Finite Group Databasebeta

← C22×#4548 #10948012

Subgroups — C22×#4548 #10948012

Order 768.

1254 conjugacy classes of subgroups; only the 331 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 331 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⋊Q8C211yesmaximal, characteristic
2C22×#836C212yesmaximal
3C2×#4548C2112yesmaximal, direct product factor
4C22×#356—31nomaximal, Sylow
5C23×C3×Q8C411yescharacteristic
6C23×C3×Q8C2211yescharacteristic
7C2×C3×C4⋊Q8—26no—
8C2×C3×C4⋊Q8C416yes—
9C2×C3×C4⋊Q8C2216yes—
10C22×C3×C4⋊C4—22no—
11C22×C3×C42C411yescharacteristic
12C22×#208—22no—
13C2×#836—26no—
14C2×#836C22112yes—
15#4548C22116yesdirect product factor
16C22×C4⋊Q8D311yescharacteristic
17C22×#101—32no—
18C2×#356—312no—
19C22×C3×Q8—23no—
20C22×C3×Q8—41no—
21C22×C3×Q8—43no—
22C22×C3×Q8—43no—
23C22×C3×Q8—41no—
24C22×C3×Q8C2×C416yes—
25C22×C3×Q8C2×C413yes—
26C22×C3×Q8D416yes—
27C22×C3×Q8C2313yes—
28C23×C3×C4—21no—
29C23×C3×C4—21no—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D412yes—
32C3×C4⋊Q8—212no—
33C3×C4⋊Q8—212no—
34C3×C4⋊Q8C2×C4112yes—
35C3×C4⋊Q8C2314yes—
36C2×C3×C4⋊C4—212no—
37C2×C3×C4⋊C4—212no—
38C2×C3×C42—23no—
39C2×C3×C42C2×C416yes—
40C2×#208—212no—
41C2×#208—46no—
42C22×C3⋊C8—42no—
43#836—212no—
44#836C2318yes—
45C23×Q8Dic311yescharacteristic
46C23×Q8C2×D311yescharacteristic
47C22×M16—62no—
48C2×C4⋊Q8—26no—
49C2×C4⋊Q8Dic316yes—
50C2×C4⋊Q8C2×D316yes—
51C22×C4⋊C4—22no—
52C22×C42Dic311yescharacteristic
53C2×#101—312no—
54C2×#101—66no—
55#356—316no—
56C24×C3C22⋊C411yescharacteristic
57C2×C3×Q8—26no—
58C2×C3×Q8—43no—
59C2×C3×Q8—43no—
60C2×C3×Q8—46no—
61C2×C3×Q8—46no—
62C2×C3×Q8—43no—
63C2×C3×Q8—43no—
64C2×C3×Q8—46no—
65C2×C3×Q8—46no—
66C2×C3×Q8—46no—
67C2×C3×Q8—46no—
68C2×C3×Q8C22⋊C416yes—
69C2×C3×Q8C22×C414yes—
70C2×C3×Q8C22×C413yes—
71C2×C3×Q8C2×D416yes—
72C2×C3×Q8C2411yescharacteristic
73C22×C3×C4—28no—
74C22×C3×C4—22no—
75C22×C3×C4—26no—
76C22×C3×C4—23no—
77C22×C3×C4—21no—
78C22×C3×C4—26no—
79C22×C3×C4—26no—
80C22×C3×C4—43no—
81C22×C3×C4—41no—
82C22×C3×C4C22⋊C413yes—
83C22×C3×C4C22⋊C416yes—
84C22×C3×C4C22×C413yes—
85C22×C3×C4C2×D416yes—
86C3×C4⋊C4—224no—
87C3×C4⋊C4—28no—
88C3×C42—26no—
89C3×C42C22×C414yes—
90#208—28no—
91#208—412no—
92C2×C3⋊C8—412no—
93C22×Q8—23no—
94C22×Q8—43no—
95C22×Q8—41no—
96C22×Q8—43no—
97C22×Q8—41no—
98C22×Q8C2×Dic316yes—
99C22×Q8C2×Dic313yes—
100C22×Q8#6716yes—
101C22×Q8C22×D313yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2×Dic311yescharacteristic
105C23×C4#6712yes—
106C2×M16—612no—
107C2×M16—126no—
108C22×C8—122no—
109C4⋊Q8—212no—
110C4⋊Q8—212no—
111C4⋊Q8C2×Dic3112yes—
112C4⋊Q8C22×D314yes—
113C2×C4⋊C4—212no—
114C2×C4⋊C4—212no—
115C2×C42—23no—
116C2×C42C2×Dic316yes—
117#101—38no—
118#101—612no—
119C23×C3—24no—
120C23×C3#9913yes—
121C23×C3#9911yescharacteristic
122C23×C3C2×C22⋊C413yes—
123C3×Q8—46no—
