Finite Group Databasebeta

← C2×#10593620 #11922863

Subgroups — C2×#10593620 #11922863

Order 1152.

728 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 113 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×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊#859C212yesmaximal
3#10593620C214yesmaximal, direct product factor
4C2×#5386—33nomaximal
5C2×#5020—31nomaximal
6C22×#634—41nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C3×C4×SL(2,𝔽3)C2212yes—
9C3⋊#859C2214yes—
10C2×C3×C4×Q8D311yescharacteristic
11C2×#869—31no—
12C2×C4×SL(2,𝔽3)D313yes—
13C2×#859—36no—
14#5386—312no—
15C2×#835—32no—
16#5020—34no—
17#5020—34no—
18C2×#808—32no—
19C2×C4⋊(C3⋊C8)—31no—
20C22×C32×C4—41no—
21C22×C3⋊Dic3—42no—
22C2×#634—44no—
23C2×#634—48no—
24C2×C3×SL(2,𝔽3)Q812yescomplemented
25C2×C3×SL(2,𝔽3)C2311yescharacteristic
26C2×#475—91noSylow
27C22×C3×Q8C2×D311yescharacteristic
28C22×Dic6—61no—
29C22×Dic6—123no—
30C22×SL(2,𝔽3)C2×D313yes—
31C3×C4×Q8—32no—
32C3×C4×Q8C2×D312yes—
33C2×C3×C4⋊C4—31no—
34C2×C3×C42—31no—
35C2×C4⋊Dic3—32no—
36C2×C4⋊Dic3—61no—
37C22×C3⋊C8—32no—
38#869—34no—
39#869—62no—
40C4×SL(2,𝔽3)C2×D316yes—
41#859—312no—
42#835—34no—
43#835—34no—
44#808—38no—
45C4⋊(C3⋊C8)—34no—
46C23×C32—41no—
47C2×C32×C4—44no—
48C2×C32×C4—42no—
49C2×C3⋊Dic3—44no—
50C2×C3⋊Dic3—48no—
51#634—416no—
52C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
53C2×C4⋊Q8—91no—
54C2×C4×Q8C3⋊D311yescharacteristic
55#475—94no—
56#475—94no—
57C2×#106—92no—
58C2×C4⋊C8—91no—
59C2×#103—92no—
60C2×C3×Q8—31no—
61C2×C3×Q8—32no—
62C2×C3×Q8Dic612yescomplemented
63C2×C3×Q8C22×D311yescharacteristic
64C22×C3×C4—31no—
65C22×C3×C4—31no—
66C22×C3×C4—43no—
67C22×C3×C4S411yescharacteristic
68C22×Dic3—62no—
69C22×Dic3—126no—
70C2×Dic6—64no—
71C2×Dic6—1212no—
72C2×Dic6—1224no—
73C2×Dic6—124no—
74C2×SL(2,𝔽3)Dic616yescomplemented
75C2×SL(2,𝔽3)C22×D313yes—
76C3×C4⋊C4—32no—
77C3×C4⋊C4—61no—
78C3×C42—32no—
79C3×C42—32no—
80C4⋊Dic3—34no—
81C4⋊Dic3—64no—
82C4⋊Dic3—62no—
83C2×C3⋊C8—34no—
84C2×C3⋊C8—64no—
85C22×C32—42no—
86C22×C32—42no—
87C22×C32—42no—
88C22×C32—41no—
89C32×C4—44no—
90C3⋊Dic3—48no—
91C22×Q8—181no—
92C22×Q8C2×C3⋊D311yescharacteristic
93C22×C8—92no—
94C4⋊Q8—94no—
95C4⋊Q8—182no—
96C4×Q8—32no—
97C4×Q8C2×C3⋊D312yes—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—92no—
100C2×C4⋊C4—181no—
101C2×C42—31no—
102#106—98no—
103C4⋊C8—94no—
104#103—94no—
105#103—94no—
106C23×C3—43no—
107C23×C3C2×S411yescharacteristic
108C3×Q8—32no—
109C3×Q8—31no—
110C3×Q8—61no—
111C3×Q8C2×Dic611yescharacteristic, complemented
