Finite Group Databasebeta

← #8329

Subgroups — #8329

Order 256.

638 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C22⋊Q8C211yesmaximal, characteristic
2#1779C211yesmaximal, characteristic
3C2×#328C214yesmaximal, complemented
4C22⋊#95C211yesmaximal, characteristic
5C23×Q8C411yescharacteristic, complemented
6C24×C4C2211yescharacteristic
7C2×C22⋊Q8—21no—
8C2×C22⋊Q8—21no—
9C2×C22⋊Q8—28no—
10C2×C22⋊Q8C412yescomplemented
11C2×C22⋊Q8C2212yes—
12C22×C4⋊C4—21no—
13C22×C4⋊C4C411yescharacteristic, complemented
14C22×C22⋊C4—21no—
15C22×C22⋊C4—21no—
16C2×C22⋊C8—21no—
17C2×C22⋊C8—21no—
18C2×C22⋊C8C2212yes—
19C2×#95—21no—
20C2×#95C2212yes—
21#328—28no—
22C25D411yescharacteristic
23C22×Q8—24no—
24C22×Q8—21no—
25C22×Q8—42no—
26C22×Q8—42no—
27C22×Q8C2×C412yes—
28C23×C4—21no—
29C23×C4—21no—
30C23×C4—24no—
31C23×C4—22no—
32C23×C4—42no—
33C23×C4—42no—
34C23×C4C2×C411yescharacteristic
35C23×C4C2×C411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4D411yescharacteristic
39C23×C4C2311yescharacteristic
40C22×C8—22no—
41C22⋊Q8—28no—
42C22⋊Q8—28no—
43C22⋊Q8—24no—
44C22⋊Q8—24no—
45C22⋊Q8—24no—
46C22⋊Q8D418yes—
47C2×C4⋊C4—28no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4C2×C412yes—
53C2×C22⋊C4—28no—
54C2×C22⋊C4—22no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—24no—
57C2×C22⋊C4—42no—
58C2×C22⋊C4—42no—
59C22⋊C8—24no—
60C22⋊C8—42no—
61C22⋊C8—42no—
62#95—24no—
63#95—42no—
64C24—22no—
65C24—22no—
66C24—22no—
67C24—22no—
68C24—41no—
69C24—41no—
70C24C22⋊C411yescharacteristic
71C24C22⋊C411yescharacteristic
72C24SD1612yes—
73C24Q1612yes—
74C24C2×D411yescharacteristic
75C2×Q8—24no—
76C2×Q8—22no—
77C2×Q8—416no—
78C2×Q8—42no—
79C2×Q8—42no—
80C2×Q8—42no—
81C2×Q8—42no—
82C2×Q8C22⋊C414yescomplemented
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—24no—
87C22×C4—24no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—24no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—416no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C22⋊C411yescharacteristic
117C22×C4C22⋊C411yescharacteristic
118C22×C4C22×C411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D414yes—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—42no—
124C4⋊C4—24no—
125C4⋊C4—28no—
126C4⋊C4—22no—
127C4⋊C4—28no—
128C4⋊C4C22⋊C414yescomplemented
129C22⋊C4—28no—
130C22⋊C4—28no—
131C22⋊C4—48no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—24no—
141C23—24no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—24no—
149C23—24no—
150C23—42no—
151C23—42no—
152C23—44no—
153C23—44no—
154C23—44no—
155C23—44no—
156C23#9912yes—
157C23#10312yes—
158C23#10312yes—
159C23#10412yes—
160C23C2×C22⋊C411yescharacteristic
161C23#12012yes—
162C23C2×SD1611yescharacteristic
163C23C2×Q1611yescharacteristic
164C23#13712yes—
165Q8—44no—
166Q8—48no—
167Q8—44no—
168C2×C4—28no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—48no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4#9912yes—
199C2×C4C2×C22⋊C412yes—
200C2×C4#12012yes—
201C8—82no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—21no—
209C22—22no—
210C22—21no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—48no—
220C22—48no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22#32814yescomplemented
226C22#37911yescharacteristic
227C22C2×#9911yescharacteristic
228C22C2×#9911yescharacteristic
229C22C2×#10311yescharacteristic
230C22#41911yescharacteristic
231C22C2×#10411yescharacteristic
232C22#42111yescharacteristic
233C22#44812yes—
234C22#45112yes—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—41no—
239C4—44no—
240C4—44no—
241C4—84no—
242C2—22no—
243C2—24no—
244C2—22no—
245C2—42no—
246C2C2×#32811yescharacteristic
247C2#150811yescharacteristic
248C2#151111yescharacteristic
249C2#152011yescharacteristic
250C2#178111yescharacteristic
251C2#178711yescharacteristic
252C2#178911yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#328C2semidirect
5. C23×Q8C4semidirect
10. C2×C22⋊Q8C4semidirect
13. C22×C4⋊C4C4semidirect
82. C2×Q8C22⋊C4semidirect
128. C4⋊C4C22⋊C4semidirect
225. C22#328semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
#17792
C2×#3282
C22⋊#952