Finite Group Databasebeta

← C2×C3×Q8×C22⋊C9 #13164847

Subgroups — C2×C3×Q8×C22⋊C9 #13164847

Order 1728.

1154 conjugacy classes of subgroups; only the 304 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 304 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
1C3×Q8×C22⋊C9C214yesmaximal, direct product factor
2C2×C3×C4×C22⋊C9C213yesmaximal
3C23×C32×Q8C311yesmaximal, characteristic
4C2×Q8×C22⋊C9C313yesmaximal, direct product factor
5C22×C3×C22⋊C9C2211yescharacteristic
6C2×C3×Q8×C9—41nomaximal
7C3×C4×C22⋊C9C2216yes—
8C22×C32×Q8—33no—
9C22×C32×Q8—34no—
10C22×C32×Q8—31no—
11C22×C32×Q8C2×C314yes—
12C23×C32×C4C2×C313yes—
13Q8×C22⋊C9C2×C3112yesdirect product factor
14C2×C4×C22⋊C9C2×C319yes—
15C2×C3×C22⋊C9Q812yesdirect product factor
16C2×C3×C22⋊C9C2311yescharacteristic
17C3×Q8×C9—44no—
18C2×C3×C4×C9—43no—
19C23×C3×Q8C913yescomplemented
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×C3×C9—41no—
37C3×C22⋊C9C2×Q811yescharacteristic, direct product factor
38C3×C4×C9—46no—
39C22×C3×Q8—39no—
40C22×C3×Q8—312no—
41C22×C3×Q8—33no—
42C22×C3×Q8—34no—
43C22×C3×Q8—33no—
44C22×C3×Q8—31no—
45C22×C3×Q8C2×C9112yescomplemented
46C22×C3×Q8C2×C3214yes—
47C23×C3×C4C2×C919yes—
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×Q816yesdirect product factor
64C2×C22⋊C9C23×C313yes—
65Q8×C9—412no—
66C2×C4×C9—49no—
67C23×Q8C3×C911yescharacteristic, complemented, Sylow
68C2×C3×C9—42no—
69C2×C3×C9—41no—
70C24×C3C22×C913yes—
71C24×C3C22×C3211yescharacteristic
72C2×C3×Q8—336no—
73C2×C3×Q8—312no—
74C2×C3×Q8—312no—
75C2×C3×Q8—336no—
76C2×C3×Q8—32no—
77C2×C3×Q8—36no—
78C2×C3×Q8—312no—
79C2×C3×Q8—312no—
80C2×C3×Q8—34no—
81C2×C3×Q8—34no—
82C2×C3×Q8—39no—
83C2×C3×Q8—33no—
84C2×C3×Q8C22⋊C913yesdirect product factor
85C2×C3×Q8C3×A411yescharacteristic
86C22×C3×C4—39no—
87C22×C3×C4—33no—
88C22×C3×C4—39no—
89C22×C3×C4—318no—
90C22×C3×C4—33no—
91C22×C3×C4—36no—
92C22×C3×C4C22×C9118yes—
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×Q813yesdirect product factor
107C4×C9—418no—
108C22×Q8—33no—
109C22×Q8—34no—
110C22×Q8—31no—
111C22×Q8C2×C3×C914yescomplemented
112C23×C4C2×C3×C913yes—
113C3×C9—41noSylow
114C23×C3—33no—
115C23×C3—31no—
116C23×C3—36no—
117C23×C3—32no—
118C23×C3—31no—
119C23×C3—33no—
120C23×C3Q8×C916yescomplemented
121C23×C3C23×C913yes—
122C23×C3C32×Q812yes—
123C23×C3C23×C3211yescharacteristic
124C3×Q8—38no—
125C3×Q8—324no—
126C3×Q8—312no—
127C3×Q8—336no—
128C3×Q8C2×C22⋊C9112yesdirect product factor
129C3×Q8C2×C3×A414yes—
130C2×C3×C4—36no—
131C2×C3×C4—318no—
132C2×C3×C4—33no—
133C2×C3×C4—39no—
134C2×C3×C4—36no—
135C2×C3×C4—36no—
136C2×C3×C4—318no—
137C2×C3×C4—318no—
138C2×C3×C4—318no—
139C2×C3×C4—36no—
140C2×C3×C4C2×C22⋊C919yes—
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×C3×C911yescharacteristic
