Finite Group Databasebeta

← #8318

Subgroups — #8318

Order 256.

529 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C4×H(5,𝔽2)C211yesmaximal, characteristic
2#1954C211yesmaximal, characteristic, complemented
3#1477C212yesmaximal, complemented
4#1476C212yesmaximal, complemented
5#1460C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)C411yescharacteristic
7#520—22no—
8#520C412yes—
9#518—22no—
10#518—22no—
11#518C411yescharacteristic
12C4×#41—22no—
13C4×#41C2212yes—
14C2×C4×D4—22no—
15C2×C4×D4—22no—
16C2×C4×D4C2211yescharacteristic
17#431C2212yescomplemented
18C2×C4⋊C8C2211yescharacteristic, complemented
19C2×C22⋊C8—22no—
20C2×C4×C8C2211yescharacteristic, complemented
21Q8⋊C8—22no—
22Q8⋊C8—22no—
23D4⋊C8—22no—
24D4⋊C8—22no—
25#323—22no—
26H(5,𝔽2)—24no—
27H(5,𝔽2)—22no—
28H(5,𝔽2)C814yescomplemented
29C2×#41—22no—
30C2×#41C2×C412yes—
31C22×D4—22no—
32C22×D4—22no—
33C22×D4C2×C411yescharacteristic
34C23×C4—22no—
35C23×C4D412yes—
36C22×C8—21no—
37C22×C8—21no—
38C4×Q8—22no—
39C4×Q8D412yes—
40C4×Q8D412yes—
41C4×D4—24no—
42C4×D4—24no—
43C4×D4—24no—
44C4×D4—24no—
45C4×D4—22no—
46C4×D4—24no—
47C4×D4—22no—
48C4×D4—22no—
49C4×D4—22no—
50C4×D4—22no—
51C4×D4—22no—
52C4×D4—22no—
53C4×D4D412yes—
54C4×D4D412yes—
55#117—24no—
56#117—22no—
57#117—22no—
58#117C2×C412yes—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4C2×C411yescharacteristic
62C2×C22⋊C4—22no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4D412yes—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42C2311yescharacteristic
69C4⋊C8—22no—
70C4⋊C8—22no—
71C22⋊C8—22no—
72C22⋊C8—44no—
73C4×C8—22no—
74C4×C8—22no—
75C24—22no—
76C24C22⋊C412yes—
77#41—28no—
78#41—28no—
79#41—22no—
80#41—22no—
81#41C2×C814yes—
82#41M1614yes—
83C2×Q8C22⋊C412yes—
84C2×D4—24no—
85C2×D4—28no—
86C2×D4—28no—
87C2×D4—28no—
88C2×D4—22no—
89C2×D4—28no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4C22⋊C412yes—
93C2×D4C2×C812yes—
94C2×D4M1612yes—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—82no—
107C22×C4—82no—
108C22×C4C22⋊C412yes—
109C22×C4C22×C411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C2×C8—22no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—42no—
116C2×C8—42no—
117C2×C8—41no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4C22⋊C412yes—
123C4⋊C4C22⋊C412yes—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—22no—
129C22⋊C4—24no—
130C42—24no—
131C42—22no—
132C42—22no—
133C42—22no—
134C42C2×D412yes—
135C42C2×D412yes—
136C23—28no—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—42no—
141C23—42no—
142C23—82no—
143C23—82no—
144C23C22⋊C814yes—
145C23C2×C22⋊C411yescharacteristic
146Q8—22no—
147Q8C22⋊C814yescomplemented
148D4—216no—
149D4—24no—
150D4—24no—
151D4—24no—
152D4—22no—
153D4—24no—
154D4C22⋊C814yescomplemented
155C2×C4—28no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—84no—
174C2×C4—82no—
175C2×C4C2×C22⋊C412yes—
176C2×C4#12012yes—
177C2×C4#12011yescharacteristic
178C2×C4#12011yescharacteristic
179C2×C4C22×C811yescharacteristic
180C2×C4C2×M1612yes—
181C2×C4C2×M1611yescharacteristic
182C8—42no—
183C8—82no—
184C8—81no—
185C22—21no—
186C22—21no—
187C22—24no—
188C22—24no—
189C22—44no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—82no—
195C22—88no—
196C22#37911yescharacteristic
197C22C2×C22⋊C811yescharacteristic
198C22#40711yescharacteristic
199C4—21no—
200C4—22no—
201C4—24no—
202C4—42no—
203C4—42no—
204C4—42no—
205C4C2×C22⋊C812yes—
206C4#40712yes—
207C4#45311yescharacteristic
208C4#45511yescharacteristic
209C2—21no—
210C2—21no—
211C2—44no—
212C2—44no—
213C2#177911yescharacteristic
214C2#179211yescharacteristic
215C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
socleC22#379
prime radicalC2×H(5,𝔽2)C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1954C2semidirect
3. #1477C2semidirect
4. #1476C2semidirect
5. #1460C2semidirect
17. #431C22semidirect
18. C2×C4⋊C8C22semidirect
20. C2×C4×C8C22semidirect
28. H(5,𝔽2)C8semidirect
147. Q8C22⋊C8semidirect
154. D4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)2
#19542
#14772
#14762
#14602