Finite Group Databasebeta

← #9188

Subgroups — #9188

Order 256.

239 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 61 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
1#2983C211yesmaximal, characteristic, complemented
2#2632C211yesmaximal, characteristic
3C4×#101C211yesmaximal, characteristic, complemented
4#1527C212yesmaximal
5#1288C212yesmaximal, complemented
6C2×#124C2211yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8#446—24no—
9#432—21no—
10#432C2212yes—
11#431—21no—
12C2×#101—21no—
13C4×M16C2212yes—
14#399—21no—
15#395C411yescharacteristic
16#395C411yescharacteristic
17#394C411yescharacteristic
18#394C411yescharacteristic
19#393—21no—
20#388—21no—
21#388—21no—
22#383C2211yescharacteristic
23#343—21no—
24#329—24no—
25C22×Q8C2×C411yescharacteristic
26C22×Q8D411yescharacteristic
27C22×D4C2×C411yescharacteristic
28C2×M16—22no—
29#124—22no—
30#124—22no—
31C4×Q8—22no—
32C4×Q8—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C42D411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42C2311yescharacteristic
47C4⋊C8—22no—
48C4⋊C8—22no—
49C4⋊C8—22no—
50#101—24no—
51C22⋊C8—22no—
52#97—22no—
53#97—22no—
54#97D414yes—
55C4×C8—22no—
56C4×C8—22no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95C2×C411yescharacteristic
62#95C2×C411yescharacteristic
63C24—21no—
64C2×Q8—22no—
65C2×Q8—22no—
66C2×Q8—22no—
67C2×Q8—41no—
68C2×Q8—81no—
69C2×D4—22no—
70C2×D4—81no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22×C411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81M16—24no—
82M16—42no—
83C2×C8—24no—
84C2×C8—42no—
85C4⋊C4—22no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C4⋊C4—81no—
89C4⋊C4—81no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—41no—
93C22⋊C4—42no—
94C22⋊C4—41no—
95C22⋊C4—81no—
96C22⋊C4—81no—
97C42—21no—
98C42—21no—
99C42—21no—
100C42—21no—
101C42C2×D412yes—
102C42#4112yes—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23C2×C22⋊C411yescharacteristic
111Q8—44no—
112Q8—82no—
113D4—82no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4#10412yes—
137C2×C4C4×D412yes—
138C2×C4#12011yescharacteristic
139C2×C4#12112yes—
140C2×C4#12311yescharacteristic
141C8—44no—
142C8—44no—
143C22—21no—
144C22—21no—
145C22—81no—
146C22—81no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22—81no—
151C22—81no—
152C22#38611yescharacteristic
153C22C2×#10411yescharacteristic
154C22#42111yescharacteristic
155C4—22no—
156C4—22no—
157C4—21no—
158C4—21no—
159C4—22no—
160C4—41no—
161C4—41no—
162C4—41no—
163C4—81no—
164C4—81no—
165C4—81no—
166C2—21no—
167C2—21no—
168C2—161no—
169C2#187111yescharacteristic
170C2#191211yescharacteristic
171C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2983C2semidirect
3. C4×#101C2semidirect
5. #1288C2semidirect

Maximal subgroups

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

subgroupindex
#29832
#26322
C4×#1012
#15272
#12882