Finite Group Databasebeta

← #30187

Subgroups — #30187

Order 256.

407 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 135 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#2857C211yesmaximal, characteristic
2#2857C211yesmaximal, characteristic
3#2100C212yesmaximal, complemented
4#2072C212yesmaximal, complemented
5#2043C212yesmaximal, complemented
6#2024C212yesmaximal, complemented
7#1892C214yesmaximal
8C2×#336C211yesmaximal, characteristic
9C23×C8C2212yes—
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8C2212yes—
14C22×C42C2211yescharacteristic
15C2×#103C2214yes—
16C2×#103C2214yes—
17#391—21no—
18#391C2212yes—
19#389—21no—
20#389—21no—
21#389—21no—
22#389C2212yes—
23#384C2211yescharacteristic
24#384C2211yescharacteristic
25#380—21no—
26#380—21no—
27#380—21no—
28#380C2212yes—
29#340C2214yes—
30#340C2214yes—
31#336C2212yes—
32#336C2212yes—
33C22×Q8—21no—
34C22×Q8C2312yes—
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8—24no—
39C22×C8—42no—
40C22×C8—42no—
41C22×C8C2314yes—
42C4⋊Q8—42no—
43C4⋊Q8—42no—
44C22⋊Q8—28no—
45C22⋊Q8—44no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C42—42no—
61C2×C42D411yescharacteristic
62C2×C42D411yescharacteristic
63C2×C42C2311yescharacteristic
64C2×C42C2311yescharacteristic
65#103—44no—
66#103—44no—
67#95—22no—
68#95—22no—
69#95—22no—
70#95—22no—
71C24C2×D411yescharacteristic
72C2×Q8—24no—
73C2×Q8—22no—
74C2×Q8—24no—
75C2×Q8—22no—
76C2×Q8—42no—
77C2×Q8—42no—
78C2×Q8—42no—
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4SD1614yes—
91C22×C4Q1614yes—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4#4114yes—
98C22×C4C2411yescharacteristic
99C2×C8—24no—
100C2×C8—24no—
101C2×C8—24no—
102C2×C8—48no—
103C4⋊C4—24no—
104C4⋊C4—22no—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—44no—
114C4⋊C4—42no—
115C4⋊C4—44no—
116C22⋊C4—44no—
117C22⋊C4—44no—
118C42—22no—
119C42—22no—
120C42—41no—
121C42—41no—
122C23—21no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23#12412yes—
127C23C2×SD1612yes—
128C23C2×Q1612yes—
129C23C22×D411yescharacteristic
130Q8—44no—
131Q8—44no—
132Q8—84no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—44no—
146C2×C4—44no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4#12412yes—
160C2×C4C2×SD1612yes—
161C2×C4C2×SD1612yes—
162C2×C4C2×Q1612yes—
163C2×C4C2×Q1612yes—
164C2×C4C2×#4112yes—
165C2×C4H(5,𝔽2)11yescharacteristic
166C2×C4H(5,𝔽2)12yes—
167C2×C42−1+411yescharacteristic
168C8—44no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22#48714yes—
176C22C2×#12411yescharacteristic
177C22#53411yescharacteristic
178C22#53711yescharacteristic
179C22C22×SD1611yescharacteristic
180C22C22×Q1611yescharacteristic
181C22#57511yescharacteristic
182C22#57511yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—22no—
188C4—82no—
189C4—82no—
190C4—82no—
191C4—82no—
192C4#53812yes—
193C4#53912yes—
194C2—24no—
195C2#263911yescharacteristic
196C2C2×#48711yescharacteristic
197C2#313511yescharacteristic
198C2#318711yescharacteristic
199C2#318911yescharacteristic
200C2#324611yescharacteristic
201C2#324911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2100C2semidirect
4. #2072C2semidirect
5. #2043C2semidirect
6. #2024C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#21002
#20722
#20432
#20242
#18922
C2×#3362