31no—
78#879—31no—
79#879—31no—
80#878—31no—
81#878—31no—
82#876—31no—
83#876—31no—
84C22×C3×D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C2×C32×C4C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3×Dic3C2311yescharacteristic
94C2×C3×Dic3C2311yescharacteristic
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C3×D12—41no—
100C3×C4×D3—41no—
101C3×Dic6—41no—
102C3×Dic6—41no—
103#627—41no—
104D3×Dic3—41no—
105#563—91noSylow
106C2×C3×Q8—31no—
107C2×C3×Q8—31no—
108C2×C3×D4—31no—
109C22×C3×C4—31no—
110C22×Dic3—31no—
111C2×D12C2×D311yescharacteristic
112C2×C4×D3—31no—
113C2×C4×D3C2×D311yescharacteristic
114C2×Dic6—31no—
115C2×Dic6—31no—
116C2×Dic6C2×D311yescharacteristic
117C2×Dic6C2×D311yescharacteristic
118C3×C4⋊C4—21no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—31no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4C2×D311yescharacteristic
124C3×C4⋊C4C2×D311yescharacteristic
125C3×C22⋊C4—31no—
126C3×C22⋊C4—31no—
127C3×C22⋊C4—31no—
128C3×C42—31no—
129#217—31no—
130#217—31no—
131#217—31no—
132#217—31no—
133#217—31no—
134#217—31no—
135#217—31no—
136(C2×D3)⋊C4—31no—
137(C2×D3)⋊C4—31no—
138(C2×D3)⋊C4—31no—
139(C2×D3)⋊C4—31no—
140(C2×D3)⋊C4—31no—
141(C2×D3)⋊C4—31no—
142(C2×D3)⋊C4—31no—
143(C2×D3)⋊C4C2×D311yescharacteristic
144(C2×D3)⋊C4C2×D311yescharacteristic
145(C2×D3)⋊C4C2×D311yescharacteristic
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—31no—
151C4⋊Dic3—61no—
152C4⋊Dic3C2×D311yescharacteristic
153Dic3⋊C4—31no—
154Dic3⋊C4—31no—
155Dic3⋊C4—31no—
156Dic3⋊C4—31no—
157Dic3⋊C4—31no—
158Dic3⋊C4—31no—
159Dic3⋊C4—31no—
160Dic3⋊C4—31no—
161Dic3⋊C4—31no—
162Dic3⋊C4—31no—
163Dic3⋊C4—31no—
164Dic3⋊C4—31no—
165Dic3⋊C4—31no—
166Dic3⋊C4—31no—
167Dic3⋊C4—31no—
168Dic3⋊C4—31no—
169Dic3⋊C4—31no—
170Dic3⋊C4—61no—
171Dic3⋊C4—61no—
172Dic3⋊C4—61no—
173Dic3⋊C4—61no—
174Dic3⋊C4C2×D311yescharacteristic
175Dic3⋊C4C2×D311yescharacteristic
176Dic3⋊C4C2×D311yescharacteristic
177C4×Dic3—31no—
178C4×Dic3—31no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—61no—
182C4×Dic3C2×D311yescharacteristic
183C22×C32C2411yescharacteristic
184C2×C3×D3—21no—
185C2×C3×D3—21no—
186C2×C3×D3—21no—
187C2×C3×D3—21no—
188C2×C3×D3—21no—
189C2×C3×D3—21no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—21no—
193C3⋊Dic3—21no—
194C3⋊Dic3—21no—
195C3⋊Dic3—21no—
196C3⋊Dic3—21no—
197C3×Dic3—21no—
198C3×Dic3—21no—
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C4⋊Q8—31no—
204C4⋊Q8—91no—
205#126—91no—
206#126—91no—
207#126—91no—
208#126—91no—
209#125—91no—
210#125—91no—
211#124—91no—
212#123—31no—
213#123—91no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C22⋊Q8—91no—
217C22⋊Q8—91no—
218C23×C3—31no—
219C23×C3—31no—
220C22×D3C22×D311yescharacteristic
221C22×D3C22×D311yescharacteristic
222C3×Q8—121no—
223C3×Q8—121no—
224C3×D4—121no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—121no—
235C2×C3×C4C22×D311yescharacteristic
236C2×C3×C4C22×D311yescharacteristic
237C2×C3×C4C22×D311yescharacteristic
238C2×C3×C4C22×D311yescharacteristic
239C2×C3×C4C22×D311yescharacteristic
240C2×C3×C4C22×D311yescharacteristic
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—121no—
254C2×Dic3C22×D311yescharacteristic
255C2×Dic3C22×D311yescharacteristic
256C2×Dic3C22×D311yescharacteristic
