Finite Group Databasebeta

← #21895

Subgroups — #21895

Order 256.

559 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 191 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×#423C211yesmaximal, characteristic, complemented
3#2283C211yesmaximal, characteristic, complemented
4#1989C212yesmaximal, complemented
5#1983C212yesmaximal, complemented
6#1895C212yesmaximal
7#1894C212yesmaximal, complemented
8#1805C212yesmaximal, complemented
9#1804C212yesmaximal, complemented
10C22×M16C2212yes—
11#529—24no—
12#529C418yescomplemented
13C2×#126—21no—
14C2×#125C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×#123—21no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C2×#117—21no—
23C22×C42C2211yescharacteristic
24#423—24no—
25C2×C4⋊C8C2212yes—
26C2×C4⋊C8C2212yes—
27C2×#103C2214yes—
28C2×#102C2214yes—
29#383—21no—
30#382C2212yes—
31#381—21no—
32#380C2212yes—
33C4×C4⋊C4C2212yes—
34C4×C22⋊C4—21no—
35#376—21no—
36#340C2214yes—
37#340C2214yes—
38C22×Q8C2311yescharacteristic
39C22×D4—21no—
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C2×M16—24no—
46C2×M16—44no—
47C22×C8C2314yes—
48#126—28no—
49#125—22no—
50#125C2×C414yes—
51#124—22no—
52#124C2×C414yes—
53#123—28no—
54C22⋊Q8—22no—
55C22⋊Q8C2×C414yes—
56#121—22no—
57#121C2×C414yes—
58C4×Q8—22no—
59C4×Q8C2×C414yes—
60C4×D4—28no—
61C4×D4—22no—
62C4×D4C2×C414yes—
63#117—28no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4C2314yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42C2×C414yes—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42D414yes—
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85C4⋊C8—24no—
86C4⋊C8—24no—
87#103—44no—
88#102—44no—
89#95—22no—
90#95—22no—
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—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—42no—
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×D411yescharacteristic
123C22×C4C2×D412yes—
124C22×C4C2411yescharacteristic
125M16—48no—
126C2×C8—24no—
127C2×C8—24no—
128C2×C8—24no—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4C22×C412yes—
138C4⋊C4C22×C412yes—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—24no—
144C22⋊C4—44no—
145C42—24no—
146C42—44no—
147C42C22⋊C414yes—
148C42C22⋊C414yes—
149C42C22×C412yes—
150C42C22×C412yes—
151C42C2×D412yes—
152C42C2×D412yes—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23C2×C22⋊C412yes—
161C23C22×D411yescharacteristic
162Q8—24no—
163Q8—41no—
164Q8—41no—
165D4—24no—
166D4—48no—
167D4—41no—
168D4—41no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—48no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4C2×C22⋊C412yes—
190C2×C4C2×C22⋊C412yes—
191C2×C4C2×C22⋊C412yes—
192C2×C4C2×C22⋊C412yes—
193C2×C4C2×C22⋊C412yes—
194C2×C4C23×C411yescharacteristic
195C2×C4C22×D411yescharacteristic
196C2×C4H(5,𝔽2)12yes—
197C2×C42−1+412yes—
198C8—44no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—84no—
207C22C22×C22⋊C411yescharacteristic
208C22#53712yes—
209C22#57512yes—
210C22#57512yes—
211C4—22no—
212C4—22no—
213C4—22no—
214C4—22no—
215C4—22no—
216C4—42no—
217C4—42no—
218C4—42no—
219C4—84no—
220C4#52012yes—
221C4#53511yescharacteristic
222C4#53611yescharacteristic
223C2—42no—
224C2—82no—
225C2#231111yescharacteristic
226C2#289611yescharacteristic
227C2#289611yescharacteristic
228C2#325614yes—
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×#423C2semidirect
3. #2283C2semidirect
4. #1989C2semidirect
5. #1983C2semidirect
7. #1894C2semidirect
8. #1805C2semidirect
9. #1804C2semidirect
12. #529C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4232
#22832
#19892
#19832
#18952
#18942
#18052
#18042