Finite Group Databasebeta

← #11755166

Subgroups — #11755166

Order 960.

580 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 138 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×Q8×D15C211yesmaximal, characteristic, complemented
2C2×#6765C211yesmaximal, characteristic, complemented
3C5×#946C211yesmaximal, characteristic, complemented
4C3×#4093C211yesmaximal, characteristic, complemented
5#91465C211yesmaximal, characteristic, complemented
6#91462C212yesmaximal, complemented
7#91461C212yesmaximal, complemented
8C4×#1203C211yesmaximal, characteristic, complemented
9#91431C212yesmaximal, complemented
10#91396C212yesmaximal, complemented
11#91333C211yesmaximal, characteristic, complemented
12#65967—31nomaximal, Hall
13C2×C3×C5×Q8C2211yescharacteristic
14Q8×D15—24no—
15C2×C4×D15C2212yes—
16C2×C4×D15C2211yescharacteristic
17C2×Dic30C2212yes—
18C2×Dic30C2211yescharacteristic
19C2×C4×C5×D3C2211yescharacteristic
20C2×C3×D20C2211yescharacteristic
21C2×#1203C2211yescharacteristic
22C2×#1202C2212yes—
23C2×#1201C2212yes—
24C2×D5×Dic3C2212yes—
25#6765—24no—
26C5×(C2×D3)⋊C4C2212yes—
27C5×C4⋊Dic3C2211yescharacteristic
28C5×Dic3⋊C4C2212yes—
29C3×#685C2212yes—
30C3×#685C2212yes—
31C3×C4×Dic5C2211yescharacteristic
32#6669C2212yes—
33#6668C2211yescharacteristic
34#6667C2211yescharacteristic
35(C2×D3)⋊Dic5C2211yescharacteristic
36#6665C2212yes—
37Dic3×Dic5C2212yes—
38#5677—51nomaximal, Hall
39C2×Q8×D5—31no—
40C2×#709—31no—
41C5×C22⋊Q8—31no—
42#4093D311yescharacteristic, complemented
43C4×C22⋊D5—31no—
44#4042—32no—
45#4041—31no—
46#4039—32no—
47#4033—31no—
48#4031—32no—
49#4030—32no—
50C22×D15C2311yescharacteristic
51C22×C5×D3C2311yescharacteristic
52C22×C3×D5C2312yes—
53C3×C5×Q8—22no—
54C2×C3×C4×C5C2312yes—
55C2×C3×C4×C5C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57C2×Dic15C2312yes—
58C4×D15—22no—
59C4×D15—24no—
60Dic30—24no—
61Dic30D414yescomplemented
62C2×C5×Dic3C2312yes—
63C2×C5×Dic3C2311yescharacteristic
64C4×C5×D3—22no—
65C2×C3×Dic5C2311yescharacteristic
66C2×C3×Dic5C2311yescharacteristic
67C3×D20—22no—
68#1203—22no—
69#1202—24no—
70#1201—24no—
71D5×Dic3—24no—
72C2×D3×Q8—51no—
73C2×#237—51no—
74C3×#124—51no—
75#946D511yescharacteristic, complemented
76#938—51no—
77#885—52no—
78#884—52no—
79#67⋊C4—52no—
80#878—52no—
81C4×D12—51no—
82C4×Dic6—51no—
83C2×C5×Q8C2×D311yescharacteristic
84C22×C4×C5—31no—
85C2×C22⋊D5—32no—
86C2×C22⋊D5—31no—
87Q8×D5—64no—
88#709—64no—
89C2×D20—32no—
90C2×D20C2×D311yescharacteristic
91C2×C4×D5—32no—
92C2×C4×D5—32no—
93C2×C4×D5—31no—
94C2×Dic10—31no—
95C2×Dic10—32no—
96C5×C4⋊C4—32no—
