Finite Group Databasebeta

← C2×Q8×C22⋊H(3,𝔽3) #13164850

Subgroups — C2×Q8×C22⋊H(3,𝔽3) #13164850

Order 1728.

1192 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1192 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×C22⋊H(3,𝔽3)C214yesmaximal, direct product factor
2C2×C4×C22⋊H(3,𝔽3)C213yesmaximal
3C23×C32×Q8C311yesmaximal, characteristic, complemented
4C2×C3×Q8×A4C313yesmaximal, complemented
5C22×C22⋊H(3,𝔽3)C2211yescharacteristic
6C2×Q8×H(3,𝔽3)—41nomaximal
7C4×C22⋊H(3,𝔽3)C2216yes—
8C22×C32×Q8—33no—
9C22×C32×Q8—34no—
10C22×C32×Q8—31no—
11C22×C32×Q8C2×C314yescomplemented
12C23×C32×C4C2×C313yes—
13C3×Q8×A4C2×C3112yescomplemented
14C2×C3×C4×A4C2×C319yes—
15C2×C22⋊H(3,𝔽3)Q812yesdirect product factor
16C2×C22⋊H(3,𝔽3)C2311yescharacteristic
17Q8×H(3,𝔽3)—44no—
18C2×C4×H(3,𝔽3)—43no—
19C23×C3×Q8—31no—
20C23×C3×Q8C3211yescharacteristic
21C2×Q8×A4—33no—
22C24×C32C22×C311yescharacteristic
23C22×C3×A4C22×C313yes—
24C2×C32×Q8—312no—
25C2×C32×Q8—32no—
26C2×C32×Q8—34no—
27C2×C32×Q8—34no—
28C2×C32×Q8—312no—
29C2×C32×Q8—33no—
30C2×C32×Q8—43no—
31C2×C32×Q8A411yescharacteristic, complemented
32C22×C32×C4—33no—
33C22×C32×C4—33no—
34C22×C32×C4—36no—
35C22×C32×C4C22×C316yes—
36C3×C4×A4C22×C3118yes—
37C22×H(3,𝔽3)—41no—
38C22⋊H(3,𝔽3)C2×Q811yescharacteristic, direct product factor
39C4×H(3,𝔽3)—46no—
40C22×C3×Q8—33no—
41C22×C3×Q8—34no—
42C22×C3×Q8—39no—
43C22×C3×Q8—312no—
44C22×C3×Q8—33no—
45C22×C3×Q8—31no—
46C22×C3×Q8—34no—
47C22×C3×Q8C2×C3214yes—
48C23×C3×C4—33no—
49C23×C3×C4C2×C3213yes—
50Q8×A4—312no—
51C2×C4×A4—39no—
52C23×C32—32no—
53C23×C32—31no—
54C23×C32—31no—
55C23×C32C3×Q812yescomplemented
56C23×C32C23×C311yescharacteristic
57C2×C3×A4C3×Q816yescomplemented
58C2×C3×A4C23×C313yes—
59C32×Q8—38no—
60C32×Q8—312no—
61C32×Q8—412no—
62C32×Q8C2×A414yescomplemented
63C2×C32×C4—36no—
64C2×C32×C4—33no—
65C2×C32×C4—36no—
66C2×C32×C4—36no—
67C2×C32×C4—36no—
68C2×C32×C4—49no—
69C2×C32×C4C2×A413yes—
70C23×Q8H(3,𝔽3)11yescharacteristic, complemented, Sylow
71C2×H(3,𝔽3)—42no—
72C2×H(3,𝔽3)—41no—
73C24×C3—31no—
74C24×C3C22×C3211yescharacteristic
75C22×A4—33no—
76C2×C3×Q8—312no—
77C2×C3×Q8—336no—
78C2×C3×Q8—336no—
79C2×C3×Q8—36no—
80C2×C3×Q8—312no—
81C2×C3×Q8—32no—
82C2×C3×Q8—312no—
83C2×C3×Q8—312no—
84C2×C3×Q8—39no—
85C2×C3×Q8—34no—
86C2×C3×Q8—34no—
87C2×C3×Q8—33no—
88C2×C3×Q8—31no—
89C2×C3×Q8—123no—
90C2×C3×Q8C3×A411yescharacteristic
91C22×C3×C4—39no—
92C22×C3×C4—39no—
93C22×C3×C4—318no—
94C22×C3×C4—33no—
95C22×C3×C4—33no—
96C22×C3×C4—36no—
97C22×C3×C4—36no—
98C22×C3×C4C22×C3216yes—
99C4×A4—318no—
100C22×C32—32no—
101C22×C32—31no—
102C22×C32—32no—
103C22×C32—32no—
104C22×C32—32no—
105C22×C32—31no—
106C22×C32—31no—
107C22×C32—43no—
108C22×C32C2×C3×Q811yescharacteristic, complemented
109C22×C32C22×A411yescharacteristic
110C3×A4C2×C3×Q813yescomplemented
111C32×C4—36no—
112C32×C4—418no—
113C32×C4C22×A416yes—
114C22×Q8—33no—
115C22×Q8—34no—
116C22×Q8—31no—
117C22×Q8C2×H(3,𝔽3)14yescomplemented
118C23×C4C2×H(3,𝔽3)13yes—
119H(3,𝔽3)—41noSylow
120C23×C3—31no—
121C23×C3—32no—
122C23×C3—33no—
123C23×C3—33no—
124C23×C3—36no—
125C23×C3—31no—
126C23×C3—32no—
127C23×C3—31no—
128C23×C3C32×Q812yes—
129C23×C3C23×C3211yescharacteristic
130C2×A4—36no—
131C2×A4—33no—
132C3×Q8—324no—
133C3×Q8—38no—