257C2×Dic3C22×D311yescharacteristic
258C2×Dic3C22×D311yescharacteristic
259C2×Dic3C22×D311yescharacteristic
260D12—41no—
261C4×D3—41no—
262C4×D3—121no—
263Dic6—41no—
264Dic6—41no—
265Dic6—121no—
266Dic6—121no—
267C2×C32H(5,𝔽2)11yescharacteristic
268C2×C322−1+411yescharacteristic
269C2×C322−1+411yescharacteristic
270C3×D3—41no—
271C3×D3—41no—
272C2×Q8—31no—
273C2×Q8—31no—
274C2×Q8—91no—
275C2×D4—31no—
276C22×C4—31no—
277C22×C4—91no—
278C4⋊C4—31no—
279C4⋊C4—31no—
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—91no—
288C4⋊C4—91no—
289C4⋊C4—91no—
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—91no—
293C4⋊C4D3211yescharacteristic
294C22⋊C4—31no—
295C22⋊C4—31no—
296C22⋊C4—31no—
297C22⋊C4—91no—
298C22⋊C4—91no—
299C22⋊C4—91no—
300C22⋊C4—91no—
301C22⋊C4—91no—
302C22⋊C4—91no—
303C22⋊C4—91no—
304C42—31no—
305C42—91no—
306C42—91no—
307C22×C3—21no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3C23×D311yescharacteristic
315C22×C3C23×D311yescharacteristic
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—21no—
319C2×D3—21no—
320C2×D3—21no—
321C2×D3—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—61no—
332C3×C4—61no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—61no—
336C3×C4—61no—
337Dic3—21no—
338Dic3—21no—
339Dic3—21no—
340Dic3—21no—
341Dic3—21no—
342Dic3—21no—
343Dic3—62no—
344Dic3—61no—
345Dic3—61no—
346Dic3—61no—
347Dic3—61no—
348Dic3—61no—
349Dic3—61no—
350Dic3—61no—
351Dic3—61no—
352Dic3—121no—
353Dic3—121no—
354Dic3—121no—
355C32#56311yescharacteristic, complemented, Sylow
356C23—31no—
357C23—31no—
358Q8—121no—
359Q8—121no—
360Q8—361no—
361D4—121no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—121no—
373C2×C4—361no—
374C2×C4C2×D3211yescharacteristic
375C2×C4C2×D3211yescharacteristic
376C2×C4C2×D3211yescharacteristic
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—121no—
381C2×C3—121no—
382C2×C3#100411yescharacteristic
383C2×C3#100411yescharacteristic
384C2×C3#100711yescharacteristic
385C2×C3#100711yescharacteristic
386C2×C3#101011yescharacteristic
387C2×C3#101011yescharacteristic
388D3—41no—
389D3—41no—
390C22—61no—
391C22—61no—
392C22—61no—
393C22—61no—
394C22—61no—
395C22—61no—
396C22C22×D3211yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—61no—
401C4—61no—
402C4—61no—
403C4—61no—
404C4—61no—
405C4—61no—
406C4—181no—
407C4—181no—
408C4—181no—
409C4—181no—
410C3—21no—
411C3#565111yescharacteristic, complemented
412C3#573511yescharacteristic, complemented
413C2—121no—
414C2—121no—
415C2#6429711yescharacteristic
416C2#6431311yescharacteristic
417C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×C3×D12C22
hypercenterC4⋊C4D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64098C2semidirect
3. C3×#888C2semidirect
4. #63899C2semidirect
5. #63887C2semidirect
7. #63870C2semidirect
9. #63861C2semidirect
10. #63859C2semidirect
11. #63859C2semidirect
12. #63858C2semidirect
13. #63856C2semidirect
14. #63851C2semidirect
15. #63839C2semidirect
75. #888D3semidirect
355. C32#563semidirect
411. C3#5651semidirect
412. C3#5735semidirect

Maximal subgroups

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

subgroupindex
#640982
C3×#8942
C3×#8882
#638992
#638872
#638842
#638702
#638632
#638612
#638592
#638582
#638562
#638512
#638392
#57353
#56513