Finite Group Databasebeta

← #10222

Subgroups — #10222

Order 256.

242 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 57 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#3230C211yesmaximal, characteristic, complemented
2#3206C211yesmaximal, characteristic
3#3135C211yesmaximal, characteristic
4#1993C211yesmaximal, characteristic, complemented
5#1518C211yesmaximal, characteristic, complemented
6#1517C211yesmaximal, characteristic
7#1281C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#500—21no—
12#492—21no—
13#491—21no—
14#490—21no—
15#489—21no—
16#489—21no—
17#487—21no—
18#487—21no—
19#484—22no—
20#477—22no—
21#461—22no—
22#448—22no—
23#431C2211yescharacteristic
24#431C2211yescharacteristic
25#416—22no—
26C2×#97C2211yescharacteristic
27#382C2211yescharacteristic
28#340—22no—
29#328—22no—
30#327—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C2×SD16—24no—
34C22×C8D412yes—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8D412yes—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#127D411yescharacteristic
43#125—21no—
44#124—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47#121—21no—
48#121—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×D4—21no—
52C4×D4—21no—
53#117—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C42C2311yescharacteristic
58#107—24no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61#103—22no—
62#103—22no—
63#103—24no—
64#103—24no—
65#102—22no—
66#102—22no—
67C22⋊C8D412yes—
68C22⋊C8D412yes—
69#97—21no—
70#97—21no—
71C4×C8—21no—
72C4×C8—21no—
73#95—21no—
74#95—21no—
75#41—42no—
76#41—42no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—22no—
80C2×Q8—21no—
81C2×Q8—41no—
82C2×Q8—82no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93SD16—84no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—22no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—22no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—42no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42—41no—
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42#4111yescharacteristic
120C23—41no—
121C23#12711yescharacteristic
122Q8—44no—
123Q8—42no—
124Q8—42no—
125Q8—81no—
126Q8—82no—
127D4—42no—
128D4—42no—
129D4—81no—
130D4—81no—
131D4—81no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4#12011yescharacteristic
144C2×C4#12011yescharacteristic
145C2×C4#12011yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#13512yes—
150C2×C4#13712yes—
151C8—42no—
152C8—42no—
153C8—42no—
154C22—21no—
155C22—21no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22#39211yescharacteristic
160C22#49511yescharacteristic
161C22#49711yescharacteristic
162C4—21no—
163C4—21no—
164C4—21no—
165C4—21no—
166C4—82no—
167C4—81no—
168C4—81no—
169C4—81no—
170C4—161no—
171C4#45212yes—
172C4#46412yes—
173C2—21no—
174C2—21no—
175C2—161no—
176C2#200811yescharacteristic
177C2#200911yescharacteristic
178C2#201511yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ 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. #3230C2semidirect
4. #1993C2semidirect
5. #1518C2semidirect
7. #1281C2semidirect

Maximal subgroups

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

subgroupindex
#32302
#32062
#31352
#19932
#15182
#15172
#12812