150C2×Q8—312no—
151C2×Q8—312no—
152C2×Q8—32no—
153C2×Q8—34no—
154C2×Q8—34no—
155C2×Q8—33no—
156C2×Q8C3×C22⋊C911yescharacteristic, direct product factor
157C22×C4—33no—
158C22×C4—33no—
159C22×C4—36no—
160C22×C4C22×C3×C916yes—
161C22×C3—36no—
162C22×C3—33no—
163C22×C3—32no—
164C22×C3—31no—
165C22×C3—36no—
166C22×C3—32no—
167C22×C3—32no—
168C22×C3—32no—
169C22×C3—31no—
170C22×C3—31no—
171C22×C3—33no—
172C22×C3—33no—
173C22×C3—36no—
174C22×C3—36no—
175C22×C3C2×Q8×C913yescomplemented
176C22×C3C22×C22⋊C913yes—
177C22×C3C2×C32×Q811yescharacteristic
178C22×C3C22×C3×A411yescharacteristic
179C3×C4—36no—
180C3×C4—318no—
181C3×C4C22×C22⋊C9118yes—
182C3×C4C22×C3×A416yes—
183C32C2×Q8×A411yescharacteristic
184C9—43no—
185C23—31no—
186C23—32no—
187C23—31no—
188C23C3×Q8×C912yescomplemented
189C23C23×C3×C911yescharacteristic
190Q8—38no—
191Q8—312no—
192Q8C2×C3×C22⋊C914yesdirect product factor
193C2×C4—36no—
194C2×C4—33no—
195C2×C4—36no—
196C2×C4—36no—
197C2×C4—36no—
198C2×C4C2×C3×C22⋊C913yes—
199C2×C3—33no—
200C2×C3—36no—
201C2×C3—33no—
202C2×C3—31no—
203C2×C3—32no—
204C2×C3—31no—
205C2×C3Q8×C22⋊C916yesdirect product factor
206C2×C3C23×C22⋊C913yes—
207C2×C3C3×Q8×A412yes—
208C2×C3C23×C3×A411yescharacteristic
209C22—32no—
210C22—31no—
211C22—32no—
212C22—31no—
213C22—31no—
214C22—32no—
215C22—32no—
216C22C2×C3×Q8×C911yescharacteristic, complemented
217C22C22×C3×C22⋊C911yescharacteristic
218C4—36no—
219C4C22×C3×C22⋊C916yes—
220C3C2×Q8×C22⋊C913yesdirect product factor
221C3C2×C3×Q8×A411yescharacteristic
222C2—31no—
223C2—32no—
224C2—31no—
225C2C3×Q8×C22⋊C912yesdirect product factor
226C2C23×C3×C22⋊C911yescharacteristic
subgroupG/H
centerC22×C32C22×A4
commutatorC23C23×C3×C9
FrattiniC2×C3C23×C3×A4
FittingC23×C32×Q8C3
socle, prime radicalC24×C32C22×C3
cosocleC23×C3C23×C32
hypercenterC2×C32×Q8A4
hypocenterC22C2×C3×Q8×C9
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C22×C32 ◁ C2×C32×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C22⋊C9C2direct (N × H)
4. C2×Q8×C22⋊C9C3direct (N × H)
13. Q8×C22⋊C9C2×C3direct (N × H)
15. C2×C3×C22⋊C9Q8direct (N × H)
19. C23×C3×Q8C9semidirect
37. C3×C22⋊C9C2×Q8direct (N × H)
45. C22×C3×Q8C2×C9semidirect
63. C2×C22⋊C9C3×Q8direct (N × H)
67. C23×Q8C3×C9semidirect
84. C2×C3×Q8C22⋊C9direct (N × H)
106. C22⋊C9C2×C3×Q8direct (N × H)
111. C22×Q8C2×C3×C9semidirect
120. C23×C3Q8×C9semidirect
128. C3×Q8C2×C22⋊C9direct (N × H)
156. C2×Q8C3×C22⋊C9direct (N × H)
175. C22×C3C2×Q8×C9semidirect
188. C23C3×Q8×C9semidirect
192. Q8C2×C3×C22⋊C9direct (N × H)
205. C2×C3Q8×C22⋊C9direct (N × H)
216. C22C2×C3×Q8×C9semidirect
220. C3C2×Q8×C22⋊C9direct (N × H)
225. C2C3×Q8×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×C22⋊C92
C2×C3×C4×C22⋊C92
C23×C32×Q83
C2×Q8×C22⋊C93
C2×C3×Q8×C94