Finite Group Databasebeta

← C2×C4×C3⋊#226 #11922864

Subgroups — C2×C4×C3⋊#226 #11922864

Order 1152.

736 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 121 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⋊#226C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×C3⋊#859C211yesmaximal, characteristic
4C4×C3⋊#226C214yesmaximal, direct product factor
5C2×C4×#226—33nomaximal
6C2×C4×#216—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊#226C414yesdirect product factor
9C2×C3⋊#226C2212yes—
10C2×C4×C3⋊Dic3—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yes—
12C3⋊#859C2212yes—
13C22×#226—33no—
14C2×C3×C4×Q8D311yescharacteristic
15C22×#216—31no—
16C2×C4×Dic6—31no—
17C2×C4×SL(2,𝔽3)D313yes—
18C2×#859—33no—
19C4×#226—312no—
20C2×#835—31no—
21C4×#216—34no—
22C4×#216—34no—
23C2×#810—31no—
24C2×#807—31no—
25C2×C4×C3⋊C8—31no—
26C22×C32×C4—41no—
27C22×C3⋊Dic3—41no—
28C22×C3⋊Dic3—41no—
29C2×C3×SL(2,𝔽3)C2×C412yes—
30C2×C3×SL(2,𝔽3)C2311yescharacteristic
31C3⋊#226C2×C414yesdirect product factor
32C4×C3⋊Dic3—44no—
33C2×C4×Q16—91noSylow
34C22×C3×Q8C2×D311yescharacteristic
35C22×Dic6—31no—
36C22×SL(2,𝔽3)C2×D313yes—
37C2×#226—312no—
38C2×#226—36no—
39C3×C4×Q8—32no—
40C3×C4×Q8C2×D312yes—
41C2×C3×C4⋊C4—31no—
42C2×C3×C42—31no—
43C2×#216—32no—
44C2×#216—34no—
45C2×#216—34no—
46C2×#216—32no—
47C2×C4⋊Dic3—31no—
48C2×Dic3⋊C4—61no—
49C2×C4×Dic3—61no—
50C2×C4×Dic3—123no—
51C22×C3⋊C8—31no—
52C22×C3⋊C8—31no—
53C4×Dic6—34no—
54C4×Dic6—62no—
55C4×SL(2,𝔽3)C2×D316yes—
56#859—36no—
57#835—32no—
58#835—32no—
59#810—34no—
60#807—34no—
61C4×C3⋊C8—34no—
62C23×C32—41no—
63C2×C32×C4—44no—
64C2×C32×C4—42no—
65C2×C3⋊Dic3—44no—
66C2×C3⋊Dic3—42no—
67C2×C3⋊Dic3—42no—
68C2×C3⋊Dic3—44no—
69C3×SL(2,𝔽3)C22×C411yescharacteristic
70C22×Q16—91no—
71C2×C4×Q8—91no—
72C2×C4×Q8C3⋊D311yescharacteristic
73C4×Q16—94no—
74C4×Q16—94no—
75C2×#107—91no—
76C2×#103—91no—
77C2×#103—91no—
78C2×C4×C8—91no—
79C2×C3×Q8—31no—
80C2×C3×Q8—32no—
81C2×C3×Q8C4×D312yes—
82C2×C3×Q8C22×D311yescharacteristic
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—43no—
86C22×C3×C4S411yescharacteristic
87C22×Dic3—61no—
88C22×Dic3—61no—
89C22×Dic3—123no—
90C22×Dic3—123no—
91C2×Dic6—32no—
92C2×Dic6—34no—
93C2×Dic6—61no—
94C2×Dic6—62no—
95C2×SL(2,𝔽3)C4×D316yes—
96C2×SL(2,𝔽3)C22×D313yes—
97#226—312no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—61no—
100C3×C42—32no—
101C3×C42—32no—
102#216—38no—
103#216—34no—
104#216—34no—
105C4⋊Dic3—32no—
106C4⋊Dic3—61no—
107Dic3⋊C4—64no—
108C4×Dic3—64no—
109C4×Dic3—1212no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—34no—
112C2×C3⋊C8—32no—
113C2×C3⋊C8—62no—
114C22×C32—42no—
115C22×C32—42no—
116C22×C32—42no—
117C22×C32—41no—
118C32×C4—44no—
119C3⋊Dic3—44no—
120C3⋊Dic3—44no—
121C22×Q8—91no—
122C22×Q8C2×C3⋊D311yescharacteristic
123C2×Q16—94no—
124C2×Q16—92no—
125C2×Q16—94no—
126C2×Q16—92no—
127C22×C8—91no—
128C22×C8—91no—
129C4×Q8—32no—
130C4×Q8—94no—
131C4×Q8—182no—
132C4×Q8C2×C3⋊D312yes—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—91no—
135C2×C4⋊C4—181no—
136C2×C42—31no—
137C2×C42—181no—
138#107—94no—
139#103—94no—
140#103—92no—
