Finite Group Databasebeta

← #7796

Subgroups — #7796

Order 256.

239 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 49 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#2733C211yesmaximal, characteristic
2#2093C211yesmaximal, characteristic, complemented
3#2072C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic, complemented
5#1272C211yesmaximal, characteristic, complemented
6#1270C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic, complemented
8C2×#124—21no—
9C2×#107—21no—
10C2×#103—21no—
11C2×#102—21no—
12C2×#102—21no—
13#394—21no—
14#392—21no—
15#392—21no—
16#391C2211yescharacteristic
17#390C2211yescharacteristic
18#390C2211yescharacteristic
19#388—21no—
20#388—21no—
21#384C2211yescharacteristic
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×D4—21no—
29C22×D4—21no—
30C22×D4D411yescharacteristic
31C22×D4D411yescharacteristic
32C22×C8—21no—
33C22×C8—21no—
34C22×C8—21no—
35#124—42no—
36#124—42no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4D411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42—21no—
52C2×C42C2311yescharacteristic
53#107—42no—
54#103—42no—
55#102—42no—
56#102—42no—
57#95—21no—
58#95—21no—
59#95—21no—
60C24—21no—
61C24—21no—
62C2×Q8—21no—
63C2×Q8—41no—
64C2×Q8—41no—
65C2×Q8D812yes—
66C2×D4—21no—
67C2×D4—21no—
68C2×D4—21no—
69C2×D4—21no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—42no—
75C2×D4—81no—
76C2×D4—81no—
77C2×D4—81no—
78C2×D4—81no—
79C2×D4—81no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—41no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—22no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C2×C8—41no—
96C2×C8—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—81no—
117C22⋊C4—81no—
118C42—41no—
119C42—42no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23#12011yescharacteristic
135Q8—42no—
136Q8—81no—
137D4—42no—
138D4—42no—
139D4—81no—
140D4—81no—
141D4—84no—
142D4—84no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
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—81no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4C2×D811yescharacteristic
165C2×C4#13511yescharacteristic
166C2×C4#13511yescharacteristic
167C2×C4AGL(1,ℤ/8)11yescharacteristic
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C2×C4#13711yescharacteristic
170C8—42no—
171C8—81no—
172C8—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22#44711yescharacteristic
188C22#44911yescharacteristic
189C22#45211yescharacteristic
190C22#45311yescharacteristic
191C22#45511yescharacteristic
192C22#45511yescharacteristic
193C22#45811yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—42no—
201C4—81no—
202C4—81no—
203C4—81no—
204C4—161no—
205C2—161no—
206C2—161no—
207C2#160711yescharacteristic
208C2#161211yescharacteristic
209C2#161811yescharacteristic
210C2#162111yescharacteristic
211C2#164111yescharacteristic
212C2#164611yescharacteristic
213C2#165911yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radical#2733C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2093C2semidirect
3. #2072C2semidirect
4. #2071C2semidirect
5. #1272C2semidirect
6. #1270C2semidirect
7. #1270C2semidirect

Maximal subgroups

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

subgroupindex
#27332
#20932
#20722
#20712
#12722
#12702