Finite Group Databasebeta

← #12139

Subgroups — #12139

Order 256.

202 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 35 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#2835C211yesmaximal, characteristic, complemented
2#2044C211yesmaximal, characteristic
3#1607C211yesmaximal, characteristic
4#1601C211yesmaximal, characteristic
5#1351C211yesmaximal, characteristic, complemented
6#1344C211yesmaximal, characteristic, complemented
7#1326C211yesmaximal, characteristic, complemented
8#534C2211yescharacteristic
9#480—21no—
10#448—21no—
11C2×#99—21no—
12#400—21no—
13#400—21no—
14#400—21no—
15#397—21no—
16#397C2211yescharacteristic
17#395—21no—
18#393C2211yescharacteristic
19#386—21no—
20#385—21no—
21#385C2211yescharacteristic
22#342C2211yescharacteristic
23#327—21no—
24#327—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×D4—21no—
29C23×C4D411yescharacteristic
30C2×SD16—41no—
31#123—41no—
32#121—21no—
33#121—21no—
34#120—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—41no—
44C2×C22⋊C4D411yescharacteristic
45C2×C22⋊C4C2311yescharacteristic
46#107—41no—
47#103—41no—
48#102—41no—
49#99—42no—
50C22⋊C8—21no—
51C22⋊C8—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—41no—
61#95D411yescharacteristic
62#95D411yescharacteristic
63C24—21no—
64C24C2×D411yescharacteristic
65C2×Q8—41no—
66C2×Q8—81no—
67C2×D4—21no—
68C2×D4—21no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—81no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87SD16—82no—
88C2×C8—41no—
89C2×C8—41no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—81no—
94C4⋊C4—81no—
95C4⋊C4—81no—
96C4⋊C4—81no—
97C22⋊C4—21no—
98C22⋊C4—21no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—81no—
107C22⋊C4—82no—
108C23—21no—
109C23—21no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—81no—
116C23—81no—
117C23#12011yescharacteristic
118C23#13511yescharacteristic
119C23AGL(1,ℤ/8)11yescharacteristic
120Q8—42no—
121Q8—81no—
122D4—81no—
123D4—81no—
124D4—81no—
125D4—81no—
126D4—81no—
127D4—81no—
128C2×C4—21no—
129C2×C4—21no—
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—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—81no—
158C8—81no—
159C8—81no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—81no—
169C22—82no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—161no—
174C22#44911yescharacteristic
175C22#45411yescharacteristic
176C22B(4,𝔽2)11yescharacteristic
177C4—41no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4—81no—
183C4—81no—
184C4—81no—
185C4—81no—
186C4—161no—
187C2—21no—
188C2—21no—
189C2—41no—
190C2—41no—
191C2—161no—
192C2#159711yescharacteristic
193C2#219911yescharacteristic
194C2#220111yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2835C2semidirect
5. #1351C2semidirect
6. #1344C2semidirect
7. #1326C2semidirect

Maximal subgroups

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

subgroupindex
#28352
#20442
#16072
#16012
#13512
#13442
#13262