124C3×Q8—43no—
125C3×Q8—46no—
126C3×Q8—41no—
127C3×Q8—44no—
128C3×Q8—412no—
129C2×C3×C4—224no—
130C2×C3×C4—24no—
131C2×C3×C4—26no—
132C2×C3×C4—26no—
133C2×C3×C4—26no—
134C2×C3×C4—26no—
135C2×C3×C4—23no—
136C2×C3×C4—23no—
137C2×C3×C4—23no—
138C2×C3×C4—23no—
139C2×C3×C4—24no—
140C2×C3×C4—46no—
141C2×C3×C4—46no—
142C2×C3×C4C2×C22⋊C413yes—
143C2×C3×C4C2×C22⋊C416yes—
144C2×C3×C4C23×C411yescharacteristic
145C2×C3×C4C22×D412yes—
146C3⋊C8—48no—
147C24#21711yescharacteristic
148C2×Q8—26no—
149C2×Q8—43no—
150C2×Q8—43no—
151C2×Q8—46no—
152C2×Q8—46no—
153C2×Q8—43no—
154C2×Q8—43no—
155C2×Q8—46no—
156C2×Q8—46no—
157C2×Q8—46no—
158C2×Q8—46no—
159C2×Q8#21716yes—
160C2×Q8C22×Dic314yes—
161C2×Q8C22×Dic313yes—
162C2×Q8C2×#6716yes—
163C2×Q8C23×D311yescharacteristic
164C22×C4—28no—
165C22×C4—26no—
166C22×C4—23no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—26no—
170C22×C4—26no—
171C22×C4—43no—
172C22×C4—41no—
173C22×C4#21713yes—
174C22×C4#21716yes—
175C22×C4C22×Dic313yes—
176C22×C4C2×#6716yes—
177M16—68no—
178M16—1212no—
179C2×C8—1212no—
180C4⋊C4—224no—
181C4⋊C4—28no—
182C42—26no—
183C42C22×Dic314yes—
184C22×C3—26no—
185C22×C3—26no—
186C22×C3#35614yescomplemented
187C22×C3C2×#9913yes—
188C22×C3C2×#9913yes—
189C22×C3C22×C22⋊C411yescharacteristic
190C3×C4—28no—
191C3×C4—22no—
192C3×C4—26no—
193C3×C4—23no—
194C3×C4—21no—
195C3×C4—44no—
196C23—24no—
197C23#83413yes—
198C23#83411yescharacteristic
199C23C2×#21713yes—
200Q8—46no—
201Q8—43no—
202Q8—46no—
203Q8—41no—
204Q8—44no—
205Q8—412no—
206C2×C4—224no—
207C2×C4—26no—
208C2×C4—26no—
209C2×C4—23no—
210C2×C4—23no—
211C2×C4—26no—
212C2×C4—26no—
213C2×C4—23no—
214C2×C4—23no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—46no—
218C2×C4—46no—
219C2×C4C2×#21713yes—
220C2×C4C2×#21716yes—
221C2×C4C23×Dic311yescharacteristic
222C2×C4C22×#6712yes—
223C8—128no—
224C2×C3—21no—
225C2×C3—23no—
226C2×C3C2×#35616yescomplemented
227C2×C3C22×#9911yescharacteristic
228C22—26no—
229C22—26no—
230C22#454814yesdirect product factor
231C22C2×#83413yes—
232C22C2×#83413yes—
233C22C22×#21711yescharacteristic
234C4—28no—
235C4—22no—
236C4—23no—
237C4—26no—
238C4—21no—
239C4—44no—
240C3C22×#35611yescharacteristic, complemented, Sylow
241C2—21no—
242C2—23no—
243C2C2×#454816yesdirect product factor
244C2C22×#83411yescharacteristic
subgroupG/H
centerC23#834
commutatorC2×C3×C4C23×C4
FrattiniC2×Q8C23×D3
FittingC22×C3×C4⋊Q8C2
socleC23×C3#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×Q8C24
hypercenterC22×C4⋊Q8D3
hypocenterC3C22×#356
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#4548C2direct (N × H)
15. #4548C22direct (N × H)
186. C22×C3#356semidirect
226. C2×C3C2×#356semidirect
230. C22#4548direct (N × H)
240. C3C22×#356semidirect
243. C2C2×#4548direct (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⋊Q82
C22×#8362
C2×#45482
C22×#3563