Finite Group Databasebeta

← #11824

Subgroups — #11824

Order 256.

232 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 49 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#2635C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic, complemented
3#1466C211yesmaximal, characteristic, complemented
4#1352C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C2×#121C411yescharacteristic, complemented
7C2×C4×D4C2211yescharacteristic
8#432—21no—
9C2×#99—21no—
10#400C411yescharacteristic, complemented
11#399C2211yescharacteristic
12#394—21no—
13#394C411yescharacteristic, complemented
14#392—21no—
15#392C2211yescharacteristic
16#388—21no—
17#386—21no—
18#381C411yescharacteristic, complemented
19#380—21no—
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C22×D4—21no—
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C22×D4D411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29#121—41no—
30#121—41no—
31C4×D4—41no—
32C4×D4—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—41no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4D411yescharacteristic
45C2×C22⋊C4C2311yescharacteristic
46C2×C42D411yescharacteristic
47C4⋊C8—41no—
48#99—42no—
49C22⋊C8—22no—
50#97—41no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—41no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24—21no—
58C24C22⋊C411yescharacteristic
59C24C2×D411yescharacteristic
60C2×D4—21no—
61C2×D4—21no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—42no—
66C2×D4—41no—
67C2×D4—41no—
68C2×D4—81no—
69C2×D4—81no—
70C2×D4—81no—
71C2×D4—81no—
72C2×D4—81no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—42no—
81C22×C4—81no—
82C22×C4—81no—
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22×C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—42no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—81no—
92C4⋊C4—82no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—42no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—81no—
101C22⋊C4—81no—
102C22⋊C4—81no—
103C22⋊C4—81no—
104C42—21no—
105C42—21no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23C2×C22⋊C411yescharacteristic
123C23AGL(1,ℤ/8)11yescharacteristic
124C23#13711yescharacteristic
125D4—42no—
126D4—42no—
127D4—82no—
128D4—82no—
129D4—82no—
130D4—84no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4—82no—
157C2×C4#9911yescharacteristic
158C2×C4#9911yescharacteristic
159C8—82no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22C2×#9911yescharacteristic
182C22#41711yescharacteristic
183C22#42111yescharacteristic
184C4—21no—
185C4—21no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—41no—
190C4—81no—
191C4—81no—
192C4—81no—
193C4—162no—
194C2—21no—
195C2—21no—
196C2—41no—
197C2—41no—
198C2—81no—
199C2—161no—
200C2#151311yescharacteristic
201C2#212111yescharacteristic
202C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2635C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1799C2semidirect
3. #1466C2semidirect
4. #1352C2semidirect
5. #1343C2semidirect
6. C2×#121C4semidirect
10. #400C4semidirect
13. #394C4semidirect
18. #381C4semidirect

Maximal subgroups

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

subgroupindex
#26352
#17992
#14662
#13522
#13432