Finite Group Databasebeta

← C2×#88672 #11722539

Subgroups — C2×#88672 #11722539

Order 864.

628 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C22×#1084C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C212yesmaximal, complemented
3C2×#6198C211yesmaximal, characteristic, complemented
4#88672C218yesmaximal, direct product factor
5C2×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
6C2×#6196C211yesmaximal, characteristic, complemented
7C2×#3804—32nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊D3)⋊C3C2212yes—
10C2×#1084C2211yescharacteristic
11C2×#1084C2211yescharacteristic
12C2×#1084C2214yescomplemented
13#638⋊C3C2218yescomplemented
14C2×(C3⋊Dic3)⋊C3C2212yes—
15#6198C2214yescomplemented
16((C3⋊D3)⋊C3)⋊C4C2218yescomplemented
17#6196C2214yescomplemented
18C2×#638D312yescomplemented
19C2×C3×#67—32no—
20C22×C3×Dic3—32no—
21C22×C3×Dic3—61no—
22C2×#627—32no—
23C2×#626—32no—
24C2×#624—32no—
25#3804—316no—
26C2×D3×Dic3—32no—
27C22×H(3,𝔽3)—22no—
28C22×H(3,𝔽3)C2312yes—
29C22×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊D3)⋊C3—24no—
31C2×(C3⋊D3)⋊C3C2314yes—
32#1084C2312yes—
33#1084C2312yes—
34(C3⋊Dic3)⋊C3C2314yes—
35C2×#237—92no—
36C2×#235—91no—
37C23×C32—21no—
38C23×C32C2×D312yes—
39C22×C3⋊D3C2×D312yes—
40C22×C3×D3—32no—
41#638C2×D318yescomplemented
42C2×C3⋊Dic3C2×D312yes—
43C3×#67—38no—
44C2×C3×Dic3—32no—
45C2×C3×Dic3—32no—
46C2×C3×Dic3—32no—
47C2×C3×Dic3—38no—
48C2×C3×Dic3—61no—
49C2×C3×Dic3—61no—
50C2×C3×Dic3—64no—
51#627—38no—
52#626—38no—
53#624—38no—
54D3×Dic3—38no—
55C2×H(3,𝔽3)—22no—
56C2×H(3,𝔽3)#4112yescomplemented
57C2×H(3,𝔽3)C2411yescharacteristic
58(C3⋊D3)⋊C3—24no—
59C2×C3×D4—92no—
60C22×C3×C4—91no—
61C2×#67—32no—
62C2×#67—92no—
63C2×#67—92no—
64C22×Dic3—32no—
65C22×Dic3—61no—
66C22×Dic3—92no—
67#237—916no—
68#235—98no—
69C2×D12—91no—
70C2×C4×D3—92no—
71C2×C4×D3—92no—
72C2×Dic6—91no—
73C2×Dic6—92no—
74C22×C32—24no—
75C22×C32—24no—
76C22×C32—22no—
77C22×C32—21no—
78C22×C32C22×D314yes—
79C22×C32C22×D312yes—
80C2×C3⋊D3—24no—
81C2×C3⋊D3C22×D314yes—
82C2×C3×D3—34no—
83C2×C3×D3—64no—
84C3⋊Dic3C22×D314yes—
85C3×Dic3—34no—
86C3×Dic3—34no—
87C3×Dic3—34no—
88C3×Dic3—62no—
89C3×Dic3—62no—
90C2×#41—271noSylow
91H(3,𝔽3)C2×#4111yescharacteristic, complemented, Sylow
92C23×C3—32no—
93C23×C3—61no—
94C23×C3—92no—
95C23×C3D3211yescharacteristic
96C22×D3—32no—
97C22×D3—92no—
98C3×D4—98no—
99C2×C3×C4—92no—
100C2×C3×C4—91no—
101C2×C3×C4—91no—
102C2×C3×C4—94no—
103#67—38no—
104#67—98no—
105#67—98no—
106C2×Dic3—32no—
107C2×Dic3—32no—
108C2×Dic3—38no—
109C2×Dic3—32no—
110C2×Dic3—64no—
111C2×Dic3—61no—
112C2×Dic3—61no—
113C2×Dic3—92no—
114C2×Dic3—98no—
115D12—94no—
116C4×D3—98no—
117C4×D3—98no—
118Dic6—94no—
119Dic6—98no—
120C2×C32—24no—
121C2×C32—24no—
122C2×C32—21no—
123C2×C32—22no—
124C2×C32#23714yescomplemented
125C2×C32C23×D312yes—
126C3⋊D3—24no—
127C3×D3—64no—
128#41—278no—
129C2×Q8—271no—
130C2×D4—92no—
131C2×D4—271no—
132C22×C4—91no—
133C22×C4—272no—
134C22×C3—22no—
135C22×C3—34no—
136C22×C3—32no—
137C22×C3—64no—
138C22×C3—64no—
139C22×C3—62no—
140C22×C3—61no—
141C22×C3—94no—
142C22×C3—184no—
143C22×C3C2×D3212yes—
144C22×C3C2×D3211yescharacteristic
145C2×D3—34no—
146C2×D3—64no—
147C2×D3—94no—
148C2×D3—184no—
149C3×C4—94no—
150C3×C4—92no—
151C3×C4—92no—
152Dic3—34no—
153Dic3—34no—
154Dic3—34no—
155Dic3—62no—
156Dic3—62no—
157Dic3—94no—
158C32—21no—
159C32C2×#23712yescomplemented
160C23—92no—
161C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
162Q8—274no—
163D4—98no—
164D4—274no—
165C2×C4—94no—
166C2×C4—91no—
167C2×C4—91no—
168C2×C4—92no—
169C2×C4—278no—
170C2×C3—22no—
171C2×C3—32no—
172C2×C3—34no—
173C2×C3—64no—
174C2×C3—64no—
175C2×C3—61no—
176C2×C3—62no—
177C2×C3—184no—
178C2×C3#380512yes—
179C2×C3C22×D3211yescharacteristic
180D3—64no—
181D3—184no—
182C22—22no—
183C22—94no—
184C22—184no—
185C22C2×((C3⋊D3)⋊C3)⋊C212yes—
186C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
187C4—92no—
188C4—92no—
189C4—94no—
190C3—32no—
191C3—61no—
192C3C2×#380511yescharacteristic
193C2—22no—
194C2—184no—
195C2#8867212yesdirect product factor
196C2C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C24
FrattiniC2×C3C22×D32
FittingC23×H(3,𝔽3)C22
socleC22×C3C2×D32
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#1084C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×#6198C2semidirect
4. #88672C2direct (N × H)
5. C2×((C3⋊D3)⋊C3)⋊C4C2semidirect
6. C2×#6196C2semidirect
12. C2×#1084C22semidirect
13. #638⋊C3C22semidirect
15. #6198C22semidirect
16. ((C3⋊D3)⋊C3)⋊C4C22semidirect
17. #6196C22semidirect
18. C2×#638D3semidirect
41. #638C2×D3semidirect
56. C2×H(3,𝔽3)#41semidirect
91. H(3,𝔽3)C2×#41semidirect
124. C2×C32#237semidirect
159. C32C2×#237semidirect
195. C2#88672direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#10842
C2×#638⋊C32
C2×#61982
#886722
C2×((C3⋊D3)⋊C3)⋊C42
C2×#61962
C2×#38043