Finite Group Databasebeta

← #11693

Subgroups — #11693

Order 256.

284 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 63 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#2619C211yesmaximal, characteristic
2#1740C211yesmaximal, characteristic, complemented
3#1458C211yesmaximal, characteristic, complemented
4#1343C214yesmaximal, complemented
5C2×#127C411yescharacteristic, complemented
6C2×#121—21no—
7C2×#117—21no—
8C2×#117C2211yescharacteristic
9#431—21no—
10#394—21no—
11#394C412yescomplemented
12#392—21no—
13#392C2212yes—
14#390—21no—
15#382C411yescharacteristic, complemented
16#342—21no—
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×D4—22no—
21C22×D4—21no—
22C22×D4—21no—
23C22×D4C2×C412yes—
24C23×C4D411yescharacteristic
25#127—41no—
26#127—41no—
27#121—42no—
28#121—42no—
29#117—24no—
30#117—41no—
31#117—41no—
32#117—42no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—41no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4C2311yescharacteristic
43C2×C42—41no—
44C2×C42C2×C411yescharacteristic
45C2×C42D411yescharacteristic
46C4⋊C8—41no—
47C22⋊C8—22no—
48C4×C8—41no—
49#95—21no—
50#95C2×C412yes—
51C24—22no—
52C24C2×D411yescharacteristic
53C2×D4—42no—
54C2×D4—42no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4—41no—
58C2×D4—41no—
59C2×D4—42no—
60C2×D4—42no—
61C2×D4—82no—
62C2×D4—82no—
63C2×D4—82no—
64C2×D4—82no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—42no—
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4D812yes—
75C22×C4SD1612yes—
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C2×C8—42no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—44no—
82C4⋊C4—42no—
83C22⋊C4—24no—
84C22⋊C4—41no—
85C22⋊C4—41no—
86C22⋊C4—42no—
87C22⋊C4—44no—
88C22⋊C4—82no—
89C22⋊C4—82no—
90C42—21no—
91C42—21no—
92C42—21no—
93C42—21no—
94C42—44no—
95C23—21no—
96C23—21no—
97C23—41no—
98C23—41no—
99C23—42no—
100C23—42no—
101C23—42no—
102C23—42no—
103C23—42no—
104C23—42no—
105C23—42no—
106C23C2×C22⋊C411yescharacteristic
107C23C2×D811yescharacteristic
108C23C2×SD1611yescharacteristic
109D4—84no—
110D4—84no—
111D4—84no—
112D4—84no—
113D4—82no—
114D4—82no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—44no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4#9911yescharacteristic
136C2×C4#9911yescharacteristic
137C2×C4#10212yes—
138C2×C4#10212yes—
139C2×C4#10412yes—
140C8—82no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—41no—
146C22—41no—
147C22—41no—
148C22—41no—
149C22—81no—
150C22—82no—
151C22—82no—
152C22—82no—
153C22—82no—
154C22—82no—
155C22—82no—
156C22—82no—
157C22C2×#9911yescharacteristic
158C22C2×#10211yescharacteristic
159C22C2×#10411yescharacteristic
160C4—22no—
161C4—41no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—82no—
166C4—84no—
167C4#32712yes—
168C4#32712yes—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—162no—
174C2C2×#32711yescharacteristic
175C2#212211yescharacteristic
176C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2619C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1740C2semidirect
3. #1458C2semidirect
4. #1343C2semidirect
5. C2×#127C4semidirect
11. #394C4semidirect
15. #382C4semidirect

Maximal subgroups

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

subgroupindex
#26192
#17402
#14582
#13432