Finite Group Databasebeta

← #7793

Subgroups — #7793

Order 256.

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

normal subgroups only

This group has 210 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#2727C211yesmaximal, characteristic
2#2093C211yesmaximal, characteristic
3#2084C211yesmaximal, characteristic, complemented
4#2083C211yesmaximal, characteristic, complemented
5#1271C211yesmaximal, characteristic, complemented
6#1271C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic
8C2×#126—21no—
9C2×#107—21no—
10C2×#107—21no—
11C2×#106—21no—
12C2×#102—21no—
13#399—21no—
14#396—21no—
15#394—21no—
16#390C2211yescharacteristic
17#389—21no—
18#389C2211yescharacteristic
19#389C2211yescharacteristic
20#388—21no—
21#384C2211yescharacteristic
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×D4D411yescharacteristic
28C22×C8—21no—
29C22×C8—21no—
30C22×C8—21no—
31#126—42no—
32#126—42no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4D411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
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#107—42no—
55#106—42no—
56#102—42no—
57#95—21no—
58#95—21no—
59#95—21no—
60C24—21no—
61C2×D4—21no—
62C2×D4—21no—
63C2×D4—41no—
64C2×D4—41no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—41no—
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C2×C8—22no—
75C2×C8—22no—
76C2×C8—22no—
77C2×C8—41no—
78C2×C8—41no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—42no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—81no—
101C4⋊C4D812yes—
102C22⋊C4—42no—
103C22⋊C4—42no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—81no—
109C22⋊C4—81no—
110C42—41no—
111C42—42no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23#12011yescharacteristic
120D4—42no—
121D4—81no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—41no—
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—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4C2×D811yescharacteristic
147C2×C4#13511yescharacteristic
148C2×C4#13511yescharacteristic
149C2×C4AGL(1,ℤ/8)11yescharacteristic
150C2×C4AGL(1,ℤ/8)11yescharacteristic
151C2×C4#13711yescharacteristic
152C8—42no—
153C8—42no—
154C8—42no—
155C22—81no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22#44711yescharacteristic
163C22#44911yescharacteristic
164C22#45211yescharacteristic
165C22#45311yescharacteristic
166C22#45611yescharacteristic
167C22#45611yescharacteristic
168C22#45811yescharacteristic
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—42no—
176C4—81no—
177C4—81no—
178C4—81no—
179C4—81no—
180C4—161no—
181C2—161no—
182C2#160911yescharacteristic
183C2#161011yescharacteristic
184C2#161811yescharacteristic
185C2#162111yescharacteristic
186C2#164711yescharacteristic
187C2#165211yescharacteristic
188C2#166011yescharacteristic
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
3. #2084C2semidirect
4. #2083C2semidirect
5. #1271C2semidirect
6. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#27272
#20932
#20842
#20832
#12712
#12702