Finite Group Databasebeta

← Q8×#1203 #11755243

Subgroups — Q8×#1203 #11755243

Order 960.

560 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 174 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×C5×D3×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×Dic5C211yesmaximal, characteristic, complemented
4C4×#1203C213yesmaximal
5#91383C213yesmaximal, complemented
6#91376C213yesmaximal, complemented
7#91365C213yesmaximal, complemented
8Q8×C22⋊D5—31nomaximal, Hall
9C2×C3×C5×Q8C2211yescharacteristic
10Q8×D15—24no—
11C2×C4×D15C2213yes—
12C2×Dic30C2213yes—
13C5×D3×Q8—24no—
14C2×C4×C5×D3C2213yes—
15C2×C5×Dic6C2213yes—
16C2×#1203C2211yescharacteristic
17C3×C4⋊Dic5C2213yes—
18C3×C4×Dic5C2213yes—
19#6668C2213yes—
20#6668C2216yes—
21#6667C2213yes—
22(C2×D3)⋊Dic5C2213yes—
23Q8×D12—51nomaximal, Hall
24C22×C5×Q8—31no—
25C2×Q8×D5—31no—
26#4092—33no—
27Q8×Dic5D311yescharacteristic, complemented
28#4090—33no—
29C4×C22⋊D5—33no—
30#4070—33no—
31C22×D15C2311yescharacteristic
32C22×C5×D3C2311yescharacteristic
33C3×C5×Q8D414yescomplemented
34C2×C3×C4×C5C2313yes—
35C2×Dic15C2313yes—
36C4×D15—26no—
37Dic30—26no—
38C2×C5×Dic3C2313yes—
39C4×C5×D3—26no—
40C5×Dic6—26no—
41C2×C3×Dic5C2313yes—
42C2×C3×Dic5C2311yescharacteristic
43#1203Q814yesdirect product factor
44C2×D3×Q8—51no—
45C2×D3×Q8D511yescharacteristic, complemented
46C3×C4×Q8—51no—
47#897—53no—
48#897—53no—
49C4×D12—53no—
50#869—53no—
51C2×C5×Q8—33no—
52C2×C5×Q8—64no—
53C2×C5×Q8C2×D311yescharacteristic
54C22×C4×C5—33no—
55C2×C22⋊D5—31no—
56Q8×D5—64no—
57C2×C4×D5—33no—
58C2×Dic10—33no—
59C22⋊Dic5—33no—
60#685—33no—
61C4⋊Dic5C2×D313yes—
62#683—36no—
63#683—33no—
64C4×Dic5C2×D313yes—
65D4×Q8—151noSylow
66C22×C3×C5C2411yescharacteristic
67C2×D15—21no—
68C2×D15—21no—
69C2×D15C2×Q812yescomplemented
70C2×C5×D3—21no—
71C2×C5×D3—21no—
72C2×C5×D3C2×Q812yescomplemented
73C3×C4×C5C2×D416yes—
74Dic15—23no—
75C3×Dic5—23no—
76C3×Dic5C2×Q812yescomplemented
77C5×Dic3—23no—
78C2×C3×Q8C2×D511yescharacteristic
79D3×Q8—24no—
80D3×Q8—104no—
81C2×D12—51no—
82C2×C4×D3—53no—
83C2×C4×D3C2×D513yes—
84C2×Dic6—53no—
85C2×Dic6C2×D513yes—
86C3×C4⋊C4—53no—
87C3×C42—53no—
88(C2×D3)⋊C4—53no—
89(C2×D3)⋊C4—53no—
90C4⋊Dic3—53no—
91C4⋊Dic3—56no—
92C23×C5—31no—
93C22×D5—31no—
94C5×Q8—66no—
95C5×Q8D1214yescomplemented
96C2×C4×C5—33no—
97C2×C4×C5—66no—
98C2×C4×C5C22×D313yes—
99C22⋊D5—34no—
100C2×Dic5—33no—
101C2×Dic5C22×D313yes—
102C2×Dic5C22×D311yescharacteristic
103C4×D5—66no—
104Dic10—66no—
105C22×Q8—31no—
106C22×Q8—151no—
107C4⋊Q8—153no—
108C22⋊Q8—153no—
109C22⋊Q8—153no—
110C4×Q8—51no—
111C4×D4—153no—
112C2×C3×C5C22×D411yescharacteristic
113C2×C3×C5C22×Q811yescharacteristic
114C2×C3×C52−1+411yescharacteristic
115D15—22no—
116C5×D3—22no—