97C5×C4⋊C4—31no—
98C5×C22⋊C4—32no—
99C22⋊Dic5—31no—
100#685—32no—
101#685—31no—
102#685C2×D312yes—
103#685C2×D312yes—
104#683—31no—
105#683—32no—
106C4×Dic5—32no—
107C4×Dic5C2×D311yescharacteristic
108#548—151noSylow
109C22×C3×C5C2411yescharacteristic
110C2×D15—21no—
111C2×D15—21no—
112C2×D15#4112yescomplemented
113C2×C5×D3—21no—
114C2×C5×D3—21no—
115C2×C5×D3—21no—
116C2×C3×D5—22no—
117C2×C3×D5—22no—
118C2×C3×D5—22no—
119C3×C4×C5—22no—
120C3×C4×C5C2×D412yes—
121Dic15—21no—
122Dic15C2×D414yes—
123C3×Dic5—21no—
124C3×Dic5—21no—
125C5×Dic3—21no—
126C5×Dic3—22no—
127C2×C3×Q8C2×D511yescharacteristic
128C2×C3×D4—51no—
129C2×#67—52no—
130C2×#67—52no—
131C22×Dic3—52no—
132D3×Q8—104no—
133#237—104no—
134C2×D12—51no—
135C2×C4×D3—51no—
136C2×C4×D3—52no—
137C2×C4×D3C2×D511yescharacteristic
138C2×Dic6—51no—
139C2×Dic6—52no—
140C3×C22⋊C4—52no—
141C3×C22⋊C4—52no—
142C3×C42—51no—
143#217—52no—
144(C2×D3)⋊C4—51no—
145(C2×D3)⋊C4—51no—
146(C2×D3)⋊C4C2×D512yes—
147C4⋊Dic3—51no—
148C4⋊Dic3C2×D511yescharacteristic
149Dic3⋊C4—52no—
150Dic3⋊C4C2×D512yes—
151C4×Dic3—52no—
152C23×C5—31no—
153C22×D5—31no—
154C22×D5C22×D312yes—
155C5×Q8—22no—
156C2×C4×C5—32no—
157C2×C4×C5—31no—
158C2×C4×C5—62no—
159C2×C4×C5C22×D312yes—
160C2×C4×C5C22×D311yescharacteristic
161C22⋊D5—62no—
162C22⋊D5—64no—
163C2×Dic5—32no—
164C2×Dic5—31no—
165C2×Dic5C22×D311yescharacteristic
166C2×Dic5C22×D311yescharacteristic
167D20—22no—
168D20—64no—
169C4×D5—64no—
170C4×D5—64no—
171C4×D5—62no—
172Dic10—34no—
173Dic10—64no—
174C2×#41—151no—
175C22×Q8—151no—
176#124—51no—
177#124—152no—
178C22⋊Q8—31no—
179C22⋊Q8—152no—
180#121—152no—
181#121—151no—
182C4×Q8—151no—
183C4×D4—151no—
184C4×D4—152no—
185C2×C3×C5C22×D411yescharacteristic
186C2×C3×C5C2×#4111yescharacteristic
187C2×C3×C52−1+411yescharacteristic
188D15—22no—
189C3×D5—42no—
190C5×D3—41no—
191C23×C3—52no—
192C22×D3—51no—
193C22×D3C22×D511yescharacteristic
194C3×Q8—22no—
195C3×D4—102no—
196C2×C3×C4—51no—
197C2×C3×C4—51no—
198C2×C3×C4C22×D512yes—
199C2×C3×C4C22×D511yescharacteristic
200#67—104no—
201#67—104no—
202C2×Dic3—52no—
203C2×Dic3—51no—
204C2×Dic3—104no—
205C2×Dic3C22×D512yes—
206C2×Dic3C22×D511yescharacteristic
207D12—102no—
208C4×D3—22no—
209C4×D3—104no—
210C4×D3—102no—
211Dic6—54no—
212Dic6—104no—
213C22×C5—61no—
214C22×C5—61no—
215C22×C5—61no—
216C22×C5C23×D311yescharacteristic
217C2×D5—22no—
218C2×D5—22no—
219C2×D5—22no—
220C2×D5—32no—
221C2×D5—61no—
222C2×D5—61no—
223C4×C5—22no—
224C4×C5—61no—
225C4×C5—62no—
226C4×C5D3×D412yes—
227Dic5—21no—
228Dic5—21no—
229Dic5—34no—
230Dic5—61no—
231#41—304no—
232C2×Q8—151no—
