Finite Group Databasebeta

← #11852

Subgroups — #11852

Order 256.

219 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 51 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#2591C211yesmaximal, characteristic
2#1821C211yesmaximal, characteristic, complemented
3C2×#324C211yesmaximal, characteristic, complemented
4#1353C212yesmaximal, complemented
5#1345C212yesmaximal, complemented
6C2×#121C411yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8#407—21no—
9C2×C22⋊C8—21no—
10#400C411yescharacteristic
11#400C2211yescharacteristic
12#394C411yescharacteristic
13#394C2211yescharacteristic
14#388—21no—
15#388—21no—
16#386—21no—
17#386—21no—
18C4×C22⋊C4C411yescharacteristic
19#343—21no—
20#343C2212yes—
21#324—21no—
22#324C2212yes—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C2×M16—22no—
29C22×C8—41no—
30#121—41no—
31#121—41no—
32C2×C4⋊C4—41no—
33C2×C4⋊C4—41no—
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C42—21no—
46C22⋊C8—22no—
47C22⋊C8—41no—
48C22⋊C8—41no—
49C22⋊C8—41no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95C2×C411yescharacteristic
55#95C2×C411yescharacteristic
56C24—21no—
57C24C22⋊C411yescharacteristic
58C2×D4—41no—
59C2×D4—42no—
60C2×D4—42no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4—81no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—22no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—41no—
75C22×C4—81no—
76C22×C4—81no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83M16—42no—
84M16—42no—
85C2×C8—42no—
86C2×C8—42no—
87C2×C8—82no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—81no—
98C22⋊C4—81no—
99C42—81no—
100C23—22no—
101C23—21no—
102C23—21no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23#9911yescharacteristic
112C23#9911yescharacteristic
113C23C2×C22⋊C411yescharacteristic
114D4—82no—
115D4—82no—
116D4—82no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4#13512yes—
146C8—82no—
147C8—82no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—42no—
156C22—42no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22#35512yes—
165C22C2×#9911yescharacteristic
166C22#41611yescharacteristic
167C22#42111yescharacteristic
168C4—22no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—81no—
175C4—81no—
176C4—81no—
177C2—21no—
178C2—21no—
179C2—22no—
180C2—41no—
181C2—161no—
182C2#151611yescharacteristic
183C2#212111yescharacteristic
184C2C2×#35511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC2×#121C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1821C2semidirect
3. C2×#324C2semidirect
4. #1353C2semidirect
5. #1345C2semidirect

Maximal subgroups

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

subgroupindex
#25912
#18212
C2×#3242
#13532
#13452