Finite Group Databasebeta

← #21862

Subgroups — #21862

Order 256.

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

normal subgroups only

This group has 569 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×#431C211yesmaximal, characteristic, complemented
3#2285C211yesmaximal, characteristic, complemented
4#1991C212yesmaximal, complemented
5#1958C212yesmaximal, complemented
6#1875C212yesmaximal
7#1873C212yesmaximal, complemented
8#1803C212yesmaximal, complemented
9#1802C212yesmaximal, complemented
10#529—24no—
11#529C418yescomplemented
12C2×#126—21no—
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×#123—21no—
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22C2×#117C2212yes—
23C22×C42C2211yescharacteristic
24#431—24no—
25C2×C4⋊C8C2212yes—
26C2×#103C2214yes—
27C2×#102C2214yes—
28C2×C22⋊C8C2212yes—
29C2×C4×C8C2212yes—
30#384C2212yes—
31#382C2212yes—
32#380—21no—
33#380—21no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36#340C2214yes—
37#340C2214yes—
38C22×Q8C2311yescharacteristic
39C22×D4—21no—
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8C2314yes—
46#126—28no—
47#125—22no—
48#125C2×C414yes—
49#124—22no—
50#124C2×C414yes—
51#123—28no—
52C22⋊Q8—22no—
53C22⋊Q8C2×C414yes—
54#121—22no—
55#121C2×C414yes—
56C4×Q8—22no—
57C4×Q8C2×C414yes—
58C4×D4—28no—
59C4×D4—22no—
60C4×D4C2×C414yes—
61#117—28no—
62#117—44no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4C2314yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C42—21no—
79C2×C42—22no—
80C2×C42C2×C414yes—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42D414yes—
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86C4⋊C8—24no—
87#103—44no—
88#102—44no—
89C22⋊C8—24no—
90C4×C8—24no—
91#95—22no—
92#95—22no—
93C24—21no—
94C24C2×D411yescharacteristic
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8C22×C412yes—
99C2×D4—24no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—44no—
104C2×D4C22×C412yes—
105C22×C4—24no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4C22⋊C414yes—
116C22×C4C22⋊C414yes—
117C22×C4C22×C412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125C2×C8—24no—
126C2×C8—24no—
127C2×C8—24no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4C22×C412yes—
139C4⋊C4C22×C412yes—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—44no—
146C42—24no—
147C42—44no—
148C42C22⋊C414yes—
149C42C22⋊C414yes—
150C42C22×C412yes—
151C42C22×C412yes—
152C42C2×D412yes—
153C42C2×D412yes—
154C23—21no—
155C23—21no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23—42no—
163C23C2×C22⋊C412yes—
164C23C22×D411yescharacteristic
165Q8—24no—
166Q8—42no—
167D4—24no—
168D4—48no—
169D4—42no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—28no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—48no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4C2×C22⋊C412yes—
191C2×C4C2×C22⋊C412yes—
192C2×C4C2×C22⋊C412yes—
193C2×C4C2×C22⋊C412yes—
194C2×C4C2×C22⋊C412yes—
195C2×C4#13514yes—
196C2×C4C23×C411yescharacteristic
197C2×C4C22×D411yescharacteristic
198C2×C4H(5,𝔽2)12yes—
199C2×C42−1+412yes—
200C8—44no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—82no—
210C22—82no—
211C22C22×C22⋊C411yescharacteristic
212C22#53511yescharacteristic
213C22#53611yescharacteristic
214C22C2×#13512yes—
215C22#57512yes—
216C4—22no—
217C4—22no—
218C4—22no—
219C4—22no—
220C4—42no—
221C4—42no—
222C4—42no—
223C4—84no—
224C4#41614yes—
225C4#52012yes—
226C4#53712yes—
227C2—42no—
228C2—82no—
229C2#231111yescharacteristic
230C2C2×#41611yescharacteristic
231C2#289611yescharacteristic
232C2#321212yes—
233C2#323312yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#431C2semidirect
3. #2285C2semidirect
4. #1991C2semidirect
5. #1958C2semidirect
7. #1873C2semidirect
8. #1803C2semidirect
9. #1802C2semidirect
11. #529C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4312
#22852
#19912
#19582
#18752
#18732
#18032
#18022