Finite Group Databasebeta

← C2×#1524 #17635

Subgroups — C2×#1524 #17635

Order 256.

583 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 199 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×#529C211yesmaximal, characteristic
2C2×C4×M16C211yesmaximal, characteristic, complemented
3C22×#97C211yesmaximal, characteristic, complemented
4#1524C218yesmaximal, direct product factor
5C2×#330C212yesmaximal, complemented
6C2×#329C212yesmaximal, complemented
7C22×M16—21no—
8C23×C8—21no—
9#529—22no—
10#529—22no—
11#529C414yes—
12#529C2214yes—
13C2×#126—21no—
14C2×#125C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×#123—21no—
17C2×C22⋊Q8C411yescharacteristic
18C2×#121C411yescharacteristic
19C2×C4×Q8C411yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C411yescharacteristic
22C2×#117—21no—
23C22×C42C2211yescharacteristic
24C4×M16—22no—
25C4×M16C2214yescomplemented
26C2×#97—22no—
27C2×#97—21no—
28C2×#97C2212yes—
29C2×#97C2212yes—
30C2×#97C2214yescomplemented
31C2×C4×C8—21no—
32#330C2218yescomplemented
33#329C2218yescomplemented
34C22×Q8C2×C411yescharacteristic
35C22×D4—21no—
36C22×D4C2×C411yescharacteristic
37C23×C4—21no—
38C23×C4C2×C411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4D411yescharacteristic
41C2×M16—28no—
42C2×M16—24no—
43C22×C8—28no—
44C22×C8—22no—
45C22×C8—24no—
46C22×C8—22no—
47#126—24no—
48#126—24no—
49#125—21no—
50#125—21no—
51#125C2×C412yes—
52#125C2312yes—
53#124—21no—
54#124—21no—
55#124C2×C412yes—
56#124C2312yes—
57#123—24no—
58#123—24no—
59C22⋊Q8—22no—
60C22⋊Q8C2×C414yes—
61#121—22no—
62#121C2×C414yes—
63C4×Q8—22no—
64C4×Q8C2×C414yes—
65C4×D4—24no—
66C4×D4—24no—
67C4×D4—22no—
68C4×D4C2×C414yes—
69#117—24no—
70#117—24no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C2×C411yescharacteristic
76C2×C4⋊C4C2×C411yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—21no—
83C2×C42C2×C412yes—
84C2×C42C2×C411yescharacteristic
85C2×C42D411yescharacteristic
86C2×C42D412yes—
87C2×C42D412yes—
88C2×C42D411yescharacteristic
89C2×C42C2311yescharacteristic
90C2×C42C2312yes—
91#97—22no—
92#97—24no—
93#97—22no—
94#97C2314yes—
95#97C2314yes—
96C4×C8—24no—
97C24—21no—
98C24C22⋊C411yescharacteristic
99C2×Q8—21no—
100C2×Q8—42no—
101C2×Q8C22×C412yes—
102C2×D4—24no—
103C2×D4—21no—
104C2×D4—42no—
105C2×D4—44no—
106C2×D4C22×C412yes—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—24no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4C22⋊C411yescharacteristic
116C22×C4C22⋊C412yes—
117C22×C4C22⋊C412yes—
118C22×C4C22⋊C411yescharacteristic
119C22×C4C22⋊C412yes—
120C22×C4C22⋊C411yescharacteristic
121C22×C4C22×C411yescharacteristic
122C22×C4C22×C412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127M16—216no—
128C2×C8—28no—
129C2×C8—216no—
130C2×C8—28no—
131C2×C8—24no—
132C2×C8—24no—
133C4⋊C4—24no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22×C412yes—
142C4⋊C4C22×C412yes—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—24no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C42—24no—
151C42C22⋊C412yes—
152C42C22⋊C412yes—
153C42C22⋊C412yes—
154C42C22×C412yes—
155C42C22×C411yescharacteristic
156C42C2×D412yes—
157C42C2×D411yescharacteristic
158C42C2×D411yescharacteristic
159C42C2×D412yes—
160C42C2411yescharacteristic
161C23—22no—
162C23—22no—
163C23—22no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23#10414yes—
168C23C2×C22⋊C412yes—
169C23C2×C22⋊C411yescharacteristic
170Q8—42no—
171Q8—42no—
172D4—42no—
173D4—42no—
174D4—48no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—48no—
193C2×C4#10414yes—
194C2×C4C2×C22⋊C411yescharacteristic
195C2×C4C2×C22⋊C412yes—
196C2×C4C2×C22⋊C411yescharacteristic
197C2×C4C2×C22⋊C412yes—
198C2×C4C2×C22⋊C411yescharacteristic
199C2×C4C2×C22⋊C412yes—
200C2×C4C23×C411yescharacteristic
201C2×C4C22×D411yescharacteristic
202C2×C4C22×D411yescharacteristic
203C8—28no—
204C8—28no—
205C22—22no—
206C22—22no—
207C22—28no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—84no—
212C22#41312yes—
213C22C2×#10414yes—
214C22C2×#10412yes—
215C22C2×#10412yes—
216C22C22×C22⋊C411yescharacteristic
217C4—22no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—22no—
224C4—21no—
225C4—42no—
226C4—42no—
227C4—42no—
228C4C2×#10414yes—
229C2—24no—
230C2—82no—
231C2#152414yesdirect product factor
232C2C2×#41311yescharacteristic
233C2C22×#10411yescharacteristic
234C2C22×#10411yescharacteristic
subgroupG/H
centerC22×C4C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC23C2×C22⋊C4
prime radicalC2×#121C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×M16C2semidirect
3. C22×#97C2semidirect
4. #1524C2direct (N × H)
5. C2×#330C2semidirect
6. C2×#329C2semidirect
25. C4×M16C22semidirect
30. C2×#97C22semidirect
32. #330C22semidirect
33. #329C22semidirect
231. C2#1524direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5292
C2×C4×M162
C22×#972
#15242
C2×#3302
C2×#3292