Finite Group Databasebeta

← #30148

Subgroups — #30148

Order 256.

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

normal subgroups only

This group has 327 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#2088C212yesmaximal
4#2084C211yesmaximal, characteristic
5#2072C211yesmaximal, characteristic
6#2068C212yesmaximal
7#2057C211yesmaximal, characteristic
8#2024C211yesmaximal, characteristic
9#1927C212yesmaximal
10#1918C212yesmaximal
11#1831C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8—21no—
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19#391—21no—
20#391C2211yescharacteristic
21#391C2211yescharacteristic
22#391C2211yescharacteristic
23#389—21no—
24#389C2212yes—
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#387—21no—
29#387—21no—
30#387—21no—
31#384C2211yescharacteristic
32#383—21no—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2212yes—
35C43C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—24no—
48C4×Q8—44no—
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⋊C4C2311yescharacteristic
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C42—22no—
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42D411yescharacteristic
65C2×C42D411yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#107—24no—
70#106—42no—
71C4⋊C8—42no—
72#103—24no—
73#103—42no—
74#95—22no—
75#95—22no—
76#95—21no—
77#95—21no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85C2×Q8—24no—
86C2×Q8—22no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—21no—
90C2×Q8—42no—
91C2×Q8—42no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2411yescharacteristic
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C4⋊C4—24no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—42no—
120C4⋊C4—44no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4#4114yes—
126C42—22no—
127C42—22no—
128C42—22no—
129C42—22no—
130C42—22no—
131C42—21no—
132C42—22no—
133C42—21no—
134C42—44no—
135C23C22×D411yescharacteristic
136Q8—48no—
137Q8—42no—
138Q8—42no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4#13512yes—
172C2×C4AGL(1,ℤ/8)12yes—
173C2×C4#13712yes—
174C2×C4#13712yes—
175C2×C4C2×#4112yes—
176C2×C4H(5,𝔽2)11yescharacteristic
177C2×C42−1+411yescharacteristic
178C2×C42−1+412yes—
179C8—42no—
180C8—42no—
181C22#52511yescharacteristic
182C22#53611yescharacteristic
183C22#53711yescharacteristic
184C22C2×#13511yescharacteristic
185C22C2×AGL(1,ℤ/8)11yescharacteristic
186C22C2×#13711yescharacteristic
187C22C2×#13711yescharacteristic
188C4—22no—
189C4—22no—
190C4—22no—
191C4—22no—
192C4—44no—
193C4—41no—
194C4—41no—
195C4—81no—
196C4—81no—
197C4—81no—
198C4—81no—
199C4—82no—
200C4#54112yes—
201C4#54212yes—
202C2#263711yescharacteristic
203C2#310511yescharacteristic
204C2#311311yescharacteristic
205C2#323111yescharacteristic
206C2#323811yescharacteristic
207C2#324811yescharacteristic
208C2#325411yescharacteristic
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
#20882
#20842
#20722
#20682
#20572
#20242
#19272
#19182
#18312