Finite Group Databasebeta

← C2×#10593625 #11922870

Subgroups — C2×#10593625 #11922870

Order 1152.

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

normal subgroups only

This group has 600 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#10593625C214yesmaximal, direct product factor
4C2×#5391—33nomaximal
5C2×#5021—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×C4⋊(C3⋊Dic3)—41nomaximal
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#5391—312no—
15C2×#835—32no—
16#5021—34no—
17#5021—34no—
18C2×#807—32no—
19C2×C4⋊(C3⋊C8)—31no—
20C22×C32×C4—41no—
21C22×C3⋊Dic3—42no—
22C2×C3×SL(2,𝔽3)Q812yescomplemented
23C2×C3×SL(2,𝔽3)C2311yescharacteristic
24C4⋊(C3⋊Dic3)—44no—
25C2×#477—91noSylow
26C22×C3×Q8C2×D311yescharacteristic
27C22×Dic6—61no—
28C22×SL(2,𝔽3)C2×D313yes—
29C3×C4×Q8—32no—
30C3×C4×Q8C2×D312yes—
31C2×C3×C4⋊C4—31no—
32C2×C3×C42—31no—
33C2×C4⋊Dic3—32no—
34C2×C4⋊Dic3—61no—
35C2×C4⋊Dic3—123no—
36C22×C3⋊C8—32no—
37#869—34no—
38#869—62no—
39C4×SL(2,𝔽3)C2×D316yes—
40#859—312no—
41#835—34no—
42#835—34no—
43#807—38no—
44C4⋊(C3⋊C8)—34no—
45C23×C32—41no—
46C2×C32×C4—44no—
47C2×C32×C4—42no—
48C2×C3⋊Dic3—44no—
49C2×C3⋊Dic3—84no—
50C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
51C2×C4⋊Q8—91no—
52C2×C4×Q8C3⋊D311yescharacteristic
53#477—94no—
54#477—94no—
55C2×#107—92no—
56C2×C4⋊C8—91no—
57C2×#103—92no—
58C2×C3×Q8—31no—
59C2×C3×Q8—32no—
60C2×C3×Q8Dic612yes—
61C2×C3×Q8C22×D311yescharacteristic
62C22×C3×C4—31no—
63C22×C3×C4—31no—
64C22×C3×C4—43no—
65C22×C3×C4S411yescharacteristic
66C22×Dic3—62no—
67C22×Dic3—126no—
68C2×Dic6—64no—
69C2×Dic6—124no—
70C2×SL(2,𝔽3)Dic616yescomplemented
71C2×SL(2,𝔽3)C22×D313yes—
72C3×C4⋊C4—32no—
73C3×C4⋊C4—61no—
74C3×C42—32no—
75C3×C42—32no—
76C4⋊Dic3—34no—
77C4⋊Dic3—62no—
78C4⋊Dic3—64no—
79C4⋊Dic3—1212no—
80C2×C3⋊C8—34no—
81C2×C3⋊C8—64no—
82C22×C32—42no—
83C22×C32—42no—
84C22×C32—42no—
85C22×C32—41no—
86C32×C4—44no—
87C3⋊Dic3—84no—
88C22×Q8—181no—
89C22×Q8C2×C3⋊D311yescharacteristic
90C22×C8—92no—
91C4⋊Q8—94no—
92C4⋊Q8—182no—
93C4×Q8—32no—
94C4×Q8C2×C3⋊D312yes—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—92no—
97C2×C4⋊C4—181no—
98C2×C42—31no—
99#107—98no—
100C4⋊C8—94no—
101#103—94no—
102#103—94no—
103C23×C3—43no—
104C23×C3C2×S411yescharacteristic
105C3×Q8—32no—
106C3×Q8—31no—
107C3×Q8—61no—
108C3×Q8C2×Dic611yescharacteristic
109C2×C3×C4—32no—
110C2×C3×C4—32no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—32no—
114C2×C3×C4—46no—
115C2×C3×C4—412no—
116C2×C3×C4—62no—
117C2×C3×C4#22614yes—
118C2×C3×C4C2×S412yes—
119C2×Dic3—64no—
120C2×Dic3—1212no—
121C2×Dic3—124no—
122C2×Dic3—2412no—
123Dic6—128no—
124SL(2,𝔽3)C2×Dic613yescomplemented
125C3⋊C8—64no—
126C2×C32—44no—
127C2×C32—41no—
128C2×C32—41no—
129C2×C32—41no—
130C2×Q8—31no—
131C2×Q8—32no—
132C2×Q8—184no—
133C2×Q8—364no—
134C2×Q8#63412yes—
135C2×Q8C22×C3⋊D311yescharacteristic
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—182no—
139C22×C4C3⋊S411yescharacteristic
140C2×C8—94no—
141C2×C8—184no—
142C4⋊C4—32no—
143C4⋊C4—61no—
144C4⋊C4—94no—
145C4⋊C4—182no—
146C4⋊C4—184no—
147C42—32no—
148C42—32no—
149C22×C3—46no—
150C22×C3—46no—
151C22×C3—46no—
152C22×C3—43no—
153C22×C3A4⋊Q812yes—
154C22×C3C2×#22612yes—
155C22×C3#98612yes—
156C22×C3C22×S411yescharacteristic
157C3×C4—31no—
158C3×C4—31no—
159C3×C4—32no—
160C3×C4—412no—
161C3×C4—62no—
162C3×C4C2×#22614yes—
163Dic3—124no—
164Dic3—2412no—
165C32—41noSylow
166C23C2×C3⋊S411yescharacteristic
167Q8—32no—
168Q8—31no—
169Q8—61no—
170Q8—368no—
171Q8C2×#63411yescharacteristic
172C2×C4—32no—
173C2×C4—32no—
174C2×C4—31no—
175C2×C4—31no—
176C2×C4—32no—
177C2×C4—62no—
178C2×C4—184no—
179C2×C4—364no—
180C2×C4C3⋊#22614yes—
181C2×C4C2×C3⋊S412yes—
182C8—184no—
183C2×C3—412no—
184C2×C3—43no—
185C2×C3—43no—
186C2×C3—43no—
187C2×C3#539114yescomplemented
188C2×C3C2×A4⋊Q811yescharacteristic
189C2×C3C22×#22611yescharacteristic
190C2×C3C2×#98611yescharacteristic
191C22#6425912yes—
192C22C2×C3⋊#22612yes—
193C22#6426712yes—
194C22C22×C3⋊S411yescharacteristic
195C4—31no—
196C4—31no—
197C4—32no—
198C4—62no—
199C4—364no—
200C4C2×C3⋊#22614yes—
201C3—43no—
202C3C2×#539111yescharacteristic, complemented
203C2#1059362514yesdirect product factor
204C2C2×#6425911yescharacteristic
205C2C22×C3⋊#22611yescharacteristic
206C2C2×#6426711yescharacteristic
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. #10593625C2direct (N × H)
22. C2×C3×SL(2,𝔽3)Q8semidirect
50. C3×SL(2,𝔽3)C2×Q8semidirect
70. C2×SL(2,𝔽3)Dic6semidirect
124. SL(2,𝔽3)C2×Dic6semidirect
187. C2×C3#5391semidirect
202. C3C2×#5391semidirect
203. C2#10593625direct (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
#105936252
C2×#53913
C2×#50213
C2×C4⋊(C3⋊Dic3)4