Finite Group Databasebeta

← #8498

Subgroups — #8498

Order 256.

346 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 71 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#3544C211yesmaximal, characteristic
2#1960C211yesmaximal, characteristic, complemented
3#1558C211yesmaximal, characteristic, complemented
4#1532C212yesmaximal, complemented
5#1512C212yesmaximal, complemented
6#563—22no—
7Q82C411yescharacteristic, complemented
8#555C411yescharacteristic, complemented
9#554C412yescomplemented
10#552—22no—
11#547—22no—
12#544—22no—
13#542—22no—
14#536—22no—
15C2×C4⋊Q8C2211yescharacteristic
16#529C2212yes—
17#432C2212yes—
18C2×#103—22no—
19C2×#97C2211yescharacteristic
20#382C2211yescharacteristic
21#333—22no—
22#330—22no—
23#329—22no—
24#328—22no—
25#327—22no—
26C2×#41—22no—
27C22×Q8D412yes—
28C22×C8—21no—
29C4⋊Q8—22no—
30C4⋊Q8—22no—
31C4⋊Q8—22no—
32C4⋊Q8—22no—
33C4⋊Q8C2×C411yescharacteristic
34C4⋊Q8C2×C411yescharacteristic
35C4⋊Q8D412yes—
36#126—24no—
37#126—22no—
38#125—24no—
39#125—22no—
40#125D412yes—
41#124—22no—
42#124—22no—
43#124D412yes—
44#123—24no—
45#123—24no—
46#123—22no—
47C22⋊Q8—24no—
48C22⋊Q8—24no—
49C22⋊Q8—24no—
50C22⋊Q8—22no—
51C22⋊Q8D412yes—
52#121—22no—
53#121D412yes—
54C4×Q8—22no—
55C4×Q8C2×C412yes—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4C2×C412yes—
59#117—22no—
60#117—22no—
61#117—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C42C2311yescharacteristic
66C4⋊C8—22no—
67#103—44no—
68C22⋊C8—22no—
69#97—22no—
70#97—22no—
71#95—22no—
72#41—82no—
73#41—82no—
74C2×Q8—22no—
75C2×Q8—22no—
76C2×Q8—42no—
77C2×Q8—42no—
78C2×Q8C22⋊C412yes—
79C2×Q8C22⋊C412yes—
80C2×Q8C22⋊C412yes—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4C22⋊C412yes—
84C22×C4—22no—
85C22×C4—24no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D412yes—
90C2×C8—22no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—42no—
94C4⋊C4—24no—
95C4⋊C4—24no—
96C4⋊C4—24no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—24no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4C22⋊C412yes—
106C4⋊C4C22⋊C412yes—
107C22⋊C4—24no—
108C22⋊C4—24no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—22no—
113C22⋊C4—22no—
114C42—22no—
115C42—22no—
116C42—22no—
117C42C22×C411yescharacteristic
118C42C2×D412yes—
119C42C2×D411yescharacteristic
120C23—22no—
121C23#12011yescharacteristic
122Q8—42no—
123Q8—42no—
124Q8—42no—
125Q8—82no—
126Q8—82no—
127D4—42no—
128D4—42no—
129D4—42no—
130D4—82no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—84no—
143C2×C4—82no—
144C2×C4—81no—
145C2×C4C2×C22⋊C412yes—
146C2×C4C2×C22⋊C411yescharacteristic
147C2×C4#12012yes—
148C2×C4#12011yescharacteristic
149C8—42no—
150C8—82no—
151C22—22no—
152C22—42no—
153C22—42no—
154C22—42no—
155C22#37911yescharacteristic
156C22#45311yescharacteristic
157C22#45511yescharacteristic
158C4—21no—
159C4—21no—
160C4—22no—
161C4—22no—
162C4—42no—
163C4—42no—
164C4—42no—
165C4—42no—
166C4—42no—
167C4—161no—
168C2—21no—
169C2—21no—
170C2—82no—
171C2#179211yescharacteristic
172C2#179612yes—
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3544C2
LCSG ▷ 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
2. #1960C2semidirect
3. #1558C2semidirect
4. #1532C2semidirect
5. #1512C2semidirect
7. Q82C4semidirect
8. #555C4semidirect
9. #554C4semidirect

Maximal subgroups

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

subgroupindex
#35442
#19602
#15582
#15322
#15122