Finite Group Databasebeta

← #7787

Subgroups — #7787

Order 256.

215 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 51 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#2732C211yesmaximal, characteristic
2#2099C211yesmaximal, characteristic, complemented
3#2072C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic, complemented
6#1271C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic
8C2×C22⋊Q8—21no—
9C2×#103—21no—
10C2×#102—21no—
11#399—21no—
12#398—21no—
13#396—21no—
14#391C2211yescharacteristic
15#390C2211yescharacteristic
16#389C2211yescharacteristic
17#386—21no—
18#385—21no—
19#384C2211yescharacteristic
20#340—21no—
21#340—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C22×C8—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22⋊Q8—42no—
34C22⋊Q8—42no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42C2311yescharacteristic
52#103—42no—
53#102—42no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24—21no—
59C2×Q8—21no—
60C2×Q8—41no—
61C2×Q8—41no—
62C2×Q8D812yes—
63C2×D4—21no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4SD1612yes—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—42no—
73C22×C4—41no—
74C22×C4—81no—
75C22×C4—81no—
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—81no—
103C22⋊C4—42no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—81no—
109C22⋊C4—81no—
110C42—41no—
111C23—22no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23#12011yescharacteristic
119Q8—42no—
120Q8—81no—
121D4—42no—
122D4—81no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—84no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4C2×D811yescharacteristic
149C2×C4C2×SD1611yescharacteristic
150C2×C4#13511yescharacteristic
151C2×C4AGL(1,ℤ/8)11yescharacteristic
152C2×C4#13711yescharacteristic
153C2×C4#13711yescharacteristic
154C8—81no—
155C8—81no—
156C8—81no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22#44711yescharacteristic
165C22#44811yescharacteristic
166C22#45211yescharacteristic
167C22#45411yescharacteristic
168C22#45511yescharacteristic
169C22#45511yescharacteristic
170C22#45811yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—42no—
178C4—81no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4—161no—
183C2—82no—
184C2#160711yescharacteristic
185C2#160811yescharacteristic
186C2#161511yescharacteristic
187C2#162711yescharacteristic
188C2#164111yescharacteristic
189C2#164311yescharacteristic
190C2#166211yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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. #2099C2semidirect
3. #2072C2semidirect
5. #1272C2semidirect
6. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#27322
#20992
#20722
#20712
#12722
#12712
#12702