Finite Group Databasebeta

← #10426

Subgroups — #10426

Order 256.

222 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 53 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#2820C211yesmaximal, characteristic
2#1992C211yesmaximal, characteristic, complemented
3#1959C211yesmaximal, characteristic
4#1836C211yesmaximal, characteristic, complemented
5C2×#330C211yesmaximal, characteristic, complemented
6C2×#329C211yesmaximal, characteristic
7#1295C211yesmaximal, characteristic, complemented
8C2×#125C2211yescharacteristic
9C2×#124C2211yescharacteristic
10C2×#103—21no—
11C2×#102—21no—
12C2×#97C2211yescharacteristic
13C2×#97C2211yescharacteristic
14C2×#97C2211yescharacteristic
15#399—21no—
16#397—21no—
17#396—21no—
18#396—21no—
19#395—21no—
20#394—21no—
21#384C2211yescharacteristic
22#382C2211yescharacteristic
23#340—21no—
24#340—21no—
25#340—21no—
26#340—21no—
27#330—22no—
28#329—22no—
29C22×Q8D411yescharacteristic
30C22×D4D411yescharacteristic
31C22×C8—21no—
32C22×C8—21no—
33C22×C8—21no—
34#125—21no—
35#125—21no—
36#125—21no—
37#125—21no—
38#124—21no—
39#124—21no—
40#124—21no—
41#124—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C4⋊C4D411yescharacteristic
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C42C2311yescharacteristic
60#103—42no—
61#102—42no—
62#97—21no—
63#97—21no—
64#97—21no—
65#97—21no—
66#97—21no—
67#97D412yes—
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72C24—21no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—41no—
80C2×D4—41no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—41no—
94C2×C8—41no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—41no—
98C2×C8—41no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—81no—
114C4⋊C4—81no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—81no—
120C22⋊C4—81no—
121C22⋊C4—81no—
122C22⋊C4—81no—
123C42C2×D411yescharacteristic
124C42#4111yescharacteristic
125C42#4111yescharacteristic
126C42#4111yescharacteristic
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23#12011yescharacteristic
135Q8—42no—
136Q8—81no—
137D4—42no—
138D4—81no—
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—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4#12111yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12111yescharacteristic
166C2×C4#12311yescharacteristic
167C2×C4#12311yescharacteristic
168C2×C4#12411yescharacteristic
169C8—42no—
170C8—42no—
171C8—42no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22#39411yescharacteristic
180C22#45311yescharacteristic
181C22#45411yescharacteristic
182C22#45411yescharacteristic
183C22#45511yescharacteristic
184C22#45511yescharacteristic
185C22#45611yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—21no—
192C4—81no—
193C4—81no—
194C4—81no—
195C4—81no—
196C4—81no—
197C4—161no—
198C2—161no—
199C2#165711yescharacteristic
200C2#166111yescharacteristic
201C2#166211yescharacteristic
202C2#166411yescharacteristic
203C2#203711yescharacteristic
204C2#203811yescharacteristic
205C2#204011yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1992C2semidirect
4. #1836C2semidirect
5. C2×#330C2semidirect
7. #1295C2semidirect

Maximal subgroups

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

subgroupindex
#28202
#19922
#19592
#18362
C2×#3302
C2×#3292
#12952