Finite Group Databasebeta

← C22×C3×C8×A4 #11919774

Subgroups — C22×C3×C8×A4 #11919774

Order 1152.

1064 conjugacy classes of subgroups; only the 304 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 304 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×C4×A4C211yesmaximal, characteristic
2C2×C3×C8×A4C216yesmaximal, direct product factor
3C24×C3×C8C311yesmaximal, characteristic, complemented
4C22×C8×A4C313yesmaximal, direct product factor
5C23×C3×A4C411yescharacteristic
6C2×C3×C4×A4C413yes—
7C2×C3×C4×A4C2213yes—
8C22×C32×C8—41nomaximal
9C3×C8×A4C2214yesdirect product factor
10C24×C3×C4C2×C311yescharacteristic
11C22×C4×A4C2×C313yes—
12C23×C3×C8—31no—
13C23×C3×C8—31no—
14C23×C3×C8—36no—
15C23×C3×C8C2×C316yescomplemented
16C2×C8×A4C2×C3118yesdirect product factor
17C22×C3×A4C814yesdirect product factor
18C22×C3×A4C2×C413yes—
19C22×C32×C4—41no—
20C3×C4×A4C2×C413yes—
21C3×C4×A4C2311yescharacteristic
22C2×C32×C8—46no—
23C24×C8C3211yescharacteristic, complemented, Sylow
24C25×C3C3×C411yescharacteristic
25C23×A4C3×C413yes—
26C23×C3×C4—31no—
27C23×C3×C4—33no—
28C23×C3×C4—31no—
29C23×C3×C4—33no—
30C23×C3×C4C3×C413yes—
31C23×C3×C4C22×C313yes—
32C2×C4×A4C3×C419yes—
33C2×C4×A4C22×C319yes—
34C22×C3×C8—36no—
35C22×C3×C8—38no—
36C22×C3×C8—31no—
37C22×C3×C8—36no—
38C22×C3×C8—36no—
39C22×C3×C8—36no—
40C22×C3×C8—312no—
41C22×C3×C8—43no—
42C22×C3×C8A411yescharacteristic, direct product factor
43C22×C3×C8C22×C314yescomplemented
44C8×A4C22×C3112yesdirect product factor
45C23×C32—41no—
46C2×C3×A4C2×C816yesdirect product factor
47C2×C3×A4C22×C411yescharacteristic
48C2×C32×C4—43no—
49C2×C32×C4—43no—
50C32×C8—44no—
51C24×C4C2×C3211yescharacteristic
52C23×C8—31no—
53C23×C8—31no—
54C23×C8—36no—
55C23×C8C2×C3216yescomplemented
56C24×C3—34no—
57C24×C3—33no—
58C24×C3—31no—
59C24×C3C3×C814yescomplemented
60C24×C3C2×C3×C413yes—
61C22×A4C3×C8112yesdirect product factor
62C22×A4C2×C3×C419yes—
63C22×C3×C4—32no—
64C22×C3×C4—36no—
65C22×C3×C4—36no—
66C22×C3×C4—31no—
67C22×C3×C4—33no—
68C22×C3×C4—33no—
69C22×C3×C4—33no—
70C22×C3×C4—33no—
71C22×C3×C4—33no—
72C22×C3×C4—33no—
73C22×C3×C4—36no—
74C22×C3×C4—33no—
75C22×C3×C4—33no—
76C22×C3×C4—43no—
77C22×C3×C4C2×C3×C413yes—
78C22×C3×C4C2×A411yescharacteristic
79C22×C3×C4C23×C311yescharacteristic
80C4×A4C2×C3×C419yes—
81C4×A4C23×C313yes—
82C2×C3×C8—312no—
83C2×C3×C8—36no—
84C2×C3×C8—312no—
85C2×C3×C8—34no—
86C2×C3×C8—34no—
87C2×C3×C8—418no—
88C2×C3×C8C2×A416yesdirect product factor
89C22×C32—44no—
90C22×C32—43no—
91C3×A4C22×C811yescharacteristic, direct product factor
92C32×C4—43no—
93C32×C4—41no—
94C25C32×C411yescharacteristic
95C23×C4—31no—
96C23×C4—33no—
97C23×C4—31no—
98C23×C4—33no—
99C23×C4C32×C413yes—
100C23×C4C22×C3213yes—
101C22×C8—36no—
102C22×C8—38no—
103C22×C8—31no—
104C22×C8—36no—
105C22×C8—36no—
106C22×C8—36no—
107C22×C8—312no—
108C22×C8C3×A411yescharacteristic, direct product factor
109C22×C8C22×C3214yescomplemented
110C23×C3—33no—
111C23×C3—312no—
112C23×C3—36no—
113C23×C3—32no—
114C23×C3—312no—
115C23×C3—34no—
116C23×C3—34no—
117C23×C3—33no—
118C23×C3—33no—
119C23×C3—43no—
120C23×C3C2×C3×C816yescomplemented
121C23×C3C4×A411yescharacteristic
122C23×C3C22×C3×C411yescharacteristic
123C2×A4C2×C3×C8118yesdirect product factor
124C2×A4C22×C3×C413yes—
125C2×C3×C4—36no—
126C2×C3×C4—36no—
127C2×C3×C4—33no—
128C2×C3×C4—33no—
129C2×C3×C4—33no—
130C2×C3×C4—36no—
131C2×C3×C4—33no—
132C2×C3×C4—33no—
133C2×C3×C4—31no—
134C2×C3×C4—33no—
135C2×C3×C4—31no—
136C2×C3×C4—49no—
137C2×C3×C4—49no—
138C2×C3×C4C4×A413yes—
139C2×C3×C4C22×A413yes—
140C3×C8—34no—
141C3×C8—412no—
