Finite Group Databasebeta

← #25683

Subgroups — #25683

Order 256.

480 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 131 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×#541C211yesmaximal, characteristic
2C2×#499C211yesmaximal, characteristic, complemented
3C2×#491C211yesmaximal, characteristic, complemented
4#2629C211yesmaximal, characteristic
5#2036C212yesmaximal
6#2035C212yesmaximal
7#1992C211yesmaximal, characteristic, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1865C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic
12#541—24no—
13#541—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—22no—
16C2×#125—22no—
17C2×#125C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8C2212yes—
20C2×C4×Q8C2212yes—
21C2×#117C2211yescharacteristic
22#499—24no—
23#491—24no—
24C2×#107C2212yes—
25C2×#106C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#398—22no—
35#396—22no—
36#389C2212yes—
37#384C2211yescharacteristic
38#380C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8C2312yes—
44C23×C4C2311yescharacteristic
45C22×C8C2311yescharacteristic
46C22×C8C2311yescharacteristic
47C22×C8C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C4⋊Q8—22no—
50C4⋊Q8D414yes—
51#126—216no—
52#125—216no—
53#125—22no—
54#125—22no—
55#125—22no—
56#125D414yes—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C4×Q8—24no—
60C4×Q8—24no—
61#117—22no—
62#117—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4D412yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4D412yes—
77C2×C42—22no—
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—24no—
81#106—24no—
82#103—24no—
83#103—24no—
84#103—24no—
85#103—24no—
86C22⋊C8—22no—
87C22⋊C8—22no—
88#97—22no—
89C4×C8—22no—
90#95—22no—
91#95—22no—
92C24C2×D411yescharacteristic
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8C2×D414yes—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2411yescharacteristic
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8#4112yes—
124C2×C8#4112yes—
125C4⋊C4—28no—
126C4⋊C4—28no—
127C4⋊C4—28no—
128C4⋊C4—28no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—44no—
136C4⋊C4—82no—
137C4⋊C4—82no—
138C4⋊C4C2×D414yes—
139C22⋊C4—28no—
140C22⋊C4—24no—
141C22⋊C4—82no—
142C42—28no—
143C42—21no—
144C42—21no—
145C42C2×D412yes—
146C42C2×D412yes—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—22no—
151C23—22no—
152C23C22×D411yescharacteristic
153Q8—24no—
154Q8—24no—
155Q8—42no—
156Q8—42no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4#12014yes—
182C2×C4C22×D411yescharacteristic
183C2×C4C22×D411yescharacteristic
184C2×C4C2×#4111yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C42−1+411yescharacteristic
187C2×C42−1+411yescharacteristic
188C8—22no—
189C8—22no—
190C8—41no—
191C8—41no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—42no—
196C22—42no—
197C22C2×#12011yescharacteristic
198C22#53211yescharacteristic
199C22#53611yescharacteristic
200C22#57712yes—
201C22#57812yes—
202C4—24no—
203C4—41no—
204C4—44no—
205C4—42no—
206C4—44no—
207C4—82no—
208C4—82no—
209C4#54711yescharacteristic
210C4#54711yescharacteristic
211C4#54811yescharacteristic
212C4#54811yescharacteristic
213C2—81no—
214C2#243211yescharacteristic
215C2#301312yes—
216C2#317411yescharacteristic
217C2#317711yescharacteristic
218C2#324212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#499C2semidirect
3. C2×#491C2semidirect
7. #1992C2semidirect
8. #1956C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4992
C2×#4912
#26292
#20362
#20352
#19922
#19562
#18652
#18632
#17502