Finite Group Databasebeta

← #97×C22⋊A4 #72247713

Subgroups — #97×C22⋊A4 #72247713

Order 1536.

5770 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 5770 conjugacy classes of subgroups — too many to draw. Only its 144 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×C8×C22⋊A4C212yesmaximal
2C42×C22⋊A4C211yesmaximal, characteristic
3C24×#97C311yesmaximal, characteristic, complemented, Sylow
4C2×C4×C22⋊A4C412yes—
5C2×C4×C22⋊A4C2211yescharacteristic
6C8×C22⋊A4C414yescomplemented
7A4×#97—45nomaximal
8C25×C8C2×C312yes—
9C24×C42C2×C311yescharacteristic
10C23×#97—35no—
11C23×#97—35no—
12C23×#97—310no—
13C22×C22⋊A4C2×C411yescharacteristic
14C4×C22⋊A4—22no—
15C4×C22⋊A4C2×C412yes—
16C2×C8×A4—410no—
17C42×A4—45no—
18C25×C4C3×C412yes—
19C25×C4C22×C311yescharacteristic
20C24×C8—310no—
21C24×C8—320no—
22C24×C8—310no—
23C24×C8C3×C414yescomplemented
24C23×C42—35no—
25C23×C42—310no—
26C23×C42—35no—
27C22×#97—360no—
28C22×#97—330no—
29C22×#97—360no—
30C22×#97—310no—
31C22×#97—310no—
32C22×#97—35no—
33C22×#97—310no—
34C22×#97A415yescomplemented
35C2×C22⋊A4C4211yescharacteristic
36C2×C22⋊A4M1612yes—
37C2×C4×A4—410no—
38C2×C4×A4—45no—
39C8×A4—420no—
40C3×#97—161no—
41C26C2×C3×C411yescharacteristic
42C24×C4—22no—
43C24×C4—35no—
44C24×C4—35no—
45C24×C4—310no—
46C24×C4—310no—
47C24×C4—310no—
48C24×C4—610no—
49C24×C4C2×C3×C412yes—
50C23×C8—360no—
51C23×C8—320no—
52C23×C8—3120no—
53C23×C8—3120no—
54C23×C8—320no—
55C23×C8—310no—
56C23×C8—320no—
57C23×C8—320no—
58C23×C8—320no—
59C23×C8C2×A4110yes—
60C22×C42—330no—
61C22×C42—310no—
62C22×C42—360no—
63C22×C42—310no—
64C22×C42—360no—
65C22×C42—310no—
66C22×C42—35no—
67C22×C42C2×A415yes—
68C2×#97—3120no—
69C2×#97—360no—
70C2×#97—330no—
71C2×#97—360no—
72C2×#97—330no—
73C2×#97—35no—
74C2×#97—310no—
75C2×#97—35no—
76C22⋊A4#9711yescharacteristic, direct product factor
77C22×A4—45no—
78C4×A4—410no—
79C4×A4—810no—
80C2×C3×C8—162no—
81C3×C42—161no—
82C25—35no—
83C25—310no—
84C25—35no—
85C25C3×C4211yescharacteristic
86C25C3×M1612yes—
87C23×C4—310no—
88C23×C4—360no—
89C23×C4—360no—
90C23×C4—310no—
91C23×C4—330no—
92C23×C4—35no—
93C23×C4—310no—
94C23×C4—310no—
95C23×C4—310no—
96C23×C4—360no—
97C23×C4—320no—
98C23×C4—310no—
99C23×C4—610no—
100C23×C4—610no—
101C23×C4—660no—
102C23×C4—660no—
103C23×C4—610no—
104C23×C4—610no—
105C23×C4C4×A4110yes—
106C23×C4C22×A415yes—
107C22×C8—3240no—
108C22×C8—360no—
109C22×C8—3120no—
110C22×C8—3120no—
111C22×C8—3120no—
112C22×C8—340no—
113C22×C8—360no—
114C22×C8—320no—
115C22×C8—310no—
116C22×C8—320no—
117C22×C8—310no—
118C22×C8C4×A4120yescomplemented
119C2×C42—330no—
120C2×C42—3120no—
121C2×C42—360no—
122C2×C42—330no—
123C2×C42—360no—
124C2×C42—35no—
125C2×C42—35no—
126C2×C42—310no—
127#97—360no—
128#97—310no—
129#97—310no—
