Finite Group Databasebeta

← #11755127

Subgroups — #11755127

Order 960.

584 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6821C211yesmaximal, characteristic, complemented
2C2×#6769C211yesmaximal, characteristic, complemented
3C5×#946C211yesmaximal, characteristic, complemented
4C3×#4090C211yesmaximal, characteristic, complemented
5C4×#1203C211yesmaximal, characteristic, complemented
6#91433C212yesmaximal, complemented
7#91420C212yesmaximal, complemented
8#91418C211yesmaximal, characteristic, complemented
9#91404C212yesmaximal, complemented
10#91390C211yesmaximal, characteristic, complemented
11#91355C212yesmaximal, complemented
12#65965—31nomaximal, Hall
13C2×C3×C5×Q8C2211yescharacteristic
14#6821—24no—
15C2×D60C2212yes—
16C2×D60C2211yescharacteristic
17C2×C4×D15C2212yes—
18C2×C4×D15C2211yescharacteristic
19C2×C4×C5×D3C2211yescharacteristic
20C2×C3×Dic10C2211yescharacteristic
21C2×#1203C2211yescharacteristic
22C2×#1203C2212yes—
23C2×#1200C2212yes—
24C2×#1200C2212yes—
25#6769—24no—
26C5×(C2×D3)⋊C4C2212yes—
27C5×C4⋊Dic3C2211yescharacteristic
28C5×Dic3⋊C4C2212yes—
29C3×#683C2212yes—
30C3×#683C2212yes—
31C3×C4×Dic5C2211yescharacteristic
32Dic3⋊Dic5C2212yes—
33#6668C2211yescharacteristic
34#6667C2212yes—
35#6667C2212yes—
36#6667C2211yescharacteristic
37(C2×D3)⋊Dic5C2211yescharacteristic
38#5726—51nomaximal, Hall
39C2×#713—31no—
40C2×#709—31no—
41C5×C22⋊Q8—31no—
42#4090D311yescharacteristic, complemented
43C4×C22⋊D5—31no—
44#4043—31no—
45#4040—32no—
46#4039—31no—
47D5×C4⋊C4—32no—
48#4030—32no—
49#4029—32no—
50C22×D15C2311yescharacteristic
51C22×D15C2312yes—
52C22×C5×D3C2311yescharacteristic
53C3×C5×Q8—22no—
54C2×C3×C4×C5C2312yes—
55C2×C3×C4×C5C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57D60—24no—
58D60D414yescomplemented
59C4×D15—24no—
60C4×D15—22no—
61C2×C5×Dic3C2312yes—
62C2×C5×Dic3C2311yescharacteristic
63C4×C5×D3—22no—
64C2×C3×Dic5C2311yescharacteristic
65C2×C3×Dic5C2312yes—
66C2×C3×Dic5C2311yescharacteristic
67C3×Dic10—22no—
68#1203—24no—
69#1203—22no—
70#1200—24no—
71#1200—24no—
72C2×#239—51no—
73C2×#239—51no—
74C3×C4⋊Q8—51no—
75#946—51no—
76#946D511yescharacteristic, complemented
77#895—52no—
78#894—52no—
79#894—52no—
80D3×C4⋊C4—52no—
81C4×D12—51no—
82C4×D12—51no—
83C2×C5×Q8C2×D311yescharacteristic
84C22×C4×C5—31no—
85C2×C22⋊D5—32no—
86C2×C22⋊D5—31no—
87#713—64no—
88#709—64no—
89C2×D20—31no—
90C2×D20—32no—
91C2×C4×D5—32no—
92C2×C4×D5—31no—
93C2×C4×D5—32no—
94C2×C4×D5—32no—
95C2×Dic10C2×D311yescharacteristic
96C5×C4⋊C4—32no—
97C5×C4⋊C4—31no—
98C5×C22⋊C4—32no—
99C22⋊Dic5—31no—
100#685—32no—
101#685—32no—