142C3×C8C22×A414yesdirect product factor
143C2×C32—41no—
144C2×C32—46no—
145C24—33no—
146C24—34no—
147C24—31no—
148C24C32×C814yescomplemented
149C24C2×C32×C413yes—
150C22×C4—32no—
151C22×C4—36no—
152C22×C4—36no—
153C22×C4—31no—
154C22×C4—33no—
155C22×C4—33no—
156C22×C4—33no—
157C22×C4—33no—
158C22×C4—33no—
159C22×C4—33no—
160C22×C4—36no—
161C22×C4—33no—
162C22×C4—33no—
163C22×C4C2×C32×C413yes—
164C22×C4C2×C3×A411yescharacteristic
165C22×C4C23×C3211yescharacteristic
166C2×C8—312no—
167C2×C8—36no—
168C2×C8—34no—
169C2×C8—312no—
170C2×C8—34no—
171C2×C8C2×C3×A416yesdirect product factor
172C22×C3—36no—
173C22×C3—31no—
174C22×C3—36no—
175C22×C3—38no—
176C22×C3—33no—
177C22×C3—31no—
178C22×C3—312no—
179C22×C3—36no—
180C22×C3—36no—
181C22×C3—49no—
182C22×C3—412no—
183C22×C3C8×A414yesdirect product factor
184C22×C3C22×C3×C811yescharacteristic, complemented
185C22×C3C2×C4×A413yes—
186A4C22×C3×C813yesdirect product factor
187C3×C4—31no—
188C3×C4—33no—
189C3×C4—49no—
190C3×C4—43no—
191C3×C4C2×C4×A413yes—
192C3×C4C23×A411yescharacteristic
193C32—41noSylow
194C23—33no—
195C23—312no—
196C23—36no—
197C23—32no—
198C23—312no—
199C23—33no—
200C23—33no—
201C23—34no—
202C23—34no—
203C23C2×C32×C816yescomplemented
204C23C3×C4×A411yescharacteristic
205C23C22×C32×C411yescharacteristic
206C2×C4—36no—
207C2×C4—36no—
208C2×C4—33no—
209C2×C4—33no—
210C2×C4—33no—
211C2×C4—33no—
212C2×C4—36no—
213C2×C4—31no—
214C2×C4—33no—
215C2×C4—33no—
216C2×C4—31no—
217C2×C4C3×C4×A413yes—
218C2×C4C22×C3×A413yes—
219C8—34no—
220C8C22×C3×A414yesdirect product factor
221C2×C3—36no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—43no—
225C2×C3—418no—
226C2×C3C2×C8×A416yesdirect product factor
227C2×C3C22×C4×A411yescharacteristic
228C22—36no—
229C22—31no—
230C22—36no—
231C22—38no—
232C22—31no—
233C22—33no—
234C22—36no—
235C22—312no—
236C22—36no—
237C22C3×C8×A414yesdirect product factor
238C22C22×C32×C811yescharacteristic, complemented
239C22C2×C3×C4×A413yes—
240C4—33no—
241C4—31no—
242C4C2×C3×C4×A413yes—
243C4C23×C3×A411yescharacteristic
244C3—43no—
245C3C22×C8×A411yescharacteristic, direct product factor
246C2—31no—
247C2—36no—
248C2—31no—
249C2C2×C3×C8×A416yesdirect product factor
250C2C22×C3×C4×A411yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C8A4
commutator, hypocenterC22C22×C32×C8
FrattiniC4C23×C3×A4
FittingC24×C3×C8C3
socleC25×C3C3×C4
prime radicalC23×C3×A4C4
cosocleC22×C4C23×C32
LCSG ▷ C22
UCSC1 ◁ C22×C3×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×A4C2direct (N × H)
3. C24×C3×C8C3semidirect
4. C22×C8×A4C3direct (N × H)
9. C3×C8×A4C22direct (N × H)
15. C23×C3×C8C2×C3semidirect
16. C2×C8×A4C2×C3direct (N × H)
17. C22×C3×A4C8direct (N × H)
23. C24×C8C32semidirect
42. C22×C3×C8A4direct (N × H)
43. C22×C3×C8C22×C3semidirect
44. C8×A4C22×C3direct (N × H)
46. C2×C3×A4C2×C8direct (N × H)
55. C23×C8C2×C32semidirect
59. C24×C3C3×C8semidirect
61. C22×A4C3×C8direct (N × H)
88. C2×C3×C8C2×A4direct (N × H)
91. C3×A4C22×C8direct (N × H)
108. C22×C8C3×A4direct (N × H)
109. C22×C8C22×C32semidirect
120. C23×C3C2×C3×C8semidirect
123. C2×A4C2×C3×C8direct (N × H)
142. C3×C8C22×A4direct (N × H)
148. C24C32×C8semidirect
171. C2×C8C2×C3×A4direct (N × H)
183. C22×C3C8×A4direct (N × H)
184. C22×C3C22×C3×C8semidirect
186. A4C22×C3×C8direct (N × H)
203. C23C2×C32×C8semidirect
220. C8C22×C3×A4direct (N × H)
226. C2×C3C2×C8×A4direct (N × H)
237. C22C3×C8×A4direct (N × H)
238. C22C22×C32×C8semidirect
245. C3C22×C8×A4direct (N × H)
249. C2C2×C3×C8×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×A42
C2×C3×C8×A42
C24×C3×C83
C22×C8×A43
C22×C32×C84