130#97—35no—
131#97C22⋊A411yescharacteristic, direct product factor
132C2×A4—45no—
133C2×A4—410no—
134C2×C3×C4—162no—
135C2×C3×C4—161no—
136C3×C8—164no—
137C24—35no—
138C24—310no—
139C24—310no—
140C24—330no—
141C24—360no—
142C24—360no—
143C24—310no—
144C24—35no—
145C24—35no—
146C24—310no—
147C24—310no—
148C24C3×#9711yescharacteristic, complemented
149C24C2×C4×A415yes—
150C22×C4—210no—
151C22×C4—330no—
152C22×C4—360no—
153C22×C4—320no—
154C22×C4—330no—
155C22×C4—3120no—
156C22×C4—360no—
157C22×C4—360no—
158C22×C4—310no—
159C22×C4—35no—
160C22×C4—35no—
161C22×C4—310no—
162C22×C4—360no—
163C22×C4—310no—
164C22×C4—310no—
165C22×C4—620no—
166C22×C4—660no—
167C22×C4—630no—
168C22×C4—660no—
169C22×C4—660no—
170C22×C4—6120no—
171C22×C4—610no—
172C22×C4—610no—
173C22×C4C2×C4×A4110yes—
174C2×C8—3120no—
175C2×C8—320no—
176C2×C8—3120no—
177C2×C8—310no—
178C2×C8—320no—
179C2×C8—320no—
180C2×C8—320no—
181C2×C8C2×C22⋊A412yes—
182C42—360no—
183C42—310no—
184C42—310no—
185C42—35no—
186C42C2×C22⋊A411yescharacteristic
187C22×C3—161no—
188A4—45no—
189C3×C4—162no—
190C3×C4—322no—
191C23—35no—
192C23—310no—
193C23—330no—
194C23—310no—
195C23—360no—
196C23—320no—
197C23—330no—
198C23—330no—
199C23—360no—
200C23—360no—
201C23—3120no—
202C23—35no—
203C23—310no—
204C23—35no—
205C23—35no—
206C23C42×A415yes—
207C23A4×M16110yes—
208C2×C4—360no—
209C2×C4—310no—
210C2×C4—360no—
211C2×C4—35no—
212C2×C4—310no—
213C2×C4—310no—
214C2×C4—310no—
215C2×C4—310no—
216C2×C4—610no—
217C2×C4—610no—
218C2×C4—660no—
219C2×C4—660no—
220C2×C4—610no—
221C2×C4—610no—
222C2×C4C4×C22⋊A412yes—
223C2×C4C22×C22⋊A411yescharacteristic
224C8—320no—
225C8C4×C22⋊A414yescomplemented
226C2×C3—161no—
227C2×C3—162no—
228C22—35no—
229C22—310no—
230C22—310no—
231C22—360no—
232C22—330no—
233C22—360no—
234C22—310no—
235C22—35no—
236C22—35no—
237C22—310no—
238C22—310no—
239C22A4×#9715yescomplemented
240C22C2×C4×C22⋊A411yescharacteristic
241C4—22no—
242C4—310no—
243C4—610no—
244C4C2×C4×C22⋊A412yes—
245C3—161noSylow
246C2—35no—
247C2—310no—
248C2—35no—
249C2C42×C22⋊A411yescharacteristic
250C2M16×C22⋊A412yes—
subgroupG/H
center, FrattiniC2×C4C22×C22⋊A4
commutatorC25C3×C42
FittingC24×#97C3
socleC26C2×C3×C4
prime radicalC22×C22⋊A4C2×C4
cosocleC25×C4C22×C3
hypercenter#97C22⋊A4
hypocenterC24C3×#97
LCSG ▷ C25 ▷ C24
UCSC1 ◁ C2×C4 ◁ #97

Complements

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

NH ≅ G/Ntype
3. C24×#97C3semidirect
6. C8×C22⋊A4C4semidirect
23. C24×C8C3×C4semidirect
34. C22×#97A4semidirect
76. C22⋊A4#97direct (N × H)
118. C22×C8C4×A4semidirect
131. #97C22⋊A4direct (N × H)
148. C24C3×#97semidirect
225. C8C4×C22⋊A4semidirect
239. C22A4×#97semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C22⋊A42
C42×C22⋊A42
C24×#973
A4×#974