Finite Group Databasebeta

← #30216

Subgroups — #30216

Order 256.

374 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 119 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#2868C211yesmaximal, characteristic, complemented
2#2861C211yesmaximal, characteristic
3#2072C212yesmaximal, complemented
4#2071C212yesmaximal
5#2024C212yesmaximal, complemented
6#2023C212yesmaximal
7#1992C212yesmaximal, complemented
8#1959C212yesmaximal
9#1745C211yesmaximal, characteristic, complemented
10C2×C4⋊Q8—21no—
11C2×C4⋊Q8—21no—
12C2×C4⋊Q8C2212yes—
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×#103C2214yes—
16C2×#102C2214yes—
17C2×#97C2212yes—
18#391C2211yescharacteristic
19#391C2211yescharacteristic
20#390C2211yescharacteristic
21#390C2211yescharacteristic
22#388—21no—
23#388—21no—
24#388—21no—
25#388—21no—
26#384—21no—
27#384—21no—
28#384C2212yes—
29#384C2212yes—
30C43C2211yescharacteristic
31#340C2214yes—
32#340C2214yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8—22no—
36C22×Q8C2311yescharacteristic
37C22×D4C2311yescharacteristic
38C22×C8C2314yes—
39C4⋊Q8—44no—
40C4⋊Q8—44no—
41C4⋊Q8—44no—
42#125—24no—
43#124—24no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4C2311yescharacteristic
52C2×C4⋊C4C2311yescharacteristic
53C2×C4⋊C4C2314yes—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C42D411yescharacteristic
61C2×C42D411yescharacteristic
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67#103—44no—
68#102—44no—
69#97—24no—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74C24—21no—
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8—22no—
78C2×Q8—44no—
79C2×Q8—44no—
80C2×Q8—44no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—22no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—24no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2411yescharacteristic
95C2×C8—24no—
96C2×C8—24no—
97C2×C8—24no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—24no—
101C4⋊C4—24no—
102C4⋊C4—44no—
103C4⋊C4—44no—
104C4⋊C4—44no—
105C4⋊C4—44no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—42no—
115C42—22no—
116C42—22no—
117C42—22no—
118C42—22no—
119C42—22no—
120C42—22no—
121C42#4114yes—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23C22×D411yescharacteristic
130Q8—44no—
131Q8—88no—
132D4—44no—
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—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—44no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4#12412yes—
158C2×C4#12412yes—
159C2×C4AGL(1,ℤ/8)12yes—
160C2×C4AGL(1,ℤ/8)12yes—
161C2×C4#13712yes—
162C2×C4#13712yes—
163C2×C4C2×#4112yes—
164C2×C4H(5,𝔽2)12yes—
165C2×C42−1+412yes—
166C8—44no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22C2×#12411yescharacteristic
175C22#53511yescharacteristic
176C22#53611yescharacteristic
177C22C2×AGL(1,ℤ/8)11yescharacteristic
178C22C2×AGL(1,ℤ/8)11yescharacteristic
179C22C2×#13711yescharacteristic
180C22C2×#13711yescharacteristic
181C4—22no—
182C4—22no—
183C4—22no—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—81no—
188C4—81no—
189C4—81no—
190C4—84no—
191C4#54214yes—
192C2—161no—
193C2#264011yescharacteristic
194C2#314111yescharacteristic
195C2#314111yescharacteristic
196C2#321711yescharacteristic
197C2#321811yescharacteristic
198C2#323811yescharacteristic
199C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2868C2semidirect
3. #2072C2semidirect
5. #2024C2semidirect
7. #1992C2semidirect
9. #1745C2semidirect

Maximal subgroups

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

subgroupindex
#28682
#28612
#20722
#20712
#20242
#20232
#19922
#19592
#17452