134C3×Q8—336no—
135C3×Q8—312no—
136C3×Q8—34no—
137C3×Q8—1212no—
138C3×Q8C2×C3×A414yes—
139C2×C3×C4—318no—
140C2×C3×C4—36no—
141C2×C3×C4—39no—
142C2×C3×C4—318no—
143C2×C3×C4—318no—
144C2×C3×C4—318no—
145C2×C3×C4—33no—
146C2×C3×C4—36no—
147C2×C3×C4—36no—
148C2×C3×C4—36no—
149C2×C3×C4—33no—
150C2×C3×C4—129no—
151C2×C3×C4C2×C3×A413yes—
152C2×C32—32no—
153C2×C32—31no—
154C2×C32—31no—
155C2×C32—46no—
156C2×C32—43no—
157C2×C32Q8×A412yescomplemented
158C2×C32C23×A411yescharacteristic
159C24C22×H(3,𝔽3)11yescharacteristic
160C2×Q8—312no—
161C2×Q8—312no—
162C2×Q8—32no—
163C2×Q8—34no—
164C2×Q8—34no—
165C2×Q8—33no—
166C2×Q8C22⋊H(3,𝔽3)11yescharacteristic, direct product factor
167C22×C4—33no—
168C22×C4—33no—
169C22×C4—36no—
170C22×C4C22×H(3,𝔽3)16yes—
171C22×C3—32no—
172C22×C3—31no—
173C22×C3—33no—
174C22×C3—33no—
175C22×C3—36no—
176C22×C3—33no—
177C22×C3—36no—
178C22×C3—32no—
179C22×C3—36no—
180C22×C3—36no—
181C22×C3—32no—
182C22×C3—32no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—31no—
186C22×C3—31no—
187C22×C3—123no—
188C22×C3C2×C32×Q811yescharacteristic
189C22×C3C22×C3×A411yescharacteristic
190A4—33no—
191C3×C4—318no—
192C3×C4—36no—
193C3×C4—36no—
194C3×C4—1218no—
195C3×C4C22×C3×A416yes—
196C32—43no—
197C32C2×Q8×A411yescharacteristic, complemented
198C23—31no—
199C23—32no—
200C23—31no—
201C23Q8×H(3,𝔽3)12yescomplemented
202C23C23×H(3,𝔽3)11yescharacteristic
203Q8—38no—
204Q8—312no—
205Q8C2×C22⋊H(3,𝔽3)14yesdirect product factor
206C2×C4—36no—
207C2×C4—33no—
208C2×C4—36no—
209C2×C4—36no—
210C2×C4—36no—
211C2×C4C2×C22⋊H(3,𝔽3)13yes—
212C2×C3—33no—
213C2×C3—36no—
214C2×C3—33no—
215C2×C3—31no—
216C2×C3—32no—
217C2×C3—31no—
218C2×C3—31no—
219C2×C3—32no—
220C2×C3—123no—
221C2×C3—126no—
222C2×C3C3×Q8×A412yes—
223C2×C3C23×C3×A411yescharacteristic
224C22—32no—
225C22—31no—
226C22—32no—
227C22—31no—
228C22—31no—
229C22—32no—
230C22—32no—
231C22C2×Q8×H(3,𝔽3)11yescharacteristic, complemented
232C22C22×C22⋊H(3,𝔽3)11yescharacteristic
233C4—36no—
234C4C22×C22⋊H(3,𝔽3)16yes—
235C3—31no—
236C3—123no—
237C3C2×C3×Q8×A411yescharacteristic
238C2—31no—
239C2—32no—
240C2—31no—
241C2Q8×C22⋊H(3,𝔽3)12yesdirect product factor
242C2C23×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC22×C3C22×C3×A4
commutator, cosocleC23×C3C23×C32
FrattiniC2×C3C23×C3×A4
FittingC23×C32×Q8C3
socleC24×C3C22×C32
prime radicalC22×C22⋊H(3,𝔽3)C22
hypercenterC2×C32×Q8A4
hypocenterC22C2×Q8×H(3,𝔽3)
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×C22⋊H(3,𝔽3)C2direct (N × H)
3. C23×C32×Q8C3semidirect
4. C2×C3×Q8×A4C3semidirect
11. C22×C32×Q8C2×C3semidirect
13. C3×Q8×A4C2×C3semidirect
15. C2×C22⋊H(3,𝔽3)Q8direct (N × H)
31. C2×C32×Q8A4semidirect
38. C22⋊H(3,𝔽3)C2×Q8direct (N × H)
55. C23×C32C3×Q8semidirect
57. C2×C3×A4C3×Q8semidirect
62. C32×Q8C2×A4semidirect
70. C23×Q8H(3,𝔽3)semidirect
108. C22×C32C2×C3×Q8semidirect
110. C3×A4C2×C3×Q8semidirect
117. C22×Q8C2×H(3,𝔽3)semidirect
157. C2×C32Q8×A4semidirect
166. C2×Q8C22⋊H(3,𝔽3)direct (N × H)
197. C32C2×Q8×A4semidirect
201. C23Q8×H(3,𝔽3)semidirect
205. Q8C2×C22⋊H(3,𝔽3)direct (N × H)
231. C22C2×Q8×H(3,𝔽3)semidirect
241. C2Q8×C22⋊H(3,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×C22⋊H(3,𝔽3)2
C2×C4×C22⋊H(3,𝔽3)2
C23×C32×Q83
C2×C3×Q8×A43
C2×Q8×H(3,𝔽3)4