Finite Group Databasebeta

← #11751148

Subgroups — #11751148

Order 960.

304 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 71 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×#6817C211yesmaximal, characteristic
2C3×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#6718C211yesmaximal, characteristic, complemented
4C2×#6717C211yesmaximal, characteristic, complemented
5#91114C211yesmaximal, characteristic, complemented
6#91094C211yesmaximal, characteristic, complemented
7#91093C211yesmaximal, characteristic, complemented
8#65336—31nomaximal, Hall
9C2×C3×C5×Q8C2211yescharacteristic
10C22×C3×C4×C5C2211yescharacteristic
11C2×#1220—21no—
12#6817—22no—
13#6817—22no—
14C2×D60C2211yescharacteristic
15C2×C4×D15—21no—
16C2×Dic30C2211yescharacteristic
17C3×C5×C4⋊C4—21no—
18C3×C5×C4⋊C4C2211yescharacteristic
19C3×C5×C22⋊C4—21no—
20#6718—22no—
21#6717—22no—
22C2×C3⋊#168C2211yescharacteristic
23C2×C3⋊#168C2211yescharacteristic
24#5047—51nomaximal, Hall
25C2×#709—31no—
26C5×C22⋊Q8D311yescharacteristic, complemented
27C2×#689—31no—
28C2×#688—31no—
29C22⋊#168—31no—
30#3942—31no—
31#3941—31no—
32C23×C3×C5D411yescharacteristic
33C22×D15—21no—
34C3×C5×Q8—21no—
35C3×C5×Q8—21no—
36C2×C3×C4×C5—21no—
37C2×C3×C4×C5—21no—
38C2×C3×C4×C5—21no—
39C2×C3×C4×C5—21no—
40C2×C3×C4×C5D411yescharacteristic
41C2×C3×C4×C5C2311yescharacteristic
42#1220—24no—
43C2×Dic15—21no—
44D60—21no—
45D60D412yes—
46C4×D15—24no—
47Dic30—21no—
48Dic30D412yes—
49C3⋊#168—21no—
50C3⋊#168—21no—
51C2×#235—51no—
52C3×C22⋊Q8D511yescharacteristic, complemented
53C2×#216—51no—
54C2×#215—51no—
55#830—51no—
56#810—51no—
57#809—51no—
58C2×C5×Q8C2×D311yescharacteristic
59C22×C4×C5C2×D311yescharacteristic
60C2×C22⋊D5—61no—
61#709—62no—
62#709—62no—
63C2×D20—31no—
64C2×C4×D5—61no—
65C2×Dic10—31no—
66C5×C4⋊C4—21no—
67C5×C4⋊C4C2×D311yescharacteristic
68C5×C22⋊C4—21no—
69#689—62no—
70#688—62no—
71C2×#168—31no—
72C2×#168—31no—
73#452—151noSylow
74C22×C3×C5—21no—
75C22×C3×C5—21no—
76C22×C3×C5—21no—
77C22×C3×C5C2×D411yescharacteristic
78C2×D15—22no—
79C2×D15—41no—
80C2×D15—41no—
81C3×C4×C5—21no—
82C3×C4×C5—41no—
83C3×C4×C5—41no—
84C3×C4×C5C2×D411yescharacteristic
85C3×C4×C5C2×D411yescharacteristic
86Dic15—22no—
87C2×C3×Q8C2×D511yescharacteristic
88C22×C3×C4C2×D511yescharacteristic
89C2×#67—101no—
90#235—102no—
91#235—102no—
92C2×D12—51no—
93C2×C4×D3—101no—
94C2×Dic6—51no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4C2×D511yescharacteristic
97C3×C22⋊C4—21no—
98#216—102no—
99#215—102no—
100C2×C3⋊C8—51no—
101C2×C3⋊C8—51no—
102C23×C5#6711yescharacteristic
103C22×D5—61no—
104C5×Q8—21no—
105C5×Q8—21no—
106C2×C4×C5—21no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5—21no—
110C2×C4×C5#6711yescharacteristic
111C2×C4×C5C22×D311yescharacteristic
112C22⋊D5—64no—
113C2×Dic5—61no—
114D20—32no—
115D20—61no—
116C4×D5—64no—
117Dic10—32no—
118Dic10—61no—
119#168—61no—
120#168—61no—
121C2×#41—151no—
122C2×Q16—151no—
123C2×SD16—151no—
124C22⋊Q8D1511yescharacteristic, complemented
125#103—151no—
126#102—151no—
127C22⋊C8—151no—
128C2×C3×C5—41no—
