Finite Group Databasebeta

← C2×C4⋊(C3⋊#226) #11922869

Subgroups — C2×C4⋊(C3⋊#226) #11922869

Order 1152.

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

normal subgroups only

This group has 696 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
1C22×C3⋊#226C212yesmaximal
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C4⋊(C3⋊#226)C214yesmaximal, direct product factor
4C2×C4⋊#226—33nomaximal
5C2×#5031—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×C3⋊#226—24no—
8C2×C3⋊#226C2214yes—
9C2×C4⋊(C3⋊Dic3)—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C22×#226—36no—
12C2×C3×C4×Q8D311yescharacteristic
13C22×#216—32no—
14C2×#869—31no—
15C2×C4×SL(2,𝔽3)D313yes—
16C4⋊#226—312no—
17#5031—34no—
18#5031—34no—
19C2×#810—32no—
20C2×C4⋊(C3⋊C8)—31no—
21C22×C32×C4—41no—
22C22×C3⋊Dic3—42no—
23C2×C3×SL(2,𝔽3)D412yes—
24C2×C3×SL(2,𝔽3)C2311yescharacteristic
25C3⋊#226—24no—
26C4⋊(C3⋊Dic3)—44no—
27C2×#462—91noSylow
28C22×C3×Q8C2×D311yescharacteristic
29C22×Dic6—32no—
30C22×SL(2,𝔽3)C2×D313yes—
31C2×#226—312no—
32C2×#226—612no—
33C3×C4×Q8—32no—
34C3×C4×Q8C2×D312yes—
35C2×C3×C4⋊C4—31no—
36C2×C3×C42—31no—
37C2×#216—34no—
38C2×#216—34no—
39C2×#216—64no—
40C2×#216—64no—
41C2×C4⋊Dic3—61no—
42C2×C4⋊Dic3—61no—
43C2×C4⋊Dic3—123no—
44C22×C3⋊C8—32no—
45#869—34no—
46#869—62no—
47C4×SL(2,𝔽3)C2×D316yes—
48#810—38no—
49C4⋊(C3⋊C8)—34no—
50C23×C32—41no—
51C2×C32×C4—44no—
52C2×C32×C4—42no—
53C2×C3⋊Dic3—44no—
54C2×C3⋊Dic3—84no—
55C3×SL(2,𝔽3)C2×D411yescharacteristic
56C22×Q16—92no—
57C2×C4⋊Q8—91no—
58C2×C4×Q8C3⋊D311yescharacteristic
59#462—94no—
60#462—94no—
61C2×C4⋊C8—91no—
62C2×#103—92no—
63C2×C3×Q8—31no—
64C2×C3×Q8—32no—
65C2×C3×Q8D1212yes—
66C2×C3×Q8C22×D311yescharacteristic
67C22×C3×C4—31no—
68C22×C3×C4—31no—
69C22×C3×C4—43no—
70C22×C3×C4S411yescharacteristic
71C22×Dic3—62no—
72C22×Dic3—126no—
73C2×Dic6—34no—
74C2×Dic6—64no—
75C2×Dic6—64no—
76C2×Dic6—62no—
77C2×SL(2,𝔽3)D1216yes—
78C2×SL(2,𝔽3)C22×D313yes—
79#226—612no—
80C3×C4⋊C4—32no—
81C3×C4⋊C4—61no—
82C3×C42—32no—
83C3×C42—32no—
84#216—68no—
85#216—64no—
86#216—64no—
87C4⋊Dic3—64no—
88C4⋊Dic3—64no—
89C4⋊Dic3—1212no—
90C2×C3⋊C8—34no—
91C2×C3⋊C8—64no—
92C22×C32—42no—
93C22×C32—42no—
94C22×C32—42no—
95C22×C32—41no—
96C32×C4—44no—
97C3⋊Dic3—84no—
98C22×Q8—92no—
99C22×Q8C2×C3⋊D311yescharacteristic
100C2×Q16—94no—
101C2×Q16—94no—
102C2×Q16—184no—
103C2×Q16—184no—
104C22×C8—92no—
105C4⋊Q8—94no—
106C4⋊Q8—182no—
107C4×Q8—32no—
108C4×Q8C2×C3⋊D312yes—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—181no—
111C2×C4⋊C4—181no—
112C2×C42—31no—
113C4⋊C8—94no—
114#103—98no—
115C23×C3—43no—
116C23×C3C2×S411yescharacteristic
117C3×Q8—32no—
