Finite Group Databasebeta

← #24698

Subgroups — #24698

Order 256.

374 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 127 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#2356C211yesmaximal, characteristic
5#2072C212yesmaximal
6#2056C212yesmaximal
7#1993C211yesmaximal, characteristic
8#1948C212yesmaximal
9#1940C211yesmaximal, characteristic
10#1927C212yesmaximal
11#1805C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#475—24no—
17#441—24no—
18C2×#106C2212yes—
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#395—21no—
28#395—21no—
29#391C2212yes—
30#391C2212yes—
31#389—22no—
32#387—22no—
33#384C2212yes—
34#383—21no—
35#382C2211yescharacteristic
36C4×C4⋊C4C2212yes—
37C4×C4⋊C4C2211yescharacteristic
38#376—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C2×Q16—22no—
47C2×Q16—22no—
48C2×Q16—22no—
49C22×C8C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D414yes—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D412yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42C2312yes—
72C2×C42C2311yescharacteristic
73#106—22no—
74#106—24no—
75C4⋊C8—24no—
76C4⋊C8—22no—
77#103—22no—
78#103—24no—
79#103—24no—
80#103—22no—
81#97—22no—
82#95—21no—
83#95—22no—
84#95—22no—
85#95—21no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95D411yescharacteristic
90#95D411yescharacteristic
91C2×Q8—21no—
92C2×Q8—21no—
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×Q8#4112yes—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D412yes—
115C22×C4C2411yescharacteristic
116Q16—28no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8#4112yes—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—82no—
138C4⋊C4—82no—
139C4⋊C4C2×D412yes—
140C42—24no—
141C42—22no—
142C42—22no—
143C42—42no—
144C42—44no—
145C42C2×D412yes—
146C42#4112yes—
147C23C22×D411yescharacteristic
148Q8—22no—
149Q8—22no—
150Q8—22no—
151Q8—22no—
152Q8—41no—
153Q8—41no—
154Q8—41no—
155Q8—41no—
156Q8—84no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4#12112yes—
181C2×C4#12112yes—
182C2×C4#13714yes—
183C2×C4C22×D411yescharacteristic
184C2×C4C2×#4111yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C2×C42−1+411yescharacteristic
188C2×C42−1+411yescharacteristic
189C8—22no—
190C8—42no—
191C8—41no—
192C22C2×#12111yescharacteristic
193C22#54711yescharacteristic
194C22#54811yescharacteristic
195C22C2×#13712yes—
196C22#57712yes—
197C4—24no—
198C4—22no—
199C4—22no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—42no—
204C4—82no—
205C4—81no—
206C4—82no—
207C4#52511yescharacteristic
208C4#53611yescharacteristic
209C4#55211yescharacteristic
210C4#55611yescharacteristic
211C2#241511yescharacteristic
212C2#303911yescharacteristic
213C2#304311yescharacteristic
214C2#334412yes—
215C2#336912yes—
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

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
#23562
#20722
#20562
#19932
#19482
#19402
#19272
#18052