Finite Group Databasebeta

← C22×C22⋊(Q8⋊C9) #11919893

Subgroups — C22×C22⋊(Q8⋊C9) #11919893

Order 1152.

2221 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 2221 conjugacy classes of subgroups — too many to draw. Only its 119 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×C22⋊(Q8⋊C9)C213yesmaximal, direct product factor
2C24×C3×Q8C311yesmaximal, characteristic
3C23×C22⋊C9—41nomaximal
4C22⋊(Q8⋊C9)C2211yescharacteristic, direct product factor
5C22×Q8⋊C9—44nomaximal
6C23×C3×Q8—312no—
7C23×C3×Q8—34no—
8C23×C3×Q8—33no—
9C23×C3×Q8C2×C313yes—
10C24×C3×C4—31no—
11C22×C22⋊C9—44no—
12C22×C22⋊C9—43no—
13C2×Q8⋊C9—412no—
14C24×Q8C911yescharacteristic, complemented, Sylow
15C25×C3A411yescharacteristic
16C22×C3×Q8—312no—
17C22×C3×Q8—324no—
18C22×C3×Q8—336no—
19C22×C3×Q8—336no—
20C22×C3×Q8—336no—
21C22×C3×Q8—38no—
22C22×C3×Q8—32no—
23C22×C3×Q8—34no—
24C22×C3×Q8—312no—
25C22×C3×Q8—312no—
26C22×C3×Q8—33no—
27C22×C3×Q8A414yes—
28C22×C3×Q8C22×C311yescharacteristic
29C23×C3×C4—318no—
30C23×C3×C4—36no—
31C23×C3×C4—33no—
32C23×C3×C4—33no—
33C23×C9—161no—
34C2×C22⋊C9—46no—
35C2×C22⋊C9—41no—
36Q8⋊C9—44no—
37C23×Q8—312no—
38C23×Q8—34no—
39C23×Q8—33no—
40C23×Q8C2×C913yescomplemented
41C24×C4—31no—
42C24×C3—34no—
43C24×C3—33no—
44C24×C3—31no—
45C24×C3SL(2,𝔽3)14yes—
46C24×C3C2×A413yes—
47C2×C3×Q8—336no—
48C2×C3×Q8—336no—
49C2×C3×Q8—372no—
50C2×C3×Q8—372no—
51C2×C3×Q8—324no—
52C2×C3×Q8—336no—
53C2×C3×Q8—312no—
54C2×C3×Q8—312no—
55C2×C3×Q8—34no—
56C2×C3×Q8—312no—
57C2×C3×Q8C2×A4112yes—
58C22×C3×C4—318no—
59C22×C3×C4—318no—
60C22×C3×C4—324no—
61C22×C3×C4—318no—
62C22×C3×C4—336no—
63C22×C3×C4—318no—
64C22×C3×C4—33no—
65C22×C3×C4—31no—
66C22×C3×C4—34no—
67C22×C9—164no—
68C22×C9—163no—
69C22⋊C9—41no—
70C25C22⋊C911yescharacteristic
71C22×Q8—312no—
72C22×Q8—324no—
73C22×Q8—336no—
74C22×Q8—336no—
75C22×Q8—336no—
76C22×Q8—38no—
77C22×Q8—32no—
78C22×Q8—34no—
79C22×Q8—312no—
80C22×Q8—312no—
81C22×Q8—33no—
82C22×Q8C22⋊C914yescomplemented
83C22×Q8C22×C911yescharacteristic, complemented
84C23×C4—318no—
85C23×C4—36no—
86C23×C4—33no—
87C23×C4—33no—
88C23×C3—33no—
89C23×C3—312no—
90C23×C3—36no—
91C23×C3—32no—
92C23×C3—312no—
93C23×C3—34no—
94C23×C3—34no—
95C23×C3—33no—
96C23×C3—33no—
97C23×C3C2×SL(2,𝔽3)16yes—
98C23×C3C22×A411yescharacteristic
99C23×C3C22⋊A411yescharacteristic
100C3×Q8—312no—
101C3×Q8—324no—
102C3×Q8—38no—
103C3×Q8—336no—
104C3×Q8—34no—
105C3×Q8C22×A414yes—
106C2×C3×C4—336no—
107C2×C3×C4—318no—
108C2×C3×C4—336no—
