Finite Group Databasebeta

← #9925

Subgroups — #9925

Order 256.

405 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 67 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×#451C211yesmaximal, characteristic
2#2627C211yesmaximal, characteristic
3C2×#395C211yesmaximal, characteristic
4#1787C211yesmaximal, characteristic
5C2×#328C211yesmaximal, characteristic
6C2×#324C211yesmaximal, characteristic
7#1294C211yesmaximal, characteristic
8C23×Q8C2211yescharacteristic
9C22×Q16—21no—
10C2×C22⋊Q8C2211yescharacteristic
11C22×C4⋊C4—21no—
12C22×C4⋊C4C2211yescharacteristic
13#451—22no—
14#451—22no—
15C2×#103—21no—
16C2×#103—22no—
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#400—21no—
20#398—22no—
21#395—21no—
22#395—22no—
23#395—22no—
24#385—21no—
25#381—21no—
26#380C2211yescharacteristic
27C2×#95—21no—
28C2×#95C2211yescharacteristic
29#328—22no—
30#324—22no—
31C22×Q8—21no—
32C22×Q8—21no—
33C22×Q8—21no—
34C22×Q8—21no—
35C22×Q8—21no—
36C22×Q8—44no—
37C22×Q8D411yescharacteristic
38C22×Q8D412yes—
39C23×C4—21no—
40C23×C4—21no—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C2×Q16—24no—
45C2×Q16—44no—
46C22×C8—21no—
47C22×C8—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C22⋊Q8—21no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—44no—
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4D412yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66#103—24no—
67#103—44no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8D412yes—
72#95—24no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—21no—
77#95—21no—
78#95D412yes—
79C24C2×D411yescharacteristic
80C2×Q8—24no—
81C2×Q8—24no—
82C2×Q8—22no—
83C2×Q8—21no—
84C2×Q8—22no—
85C2×Q8—21no—
86C2×Q8—44no—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×Q8—44no—
90C2×Q8—42no—
91C2×Q8—44no—
92C2×Q8—44no—
93C2×Q8—41no—
94C2×Q8—41no—
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4Q1612yescomplemented
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4#4111yescharacteristic
126Q16—48no—
127C2×C8—22no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—41no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—42no—
134C4⋊C4—48no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—82no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C22⋊C4—41no—
145C22⋊C4—41no—
146C23—22no—
147C23—22no—
148C23#12011yescharacteristic
149C23#12011yescharacteristic
150C23#12111yescharacteristic
151C23C2×Q1611yescharacteristic
152C23#13511yescharacteristic
153C23#13711yescharacteristic
154C23#13711yescharacteristic
155Q8—24no—
156Q8—24no—
157Q8—42no—
158Q8—44no—
159Q8—42no—
160Q8—44no—
161Q8—42no—
162Q8—81no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—81no—
197C2×C4#12011yescharacteristic
198C2×C4#12111yescharacteristic
199C2×C4#12111yescharacteristic
200C2×C4#12711yescharacteristic
201C8—42no—
202C8—81no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22#39211yescharacteristic
212C22#45111yescharacteristic
213C22#45111yescharacteristic
214C22#45211yescharacteristic
215C22#45211yescharacteristic
216C22#45612yes—
217C22B(4,𝔽2)12yes—
218C22#46711yescharacteristic
219C22#47011yescharacteristic
220C4—21no—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—44no—
225C4—42no—
226C4—44no—
227C4—82no—
228C4—81no—
229C4—81no—
230C4—81no—
231C2—21no—
232C2—21no—
233C2—21no—
234C2—21no—
235C2#160312yes—
236C2#160512yes—
237C2#200111yescharacteristic
238C2#200511yescharacteristic
239C2#202111yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C23×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
119. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
#26272
C2×#3952
#17872
C2×#3282
C2×#3242
#12942