Finite Group Databasebeta

← #28729

Subgroups — #28729

Order 256.

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

normal subgroups only

This group has 363 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×#491C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic
3#2660C211yesmaximal, characteristic
4#2327C211yesmaximal, characteristic
5#2074C212yesmaximal
6#2072C212yesmaximal
7#1993C211yesmaximal, characteristic
8#1991C211yesmaximal, characteristic
9#1805C212yesmaximal
10#1803C212yesmaximal
11#1768C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C4×Q8C2212yes—
16#491—24no—
17#487—24no—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#97C2211yescharacteristic
24C2×C4×C8C2211yescharacteristic
25#398—22no—
26#395—22no—
27#391C2212yes—
28#387—22no—
29#387—22no—
30#384C2211yescharacteristic
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33#382C2211yescharacteristic
34#382C2211yescharacteristic
35#376C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8C2312yes—
43C22×C8C2312yes—
44C22×C8C2312yes—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8D414yes—
49#125—22no—
50#125—22no—
51C4×Q8—28no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C42—22no—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69C4⋊C8—24no—
70#103—24no—
71#103—24no—
72#103—24no—
73#103—24no—
74#97—22no—
75C4×C8—22no—
76#95—21no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95—24no—
81#95D412yes—
82#95D412yes—
83C2×Q8—22no—
84C2×Q8—24no—
85C2×Q8—24no—
86C2×Q8—42no—
87C2×Q8—82no—
88C2×Q8C2×D414yes—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4C2×D412yes—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D412yes—
99C22×C4C2411yescharacteristic
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—24no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—44no—
121C4⋊C4—82no—
122C4⋊C4—82no—
123C42—22no—
124C42—44no—
125C42C2×D412yes—
126C42#4112yes—
127C42#4114yes—
128C23C22×D411yescharacteristic
129Q8—28no—
130Q8—44no—
131Q8—84no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4#12112yes—
154C2×C4#12112yes—
155C2×C4#12414yes—
156C2×C4C22×D411yescharacteristic
157C2×C4C2×#4112yes—
158C2×C4C2×#4111yescharacteristic
159C2×C42−1+412yes—
160C8—42no—
161C8—42no—
162C22C2×#12111yescharacteristic
163C22C2×#12411yescharacteristic
164C22#53611yescharacteristic
165C22#57712yes—
166C22#57812yes—
167C4—24no—
168C4—22no—
169C4—22no—
170C4—42no—
171C4—44no—
172C4—82no—
173C4—81no—
174C4—81no—
175C4—81no—
176C4—81no—
177C4#54712yes—
178C4#55612yes—
179C2#254711yescharacteristic
180C2#304311yescharacteristic
181C2#304411yescharacteristic
182C2#313712yes—
183C2#324212yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ 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×#4912
C2×#4872
#26602
#23272
#20742
#20722
#19932
#19912
#18052
#18032
#17682