Finite Group Databasebeta

← #11755129

Subgroups — #11755129

Order 960.

432 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#91836C211yesmaximal, characteristic, complemented
2C5×#946C211yesmaximal, characteristic, complemented
3C3×#4090C211yesmaximal, characteristic, complemented
4#91446C211yesmaximal, characteristic, complemented
5#91433C211yesmaximal, characteristic, complemented
6#91420C211yesmaximal, characteristic
7#91419C211yesmaximal, characteristic, complemented
8#91411C211yesmaximal, characteristic, complemented
9#91402C211yesmaximal, characteristic, complemented
10#91367C211yesmaximal, characteristic, complemented
11#91351C211yesmaximal, characteristic, complemented
12#91350C211yesmaximal, characteristic, complemented
13#91337C211yesmaximal, characteristic, complemented
14#91326C211yesmaximal, characteristic, complemented
15#91325C211yesmaximal, characteristic, complemented
16#65974—31nomaximal, Hall
17C2×C3×C5×Q8C2211yescharacteristic
18C2×D60C2211yescharacteristic
19C2×C4×C5×D3C2211yescharacteristic
20C2×C3×Dic10C2211yescharacteristic
21C2×#1204C2211yescharacteristic
22C2×#1203C2211yescharacteristic
23C2×#1203C2211yescharacteristic
24C2×#1200C2211yescharacteristic
25C2×#1200C2211yescharacteristic
26#6714C2211yescharacteristic
27#6714C2211yescharacteristic
28#6714C2211yescharacteristic
29#6714C2211yescharacteristic
30C4×Dic15C2211yescharacteristic
31C5×(C2×D3)⋊C4C2211yescharacteristic
32C5×(C2×D3)⋊C4C2211yescharacteristic
33C5×C4⋊Dic3C2211yescharacteristic
34C5×Dic3⋊C4C2211yescharacteristic
35C5×Dic3⋊C4C2211yescharacteristic
36C3×#683C2211yescharacteristic
37C3×#683C2211yescharacteristic
38C3×#683C2211yescharacteristic
39C3×#683C2211yescharacteristic
40C3×C4×Dic5C2211yescharacteristic
41Dic3⋊Dic5C2211yescharacteristic
42#6669C2211yescharacteristic
43#6668C2211yescharacteristic
44#6667C2211yescharacteristic
45#6667C2211yescharacteristic
46#6667C2211yescharacteristic
47#6667C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic
49(C2×D3)⋊Dic5C2211yescharacteristic
50Dic3×Dic5C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic
52#5733—51nomaximal, Hall
53C5×C22⋊Q8—31no—
54#4093—31no—
55#4090D311yescharacteristic, complemented
56#4075—31no—
57C4×C22⋊D5—31no—
58#4044—31no—
59#4042—31no—
60#4042—31no—
61#4040—31no—
62#4038—31no—
63#4033—31no—
64#4031—31no—
65#4031—31no—
66#4030—31no—
67#4025—31no—
68C22×D15C2311yescharacteristic
69C22×D15C2311yescharacteristic
70C22×C5×D3C2311yescharacteristic
71C3×C5×Q8—41no—
72C2×C3×C4×C5C2311yescharacteristic
73C2×C3×C4×C5C2311yescharacteristic
74C2×C3×C4×C5C2311yescharacteristic
75C2×Dic15C2311yescharacteristic
76C2×Dic15C2311yescharacteristic
77D60—41no—
78C2×C5×Dic3C2311yescharacteristic
79C2×C5×Dic3C2311yescharacteristic
80C2×C5×Dic3C2311yescharacteristic
81C4×C5×D3—41no—
82C2×C3×Dic5C2311yescharacteristic
83C2×C3×Dic5C2311yescharacteristic
84C2×C3×Dic5C2311yescharacteristic
85C2×C3×Dic5C2311yescharacteristic
86C3×Dic10—41no—
87#1204—22no—
88#1203—22no—
89#1203—41no—
90#1200—22no—
91#1200—41no—
92C3×C4⋊Q8—51no—
93#947—51no—
94#947—51no—
95#946—51no—
96#946D511yescharacteristic, complemented
97#898—51no—
98#898—51no—
99#897—51no—
100#895—51no—
101#894—51no—
102#894—51no—
103#893—51no—
104#887—51no—
105#875—51no—
106#872—51no—
107C2×C5×Q8C2×D311yescharacteristic
108C22×C4×C5—31no—
