Finite Group Databasebeta

← #8478

Subgroups — #8478

Order 256.

328 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 75 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#3504C211yesmaximal, characteristic
2#1544C211yesmaximal, characteristic, complemented
3#1537C211yesmaximal, characteristic, complemented
4#1488C212yesmaximal
5#1470C212yesmaximal, complemented
6#557—22no—
7#554—22no—
8#554—24no—
9#544—22no—
10#544C412yes—
11#533—21no—
12#533C2211yescharacteristic
13C2×#125—21no—
14C2×#125C2211yescharacteristic
15#520C412yes—
16C4×#41C2211yescharacteristic
17#432C2212yes—
18#431C2212yes—
19#332—22no—
20#330—21no—
21#330—21no—
22Q8⋊C8—22no—
23D4⋊C8—22no—
24#324—24no—
25C2×#41C2×C411yescharacteristic
26C4⋊Q8—24no—
27C4⋊Q8—22no—
28C4⋊Q8—22no—
29C4⋊Q8D412yes—
30#125—24no—
31#125—24no—
32#125—22no—
33#125—22no—
34#125—21no—
35#125—21no—
36#125—22no—
37#125D411yescharacteristic
38#125D411yescharacteristic
39C22⋊Q8—24no—
40C22⋊Q8—24no—
41C22⋊Q8—24no—
42C22⋊Q8—24no—
43C22⋊Q8—22no—
44C4×Q8—24no—
45C4×Q8—22no—
46C4×Q8—22no—
47C4×Q8D412yes—
48C4×D4—22no—
49C4×D4—24no—
50C4×D4—22no—
51C4×D4D412yes—
52#117—22no—
53#117—21no—
54#117C2×C411yescharacteristic
55#117C2×C412yes—
56C2×C4⋊C4—24no—
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—22no—
69#41—22no—
70C2×Q8—24no—
71C2×Q8—22no—
72C2×Q8C22⋊C411yescharacteristic
73C2×D4—21no—
74C2×D4C22⋊C411yescharacteristic
75C22×C4—21no—
76C22×C4—24no—
77C22×C4—22no—
78C22×C4C22⋊C414yes—
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D412yes—
81C2×C8—42no—
82C2×C8—42no—
83C4⋊C4—24no—
84C4⋊C4—24no—
85C4⋊C4—24no—
86C4⋊C4—24no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4C22⋊C412yes—
96C4⋊C4C22⋊C412yes—
97C4⋊C4C22⋊C412yes—
98C22⋊C4—24no—
99C22⋊C4—22no—
100C22⋊C4—21no—
101C22⋊C4—22no—
102C42—24no—
103C42—21no—
104C42—22no—
105C42—22no—
106C42C2×D412yes—
107C42C2×D411yescharacteristic
108C42C2×D411yescharacteristic
109C23—21no—
110C23C2×C22⋊C411yescharacteristic
111Q8—21no—
112Q8—82no—
113Q8—84no—
114Q8#10112yes—
115D4—24no—
116D4—21no—
117D4#10112yes—
118C2×C4—24no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—42no—
127C2×C4—44no—
128C2×C4—82no—
129C2×C4—84no—
130C2×C4C2×C22⋊C411yescharacteristic
131C2×C4C2×C22⋊C411yescharacteristic
132C2×C4#12012yes—
133C2×C4#12012yes—
134C8—82no—
135C8—82no—
136C22—22no—
137C22—21no—
138C22—22no—
139C22—42no—
140C22#37911yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—22no—
145C4—44no—
146C4—42no—
147C4—42no—
148C4—42no—
149C4C2×#10111yescharacteristic
150C4C2×#10111yescharacteristic
151C2—41no—
152C2—42no—
153C2#178411yescharacteristic
154C2#179412yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#41C2×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
2. #1544C2semidirect
3. #1537C2semidirect
5. #1470C2semidirect

Maximal subgroups

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

subgroupindex
#35042
#15442
#15372
#14882
#14702