Finite Group Databasebeta

← #8405

Subgroups — #8405

Order 256.

510 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 95 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×#543C211yesmaximal, characteristic
2#1872C212yesmaximal, complemented
3C2×#328C212yesmaximal, complemented
4C2×#323C211yesmaximal, characteristic
5#1439C211yesmaximal, characteristic
6#543—24no—
7#543—22no—
8#543C414yescomplemented
9C2×C22⋊Q8—22no—
10C2×C22⋊Q8—21no—
11C2×C22⋊Q8—22no—
12C2×C22⋊Q8C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4—21no—
15C22×C22⋊C4—21no—
16C22×C22⋊C4C2211yescharacteristic
17C2×#103—22no—
18C2×C22⋊C8C2212yes—
19C2×#95—21no—
20C2×#95C2211yescharacteristic
21#340—22no—
22#328—24no—
23#323—22no—
24C25D411yescharacteristic
25C22×Q8—21no—
26C22×Q8D412yes—
27C23×C4—21no—
28C23×C4—21no—
29C23×C4—21no—
30C23×C4—21no—
31C23×C4D411yescharacteristic
32C23×C4C2311yescharacteristic
33C22×C8—22no—
34C22⋊Q8—28no—
35C22⋊Q8—28no—
36C22⋊Q8—24no—
37C22⋊Q8—28no—
38C22⋊Q8—24no—
39C22⋊Q8—22no—
40C22⋊Q8—22no—
41C22⋊Q8—28no—
42C22⋊Q8C2×C414yes—
43C22⋊Q8D414yes—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4D412yes—
50C2×C22⋊C4—22no—
51C2×C22⋊C4—24no—
52C2×C22⋊C4—24no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—42no—
57C2×C22⋊C4—42no—
58C2×C22⋊C4—42no—
59C2×C22⋊C4C2×C412yes—
60C2×C22⋊C4D412yes—
61#103—44no—
62C22⋊C8—22no—
63C22⋊C8—22no—
64#95—22no—
65#95—42no—
66C24—21no—
67C24—21no—
68C24—21no—
69C24—21no—
70C24—41no—
71C24—41no—
72C24—41no—
73C24—41no—
74C24C22⋊C412yes—
75C24SD1612yes—
76C24Q1612yes—
77C24C2×D411yescharacteristic
78C2×Q8—24no—
79C2×Q8—22no—
80C2×Q8—42no—
81C2×Q8—42no—
82C2×Q8—81no—
83C2×Q8—81no—
84C2×Q8C22⋊C414yescomplemented
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—24no—
89C22×C4—24no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—81no—
110C22×C4—81no—
111C22×C4—81no—
112C22×C4—81no—
113C22×C4C22⋊C412yes—
114C22×C4C22×C411yescharacteristic
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C2×C8—42no—
119C2×C8—42no—
120C2×C8—42no—
121C4⋊C4—24no—
122C4⋊C4—28no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—24no—
127C4⋊C4—42no—
128C4⋊C4C22⋊C414yescomplemented
129C22⋊C4—28no—
130C22⋊C4—24no—
131C22⋊C4—24no—
132C22⋊C4—24no—
133C22⋊C4—24no—
134C22⋊C4—48no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—22no—
141C23—22no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—81no—
156C23—81no—
157C23—81no—
158C23—81no—
159C23—81no—
160C23—81no—
161C23#10314yescomplemented
162C23C2×C22⋊C411yescharacteristic
163C23#12011yescharacteristic
164C23#12011yescharacteristic
165C23C2×SD1611yescharacteristic
166C23C2×Q1611yescharacteristic
167C23#13711yescharacteristic
168C23#13711yescharacteristic
169Q8—44no—
170Q8—82no—
171Q8—84no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—84no—
195C2×C4—84no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4C2×C22⋊C412yes—
205C2×C4#12012yes—
206C8—82no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—24no—
220C22—44no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22#37911yescharacteristic
231C22C2×#10311yescharacteristic
232C22#41911yescharacteristic
233C22#44811yescharacteristic
234C22#44811yescharacteristic
235C22#45111yescharacteristic
236C22#45111yescharacteristic
237C22#45311yescharacteristic
238C22#45511yescharacteristic
239C22B(4,𝔽2)11yescharacteristic
240C22#45811yescharacteristic
241C4—22no—
242C4—22no—
243C4—42no—
244C4—44no—
245C4—44no—
246C4—82no—
247C4—82no—
248C2—21no—
249C2—21no—
250C2—21no—
251C2—21no—
252C2—44no—
253C2#159611yescharacteristic
254C2#160211yescharacteristic
255C2#160811yescharacteristic
256C2#161411yescharacteristic
257C2#178711yescharacteristic
258C2#179011yescharacteristic
259C2#179211yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1872C2semidirect
3. C2×#328C2semidirect
8. #543C4semidirect
84. C2×Q8C22⋊C4semidirect
128. C4⋊C4C22⋊C4semidirect
161. C23#103semidirect

Maximal subgroups

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

subgroupindex
C2×#5432
#18722
C2×#3282
C2×#3232
#14392