Finite Group Databasebeta

← #25849

Subgroups — #25849

Order 256.

500 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 153 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×#501C211yesmaximal, characteristic
3C2×#477C211yesmaximal, characteristic
4#2546C211yesmaximal, characteristic
5#2059C212yesmaximal
6#2057C212yesmaximal
7#1989C211yesmaximal, characteristic
8#1985C211yesmaximal, characteristic
9#1865C212yesmaximal
10#1863C212yesmaximal
11#1763C211yesmaximal, characteristic
12#557—24no—
13#557—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8—21no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—21no—
19C2×#125—21no—
20C2×#125C2211yescharacteristic
21C2×#125C2211yescharacteristic
22C2×C4×Q8—21no—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27#501—24no—
28#477—24no—
29C2×#107C2212yes—
30C2×#107C2212yes—
31C2×#106C2212yes—
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#97C2211yescharacteristic
36#398—21no—
37#398—21no—
38#391—21no—
39#389C2212yes—
40#389C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8D411yescharacteristic
50C22×Q8C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—28no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—44no—
59#125—28no—
60#125—28no—
61#125—22no—
62#125—22no—
63#125—22no—
64#125D414yes—
65C4×Q8—28no—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×Q8D414yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2312yes—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—24no—
101#107—24no—
102#106—24no—
103C4⋊C8—22no—
104#103—24no—
105#103Q818yescomplemented
106#97—22no—
107#95—21no—
108#95—22no—
109#95—22no—
110C2×Q8—24no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—41no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8C2×D412yes—
122C2×Q8C2×Q814yes—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—24no—
140C2×C8C2×Q814yes—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—44no—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×Q814yes—
165C42—24no—
166C42—24no—
167C42—24no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—42no—
173C42C2×D412yes—
174C42C2×D412yes—
175C23C22×D411yescharacteristic
176Q8—22no—
177Q8—24no—
178Q8—44no—
179Q8—48no—
180Q8—84no—
181Q8C4⋊Q814yescomplemented
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—24no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4#12012yes—
211C2×C4#12012yes—
212C2×C4AGL(1,ℤ/8)12yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C22×Q812yes—
216C2×C42−1+412yes—
217C8—24no—
218C8—42no—
219C22C2×#12011yescharacteristic
220C22D4×Q812yes—
221C22C2×AGL(1,ℤ/8)11yescharacteristic
222C22#57511yescharacteristic
223C22#57812yes—
224C4—24no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—44no—
229C4—41no—
230C4—42no—
231C4—42no—
232C4—41no—
233C4—41no—
234C4—42no—
235C4—82no—
236C4C2×C4⋊Q811yescharacteristic
237C4#53611yescharacteristic
238C4D4×Q812yes—
239C2#243411yescharacteristic
240C2#300611yescharacteristic
241C2#301111yescharacteristic
242C2#338612yes—
243C2#339012yes—
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
105. #103Q8semidirect
181. Q8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#5012
C2×#4772
#25462
#20592
#20572
#19892
#19852
#18652
#18632
#17632