Finite Group Databasebeta

← #8477

Subgroups — #8477

Order 256.

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

normal subgroups only

This group has 341 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#3510C211yesmaximal, characteristic
2#1544C211yesmaximal, characteristic, complemented
3#1537C211yesmaximal, characteristic, complemented
4#1488C212yesmaximal, complemented
5#1469C212yesmaximal, complemented
6#559—22no—
7#556—22no—
8#551—24no—
9#542—22no—
10#536C411yescharacteristic
11#535C411yescharacteristic
12#533C2211yescharacteristic
13#529—22no—
14C2×#124C2211yescharacteristic
15#519C411yescharacteristic
16#518C411yescharacteristic
17C4×#41C2211yescharacteristic
18#432C2212yes—
19#431C2212yes—
20#332—22no—
21#330—22no—
22Q8⋊C8—22no—
23D4⋊C8—22no—
24#323—24no—
25C2×#41C2×C411yescharacteristic
26C22×Q8C2×C411yescharacteristic
27C22×D4C2×C411yescharacteristic
28C4⋊Q8—21no—
29C4⋊Q8D412yes—
30#127—21no—
31#126—24no—
32#126—24no—
33#125D412yes—
34#124—24no—
35#124—21no—
36#124—21no—
37#124—21no—
38#124—21no—
39#124—22no—
40#123—24no—
41#123—24no—
42#123—22no—
43C22⋊Q8—24no—
44C22⋊Q8—22no—
45C22⋊Q8—22no—
46#121—22no—
47#120—22no—
48C4×Q8—22no—
49C4×Q8—22no—
50C4×Q8—22no—
51C4×Q8D412yes—
52C4×D4—24no—
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4D412yes—
57#117—24no—
58#117—22no—
59#117—21no—
60#117C2×C411yescharacteristic
61#117C2×C411yescharacteristic
62#117C2×C411yescharacteristic
63C2×C22⋊C4—22no—
64C2×C22⋊C4D414yes—
65C2×C42—21no—
66C2×C42C2311yescharacteristic
67C4⋊C8—22no—
68C4⋊C8—22no—
69C22⋊C8—22no—
70C22⋊C8—22no—
71#97—22no—
72C4×C8—22no—
73C24C22⋊C412yes—
74#41—42no—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8—22no—
78C2×Q8—42no—
79C2×Q8C22⋊C411yescharacteristic
80C2×D4—21no—
81C2×D4—24no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—42no—
85C2×D4C22⋊C411yescharacteristic
86C22×C4—24no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4C22⋊C412yes—
90C22×C4C22×C411yescharacteristic
91C22×C4C2×D412yes—
92C2×C8—42no—
93C2×C8—42no—
94C4⋊C4—24no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4C22⋊C412yes—
101C4⋊C4C22⋊C412yes—
102C4⋊C4C22⋊C412yes—
103C22⋊C4—24no—
104C22⋊C4—24no—
105C22⋊C4—24no—
106C22⋊C4—21no—
107C22⋊C4—21no—
108C22⋊C4—21no—
109C22⋊C4—24no—
110C22⋊C4—24no—
111C22⋊C4—22no—
112C42—24no—
113C42—21no—
114C42—21no—
115C42—21no—
116C42—22no—
117C42C2×D412yes—
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C23—22no—
121C23—21no—
122C23—42no—
123C23—44no—
124C23C2×C22⋊C411yescharacteristic
125Q8—22no—
126Q8—42no—
127Q8—42no—
128Q8—82no—
129D4—22no—
130D4—42no—
131D4—42no—
132D4—42no—
133D4—84no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—82no—
144C2×C4—84no—
145C2×C4C2×C22⋊C411yescharacteristic
146C2×C4C2×C22⋊C411yescharacteristic
147C2×C4#12012yes—
148C2×C4#12012yes—
149C8—82no—
150C8—82no—
151C22—22no—
152C22—21no—
153C22—42no—
154C22—44no—
155C22—41no—
156C22—42no—
157C22—162no—
158C22#37911yescharacteristic
159C4—22no—
160C4—22no—
161C4—21no—
162C4—42no—
163C4—42no—
164C4—42no—
165C4—41no—
166C4—42no—
167C4#41312yes—
168C2—41no—
169C2—81no—
170C2—82no—
171C2#178511yescharacteristic
172C2#179412yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#535C4
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
4. #1488C2semidirect
5. #1469C2semidirect

Maximal subgroups

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

subgroupindex
#35102
#15442
#15372
#14882
#14692