Finite Group Databasebeta

← #28660

Subgroups — #28660

Order 256.

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

normal subgroups only

This group has 365 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×#501C211yesmaximal, characteristic
2C2×#491C211yesmaximal, characteristic
3#2862C211yesmaximal, characteristic
4C4×#125C211yesmaximal, characteristic
5#2068C212yesmaximal
6#2066C212yesmaximal
7#1993C211yesmaximal, characteristic
8#1960C211yesmaximal, characteristic
9#1928C212yesmaximal
10#1927C212yesmaximal
11#1767C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15#501—24no—
16#491—24no—
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#97C2211yescharacteristic
25C2×#97C2211yescharacteristic
26#391—21no—
27#391—21no—
28#391C2212yes—
29#391C2212yes—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#387—21no—
33#387—21no—
34#382—21no—
35#382—21no—
36#382C2211yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2211yescharacteristic
40C4×C4⋊C4C2211yescharacteristic
41C43C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8C2312yes—
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53#125—22no—
54#125D414yes—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C42—21no—
68C2×C42—21no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—24no—
77#106—24no—
78C4⋊C8—24no—
79#103—24no—
80#103—24no—
81#103—24no—
82#103—24no—
83#97—22no—
84#97—22no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8#4114yes—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4C2×D412yes—
137C4⋊C4C2×D412yes—
138C42—22no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—41no—
143C42—41no—
144C42—42no—
145C42—42no—
146C42C2×D412yes—
147C42#4112yes—
148C23C22×D411yescharacteristic
149Q8—42no—
150Q8—42no—
151Q8—42no—
152Q8—42no—
153Q8—84no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4#12112yes—
187C2×C4#12112yes—
188C2×C4#13712yes—
189C2×C4#13712yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C2×#4112yes—
192C2×C4C2×#4111yescharacteristic
193C2×C42−1+412yes—
194C8—24no—
195C8—42no—
196C22C2×#12111yescharacteristic
197C22#53211yescharacteristic
198C22#53611yescharacteristic
199C22C2×#13711yescharacteristic
200C22C2×#13711yescharacteristic
201C22#57511yescharacteristic
202C22#57511yescharacteristic
203C4—22no—
204C4—22no—
205C4—22no—
206C4—22no—
207C4—41no—
208C4—41no—
209C4—42no—
210C4—42no—
211C4—81no—
212C4—81no—
213C4—82no—
214C4—82no—
215C4#54812yes—
216C4#55612yes—
217C2#254511yescharacteristic
218C2#303911yescharacteristic
219C2#304011yescharacteristic
220C2#317211yescharacteristic
221C2#318111yescharacteristic
222C2#323611yescharacteristic
223C2#323711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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×#5012
C2×#4912
#28622
C4×#1252
#20682
#20662
#19932
#19602
#19282
#19272
#17672