Finite Group Databasebeta

← #30080

Subgroups — #30080

Order 256.

328 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#2864C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic
3#2097C212yesmaximal
4#2084C211yesmaximal, characteristic
5#2083C211yesmaximal, characteristic
6#2074C212yesmaximal
7#2043C211yesmaximal, characteristic
8#2043C211yesmaximal, characteristic
9#1865C212yesmaximal
10#1863C212yesmaximal
11#1839C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8C2212yes—
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20#391C2212yes—
21#389—21no—
22#389—21no—
23#389C2211yescharacteristic
24#389C2211yescharacteristic
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#387—21no—
28#387—21no—
29#387—21no—
30#387—21no—
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33#382C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2312yes—
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—24no—
47C4×Q8—42no—
48C4×Q8—42no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
63C2×C42—22no—
64C2×C42D411yescharacteristic
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—42no—
72#106—42no—
73#103—24no—
74#103—24no—
75#103—42no—
76#103—42no—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8#4114yes—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2411yescharacteristic
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—44no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C42—22no—
130C42—22no—
131C42—22no—
132C42—22no—
133C42—21no—
134C42—21no—
135C42—22no—
136C42—41no—
137C42—44no—
138C23C22×D411yescharacteristic
139Q8—24no—
140Q8—24no—
141Q8—42no—
142Q8—42no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—24no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4#13512yes—
177C2×C4#13712yes—
178C2×C4#13712yes—
179C2×C4C2×#4112yes—
180C2×C4H(5,𝔽2)11yescharacteristic
181C2×C4H(5,𝔽2)11yescharacteristic
182C2×C42−1+412yes—
183C8—42no—
184C8—42no—
185C22#52511yescharacteristic
186C22#53511yescharacteristic
187C22#53611yescharacteristic
188C22C2×#13511yescharacteristic
189C22C2×#13711yescharacteristic
190C22C2×#13711yescharacteristic
191C22#57511yescharacteristic
192C4—22no—
193C4—22no—
194C4—22no—
195C4—44no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4—81no—
201C4—82no—
202C4—81no—
203C4—81no—
204C4#53912yes—
205C4#53912yes—
206C2#263311yescharacteristic
207C2#310811yescharacteristic
208C2#311711yescharacteristic
209C2#321211yescharacteristic
210C2#321811yescharacteristic
211C2#323111yescharacteristic
212C2#323711yescharacteristic
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

Maximal subgroups

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

subgroupindex
#28642
C4×C4⋊Q82
#20972
#20842
#20832
#20742
#20432
#18652
#18632
#18392