233C2×Q8—152no—
234C2×Q8—304no—
235C2×Q8D3×D511yescharacteristic
236C2×D4—51no—
237C2×D4—152no—
238C2×D4—151no—
239C2×D4—152no—
240C22×C4—31no—
241C22×C4—152no—
242C22×C4—152no—
243C22×C4—151no—
244C4⋊C4—32no—
245C4⋊C4—31no—
246C4⋊C4—151no—
247C4⋊C4—152no—
248C22⋊C4—32no—
249C22⋊C4—52no—
250C22⋊C4—52no—
251C22⋊C4—151no—
252C22⋊C4—152no—
253C22⋊C4—151no—
254C42—51no—
255C42—152no—
256C3×C5#54811yescharacteristic, complemented, Hall
257C22×C3—102no—
258C22×C3—102no—
259C22×C3—102no—
260C22×C3C23×D511yescharacteristic
261C2×D3—21no—
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—52no—
265C2×D3—101no—
266C2×D3—101no—
267C3×C4—22no—
268C3×C4—101no—
269C3×C4—101no—
270C3×C4D4×D512yes—
271Dic3—21no—
272Dic3—22no—
273Dic3—54no—
274Dic3—101no—
275C2×C5—121no—
276C2×C5C2×D3×D411yescharacteristic
277C2×C5D3×#4111yescharacteristic
278C2×C5#101011yescharacteristic
279D5—42no—
280D5—62no—
281C23—31no—
282C23—52no—
283C23—151no—
284Q8—22no—
285Q8—154no—
286Q8—304no—
287D4—102no—
288D4—304no—
289D4—302no—
290D4—304no—
291C2×C4—32no—
292C2×C4—31no—
293C2×C4—51no—
294C2×C4—51no—
295C2×C4—62no—
296C2×C4—152no—
297C2×C4—151no—
298C2×C4—304no—
299C2×C4—304no—
300C2×C4—302no—
301C2×C4C2×D3×D512yes—
302C2×C4C2×D3×D511yescharacteristic
303C2×C3—202no—
304C2×C3C2×D4×D511yescharacteristic
305C2×C3#414711yescharacteristic
306C2×C3D5×#4111yescharacteristic
307D3—41no—
308D3—102no—
309C5#567711yescharacteristic, complemented, Sylow
310C22—61no—
311C22—61no—
312C22—61no—
313C22—102no—
314C22—102no—
315C22—102no—
316C22—152no—
317C22—301no—
318C22—301no—
319C22C22×D3×D511yescharacteristic
320C4—22no—
321C4—62no—
322C4—61no—
323C4—101no—
324C4—101no—
325C4—154no—
326C4—301no—
327C4#679012yes—
328C3#6596711yescharacteristic, complemented, Sylow
329C2—121no—
330C2—202no—
331C2—302no—
332C2#9202811yescharacteristic
333C2#9203411yescharacteristic
334C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×Q8C22
hypercenterC2×Q8D3×D5
hypocenterC3×C5#548
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×Q8×D15C2semidirect
2. C2×#6765C2semidirect
3. C5×#946C2semidirect
4. C3×#4093C2semidirect
5. #91465C2semidirect
6. #91462C2semidirect
7. #91461C2semidirect
8. C4×#1203C2semidirect
9. #91431C2semidirect
10. #91396C2semidirect
11. #91333C2semidirect
42. #4093D3semidirect
61. Dic30D4semidirect
75. #946D5semidirect
112. C2×D15#41semidirect
256. C3×C5#548semidirect
309. C5#5677semidirect
328. C3#65967semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D152
C2×#67652
C5×#9462
C3×#40932
#914652
#914622
#914612
C4×#12032
#914312
#913962
#913332
#659673
#56775