109C2×C22⋊D5—31no—
110C2×C22⋊D5—31no—
111C2×D20—31no—
112C2×C4×D5—31no—
113C2×C4×D5—31no—
114C2×Dic10—31no—
115C2×Dic10C2×D311yescharacteristic
116C5×C4⋊C4—31no—
117C5×C4⋊C4—31no—
118C5×C4⋊C4—31no—
119C5×C22⋊C4—31no—
120C5×C22⋊C4—31no—
121C22⋊Dic5—31no—
122C22⋊Dic5—31no—
123#685—31no—
124#685—31no—
125#685—31no—
126#685—31no—
127#685—31no—
128#685—31no—
129#685—31no—
130#685—31no—
131C4⋊Dic5—31no—
132#683—31no—
133#683—31no—
134#683C2×D311yescharacteristic
135#683C2×D311yescharacteristic
136#683C2×D311yescharacteristic
137#683C2×D311yescharacteristic
138C4×Dic5—31no—
139C4×Dic5—31no—
140C4×Dic5—31no—
141C4×Dic5C2×D311yescharacteristic
142#542—151noSylow
143C22×C3×C5C2411yescharacteristic
144C2×D15—21no—
145C2×D15—21no—
146C2×D15—21no—
147C2×D15—21no—
148C2×D15—21no—
149C2×D15—21no—
150C2×C5×D3—21no—
151C2×C5×D3—21no—
152C2×C5×D3—21no—
153C3×C4×C5—21no—
154C3×C4×C5—21no—
155C3×C4×C5—21no—
156Dic15—21no—
157Dic15—21no—
158C3×Dic5—21no—
159C3×Dic5—21no—
160C3×Dic5—21no—
161C3×Dic5#4112yes—
162C5×Dic3—21no—
163C5×Dic3—21no—
164C5×Dic3—21no—
165C2×C3×Q8—51no—
166C2×C3×Q8C2×D511yescharacteristic
167C2×D12—51no—
168C2×D12—51no—
169C2×D12—51no—
170C2×C4×D3—51no—
171C2×C4×D3—51no—
172C2×C4×D3C2×D511yescharacteristic
173C2×Dic6—51no—
174C3×C4⋊C4—51no—
175C3×C4⋊C4—51no—
176C3×C4⋊C4—51no—
177C3×C4⋊C4—51no—
178C3×C42—51no—
179(C2×D3)⋊C4—51no—
180(C2×D3)⋊C4—51no—
181(C2×D3)⋊C4—51no—
182(C2×D3)⋊C4—51no—
183(C2×D3)⋊C4—51no—
184(C2×D3)⋊C4—51no—
185(C2×D3)⋊C4—51no—
186(C2×D3)⋊C4—51no—
187(C2×D3)⋊C4—51no—
188(C2×D3)⋊C4—51no—
189(C2×D3)⋊C4C2×D511yescharacteristic
190(C2×D3)⋊C4C2×D511yescharacteristic
191C4⋊Dic3—51no—
192C4⋊Dic3C2×D511yescharacteristic
193Dic3⋊C4—51no—
194Dic3⋊C4—51no—
195Dic3⋊C4C2×D511yescharacteristic
196Dic3⋊C4C2×D511yescharacteristic
197C4×Dic3—51no—
198C4×Dic3—51no—
199C4×Dic3—51no—
200C23×C5—31no—
201C22×D5—31no—
202C22×D5—31no—
203C5×Q8—41no—
204C2×C4×C5—31no—
205C2×C4×C5—31no—
206C2×C4×C5—31no—
207C2×C4×C5—121no—
208C2×C4×C5C22×D311yescharacteristic
209C2×C4×C5C22×D311yescharacteristic
210C2×C4×C5C22×D311yescharacteristic
211C22⋊D5—62no—
212C22⋊D5—121no—
213C2×Dic5—31no—
214C2×Dic5—31no—
215C2×Dic5C22×D311yescharacteristic
216C2×Dic5C22×D311yescharacteristic
217C2×Dic5C22×D311yescharacteristic
218C2×Dic5C22×D311yescharacteristic
219D20—121no—
220C4×D5—62no—
221C4×D5—121no—
222Dic10—41no—
223Dic10—62no—
224C4⋊Q8—51no—
225#126—151no—
226#126—151no—
227#124—151no—
228#124—151no—
229#124—151no—
230#123—151no—
231#123—151no—
232C22⋊Q8—31no—
233C22⋊Q8—151no—
234C22⋊Q8—151no—
235#121—151no—
236C4×Q8—151no—
237C4×D4—151no—
238#117—151no—
239C2×C3×C5C2×#4111yescharacteristic
240C2×C3×C5H(5,𝔽2)11yescharacteristic
241C2×C3×C52−1+411yescharacteristic
242D15—41no—
243D15—41no—
244C5×D3—41no—
245C22×D3—51no—
246C22×D3—51no—
247C22×D3C22×D511yescharacteristic
248C3×Q8—41no—
249C3×Q8—201no—
250C2×C3×C4—51no—
251C2×C3×C4—51no—
252C2×C3×C4—51no—
253C2×C3×C4—51no—
254C2×C3×C4C22×D511yescharacteristic
255C2×C3×C4C22×D511yescharacteristic
256C2×C3×C4C22×D511yescharacteristic
257C2×Dic3—51no—
258C2×Dic3—51no—
259C2×Dic3C22×D511yescharacteristic
260C2×Dic3C22×D511yescharacteristic
261C2×Dic3C22×D511yescharacteristic
262D12—102no—
263D12—201no—
264D12—201no—
265C4×D3—41no—
266C4×D3—102no—
267C4×D3—201no—
268Dic6—102no—
269C22×C5—61no—
270C22×C5—61no—
271C22×C5—61no—
