Finite Group Databasebeta

← #7765

Subgroups — #7765

Order 256.

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

normal subgroups only

This group has 223 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#2057C211yesmaximal, characteristic, complemented
3#2043C211yesmaximal, characteristic, complemented
4#2023C211yesmaximal, characteristic
5#1271C211yesmaximal, characteristic, complemented
6#1271C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic
8C2×C4⋊Q8C2211yescharacteristic
9C2×#126—21no—
10C2×#107—21no—
11C2×#103—21no—
12C2×#103—21no—
13C2×#102—21no—
14#399—21no—
15#396—21no—
16#394—21no—
17#390C2211yescharacteristic
18#389—21no—
19#389C2211yescharacteristic
20#389C2211yescharacteristic
21#388—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8—21no—
28C22×D4D411yescharacteristic
29C22×C8—21no—
30C22×C8—21no—
31C22×C8—21no—
32C4⋊Q8—42no—
33#126—42no—
34#126—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⋊C4D411yescharacteristic
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×C22⋊C4—21no—
52C2×C42—21no—
53C2×C42C2311yescharacteristic
54#107—42no—
55#103—42no—
56#103—42no—
57#102—42no—
58#95—21no—
59#95—21no—
60#95—21no—
61C24—21no—
62C2×Q8—22no—
63C2×Q8—41no—
64C2×Q8—41no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×D4—21no—
69C2×D4—21no—
70C2×D4—41no—
71C2×D4—41no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—41no—
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C2×C8—22no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C2×C8—41no—
89C2×C8—41no—
90C4⋊C4—21no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—42no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—81no—
106C4⋊C4D812yes—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—81no—
114C22⋊C4—81no—
115C42—41no—
116C42—42no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23#12011yescharacteristic
125Q8—42no—
126Q8—42no—
127Q8—81no—
128Q8—81no—
129D4—42no—
130D4—81no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—81no—
154C2×C4—81no—
155C2×C4C2×D811yescharacteristic
156C2×C4#13511yescharacteristic
157C2×C4#13511yescharacteristic
158C2×C4AGL(1,ℤ/8)11yescharacteristic
159C2×C4#13711yescharacteristic
160C2×C4#13711yescharacteristic
161C8—42no—
162C8—81no—
163C8—81no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22#44711yescharacteristic
172C22#45211yescharacteristic
173C22#45211yescharacteristic
174C22#45411yescharacteristic
175C22#45511yescharacteristic
176C22#45611yescharacteristic
177C22B(4,𝔽2)11yescharacteristic
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—42no—
185C4—81no—
186C4—82no—
187C4—81no—
188C4—81no—
189C4—81no—
190C2—161no—
191C2#159811yescharacteristic
192C2#160311yescharacteristic
193C2#160611yescharacteristic
194C2#163811yescharacteristic
195C2#164111yescharacteristic
196C2#164711yescharacteristic
197C2#166411yescharacteristic
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. #2057C2semidirect
3. #2043C2semidirect
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
#20572
#20432
#20232
#12712
#12702