Finite Group Databasebeta

← #24191

Subgroups — #24191

Order 256.

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

normal subgroups only

This group has 525 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×#557C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic
3C2×C4×Q16C211yesmaximal, characteristic
4#2582C211yesmaximal, characteristic
5#2043C212yesmaximal
6#2016C212yesmaximal
7#1984C211yesmaximal, characteristic
8#1956C211yesmaximal, characteristic
9#1878C212yesmaximal
10#1863C212yesmaximal
11#1803C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C22×Q16C2212yes—
14#557—24no—
15#557—24no—
16C2×C4⋊Q8C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—21no—
19C2×#125—22no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2212yes—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25#462—24no—
26C4×Q16—24no—
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×C4×C8C2211yescharacteristic
36#398—22no—
37#391—21no—
38#391C2212yes—
39#389—21no—
40#389C2212yes—
41#384C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8D411yescharacteristic
46C22×Q8C2312yes—
47C22×Q8C2311yescharacteristic
48C22×Q8C2311yescharacteristic
49C2×Q16—22no—
50C2×Q16—22no—
51C2×Q16—24no—
52C2×Q16—24no—
53C2×Q16—22no—
54C2×Q16—42no—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C4⋊Q8—24no—
59C4⋊Q8—24no—
60C4⋊Q8—22no—
61C4⋊Q8D414yes—
62#125—216no—
63#125—28no—
64C4×Q8—28no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8D414yes—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D412yes—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C42—22no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C2312yes—
92C2×C42C2311yescharacteristic
93#107—22no—
94C4⋊C8—22no—
95#103—22no—
96#103—24no—
97#103—24no—
98#103—22no—
99#103—42no—
100#103—42no—
101C4×C8—22no—
102#95—22no—
103#95—22no—
104#95—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8Q1614yescomplemented
123C2×Q8C2×D414yes—
124C2×Q8C2×D412yes—
125C2×Q8#4112yes—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2411yescharacteristic
137Q16—28no—
138Q16—48no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—21no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8#4112yes—
148C4⋊C4—28no—
149C4⋊C4—28no—
150C4⋊C4—28no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—81no—
164C4⋊C4—82no—
165C4⋊C4C2×D412yes—
166C42—28no—
167C42—24no—
168C42—22no—
169C42—22no—
170C42—42no—
171C42C2×D412yes—
172C42C2×D412yes—
173C23C22×D411yescharacteristic
174Q8—28no—
175Q8—22no—
176Q8—22no—
177Q8—22no—
178Q8—22no—
179Q8—24no—
180Q8—24no—
181Q8—42no—
182Q8—42no—
183Q8—41no—
184Q8—41no—
185Q8—41no—
186Q8—41no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4#12012yes—
213C2×C4#12012yes—
214C2×C4C2×Q1614yes—
215C2×C4C2×Q1612yes—
216C2×C4#13712yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C2×C4C2×#4111yescharacteristic
220C2×C4C2×#4111yescharacteristic
221C2×C4H(5,𝔽2)11yescharacteristic
222C2×C42−1+411yescharacteristic
223C8—22no—
224C8—41no—
225C8—42no—
226C22C2×#12011yescharacteristic
227C22#54811yescharacteristic
228C22#55011yescharacteristic
229C22C22×Q1611yescharacteristic
230C22C2×#13711yescharacteristic
231C22#57711yescharacteristic
232C22#57811yescharacteristic
233C4—24no—
234C4—22no—
235C4—42no—
236C4—44no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4—81no—
242C4#45114yes—
243C4#52511yescharacteristic
244C4#53711yescharacteristic
245C4#54611yescharacteristic
246C4#54711yescharacteristic
247C2#240711yescharacteristic
248C2C2×#45111yescharacteristic
249C2#301311yescharacteristic
250C2#336311yescharacteristic
251C2#337411yescharacteristic
252C2#339511yescharacteristic
253C2#340011yescharacteristic
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
122. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4622
C2×C4×Q162
#25822
#20432
#20162
#19842
#19562
#18782
#18632
#18032