272C22×C5C23×D311yescharacteristic
273C2×D5—61no—
274C2×D5—61no—
275C2×D5—61no—
276C2×D5—61no—
277C2×D5—61no—
278C2×D5—61no—
279C4×C5—21no—
280C4×C5—21no—
281C4×C5—21no—
282C4×C5—61no—
283C4×C5—61no—
284C4×C5—61no—
285Dic5—21no—
286Dic5—21no—
287Dic5—21no—
288Dic5—61no—
289Dic5—61no—
290Dic5#23912yes—
291C2×Q8—51no—
292C2×Q8—151no—
293C2×Q8D3×D511yescharacteristic
294C2×D4—151no—
295C2×D4—151no—
296C2×D4—151no—
297C22×C4—31no—
298C22×C4—151no—
299C22×C4—151no—
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—51no—
304C4⋊C4—51no—
305C4⋊C4—51no—
306C4⋊C4—51no—
307C4⋊C4—151no—
308C4⋊C4—151no—
309C4⋊C4—151no—
310C22⋊C4—31no—
311C22⋊C4—31no—
312C22⋊C4—151no—
313C22⋊C4—151no—
314C22⋊C4—151no—
315C22⋊C4—151no—
316C22⋊C4—151no—
317C22⋊C4—151no—
318C22⋊C4—151no—
319C22⋊C4—151no—
320C22⋊C4—151no—
321C22⋊C4—151no—
322C42—51no—
323C42—151no—
324C42—151no—
325C42—151no—
326C3×C5#54211yescharacteristic, complemented, Hall
327C22×C3C23×D511yescharacteristic
328C2×D3—21no—
329C2×D3—21no—
330C2×D3—21no—
331C2×D3—101no—
332C2×D3—101no—
333C2×D3—101no—
334C2×D3—101no—
335C2×D3—101no—
336C2×D3—101no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—52no—
341C3×C4—101no—
342C3×C4—101no—
343C3×C4—101no—
344Dic3—21no—
345Dic3—21no—
346Dic3—21no—
347Dic3—101no—
348Dic3—101no—
349C2×C5—121no—
350C2×C5#100411yescharacteristic
351C2×C5C2×#23911yescharacteristic
352C2×C5#100711yescharacteristic
353D5—121no—
354D5—121no—
355C23—31no—
356C23—151no—
357C23—151no—
358Q8—41no—
359Q8—201no—
360Q8—302no—
361D4—302no—
362D4—601no—
363D4—601no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—51no—
368C2×C4—51no—
369C2×C4—51no—
370C2×C4—51no—
371C2×C4—121no—
372C2×C4—151no—
373C2×C4—151no—
374C2×C4—302no—
375C2×C4—601no—
376C2×C4C2×D3×D511yescharacteristic
377C2×C4C2×D3×D511yescharacteristic
378C2×C4C2×D3×D511yescharacteristic
379C2×C3#414411yescharacteristic
380C2×C3#414711yescharacteristic
381C2×C3D5×#4111yescharacteristic
382D3—41no—
383D3—201no—
384D3—201no—
385C5#573311yescharacteristic, complemented, Sylow
386C22—61no—
387C22—61no—
388C22—61no—
389C22—301no—
390C22—301no—
391C22—301no—
392C22—301no—
393C22—301no—
394C22—301no—
395C22C22×D3×D511yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—52no—
400C4—61no—
401C4—61no—
402C4—61no—
403C4—101no—
404C4—101no—
405C4—101no—
406C4—301no—
407C4—301no—
408C3#6597411yescharacteristic, complemented, Sylow
409C2—121no—
410C2—601no—
411C2—601no—
412C2#9202911yescharacteristic
413C2D5×#23911yescharacteristic
414C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×Q8C22
prime radical#91420C2
hypercenterC2×Q8D3×D5
hypocenterC3×C5#542
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #91836C2semidirect
2. C5×#946C2semidirect
3. C3×#4090C2semidirect
4. #91446C2semidirect
5. #91433C2semidirect
7. #91419C2semidirect
8. #91411C2semidirect
9. #91402C2semidirect
10. #91367C2semidirect
11. #91351C2semidirect
12. #91350C2semidirect
13. #91337C2semidirect
14. #91326C2semidirect
15. #91325C2semidirect
55. #4090D3semidirect
96. #946D5semidirect
326. C3×C5#542semidirect
385. C5#5733semidirect
408. C3#65974semidirect

Maximal subgroups

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

subgroupindex
#918362
C5×#9462
C3×#40902
#914462
#914332
#914202
#914192
#914112
#914022
#913672
#913512
#913502
#913372
#913262
#913252
#659743
#57335