Finite Group Databasebeta

← #24699

Subgroups — #24699

Order 256.

366 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 123 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×#477C211yesmaximal, characteristic
2C2×#441C211yesmaximal, characteristic
3#2568C211yesmaximal, characteristic
4#2356C211yesmaximal, characteristic
5#2074C212yesmaximal
6#2056C212yesmaximal
7#1993C211yesmaximal, characteristic
8#1949C212yesmaximal
9#1940C211yesmaximal, characteristic
10#1928C212yesmaximal
11#1805C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#477—24no—
17#441—24no—
18C2×#107C2212yes—
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#400—21no—
28#400—21no—
29#391—22no—
30#391C2212yes—
31#391C2212yes—
32#387—22no—
33#383—21no—
34#382C2212yes—
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⋊C4D412yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C42—21no—
67C2×C42—21no—
68C2×C42—22no—
69C2×C42C2312yes—
70C2×C42C2311yescharacteristic
71#107—24no—
72#106—22no—
73C4⋊C8—24no—
74C4⋊C8—22no—
75#103—22no—
76#103—24no—
77#103—24no—
78#103—22no—
79#97—22no—
80#95—21no—
81#95—22no—
82#95—22no—
83#95—21no—
84#95—22no—
85#95—22no—
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—44no—
102C2×Q8—42no—
103C2×Q8C2×D412yes—
104C2×Q8#4112yes—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4C2×D412yes—
112C22×C4C2×D412yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
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—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4C2×D412yes—
138C42—24no—
139C42—22no—
140C42—22no—
141C42—44no—
142C42—42no—
143C42C2×D412yes—
144C42#4112yes—
145C23C22×D411yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—22no—
149Q8—22no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154Q8—84no—
155C2×C4—22no—
156C2×C4—22no—
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—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4#12112yes—
179C2×C4#12112yes—
180C2×C4C22×D411yescharacteristic
181C2×C4C2×#4111yescharacteristic
182C2×C4C2×#4111yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C2×C42−1+411yescharacteristic
185C2×C42−1+411yescharacteristic
186C8—22no—
187C8—42no—
188C8—41no—
189C22C2×#12111yescharacteristic
190C22#54711yescharacteristic
191C22#54811yescharacteristic
192C22#57512yes—
193C22#57812yes—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—42no—
198C4—42no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4—82no—
203C4—81no—
204C4#52511yescharacteristic
205C4#53611yescharacteristic
206C4#55211yescharacteristic
207C4#55611yescharacteristic
208C2#241511yescharacteristic
209C2#304011yescharacteristic
210C2#304411yescharacteristic
211C2#334912yes—
212C2#337712yes—
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×#4772
C2×#4412
#25682
#23562
#20742
#20562
#19932
#19492
#19402
#19282
#18052