Finite Group Databasebeta

← #8379

Subgroups — #8379

Order 256.

517 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 93 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×D4×Q8C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1787C212yesmaximal, complemented
4#1520C212yesmaximal, complemented
5#1459C211yesmaximal, characteristic
6C23×Q8C411yescharacteristic, complemented
7C23×Q8C2211yescharacteristic
8D4×Q8—24no—
9D4×Q8—24no—
10C2×C4⋊Q8—21no—
11C2×C4⋊Q8C411yescharacteristic, complemented
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C411yescharacteristic, complemented
15C2×C22⋊Q8C2211yescharacteristic
16C2×C4×Q8C411yescharacteristic, complemented
17C2×C4×D4—21no—
18C2×C4×D4C2211yescharacteristic
19C2×#103—24no—
20C2×#99—21no—
21C2×C22⋊C8C2212yes—
22#380C2212yes—
23#342—21no—
24#328—24no—
25#324—21no—
26#323—21no—
27C22×Q8—22no—
28C22×Q8—22no—
29C22×Q8—21no—
30C22×Q8—21no—
31C22×Q8—22no—
32C22×Q8—44no—
33C22×Q8—44no—
34C22×Q8C2×C411yescharacteristic
35C22×Q8C2×C411yescharacteristic
36C22×Q8C2×C411yescharacteristic
37C22×Q8D414yes—
38C22×D4D411yescharacteristic
39C23×C4—21no—
40C23×C4—21no—
41C23×C4C2×C411yescharacteristic
42C23×C4C2311yescharacteristic
43C22×C8—22no—
44C4⋊Q8—28no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C22⋊Q8—28no—
48C22⋊Q8—28no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C22⋊Q8D414yes—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×D4—28no—
56C4×D4—22no—
57C4×D4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4C2×C411yescharacteristic
65C2×C4⋊C4D411yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4D411yescharacteristic
70C2×C22⋊C4D411yescharacteristic
71C2×C42—21no—
72C2×C42C2×C411yescharacteristic
73#103—48no—
74#99—42no—
75C22⋊C8—22no—
76C22⋊C8—22no—
77#95—22no—
78C24C22⋊C411yescharacteristic
79C24C2×D411yescharacteristic
80C2×Q8—24no—
81C2×Q8—24no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—24no—
86C2×Q8—28no—
87C2×Q8—42no—
88C2×Q8—44no—
89C2×Q8—42no—
90C2×Q8—44no—
91C2×Q8—44no—
92C2×Q8—48no—
93C2×Q8—48no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8C22⋊C412yes—
97C2×Q8C22⋊C412yes—
98C2×Q8C22⋊C412yes—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4SD1612yes—
102C2×D4Q1612yes—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C22⋊C411yescharacteristic
117C22×C4C22⋊C411yescharacteristic
118C22×C4C22⋊C411yescharacteristic
119C22×C4C22×C411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C2×C8—44no—
124C2×C8—42no—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—21no—
130C4⋊C4—24no—
131C4⋊C4—22no—
132C4⋊C4—44no—
133C4⋊C4C22⋊C412yes—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—24no—
141C42—21no—
142C42—21no—
143C23—21no—
144C23—21no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23#10312yescomplemented
150C23C2×C22⋊C411yescharacteristic
151C23#12012yes—
152C23C2×SD1611yescharacteristic
153C23C2×Q1611yescharacteristic
154C23#13712yes—
155Q8—24no—
156Q8—22no—
157Q8—24no—
158Q8—48no—
159Q8—42no—
160Q8—48no—
161Q8—42no—
162Q8—42no—
163Q8#9914yescomplemented
164D4—24no—
165D4—24no—
166C2×C4—24no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—48no—
180C2×C4—48no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4#10312yescomplemented
192C2×C4C2×C22⋊C411yescharacteristic
193C2×C4C2×C22⋊C411yescharacteristic
194C2×C4#12012yes—
195C8—82no—
196C22—21no—
197C22—22no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—41no—
207C22—41no—
208C22#37911yescharacteristic
209C22C2×#10311yescharacteristic
210C22#41911yescharacteristic
211C22#44812yes—
212C22#45112yes—
213C4—22no—
214C4—24no—
215C4—21no—
216C4—22no—
217C4—41no—
218C4—42no—
219C4—42no—
220C4—42no—
221C4—41no—
222C4—162no—
223C4C2×#9912yes—
224C2—22no—
225C2—21no—
226C2—21no—
227C2—22no—
228C2—42no—
229C2#178111yescharacteristic
230C2#178711yescharacteristic
231C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1787C2semidirect
4. #1520C2semidirect
6. C23×Q8C4semidirect
11. C2×C4⋊Q8C4semidirect
14. C2×C22⋊Q8C4semidirect
16. C2×C4×Q8C4semidirect
149. C23#103semidirect
163. Q8#99semidirect
191. C2×C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
#17992
#17872
#15202
#14592