117C22×D3—51no—
118C22×D3C22×D511yescharacteristic
119C3×Q8C22⋊D514yescomplemented
120C2×C3×C4—51no—
121C2×C3×C4—53no—
122C2×C3×C4C22×D513yes—
123C2×Dic3—53no—
124C2×Dic3C22×D513yes—
125D12—54no—
126C4×D3—26no—
127C4×D3—106no—
128Dic6—26no—
129Dic6—106no—
130C22×C5—32no—
131C22×C5—61no—
132C22×C5—61no—
133C22×C5C23×D311yescharacteristic
134C2×D5—32no—
135C2×D5—61no—
136C2×D5—61no—
137C4×C5—63no—
138C4×C5C2×D1216yes—
139Dic5—23no—
140Dic5—63no—
141Dic5D3×Q812yescomplemented
142C2×Q8—33no—
143C2×Q8—64no—
144C2×Q8—153no—
145C2×Q8—304no—
146C2×Q8D3×D511yescharacteristic
147C2×D4—151no—
148C22×C4—33no—
149C22×C4—153no—
150C4⋊C4—53no—
151C4⋊C4—153no—
152C4⋊C4—156no—
153C22⋊C4—153no—
154C22⋊C4—153no—
155C42—53no—
156C3×C5D4×Q811yescharacteristic, complemented, Hall
157C22×C3C23×D511yescharacteristic
158C2×D3—21no—
159C2×D3—21no—
160C2×D3—52no—
161C2×D3—101no—
162C2×D3—101no—
163C2×D3Q8×D512yescomplemented
164C3×C4—52no—
165C3×C4—103no—
166C3×C4C2×C22⋊D516yes—
167Dic3—23no—
168Dic3—103no—
169C2×C5—62no—
170C2×C5C22×D1211yescharacteristic
171C2×C5C2×D3×Q811yescharacteristic
172C2×C5#101011yescharacteristic
173D5—62no—
174C23—31no—
175C23—151no—
176Q8—66no—
177Q8—306no—
178Q8#120314yesdirect product factor
179D4—154no—
180C2×C4—33no—
181C2×C4—51no—
182C2×C4—53no—
183C2×C4—66no—
184C2×C4—153no—
185C2×C4—306no—
186C2×C4C2×D3×D513yes—
187C2×C3C2×Q8×D511yescharacteristic
188C2×C3#414711yescharacteristic
189C2×C3C22×C22⋊D511yescharacteristic
190D3—22no—
191D3—102no—
192C5Q8×D1211yescharacteristic, complemented, Sylow
193C22—32no—
194C22—61no—
195C22—61no—
196C22—152no—
197C22—301no—
198C22—301no—
199C22C22×D3×D511yescharacteristic
200C4—52no—
201C4—63no—
202C4—103no—
203C4—303no—
204C4C2×#120316yes—
205C3Q8×C22⋊D511yescharacteristic, complemented, Sylow
206C2—62no—
207C2—302no—
208C2#9202811yescharacteristic
209C2D3×Q8×D511yescharacteristic
210C2C22×#120311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×Q8C22
prime radicalC2×#1203C22
hypercenterC2×Q8D3×D5
hypocenterC3×C5D4×Q8
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×C5×D3×Q8C2semidirect
3. C3×Q8×Dic5C2semidirect
5. #91383C2semidirect
6. #91376C2semidirect
7. #91365C2semidirect
27. Q8×Dic5D3semidirect
33. C3×C5×Q8D4semidirect
43. #1203Q8direct (N × H)
45. C2×D3×Q8D5semidirect
69. C2×D15C2×Q8semidirect
72. C2×C5×D3C2×Q8semidirect
76. C3×Dic5C2×Q8semidirect
95. C5×Q8D12semidirect
119. C3×Q8C22⋊D5semidirect
141. Dic5D3×Q8semidirect
156. C3×C5D4×Q8semidirect
163. C2×D3Q8×D5semidirect
178. Q8#1203direct (N × H)
192. C5Q8×D12semidirect
205. C3Q8×C22⋊D5semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D152
C2×C5×D3×Q82
C3×Q8×Dic52
C4×#12032
#913832
#913762
#913652
Q8×C22⋊D53
Q8×D125