Finite Group Databasebeta

← #8476

Subgroups — #8476

Order 256.

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

normal subgroups only

This group has 339 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#3512C211yesmaximal, characteristic
2#1543C211yesmaximal, characteristic, complemented
3#1536C211yesmaximal, characteristic, complemented
4#1488C212yesmaximal, complemented
5#1470C212yesmaximal, complemented
6#553—22no—
7#553—24no—
8#552—22no—
9#541—21no—
10#540—21no—
11#537C412yes—
12#532C2211yescharacteristic
13#529—22no—
14C2×#125C2211yescharacteristic
15#520C412yes—
16C4×#41C2211yescharacteristic
17#432C2212yes—
18#431C2212yes—
19#333—21no—
20#331—21no—
21#329—22no—
22Q8⋊C8—22no—
23D4⋊C8—22no—
24#324—24no—
25C2×#41C2×C411yescharacteristic
26C2×#41C2×C412yes—
27C4⋊Q8D411yescharacteristic
28#127D411yescharacteristic
29#126—24no—
30#126—24no—
31#125—24no—
32#125—21no—
33#125—21no—
34#125—22no—
35#124D412yes—
36#123—24no—
37#123—24no—
38C22⋊Q8—22no—
39C22⋊Q8—22no—
40#121—24no—
41#121—22no—
42#121—22no—
43#121—24no—
44#121—22no—
45C4×Q8—22no—
46C4×Q8D412yes—
47C4×D4—24no—
48C4×D4—22no—
49C4×D4—24no—
50C4×D4—22no—
51C4×D4—22no—
52C4×D4D412yes—
53#117—24no—
54#117—22no—
55#117—21no—
56#117C2×C411yescharacteristic
57C2×C4⋊C4—22no—
58C2×C4⋊C4C2×C412yes—
59C2×C4⋊C4D414yes—
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62C4⋊C8—22no—
63C4⋊C8—22no—
64C22⋊C8—22no—
65C22⋊C8—22no—
66#97—22no—
67C4×C8—22no—
68#41—42no—
69#41—44no—
70C2×Q8C22⋊C411yescharacteristic
71C2×Q8C22⋊C412yes—
72C2×D4—22no—
73C2×D4—21no—
74C2×D4—24no—
75C2×D4—22no—
76C2×D4C22⋊C411yescharacteristic
77C2×D4C22⋊C412yes—
78C22×C4—24no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4C22⋊C414yes—
83C22×C4C22×C411yescharacteristic
84C22×C4C2×D412yes—
85C2×C8—42no—
86C2×C8—42no—
87C4⋊C4—24no—
88C4⋊C4—24no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—24no—
92C4⋊C4—24no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4C22⋊C412yes—
96C22⋊C4—24no—
97C22⋊C4—24no—
98C22⋊C4—24no—
99C22⋊C4—21no—
100C22⋊C4—22no—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42—24no—
104C42—21no—
105C42—22no—
106C42C2×D412yes—
107C42C2×D411yescharacteristic
108C42C2×D411yescharacteristic
109C23—22no—
110C23—21no—
111C23C2×C22⋊C411yescharacteristic
112Q8—22no—
113Q8—42no—
114D4—22no—
115D4—42no—
116D4—42no—
117D4—44no—
118D4—82no—
119D4—84no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—44no—
128C2×C4—44no—
129C2×C4—82no—
130C2×C4—84no—
131C2×C4C2×C22⋊C411yescharacteristic
132C2×C4C2×C22⋊C411yescharacteristic
133C2×C4#12012yes—
134C2×C4#12012yes—
135C8—82no—
136C8—82no—
137C22—22no—
138C22—21no—
139C22—42no—
140C22—42no—
141C22—42no—
142C22—41no—
143C22—42no—
144C22#37911yescharacteristic
145C4—22no—
146C4—22no—
147C4—21no—
148C4—44no—
149C4—42no—
150C4—41no—
151C4—42no—
152C4#41312yes—
153C2—41no—
154C2—82no—
155C2—81no—
156C2#178511yescharacteristic
157C2#179412yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3512C2
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
2. #1543C2semidirect
3. #1536C2semidirect
4. #1488C2semidirect
5. #1470C2semidirect

Maximal subgroups

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

subgroupindex
#35122
#15432
#15362
#14882
#14702