Finite Group Databasebeta

← C2×Q8×#1094 #13164848

Subgroups — C2×Q8×#1094 #13164848

Order 1728.

1078 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 190 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
1Q8×#1094C214yesmaximal, direct product factor
2C2×C4×#1094C213yesmaximal
3C23×C32×Q8C311yesmaximal, characteristic
4C2×Q8×C22⋊C9C313yesmaximal, complemented
5C22×#1094C2211yescharacteristic
6C2×Q8×3−1+2—41nomaximal
7C4×#1094C2216yes—
8C22×C32×Q8—33no—
9C22×C32×Q8—34no—
10C22×C32×Q8—31no—
11C22×C32×Q8C2×C314yes—
12C23×C32×C4C2×C313yes—
13Q8×C22⋊C9C2×C3112yescomplemented
14C2×C4×C22⋊C9C2×C319yes—
15C2×#1094Q812yesdirect product factor
16C2×#1094C2311yescharacteristic
17Q8×3−1+2—44no—
18C2×C4×3−1+2—43no—
19C23×C3×Q8—31no—
20C23×C3×Q8C3211yescharacteristic
21C24×C32C22×C311yescharacteristic
22C2×C32×Q8—312no—
23C2×C32×Q8—32no—
24C2×C32×Q8—34no—
25C2×C32×Q8—34no—
26C2×C32×Q8—312no—
27C2×C32×Q8—33no—
28C2×C32×Q8A411yescharacteristic
29C22×C32×C4—33no—
30C22×C32×C4—33no—
31C22×C32×C4—36no—
32C22×C32×C4C22×C316yes—
33C22×C22⋊C9C22×C313yes—
34C2×Q8×C9—43no—
35C4×C22⋊C9C22×C3118yes—
36C22×3−1+2—41no—
37#1094C2×Q811yescharacteristic, direct product factor
38C4×3−1+2—46no—
39C22×C3×Q8—33no—
40C22×C3×Q8—34no—
41C22×C3×Q8—39no—
42C22×C3×Q8—312no—
43C22×C3×Q8—33no—
44C22×C3×Q8—31no—
45C22×C3×Q8—34no—
46C22×C3×Q8C2×C3214yes—
47C23×C3×C4—33no—
48C23×C3×C4C2×C3213yes—
49C23×C32—32no—
50C23×C32—31no—
51C23×C32—31no—
52C23×C32C3×Q812yes—
53C23×C32C23×C311yescharacteristic
54C32×Q8—38no—
55C32×Q8—312no—
56C32×Q8C2×A414yes—
57C2×C32×C4—36no—
58C2×C32×C4—33no—
59C2×C32×C4—36no—
60C2×C32×C4—36no—
61C2×C32×C4—36no—
62C2×C32×C4C2×A413yes—
63C2×C22⋊C9C3×Q816yescomplemented
64C2×C22⋊C9C23×C313yes—
65Q8×C9—412no—
66C2×C4×C9—49no—
67C23×Q83−1+211yescharacteristic, complemented, Sylow
68C2×3−1+2—42no—
69C2×3−1+2—41no—
70C24×C3—31no—
71C24×C3C22×C3211yescharacteristic
72C2×C3×Q8—312no—
73C2×C3×Q8—336no—
74C2×C3×Q8—336no—
75C2×C3×Q8—36no—
76C2×C3×Q8—312no—
77C2×C3×Q8—32no—
78C2×C3×Q8—312no—
79C2×C3×Q8—312no—
80C2×C3×Q8—39no—
81C2×C3×Q8—34no—
82C2×C3×Q8—34no—
83C2×C3×Q8—33no—
84C2×C3×Q8—31no—
85C2×C3×Q8C3×A411yescharacteristic
86C22×C3×C4—39no—
87C22×C3×C4—39no—
88C22×C3×C4—318no—
89C22×C3×C4—33no—
90C22×C3×C4—33no—
91C22×C3×C4—36no—
92C22×C3×C4—36no—
93C22×C3×C4C22×C3216yes—
94C22×C32—32no—
95C22×C32—31no—
96C22×C32—32no—
97C22×C32—32no—
98C22×C32—32no—
99C22×C32—31no—
100C22×C32—31no—
101C22×C32C2×C3×Q811yescharacteristic
102C22×C32C22×A411yescharacteristic
103C32×C4—36no—
104C32×C4C22×A416yes—
105C22×C9—43no—
106C22⋊C9C2×C3×Q813yescomplemented
107C4×C9—418no—
108C22×Q8—33no—
109C22×Q8—34no—
110C22×Q8—31no—
111C22×Q8C2×3−1+214yescomplemented
112C23×C4C2×3−1+213yes—
1133−1+2—41noSylow
114C23×C3—31no—
115C23×C3—32no—
116C23×C3—33no—
117C23×C3—33no—
118C23×C3—36no—
119C23×C3—31no—
120C23×C3—32no—
121C23×C3—31no—
122C23×C3C32×Q812yes—