141#103—92no—
142C4×C8—94no—
143C23×C3—43no—
144C23×C3C2×S411yescharacteristic
145C3×Q8—32no—
146C3×Q8—31no—
147C3×Q8—61no—
148C3×Q8C2×C4×D311yescharacteristic
149C2×C3×C4—32no—
150C2×C3×C4—32no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—32no—
154C2×C3×C4—46no—
155C2×C3×C4—412no—
156C2×C3×C4—62no—
157C2×C3×C4#22614yescomplemented
158C2×C3×C4C2×S412yes—
159C2×Dic3—64no—
160C2×Dic3—62no—
161C2×Dic3—62no—
162C2×Dic3—126no—
163C2×Dic3—126no—
164C2×Dic3—1212no—
165C2×Dic3—1212no—
166C2×Dic3—122no—
167Dic6—34no—
168Dic6—62no—
169Dic6—64no—
170SL(2,𝔽3)C2×C4×D313yes—
171C3⋊C8—34no—
172C3⋊C8—62no—
173C2×C32—44no—
174C2×C32—41no—
175C2×C32—41no—
176C2×C32—41no—
177C2×Q8—31no—
178C2×Q8—32no—
179C2×Q8—92no—
180C2×Q8—94no—
181C2×Q8—181no—
182C2×Q8—182no—
183C2×Q8C4×C3⋊D312yes—
184C2×Q8C22×C3⋊D311yescharacteristic
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—181no—
188C22×C4—181no—
189C22×C4C3⋊S411yescharacteristic
190Q16—98no—
191Q16—94no—
192Q16—94no—
193C2×C8—92no—
194C2×C8—94no—
195C2×C8—92no—
196C2×C8—182no—
197C4⋊C4—32no—
198C4⋊C4—61no—
199C4⋊C4—92no—
200C4⋊C4—181no—
201C4⋊C4—184no—
202C42—32no—
203C42—32no—
204C42—184no—
205C22×C3—46no—
206C22×C3—46no—
207C22×C3—46no—
208C22×C3—43no—
209C22×C3C4×S412yes—
210C22×C3C2×#22612yes—
211C22×C3#98512yes—
212C22×C3C22×S411yescharacteristic
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—32no—
216C3×C4—412no—
217C3×C4—62no—
218C3×C4C2×#22614yescomplemented
219Dic3—64no—
220Dic3—1212no—
221Dic3—1212no—
222Dic3—122no—
223C32—41noSylow
224C23C2×C3⋊S411yescharacteristic
225Q8—32no—
226Q8—31no—
227Q8—61no—
228Q8—94no—
229Q8—182no—
230Q8—184no—
231Q8C2×C4×C3⋊D311yescharacteristic
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—32no—
237C2×C4—62no—
238C2×C4—184no—
239C2×C4—182no—
240C2×C4—182no—
241C2×C4—362no—
242C2×C4C3⋊#22614yesdirect product factor
243C2×C4C2×C3⋊S412yes—
244C8—94no—
245C8—182no—
246C2×C3—412no—
247C2×C3—43no—
248C2×C3—43no—
249C2×C3—43no—
250C2×C3C4×#22614yescomplemented
251C2×C3C2×C4×S411yescharacteristic
252C2×C3C22×#22611yescharacteristic
253C2×C3C2×#98511yescharacteristic
254C22C4×C3⋊S412yes—
255C22C2×C3⋊#22612yes—
256C22#6426612yes—
257C22C22×C3⋊S411yescharacteristic
258C4—31no—
259C4—31no—
260C4—32no—
261C4—62no—
262C4—184no—
263C4—362no—
264C4C2×C3⋊#22614yesdirect product factor
265C3—43no—
266C3C2×C4×#22611yescharacteristic, complemented
267C2C4×C3⋊#22614yesdirect product factor
268C2C2×C4×C3⋊S411yescharacteristic
269C2C22×C3⋊#22611yescharacteristic
270C2C2×#6426611yescharacteristic
subgroupG/H
center, hypercenterC22×C4C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radicalC22×C3×SL(2,𝔽3)C22
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C4×C3⋊#226C2direct (N × H)
8. C2×C3⋊#226C4direct (N × H)
31. C3⋊#226C2×C4direct (N × H)
157. C2×C3×C4#226semidirect
218. C3×C4C2×#226semidirect
242. C2×C4C3⋊#226direct (N × H)
250. C2×C3C4×#226semidirect
264. C4C2×C3⋊#226direct (N × H)
266. C3C2×C4×#226semidirect
267. C2C4×C3⋊#226direct (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
C2×C3⋊#8592
C4×C3⋊#2262
C2×C4×#2263
C2×C4×#2163
C2×C4×C3⋊Dic34