Finite Group Databasebeta

← #8463

Subgroups — #8463

Order 256.

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

normal subgroups only

This group has 359 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#3493C211yesmaximal, characteristic
2#1820C211yesmaximal, characteristic, complemented
3#1818C211yesmaximal, characteristic, complemented
4#1516C212yesmaximal, complemented
5#1516C212yesmaximal, complemented
6#556—22no—
7#555—22no—
8#553—22no—
9#552—22no—
10#544C411yescharacteristic, complemented
11#538—22no—
12#537C411yescharacteristic, complemented
13#529—22no—
14#525C2212yes—
15#520C412yescomplemented
16C2×#117C2211yescharacteristic
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#381C2212yes—
20#342—22no—
21#336—22no—
22#328—22no—
23#328—22no—
24#327—22no—
25#327—22no—
26C2×#41C2×C412yes—
27C23×C4C2311yescharacteristic
28C22×C8—21no—
29C22×C8—21no—
30C4⋊Q8—22no—
31#126—24no—
32#126—24no—
33#125—22no—
34#125—22no—
35#124—22no—
36#124—24no—
37#123—24no—
38#123—22no—
39C22⋊Q8—22no—
40C22⋊Q8—22no—
41C22⋊Q8D412yes—
42C22⋊Q8D412yes—
43#121—24no—
44#121—22no—
45#121—22no—
46#121D412yes—
47#121D412yes—
48#120—22no—
49C4×Q8—22no—
50C4×Q8—22no—
51C4×D4—22no—
52C4×D4—22no—
53C4×D4—24no—
54C4×D4—22no—
55C4×D4—22no—
56#117—24no—
57#117—22no—
58#117—22no—
59#117—22no—
60#117C2×C412yes—
61C2×C4⋊C4—22no—
62C2×C4⋊C4C2×C411yescharacteristic
63C2×C4⋊C4C2×C411yescharacteristic
64C2×C22⋊C4—22no—
65C2×C22⋊C4D412yes—
66C2×C42D412yes—
67C22⋊C8—22no—
68C22⋊C8—22no—
69#95—22no—
70#95—22no—
71C24C2×D411yescharacteristic
72#41—44no—
73C2×Q8—22no—
74C2×Q8C22⋊C412yes—
75C2×D4—22no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—22no—
79C2×D4C22⋊C412yes—
80C22×C4—24no—
81C22×C4—22no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4C22⋊C412yes—
85C22×C4C22⋊C412yes—
86C22×C4C22×C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D412yes—
89C22×C4C2×D412yes—
90C2×C8—42no—
91C2×C8—42no—
92C2×C8—41no—
93C2×C8—41no—
94C4⋊C4—24no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊C412yes—
103C4⋊C4C22⋊C412yes—
104C22⋊C4—24no—
105C22⋊C4—24no—
106C22⋊C4—22no—
107C22⋊C4—24no—
108C22⋊C4—22no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—44no—
113C42—24no—
114C42—22no—
115C42—22no—
116C42—22no—
117C23—22no—
118C23—22no—
119C23—41no—
120C23—41no—
121C23C2×C22⋊C411yescharacteristic
122C23#12012yes—
123Q8—42no—
124Q8—82no—
125D4—42no—
126D4—44no—
127D4—82no—
128D4—82no—
129D4—82no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—84no—
143C2×C4—82no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4C2×C22⋊C412yes—
147C2×C4#12011yescharacteristic
148C2×C4#12011yescharacteristic
149C8—81no—
150C8—81no—
151C22—21no—
152C22—21no—
153C22—22no—
154C22—22no—
155C22—22no—
156C22—42no—
157C22—42no—
158C22—42no—
159C22—82no—
160C22#37911yescharacteristic
161C22#41012yes—
162C4—21no—
163C4—21no—
164C4—22no—
165C4—42no—
166C4—42no—
167C4—42no—
168C4—42no—
169C4—42no—
170C4—162no—
171C2—22no—
172C2—42no—
173C2—82no—
174C2#178211yescharacteristic
175C2#179411yescharacteristic
176C2#179411yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3493C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1820C2semidirect
3. #1818C2semidirect
4. #1516C2semidirect
5. #1516C2semidirect
10. #544C4semidirect
12. #537C4semidirect
15. #520C4semidirect

Maximal subgroups

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

subgroupindex
#34932
#18202
#18182
#15162