112C2×C3×C4—32no—
113C2×C3×C4—32no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—32no—
117C2×C3×C4—46no—
118C2×C3×C4—412no—
119C2×C3×C4—62no—
120C2×C3×C4GL(2,𝔽3)14yes—
121C2×C3×C4C2×S412yes—
122C2×Dic3—64no—
123C2×Dic3—1212no—
124C2×Dic3—1224no—
125C2×Dic3—124no—
126Dic6—1248no—
127Dic6—128no—
128SL(2,𝔽3)C2×Dic613yescomplemented
129C3⋊C8—64no—
130C2×C32—44no—
131C2×C32—41no—
132C2×C32—41no—
133C2×C32—41no—
134C2×Q8—31no—
135C2×Q8—32no—
136C2×Q8—184no—
137C2×Q8—364no—
138C2×Q8#63412yescomplemented
139C2×Q8C22×C3⋊D311yescharacteristic
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—182no—
143C22×C4C3⋊S411yescharacteristic
144C2×C8—94no—
145C2×C8—184no—
146C4⋊C4—32no—
147C4⋊C4—61no—
148C4⋊C4—94no—
149C4⋊C4—182no—
150C4⋊C4—184no—
151C42—32no—
152C42—32no—
153C22×C3—46no—
154C22×C3—46no—
155C22×C3—46no—
156C22×C3—43no—
157C22×C3A4⋊Q812yes—
158C22×C3C2×GL(2,𝔽3)12yes—
159C22×C3#98412yes—
160C22×C3C22×S411yescharacteristic
161C3×C4—31no—
162C3×C4—31no—
163C3×C4—32no—
164C3×C4—412no—
165C3×C4—62no—
166C3×C4C2×GL(2,𝔽3)14yes—
167Dic3—1224no—
168Dic3—124no—
169C32—41noSylow
170C23C2×C3⋊S411yescharacteristic
171Q8—32no—
172Q8—31no—
173Q8—61no—
174Q8—368no—
175Q8C2×#63411yescharacteristic, complemented
176C2×C4—32no—
177C2×C4—32no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—32no—
181C2×C4—62no—
182C2×C4—184no—
183C2×C4—364no—
184C2×C4C3⋊GL(2,𝔽3)14yes—
185C2×C4C2×C3⋊S412yes—
186C8—184no—
187C2×C3—412no—
188C2×C3—43no—
189C2×C3—43no—
190C2×C3—43no—
191C2×C3#538614yescomplemented
192C2×C3C2×A4⋊Q811yescharacteristic
193C2×C3C22×GL(2,𝔽3)11yescharacteristic
194C2×C3C2×#98411yescharacteristic
195C22#6425912yes—
196C22C2×C3⋊GL(2,𝔽3)12yes—
197C22#6426512yes—
198C22C22×C3⋊S411yescharacteristic
199C4—31no—
200C4—31no—
201C4—32no—
202C4—62no—
203C4—364no—
204C4C2×C3⋊GL(2,𝔽3)14yes—
205C3—43no—
206C3C2×#538611yescharacteristic, complemented
207C2#1059362014yesdirect product factor
208C2C2×#6425911yescharacteristic
209C2C22×C3⋊GL(2,𝔽3)11yescharacteristic
210C2C2×#6426511yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #10593620C2direct (N × H)
24. C2×C3×SL(2,𝔽3)Q8semidirect
52. C3×SL(2,𝔽3)C2×Q8semidirect
62. C2×C3×Q8Dic6semidirect
74. C2×SL(2,𝔽3)Dic6semidirect
111. C3×Q8C2×Dic6semidirect
128. SL(2,𝔽3)C2×Dic6semidirect
138. C2×Q8#634semidirect
175. Q8C2×#634semidirect
191. C2×C3#5386semidirect
206. C3C2×#5386semidirect
207. C2#10593620direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C2×C3⋊#8592
#105936202
C2×#53863
C2×#50203
C22×#6344