Finite Group Databasebeta

← #8324

Subgroups — #8324

Order 256.

503 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#520C211yesmaximal, characteristic
2#1479C212yesmaximal, complemented
3#1478C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic, complemented
5#1458C211yesmaximal, characteristic, complemented
6C22×#41C411yescharacteristic
7#520—24no—
8#520—24no—
9C2×C4×Q8C411yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11C2×C4×D4—21no—
12C2×C4×D4—21no—
13C2×C4×D4C411yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#117—21no—
16C2×#117C2211yescharacteristic
17C22×C4⋊C4—21no—
18C22×C4⋊C4C411yescharacteristic
19#431C2212yes—
20#431C2212yes—
21Q8⋊C8—24no—
22D4⋊C8—24no—
23#324—21no—
24#324—21no—
25#323—21no—
26#323—21no—
27C2×#41—22no—
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—22no—
31C2×#41—22no—
32C2×#41C814yescomplemented
33C22×Q8C2×C411yescharacteristic
34C22×D4—21no—
35C22×D4C2×C411yescharacteristic
36C23×C4—21no—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4—21no—
40C23×C4C2×C411yescharacteristic
41C4×Q8—22no—
42C4×Q8—22no—
43C4×Q8—22no—
44C4×Q8D414yes—
45C4×D4—28no—
46C4×D4—28no—
47C4×D4—22no—
48C4×D4—22no—
49C4×D4—22no—
50C4×D4D414yes—
51#117—28no—
52#117—22no—
53#117—22no—
54C2×C4⋊C4—28no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4C2×C411yescharacteristic
61C2×C4⋊C4C2×C411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4D411yescharacteristic
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4D411yescharacteristic
67C2×C22⋊C4D411yescharacteristic
68C2×C42—21no—
69C2×C42—21no—
70C2×C42C2×C411yescharacteristic
71C2×C42C2311yescharacteristic
72C4⋊C8—22no—
73C4⋊C8—22no—
74C22⋊C8—22no—
75C22⋊C8—22no—
76C4×C8—22no—
77C4×C8—22no—
78C24—21no—
79C24C22⋊C411yescharacteristic
80#41—28no—
81#41—28no—
82#41—28no—
83#41—28no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8C22⋊C412yes—
88C2×Q8C2×C812yes—
89C2×Q8M1612yes—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4C22⋊C412yes—
97C2×D4C2×C812yes—
98C2×D4M1612yes—
99C22×C4—24no—
100C22×C4—24no—
101C22×C4—21no—
102C22×C4—24no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—82no—
113C22×C4—82no—
114C22×C4—82no—
115C22×C4C22⋊C411yescharacteristic
116C22×C4C22⋊C411yescharacteristic
117C22×C4C22⋊C411yescharacteristic
118C22×C4C2×C812yes—
119C22×C4M1612yes—
120C22×C4C22×C411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C2×C8—42no—
124C2×C8—42no—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—24no—
131C4⋊C4—22no—
132C4⋊C4C22⋊C412yes—
133C4⋊C4C22⋊C412yes—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C42—24no—
139C42—21no—
140C42—21no—
141C42—24no—
142C42C2×D412yes—
143C42C2×D412yes—
144C23—21no—
145C23—22no—
146C23—22no—
147C23—21no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23C22⋊C812yes—
154C23C2×C22⋊C411yescharacteristic
155Q8—22no—
156Q8—22no—
157Q8—22no—
158Q8C22⋊C814yes—
159D4—216no—
160D4—22no—
161D4—22no—
162D4—22no—
163D4C22⋊C814yes—
164C2×C4—216no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—84no—
185C2×C4—84no—
186C2×C4—84no—
187C2×C4C22⋊C812yes—
188C2×C4C2×C22⋊C411yescharacteristic
189C2×C4C2×C22⋊C411yescharacteristic
190C2×C4#12012yes—
191C2×C4#12012yes—
192C2×C4C22×C811yescharacteristic
193C2×C4C2×M1611yescharacteristic
194C2×C4C2×M1611yescharacteristic
195C2×C4C2×M1611yescharacteristic
196C8—82no—
197C8—82no—
198C22—21no—
199C22—21no—
200C22—22no—
201C22—24no—
202C22—42no—
203C22—44no—
204C22—42no—
205C22—44no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22—42no—
210C22#37911yescharacteristic
211C22C2×C22⋊C811yescharacteristic
212C22#40711yescharacteristic
213C4—22no—
214C4—22no—
215C4—22no—
216C4—24no—
217C4—42no—
218C4—42no—
219C4—41no—
220C4—41no—
221C4C2×C22⋊C811yescharacteristic
222C4C2×C22⋊C811yescharacteristic
223C4#40711yescharacteristic
224C4#40711yescharacteristic
225C2—21no—
226C2—21no—
227C2—42no—
228C2—44no—
229C2#177911yescharacteristic
230C2#179511yescharacteristic
231C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC22×#41C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1479C2semidirect
3. #1478C2semidirect
4. #1458C2semidirect
5. #1458C2semidirect
32. C2×#41C8semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
#14792
#14782
#14582