Finite Group Databasebeta

← #24831

Subgroups — #24831

Order 256.

396 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 139 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×#475C211yesmaximal, characteristic
2C2×#441C211yesmaximal, characteristic
3#2567C211yesmaximal, characteristic
4#2351C211yesmaximal, characteristic
5#2084C212yesmaximal
6#2066C212yesmaximal
7#1993C211yesmaximal, characteristic
8#1940C211yesmaximal, characteristic
9#1878C212yesmaximal
10#1805C211yesmaximal, characteristic
11#1803C212yesmaximal
12C22×Q16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17#475—24no—
18#441—24no—
19C2×#106C2212yes—
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×#97C2211yescharacteristic
28#395—21no—
29#395—21no—
30#391C2212yes—
31#389—22no—
32#389—22no—
33#389C2212yes—
34#384C2212yes—
35#382—21no—
36#382C2211yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C2×Q16—22no—
47C2×Q16—22no—
48C2×Q16Q814yescomplemented
49C22×C8C2312yes—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4×Q8—22no—
55C4×Q8—24no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×Q8—42no—
59C4×Q8D414yes—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4D412yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C42—21no—
73C2×C42—22no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#106—22no—
78#106—24no—
79C4⋊C8—22no—
80C4⋊C8—42no—
81#103—22no—
82#103—24no—
83#103—22no—
84#103—42no—
85#97—22no—
86#95—21no—
87#95—22no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95D411yescharacteristic
92#95D411yescharacteristic
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—42no—
102C2×Q8—82no—
103C2×Q8C2×D412yes—
104C2×Q8C2×Q812yes—
105C2×Q8C2×Q812yes—
106C2×Q8#4112yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2411yescharacteristic
118Q16—28no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8C2×Q812yes—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—44no—
141C4⋊C4—82no—
142C4⋊C4—82no—
143C4⋊C4C2×D412yes—
144C42—24no—
145C42—22no—
146C42—22no—
147C42—44no—
148C42—42no—
149C42C2×D412yes—
150C42#4112yes—
151C23C22×D411yescharacteristic
152Q8—22no—
153Q8—22no—
154Q8—22no—
155Q8—22no—
156Q8—22no—
157Q8—41no—
158Q8—41no—
159Q8—84no—
160Q8C22⋊Q814yes—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4#12112yes—
185C2×C4#12112yes—
186C2×C4#13714yes—
187C2×C4C22×D411yescharacteristic
188C2×C4C22×Q811yescharacteristic
189C2×C4C2×#4111yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4H(5,𝔽2)11yescharacteristic
192C2×C42−1+411yescharacteristic
193C8—22no—
194C8—41no—
195C8—42no—
196C22C2×#12111yescharacteristic
197C22#54611yescharacteristic
198C22D4×Q811yescharacteristic
199C22C2×#13712yes—
200C22#57712yes—
201C4—24no—
202C4—22no—
203C4—22no—
204C4—24no—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—81no—
209C4—82no—
210C4—82no—
211C4C2×C22⋊Q811yescharacteristic
212C4#53711yescharacteristic
213C4#55211yescharacteristic
214C4#55411yescharacteristic
215C2#241711yescharacteristic
216C2#303911yescharacteristic
217C2#304311yescharacteristic
218C2#331612yes—
219C2#338712yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
48. C2×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4412
#25672
#23512
#20842
#20662
#19932
#19402
#18782
#18052
#18032