Finite Group Databasebeta

← #23782

Subgroups — #23782

Order 256.

366 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C4×C4⋊Q8C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic
4#2095C211yesmaximal, characteristic
5#2086C212yesmaximal
6#2066C212yesmaximal
7#2058C211yesmaximal, characteristic
8#2024C211yesmaximal, characteristic
9#1776C212yesmaximal
10#1767C211yesmaximal, characteristic
11#1763C212yesmaximal
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8—21no—
14C2×C4×Q8C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#97C2212yes—
21#391—21no—
22#391C2211yescharacteristic
23#391C2211yescharacteristic
24#389—21no—
25#389C2212yes—
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#387—21no—
29#387—21no—
30#384C2211yescharacteristic
31C4×C4⋊C4—21no—
32C4×C4⋊C4—21no—
33C4×C4⋊C4C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35#376—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—24no—
48C4×Q8—22no—
49C4×Q8—44no—
50C4×Q8Q814yescomplemented
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C42—21no—
64C2×C42—22no—
65C2×C42—21no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—24no—
74#106—24no—
75C4⋊C8—24no—
76#103—24no—
77#103—24no—
78#97—24no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—22no—
86#95—21no—
87#95—22no—
88C2×Q8—24no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8C2×Q812yes—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8#4114yes—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C4⋊C4—42no—
127C4⋊C4C2×Q812yes—
128C4⋊C4#4114yes—
129C42—24no—
130C42—22no—
131C42—22no—
132C42—22no—
133C42—22no—
134C42—22no—
135C42—41no—
136C42—42no—
137C42—44no—
138C42C2×Q812yes—
139C42#4112yes—
140C23C22×D411yescharacteristic
141Q8—24no—
142Q8—48no—
143Q8—42no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4C22⋊Q812yes—
176C2×C4C22⋊Q812yes—
177C2×C4AGL(1,ℤ/8)12yes—
178C2×C4#13712yes—
179C2×C4#13712yes—
180C2×C4C22×Q811yescharacteristic
181C2×C4C2×#4111yescharacteristic
182C2×C4C2×#4112yes—
183C2×C4C2×#4112yes—
184C8—24no—
185C8—42no—
186C22C2×C22⋊Q811yescharacteristic
187C22#52511yescharacteristic
188C22#53211yescharacteristic
189C22C2×AGL(1,ℤ/8)11yescharacteristic
190C22C2×#13711yescharacteristic
191C22C2×#13711yescharacteristic
192C22#57511yescharacteristic
193C4—22no—
194C4—22no—
195C4—22no—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—44no—
200C4—42no—
201C4—41no—
202C4—81no—
203C4—82no—
204C4—81no—
205C4#52912yes—
206C4#53312yes—
207C2#239611yescharacteristic
208C2#307811yescharacteristic
209C2#307911yescharacteristic
210C2#311011yescharacteristic
211C2#311711yescharacteristic
212C2#318011yescharacteristic
213C2#318511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
50. C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊Q82
#22952
#21002
#20952
#20862
#20662
#20582
#20242
#17762
#17672
#17632