Finite Group Databasebeta

← #10427

Subgroups — #10427

Order 256.

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

normal subgroups only

This group has 230 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#1847C211yesmaximal, characteristic, complemented
5C2×#330C211yesmaximal, characteristic, complemented
6C2×#329C211yesmaximal, characteristic
7#1295C211yesmaximal, characteristic, complemented
8C2×#125C2211yescharacteristic
9C2×#124C2211yescharacteristic
10C2×#107—21no—
11C2×#106—21no—
12C2×#103—21no—
13C2×#102—21no—
14C2×#97C2211yescharacteristic
15C2×#97C2211yescharacteristic
16C2×#97C2211yescharacteristic
17#399—21no—
18#397—21no—
19#396—21no—
20#396—21no—
21#395—21no—
22#394—21no—
23#384C2211yescharacteristic
24#382C2211yescharacteristic
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#107—42no—
61#106—42no—
62#103—42no—
63#102—42no—
64#97—21no—
65#97—21no—
66#97—21no—
67#97—21no—
68#97—21no—
69#97D412yes—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74C24—21no—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8—41no—
78C2×Q8—41no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—41no—
82C2×D4—41no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—41no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—42no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—81no—
117C4⋊C4—81no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—81no—
123C22⋊C4—81no—
124C22⋊C4—81no—
125C22⋊C4—81no—
126C42C2×D411yescharacteristic
127C42#4111yescharacteristic
128C42#4111yescharacteristic
129C42#4111yescharacteristic
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23#12011yescharacteristic
138Q8—42no—
139Q8—81no—
140D4—42no—
141D4—81no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
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—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4#12111yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12311yescharacteristic
170C2×C4#12311yescharacteristic
171C2×C4#12411yescharacteristic
172C8—24no—
173C8—42no—
174C8—42no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22#39411yescharacteristic
183C22#45311yescharacteristic
184C22#45311yescharacteristic
185C22#45411yescharacteristic
186C22#45511yescharacteristic
187C22#45611yescharacteristic
188C22#45611yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—81no—
196C4—81no—
197C4—81no—
198C4—81no—
199C4—81no—
200C4—161no—
201C2—161no—
202C2#165611yescharacteristic
203C2#166011yescharacteristic
204C2#166111yescharacteristic
205C2#166411yescharacteristic
206C2#203711yescharacteristic
207C2#203811yescharacteristic
208C2#203911yescharacteristic
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. #1847C2semidirect
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
#18472
C2×#3302
C2×#3292
#12952