129C2×C3×C5#12011yescharacteristic
130C2×C3×C5#13511yescharacteristic
131C2×C3×C5#13711yescharacteristic
132D15—42no—
133C23×C3C22⋊D511yescharacteristic
134C22×D3—101no—
135C3×Q8—21no—
136C3×Q8—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4C22⋊D511yescharacteristic
142C2×C3×C4C22×D511yescharacteristic
143#67—104no—
144C2×Dic3—101no—
145D12—52no—
146D12—101no—
147C4×D3—104no—
148Dic6—52no—
149Dic6—101no—
150C3⋊C8—101no—
151C3⋊C8—101no—
152C22×C5—21no—
153C22×C5—21no—
154C22×C5—21no—
155C22×C5C2×#6711yescharacteristic
156C2×D5—62no—
157C2×D5—121no—
158C2×D5—121no—
159C4×C5—21no—
160C4×C5—41no—
161C4×C5—41no—
162C4×C5D3×D411yescharacteristic
163C4×C5D3×D411yescharacteristic
164Dic5—62no—
165#41—302no—
166#41—302no—
167C2×Q8—151no—
168C2×Q8C2×D1511yescharacteristic
169C2×D4—151no—
170C2×D4—301no—
171C22×C4—301no—
172C22×C4C2×D1511yescharacteristic
173Q16—302no—
174SD16—302no—
175C2×C8—151no—
176C2×C8—151no—
177C4⋊C4—21no—
178C4⋊C4C2×D1511yescharacteristic
179C22⋊C4—21no—
180C3×C5#45211yescharacteristic, complemented, Hall
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—21no—
184C22×C3C2×C22⋊D511yescharacteristic
185C2×D3—102no—
186C2×D3—201no—
187C2×D3—201no—
188C3×C4—21no—
189C3×C4—41no—
190C3×C4—41no—
191C3×C4D4×D511yescharacteristic
192C3×C4D4×D511yescharacteristic
193Dic3—102no—
194C2×C5—41no—
195C2×C5#93711yescharacteristic
196C2×C5#94211yescharacteristic
197C2×C5#95011yescharacteristic
198D5—122no—
199C23—301no—
200C23#122011yescharacteristic
201Q8—21no—
202Q8—21no—
203Q8—152no—
204Q8—301no—
205D4—152no—
206D4—304no—
207D4—301no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—304no—
213C2×C4—301no—
214C2×C4#122011yescharacteristic
215C2×C4C22×D1511yescharacteristic
216C8—301no—
217C8—301no—
218C2×C3—41no—
219C2×C3#408311yescharacteristic
220C2×C3#408811yescharacteristic
221C2×C3#409611yescharacteristic
222D3—202no—
223C5#504711yescharacteristic, complemented, Sylow
224C22—21no—
225C22—21no—
226C22—21no—
227C22—302no—
228C22—601no—
229C22—601no—
230C22C2×#122011yescharacteristic
231C4—21no—
232C4—41no—
233C4—41no—
234C4—302no—
235C4D4×D1511yescharacteristic
236C4D4×D1511yescharacteristic
237C3#6533611yescharacteristic, complemented, Sylow
238C2—41no—
239C2—602no—
240C2#9182611yescharacteristic
241C2#9183111yescharacteristic
242C2#9183911yescharacteristic
subgroupG/H
centerC22C2×#1220
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C22×D15
FittingC3×C5×C22⋊Q8C2
socleC22×C3×C5C2×D4
prime radicalC2×#6817C2
hypercenterC22⋊Q8D15
hypocenterC3×C5#452
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×C5×C22⋊Q8C2semidirect
3. C2×#6718C2semidirect
4. C2×#6717C2semidirect
5. #91114C2semidirect
6. #91094C2semidirect
7. #91093C2semidirect
26. C5×C22⋊Q8D3semidirect
52. C3×C22⋊Q8D5semidirect
124. C22⋊Q8D15semidirect
180. C3×C5#452semidirect
223. C5#5047semidirect
237. C3#65336semidirect

Maximal subgroups

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

subgroupindex
C2×#68172
C3×C5×C22⋊Q82
C2×#67182
C2×#67172
#911142
#910942
#910932
#653363
#50475