Finite Group Databasebeta

← #22563

Subgroups — #22563

Order 256.

495 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 175 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×#103C211yesmaximal, characteristic
2#2344C211yesmaximal, characteristic
3Q8×C22⋊C4C211yesmaximal, characteristic
4#1935C212yesmaximal, complemented
5#1928C212yesmaximal, complemented
6#1863C212yesmaximal, complemented
7#1823C211yesmaximal, characteristic
8#1808C211yesmaximal, characteristic
9#1787C211yesmaximal, characteristic
10#1763C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C4×Q8C2212yes—
15C2×#117—21no—
16C22×C4⋊C4C2211yescharacteristic
17C2×#107C2212yes—
18C2×#103C418yescomplemented
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2212yes—
27#391—22no—
28#391C2212yes—
29#389C2212yes—
30#387—22no—
31#381—21no—
32#380—21no—
33#380C2211yescharacteristic
34#380C2211yescharacteristic
35C4×C4⋊C4C2212yes—
36C4×C22⋊C4—21no—
37C4×C22⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42C22×Q8—21no—
43C22×Q8—22no—
44C22×Q8—24no—
45C22×Q8—42no—
46C22×Q8C2×C414yes—
47C22×Q8C2311yescharacteristic
48C22×Q8C2311yescharacteristic
49C23×C4—21no—
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52C22×C8—42no—
53C22×C8C2×C414yes—
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C4×Q8—24no—
58C4×Q8—42no—
59#117—44no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4C2×C414yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4D411yescharacteristic
72C2×C22⋊C4D411yescharacteristic
73C2×C42—22no—
74C2×C42—22no—
75C2×C42C2312yes—
76#107—42no—
77#103—24no—
78#103—42no—
79#103C2×C414yes—
80#103C2×C414yes—
81#103C2×C414yes—
82#103C2×C414yes—
83C22⋊C8—22no—
84#97—42no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95D412yes—
92C24C2×D411yescharacteristic
93C2×Q8—24no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—28no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—44no—
101C2×Q8—44no—
102C2×Q8—48no—
103C2×Q8C22×C412yes—
104C2×Q8C22×C412yes—
105C2×Q8#4114yes—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C22×C412yes—
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C22×C4#4112yes—
124C22×C4C2411yescharacteristic
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—44no—
131C2×C8C22×C412yes—
132C2×C8C22×C412yes—
133C4⋊C4—24no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4C22×C412yes—
143C4⋊C4C22×C412yes—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C42—22no—
149C42—22no—
150C42—44no—
151C42—44no—
152C23—22no—
153C23—22no—
154C23C4×D412yes—
155C23AGL(1,ℤ/8)12yes—
156C23#13712yes—
157C23C22×D411yescharacteristic
158Q8—22no—
159Q8—24no—
160Q8—22no—
161Q8—22no—
162Q8—48no—
163Q8—42no—
164Q8#11714yescomplemented
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—48no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4C4×D412yes—
194C2×C4C23×C411yescharacteristic
195C2×C4C2×#4112yes—
196C2×C4C2×#4111yescharacteristic
197C2×C4H(5,𝔽2)12yes—
198C8—22no—
199C8—22no—
200C8—42no—
201C22—22no—
202C22—22no—
203C22—24no—
204C22—24no—
205C22#44014yes—
206C22C2×C4×D411yescharacteristic
207C22#54612yes—
208C22C2×AGL(1,ℤ/8)11yescharacteristic
209C22C2×#13711yescharacteristic
210C22#57812yes—
211C4—22no—
212C4—24no—
213C4—42no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—42no—
218C4—82no—
219C4—82no—
220C4C2×#11711yescharacteristic
221C4#51811yescharacteristic
222C4#55112yes—
223C2—24no—
224C2#234211yescharacteristic
225C2C2×#44011yescharacteristic
226C2#295011yescharacteristic
227C2#330812yes—
228C2#331212yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1935C2semidirect
5. #1928C2semidirect
6. #1863C2semidirect
10. #1763C2semidirect
18. C2×#103C4semidirect
164. Q8#117semidirect

Maximal subgroups

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

subgroupindex
C22×#1032
#23442
Q8×C22⋊C42
#19352
#19282
#18632
#18232
#18082
#17872
#17632
#17522