Finite Group Databasebeta

← #24345

Subgroups — #24345

Order 256.

510 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 147 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×Q82C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic
3C2×#441C211yesmaximal, characteristic
4#2581C211yesmaximal, characteristic
5#2058C212yesmaximal
6#2024C212yesmaximal
7#1993C211yesmaximal, characteristic
8#1985C211yesmaximal, characteristic
9#1878C212yesmaximal
10#1865C212yesmaximal
11#1805C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13Q82—24no—
14Q82—24no—
15C2×C4⋊Q8—22no—
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2212yes—
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4×Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24#475—24no—
25#441—24no—
26C2×#106C2212yes—
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#400—21no—
35#395—21no—
36#391C2212yes—
37#389C2212yes—
38#387—22no—
39#382C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—22no—
46C22×Q8—21no—
47C22×Q8D411yescharacteristic
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50C2×Q16—22no—
51C2×Q16—22no—
52C2×Q16Q814yescomplemented
53C22×C8C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C4⋊Q8—216no—
57C4⋊Q8—28no—
58C4⋊Q8—24no—
59C4⋊Q8—24no—
60C4⋊Q8—22no—
61C4⋊Q8D414yes—
62C4×Q8—28no—
63C4×Q8—24no—
64C4×Q8—24no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8D414yes—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4D412yes—
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2312yes—
82C2×C4⋊C4C2311yescharacteristic
83C2×C42—22no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2312yes—
87C2×C42C2311yescharacteristic
88#106—22no—
89#106—24no—
90C4⋊C8—22no—
91#103—22no—
92#103—24no—
93#103—22no—
94#103—24no—
95#97—22no—
96#95—21no—
97#95—21no—
98#95—22no—
99#95—22no—
100#95—22no—
101C2×Q8—28no—
102C2×Q8—22no—
103C2×Q8—24no—
104C2×Q8—24no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—82no—
114C2×Q8—84no—
115C2×Q8C2×D412yes—
116C2×Q8C2×Q812yes—
117C2×Q8C2×Q812yes—
118C2×Q8#4112yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D412yes—
129C22×C4C2411yescharacteristic
130Q16—28no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8C2×Q812yes—
140C4⋊C4—28no—
141C4⋊C4—28no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4C2×D414yes—
155C4⋊C4C2×D412yes—
156C42—28no—
157C42—24no—
158C42—22no—
159C42—22no—
160C42—42no—
161C42C2×D412yes—
162C42C2×D412yes—
163C23C22×D411yescharacteristic
164Q8—28no—
165Q8—22no—
166Q8—22no—
167Q8—22no—
168Q8—22no—
169Q8—22no—
170Q8—41no—
171Q8—41no—
172Q8—84no—
173Q8—88no—
174Q8C22⋊Q814yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4#12012yes—
197C2×C4#12012yes—
198C2×C4#13714yes—
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C2×C4C22×Q811yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C42−1+411yescharacteristic
204C2×C42−1+411yescharacteristic
205C8—22no—
206C8—41no—
207C8—42no—
208C22C2×#12011yescharacteristic
209C22#54811yescharacteristic
210C22D4×Q811yescharacteristic
211C22C2×#13712yes—
212C22#57712yes—
213C4—24no—
214C4—22no—
215C4—24no—
216C4—22no—
217C4—42no—
218C4—42no—
219C4—44no—
220C4—42no—
221C4—82no—
222C4—81no—
223C4C2×C22⋊Q811yescharacteristic
224C4#53611yescharacteristic
225C4#54811yescharacteristic
226C4D4×Q811yescharacteristic
227C2#240911yescharacteristic
228C2#300411yescharacteristic
229C2#300911yescharacteristic
230C2#336912yes—
231C2#338712yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
52. C2×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q822
C2×#4752
C2×#4412
#25812
#20582
#20242
#19932
#19852
#18782
#18652
#18052