102#685—31no—
103C4⋊Dic5—32no—
104#683—31no—
105#683C2×D312yes—
106#683C2×D312yes—
107C4×Dic5C2×D311yescharacteristic
108#547—151noSylow
109C22×C3×C5C2411yescharacteristic
110C2×D15—22no—
111C2×D15—22no—
112C2×D15—21no—
113C2×D15—21no—
114C2×D15—21no—
115C2×D15C2×D414yes—
116C2×C5×D3—21no—
117C2×C5×D3—21no—
118C2×C5×D3—21no—
119C3×C4×C5—22no—
120C3×C4×C5C2×D412yes—
121Dic15—21no—
122C3×Dic5—22no—
123C3×Dic5—21no—
124C3×Dic5#4112yescomplemented
125C5×Dic3—21no—
126C5×Dic3—22no—
127C2×C3×Q8—51no—
128C2×C3×Q8C2×D511yescharacteristic
129#239—104no—
130#239—104no—
131C2×D12—51no—
132C2×D12—51no—
133C2×D12—52no—
134C2×D12—52no—
135C2×C4×D3—51no—
136C2×C4×D3—52no—
137C2×C4×D3—52no—
138C2×C4×D3—52no—
139C2×C4×D3C2×D511yescharacteristic
140C3×C4⋊C4—52no—
141C3×C4⋊C4—52no—
142C3×C42—51no—
143(C2×D3)⋊C4—51no—
144(C2×D3)⋊C4—52no—
145(C2×D3)⋊C4—51no—
146(C2×D3)⋊C4—52no—
147(C2×D3)⋊C4C2×D512yes—
148C4⋊Dic3—51no—
149C4⋊Dic3C2×D511yescharacteristic
150Dic3⋊C4—52no—
151Dic3⋊C4C2×D512yes—
152C23×C5—31no—
153C22×D5—32no—
154C22×D5—31no—
155C5×Q8—22no—
156C2×C4×C5—31no—
157C2×C4×C5—32no—
158C2×C4×C5—62no—
159C2×C4×C5C22×D312yes—
160C2×C4×C5C22×D311yescharacteristic
161C22⋊D5—64no—
162C22⋊D5—62no—
163C2×Dic5—31no—
164C2×Dic5C22×D311yescharacteristic
165C2×Dic5C22×D312yes—
166C2×Dic5C22×D311yescharacteristic
167D20—34no—
168D20—64no—
169C4×D5—64no—
170C4×D5—64no—
171C4×D5—62no—
172C4×D5—64no—
173Dic10—22no—
174C2×#41—151no—
175C2×#41—151no—
176C4⋊Q8—51no—
177#123—152no—
178#123—152no—
179C22⋊Q8—31no—
180C22⋊Q8—151no—
181#121—152no—
182C4×D4—151no—
183C4×D4—151no—
184C2×C4⋊C4—152no—
185C2×C3×C5C22×D411yescharacteristic
186C2×C3×C5C2×#4111yescharacteristic
187C2×C3×C52−1+411yescharacteristic
188D15—24no—
189D15—41no—
190C5×D3—41no—
191C22×D3—52no—
192C22×D3—51no—
193C22×D3C22×D511yescharacteristic
194C3×Q8—22no—
195C3×Q8—102no—
196C2×C3×C4—51no—
197C2×C3×C4—52no—
198C2×C3×C4—51no—
199C2×C3×C4C22×D512yes—
200C2×C3×C4C22×D511yescharacteristic
201C2×Dic3—51no—
202C2×Dic3C22×D512yes—
203C2×Dic3C22×D511yescharacteristic
204D12—54no—
205D12—104no—
206D12—102no—
207D12—104no—
208C4×D3—22no—
209C4×D3—104no—
210C4×D3—104no—
211C4×D3—104no—
212C4×D3—102no—
213C22×C5—61no—
214C22×C5—61no—
215C22×C5—61no—
216C22×C5C23×D311yescharacteristic
217C2×D5—34no—
218C2×D5—62no—
219C2×D5—62no—
220C2×D5—61no—
221C2×D5—61no—
222C2×D5—61no—
223C4×C5—22no—
224C4×C5—61no—
225C4×C5—62no—
226C4×C5D3×D412yes—
227Dic5—22no—
228Dic5—21no—
229Dic5—61no—
230Dic5#23912yescomplemented
231#41—304no—
232#41—304no—
233C2×Q8—51no—
234C2×Q8D3×D511yescharacteristic
235C2×D4—151no—
236C2×D4—152no—
