Finite Group Databasebeta

← #30067

Subgroups — #30067

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#2862C211yesmaximal, characteristic
2C4×#125C211yesmaximal, characteristic
3#2097C212yesmaximal
4#2058C211yesmaximal, characteristic
5#2057C211yesmaximal, characteristic
6#2043C211yesmaximal, characteristic
7#2043C211yesmaximal, characteristic
8#2026C212yesmaximal
9#1928C212yesmaximal
10#1927C212yesmaximal
11#1839C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21#391—21no—
22#391—21no—
23#391C2212yes—
24#391C2212yes—
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#387—21no—
28#387—21no—
29#382—21no—
30#382—21no—
31#382C2211yescharacteristic
32C4×C4⋊C4—21no—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—42no—
48C4⋊Q8—42no—
49#125—22no—
50#125—22no—
51C2×C4⋊C4—21no—
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—21no—
64C2×C42—21no—
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72#107—42no—
73#106—42no—
74#103—24no—
75#103—24no—
76#103—42no—
77#103—42no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—42no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2411yescharacteristic
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4#4112yes—
130C4⋊C4#4112yes—
131C42—22no—
132C42—22no—
133C42—22no—
134C42—21no—
135C42—21no—
136C42—22no—
137C42—22no—
138C42—41no—
139C42—42no—
140C42—42no—
141C23C22×D411yescharacteristic
142Q8—42no—
143Q8—42no—
144Q8—42no—
145Q8—42no—
146Q8—84no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4#13512yes—
181C2×C4#13712yes—
182C2×C4#13712yes—
183C2×C4C2×#4111yescharacteristic
184C2×C4C2×#4111yescharacteristic
185C2×C4H(5,𝔽2)12yes—
186C2×C42−1+412yes—
187C8—42no—
188C8—42no—
189C22#52511yescharacteristic
190C22#53511yescharacteristic
191C22#53611yescharacteristic
192C22C2×#13511yescharacteristic
193C22C2×#13711yescharacteristic
194C22C2×#13711yescharacteristic
195C22#57511yescharacteristic
196C4—22no—
197C4—22no—
198C4—22no—
199C4—41no—
200C4—42no—
201C4—41no—
202C4—42no—
203C4—41no—
204C4—82no—
205C4—81no—
206C4—82no—
207C4—81no—
208C4#54212yes—
209C4#54212yes—
210C2#263211yescharacteristic
211C2#310711yescharacteristic
212C2#311611yescharacteristic
213C2#321111yescharacteristic
214C2#321811yescharacteristic
215C2#323111yescharacteristic
216C2#323611yescharacteristic
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
#28622
C4×#1252
#20972
#20582
#20572
#20432
#20262
#19282
#19272
#18392