Finite Group Databasebeta

← #30132

Subgroups — #30132

Order 256.

351 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 115 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#2863C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic
4#2096C211yesmaximal, characteristic
5#2066C212yesmaximal
6#2056C212yesmaximal
7#2043C211yesmaximal, characteristic
8#2016C211yesmaximal, characteristic
9#1878C212yesmaximal
10#1865C212yesmaximal
11#1831C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21#391C2212yes—
22#391C2211yescharacteristic
23#391C2211yescharacteristic
24#391C2211yescharacteristic
25#389—21no—
26#389—21no—
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#387—21no—
31#387—21no—
32#387—21no—
33#384C2211yescharacteristic
34#383—21no—
35C4×C4⋊C4—21no—
36C4×C4⋊C4—21no—
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×Q8C2312yes—
45C2×Q16—24no—
46C2×Q16—42no—
47C2×Q16—42no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—24no—
51C4×Q8—42no—
52C4×Q8—42no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C42—22no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#106—42no—
74C4⋊C8—42no—
75#103—24no—
76#103—42no—
77#103—42no—
78#103—42no—
79#95—22no—
80#95—22no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—21no—
99C2×Q8#4114yes—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2411yescharacteristic
113Q16—48no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—44no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C42—22no—
135C42—22no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—21no—
140C42—22no—
141C42—21no—
142C42—44no—
143C23C22×D411yescharacteristic
144Q8—24no—
145Q8—24no—
146Q8—42no—
147Q8—42no—
148Q8—42no—
149Q8—42no—
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—24no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4#13512yes—
183C2×C4#13712yes—
184C2×C4#13712yes—
185C2×C4#13712yes—
186C2×C4C2×#4112yes—
187C2×C4H(5,𝔽2)11yescharacteristic
188C2×C4H(5,𝔽2)12yes—
189C2×C42−1+411yescharacteristic
190C8—42no—
191C8—42no—
192C22#52511yescharacteristic
193C22#53511yescharacteristic
194C22#53711yescharacteristic
195C22C2×#13511yescharacteristic
196C22C2×#13711yescharacteristic
197C22C2×#13711yescharacteristic
198C22C2×#13711yescharacteristic
199C4—22no—
200C4—22no—
201C4—22no—
202C4—22no—
203C4—44no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—81no—
208C4—82no—
209C4—81no—
210C4—81no—
211C4#53812yes—
212C4#53912yes—
213C2#263611yescharacteristic
214C2#310811yescharacteristic
215C2#311311yescharacteristic
216C2#321011yescharacteristic
217C2#321811yescharacteristic
218C2#324811yescharacteristic
219C2#325511yescharacteristic
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
#28632
C4×C4⋊Q82
#20992
#20962
#20662
#20562
#20432
#20162
#18782
#18652
#18312