109C2×C3×C4—36no—
110C2×C3×C4—312no—
111C2×C3×C4—312no—
112C2×C9—166no—
113C2×C9—161no—
114C24—33no—
115C24—34no—
116C24—31no—
117C24Q8⋊C914yescomplemented
118C24C2×C22⋊C913yes—
119C2×Q8—336no—
120C2×Q8—336no—
121C2×Q8—372no—
122C2×Q8—372no—
123C2×Q8—324no—
124C2×Q8—312no—
125C2×Q8—336no—
126C2×Q8—34no—
127C2×Q8—312no—
128C2×Q8—312no—
129C2×Q8C2×C22⋊C9112yescomplemented
130C22×C4—318no—
131C22×C4—318no—
132C22×C4—324no—
133C22×C4—318no—
134C22×C4—336no—
135C22×C4—318no—
136C22×C4—31no—
137C22×C4—33no—
138C22×C4—34no—
139C22×C3—36no—
140C22×C3—31no—
141C22×C3—36no—
142C22×C3—38no—
143C22×C3—312no—
144C22×C3—36no—
145C22×C3—36no—
146C22×C3—33no—
147C22×C3—31no—
148C22×C3C22×SL(2,𝔽3)11yescharacteristic
149C22×C3C22⋊SL(2,𝔽3)14yes—
150C22×C3C2×C22⋊A413yes—
151C3×C4—312no—
152C3×C4—34no—
153C9—161noSylow
154C23—33no—
155C23—312no—
156C23—36no—
157C23—32no—
158C23—312no—
159C23—33no—
160C23—33no—
161C23—34no—
162C23—34no—
163C23C2×Q8⋊C916yescomplemented
164C23C22×C22⋊C911yescharacteristic
165C23C22⋊(C22⋊C9)11yescharacteristic
166Q8—312no—
167Q8—34no—
168Q8—336no—
169Q8—324no—
170Q8—38no—
171Q8C22×C22⋊C914yescomplemented
172C2×C4—36no—
173C2×C4—336no—
174C2×C4—318no—
175C2×C4—336no—
176C2×C4—312no—
177C2×C4—312no—
178C2×C3—36no—
179C2×C3—31no—
180C2×C3—31no—
181C2×C3C2×C22⋊SL(2,𝔽3)16yes—
182C2×C3C22×C22⋊A411yescharacteristic
183C22—36no—
184C22—31no—
185C22—36no—
186C22—38no—
187C22—31no—
188C22—33no—
189C22—36no—
190C22—312no—
191C22—36no—
192C22C22×Q8⋊C911yescharacteristic, complemented
193C22C22⋊(Q8⋊C9)14yesdirect product factor
194C22C2×C22⋊(C22⋊C9)13yes—
195C4—34no—
196C4—312no—
197C3C22×C22⋊SL(2,𝔽3)11yescharacteristic
198C2—31no—
199C2—36no—
200C2—31no—
201C2C2×C22⋊(Q8⋊C9)16yesdirect product factor
202C2C22×C22⋊(C22⋊C9)11yescharacteristic
subgroupG/H
center, hypercenterC23×C3C22⋊A4
commutator, hypocenterC22×Q8C22×C9
FrattiniC2×C3C22×C22⋊A4
FittingC24×C3×Q8C3
socle, prime radicalC25×C3A4
cosocleC22×C3×Q8C22×C3
LCSG ▷ C22×Q8
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(Q8⋊C9)C2direct (N × H)
4. C22⋊(Q8⋊C9)C22direct (N × H)
14. C24×Q8C9semidirect
40. C23×Q8C2×C9semidirect
82. C22×Q8C22⋊C9semidirect
83. C22×Q8C22×C9semidirect
117. C24Q8⋊C9semidirect
129. C2×Q8C2×C22⋊C9semidirect
163. C23C2×Q8⋊C9semidirect
171. Q8C22×C22⋊C9semidirect
192. C22C22×Q8⋊C9semidirect
193. C22C22⋊(Q8⋊C9)direct (N × H)
201. C2C2×C22⋊(Q8⋊C9)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊(Q8⋊C9)2
C24×C3×Q83
C23×C22⋊C94
C22×Q8⋊C94