237C2×D4—151no—
238C2×D4—152no—
239C22×C4—31no—
240C22×C4—152no—
241C22×C4—152no—
242C22×C4—152no—
243C22×C4—151no—
244C4⋊C4—32no—
245C4⋊C4—31no—
246C4⋊C4—52no—
247C4⋊C4—52no—
248C4⋊C4—151no—
249C4⋊C4—152no—
250C22⋊C4—32no—
251C22⋊C4—152no—
252C22⋊C4—151no—
253C22⋊C4—152no—
254C22⋊C4—151no—
255C42—51no—
256C3×C5#54711yescharacteristic, complemented, Hall
257C22×C3C23×D511yescharacteristic
258C2×D3—21no—
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—54no—
262C2×D3—102no—
263C2×D3—102no—
264C2×D3—101no—
265C2×D3—101no—
266C2×D3—101no—
267C3×C4—22no—
268C3×C4—52no—
269C3×C4—102no—
270C3×C4—101no—
271C3×C4D4×D512yes—
272Dic3—21no—
273Dic3—22no—
274Dic3—101no—
275C2×C5—121no—
276C2×C5C2×D3×D411yescharacteristic
277C2×C5C2×#23911yescharacteristic
278C2×C5#100711yescharacteristic
279D5—64no—
280D5—121no—
281C23—31no—
282C23—152no—
283C23—151no—
284Q8—22no—
285Q8—102no—
286D4—154no—
287D4—304no—
288D4—302no—
289D4—304no—
290C2×C4—32no—
291C2×C4—31no—
292C2×C4—51no—
293C2×C4—51no—
294C2×C4—52no—
295C2×C4—62no—
296C2×C4—151no—
297C2×C4—304no—
298C2×C4—304no—
299C2×C4—304no—
300C2×C4—302no—
301C2×C4C2×D3×D512yes—
302C2×C4C2×D3×D511yescharacteristic
303C2×C3C2×D4×D511yescharacteristic
304C2×C3#414711yescharacteristic
305C2×C3D5×#4111yescharacteristic
306D3—41no—
307D3—104no—
308D3—201no—
309C5#572611yescharacteristic, complemented, Sylow
310C22—61no—
311C22—61no—
312C22—61no—
313C22—154no—
314C22—302no—
315C22—302no—
316C22—301no—
317C22—301no—
318C22—301no—
319C22C22×D3×D511yescharacteristic
320C4—22no—
321C4—52no—
322C4—62no—
323C4—61no—
324C4—101no—
325C4—102no—
326C4—301no—
327C4#679012yes—
328C3#6596511yescharacteristic, complemented, Sylow
329C2—121no—
330C2—304no—
331C2—601no—
332C2#9202911yescharacteristic
333C2D5×#23911yescharacteristic
334C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×Q8C22
hypercenterC2×Q8D3×D5
hypocenterC3×C5#547
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. C2×#6821C2semidirect
2. C2×#6769C2semidirect
3. C5×#946C2semidirect
4. C3×#4090C2semidirect
5. C4×#1203C2semidirect
6. #91433C2semidirect
7. #91420C2semidirect
8. #91418C2semidirect
9. #91404C2semidirect
10. #91390C2semidirect
11. #91355C2semidirect
42. #4090D3semidirect
58. D60D4semidirect
76. #946D5semidirect
124. C3×Dic5#41semidirect
230. Dic5#239semidirect
256. C3×C5#547semidirect
309. C5#5726semidirect
328. C3#65965semidirect

Maximal subgroups

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

subgroupindex
C2×#68212
C2×#67692
C5×#9462
C3×#40902
C4×#12032
#914332
#914202
#914182
#914042
#913902
#913552
#659653
#57265