123C23×C3C23×C3211yescharacteristic
124C3×Q8—324no—
125C3×Q8—38no—
126C3×Q8—336no—
127C3×Q8—312no—
128C3×Q8—34no—
129C3×Q8C2×C3×A414yes—
130C2×C3×C4—318no—
131C2×C3×C4—36no—
132C2×C3×C4—39no—
133C2×C3×C4—318no—
134C2×C3×C4—318no—
135C2×C3×C4—318no—
136C2×C3×C4—33no—
137C2×C3×C4—36no—
138C2×C3×C4—36no—
139C2×C3×C4—36no—
140C2×C3×C4—33no—
141C2×C3×C4C2×C3×A413yes—
142C2×C32—32no—
143C2×C32—31no—
144C2×C32—31no—
145C2×C32Q8×A412yes—
146C2×C32C23×A411yescharacteristic
147C2×C9—46no—
148C2×C9—43no—
149C24C22×3−1+211yescharacteristic
150C2×Q8—312no—
151C2×Q8—312no—
152C2×Q8—32no—
153C2×Q8—34no—
154C2×Q8—34no—
155C2×Q8—33no—
156C2×Q8#109411yescharacteristic, direct product factor
157C22×C4—33no—
158C22×C4—33no—
159C22×C4—36no—
160C22×C4C22×3−1+216yes—
161C22×C3—32no—
162C22×C3—31no—
163C22×C3—33no—
164C22×C3—33no—
165C22×C3—36no—
166C22×C3—33no—
167C22×C3—36no—
168C22×C3—32no—
169C22×C3—36no—
170C22×C3—36no—
171C22×C3—32no—
172C22×C3—32no—
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—31no—
176C22×C3—31no—
177C22×C3C2×C32×Q811yescharacteristic
178C22×C3C22×C3×A411yescharacteristic
179C3×C4—318no—
180C3×C4—36no—
181C3×C4—36no—
182C3×C4C22×C3×A416yes—
183C32C2×Q8×A411yescharacteristic
184C9—43no—
185C23—31no—
186C23—32no—
187C23—31no—
188C23Q8×3−1+212yescomplemented
189C23C23×3−1+211yescharacteristic
190Q8—38no—
191Q8—312no—
192Q8C2×#109414yesdirect product factor
193C2×C4—36no—
194C2×C4—33no—
195C2×C4—36no—
196C2×C4—36no—
197C2×C4—36no—
198C2×C4C2×#109413yes—
199C2×C3—33no—
200C2×C3—36no—
201C2×C3—33no—
202C2×C3—31no—
203C2×C3—32no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—32no—
207C2×C3C3×Q8×A412yes—
208C2×C3C23×C3×A411yescharacteristic
209C22—32no—
210C22—31no—
211C22—32no—
212C22—31no—
213C22—31no—
214C22—32no—
215C22—32no—
216C22C2×Q8×3−1+211yescharacteristic, complemented
217C22C22×#109411yescharacteristic
218C4—36no—
219C4C22×#109416yes—
220C3—31no—
221C3C2×C3×Q8×A411yescharacteristic
222C2—31no—
223C2—32no—
224C2—31no—
225C2Q8×#109412yesdirect product factor
226C2C23×#109411yescharacteristic
subgroupG/H
centerC22×C3C22×C3×A4
commutator, cosocleC23×C3C23×C32
FrattiniC2×C3C23×C3×A4
FittingC23×C32×Q8C3
socleC24×C3C22×C32
prime radicalC24×C32C22×C3
hypercenterC2×C32×Q8A4
hypocenterC22C2×Q8×3−1+2
LCSG ▷ C23×C3 ▷ C22
UCSC1 ◁ C22×C3 ◁ C2×C32×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#1094C2direct (N × H)
4. C2×Q8×C22⋊C9C3semidirect
13. Q8×C22⋊C9C2×C3semidirect
15. C2×#1094Q8direct (N × H)
37. #1094C2×Q8direct (N × H)
63. C2×C22⋊C9C3×Q8semidirect
67. C23×Q83−1+2semidirect
106. C22⋊C9C2×C3×Q8semidirect
111. C22×Q8C2×3−1+2semidirect
156. C2×Q8#1094direct (N × H)
188. C23Q8×3−1+2semidirect
192. Q8C2×#1094direct (N × H)
216. C22C2×Q8×3−1+2semidirect
225. C2Q8×#1094direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#10942
C2×C4×#10942
C23×C32×Q83
C2×Q8×C22⋊C93
C2×Q8×3−1+24