118C3×Q8—31no—
119C3×Q8—61no—
120C3×Q8C2×D1211yescharacteristic
121C2×C3×C4—32no—
122C2×C3×C4—32no—
123C2×C3×C4—31no—
124C2×C3×C4—31no—
125C2×C3×C4—32no—
126C2×C3×C4—46no—
127C2×C3×C4—412no—
128C2×C3×C4—62no—
129C2×C3×C4#22614yescomplemented
130C2×C3×C4C2×S412yes—
131C2×Dic3—64no—
132C2×Dic3—1212no—
133C2×Dic3—124no—
134C2×Dic3—2412no—
135Dic6—64no—
136Dic6—64no—
137Dic6—124no—
138SL(2,𝔽3)C2×D1213yes—
139C3⋊C8—64no—
140C2×C32—44no—
141C2×C32—41no—
142C2×C32—41no—
143C2×C32—41no—
144C2×Q8—31no—
145C2×Q8—32no—
146C2×Q8—94no—
147C2×Q8—182no—
148C2×Q8—184no—
149C2×Q8—184no—
150C2×Q8C4⋊(C3⋊D3)12yes—
151C2×Q8C22×C3⋊D311yescharacteristic
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—182no—
155C22×C4C3⋊S411yescharacteristic
156Q16—188no—
157Q16—184no—
158Q16—184no—
159C2×C8—94no—
160C2×C8—184no—
161C4⋊C4—32no—
162C4⋊C4—61no—
163C4⋊C4—184no—
164C4⋊C4—184no—
165C42—32no—
166C42—32no—
167C22×C3—46no—
168C22×C3—46no—
169C22×C3—46no—
170C22×C3—43no—
171C22×C3C4⋊S412yes—
172C22×C3C2×#22612yes—
173C22×C3#98412yes—
174C22×C3C22×S411yescharacteristic
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—32no—
178C3×C4—412no—
179C3×C4—62no—
180C3×C4C2×#22614yescomplemented
181Dic3—124no—
182Dic3—2412no—
183C32—41noSylow
184C23C2×C3⋊S411yescharacteristic
185Q8—32no—
186Q8—31no—
187Q8—61no—
188Q8—184no—
189Q8—184no—
190Q8—364no—
191Q8C2×C4⋊(C3⋊D3)11yescharacteristic
192C2×C4—32no—
193C2×C4—32no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—32no—
197C2×C4—62no—
198C2×C4—184no—
199C2×C4—364no—
200C2×C4C3⋊#22614yescomplemented
201C2×C4C2×C3⋊S412yes—
202C8—184no—
203C2×C3—412no—
204C2×C3—43no—
205C2×C3—43no—
206C2×C3—43no—
207C2×C3C4⋊#22614yescomplemented
208C2×C3C2×C4⋊S411yescharacteristic
209C2×C3C22×#22611yescharacteristic
210C2×C3C2×#98411yescharacteristic
211C22C4⋊(C3⋊S4)12yes—
212C22C2×C3⋊#22612yes—
213C22#6426512yes—
214C22C22×C3⋊S411yescharacteristic
215C4—31no—
216C4—31no—
217C4—32no—
218C4—62no—
219C4—364no—
220C4C2×C3⋊#22614yescomplemented
221C3—43no—
222C3C2×C4⋊#22611yescharacteristic, complemented
223C2C4⋊(C3⋊#226)14yesdirect product factor
224C2C2×C4⋊(C3⋊S4)11yescharacteristic
225C2C22×C3⋊#22611yescharacteristic
226C2C2×#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×D4
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. C4⋊(C3⋊#226)C2direct (N × H)
129. C2×C3×C4#226semidirect
180. C3×C4C2×#226semidirect
200. C2×C4C3⋊#226semidirect
207. C2×C3C4⋊#226semidirect
220. C4C2×C3⋊#226semidirect
222. C3C2×C4⋊#226semidirect
223. C2C4⋊(C3⋊#226)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊#2262
C2×C3×C4×SL(2,𝔽3)2
C4⋊(C3⋊#226)2
C2×C4⋊#2263
C2×#50313
C2×C4⋊(C3⋊Dic3)4