Finite Group Databasebeta

← #12267

Subgroups — #12267

Order 256.

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

normal subgroups only

This group has 207 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#2752C211yesmaximal, characteristic
2#2030C211yesmaximal, characteristic, complemented
3#1614C211yesmaximal, characteristic, complemented
4#1596C211yesmaximal, characteristic
5#1352C211yesmaximal, characteristic, complemented
6#1343C211yesmaximal, characteristic
7#1317C211yesmaximal, characteristic, complemented
8#543C2211yescharacteristic
9#484—21no—
10#450—21no—
11#399C2211yescharacteristic
12#397—21no—
13#394—21no—
14#392C2211yescharacteristic
15#390—21no—
16#388—21no—
17#387—21no—
18#386—21no—
19#381—21no—
20#381C2211yescharacteristic
21#342—21no—
22#342C2211yescharacteristic
23#328—21no—
24#328—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×D4—21no—
28C22×D4D411yescharacteristic
29C23×C4D411yescharacteristic
30C2×SD16—41no—
31C22⋊Q8—21no—
32C22⋊Q8—21no—
33C22⋊Q8—41no—
34C22⋊Q8—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4D411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4D411yescharacteristic
47C2×C22⋊C4C2311yescharacteristic
48C2×C42—21no—
49#107—41no—
50#103—41no—
51#102—41no—
52C22⋊C8—21no—
53C22⋊C8—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—41no—
59#95—41no—
60#95D411yescharacteristic
61#95D411yescharacteristic
62C24—21no—
63C24C2×D411yescharacteristic
64C2×Q8—21no—
65C2×Q8—21no—
66C2×D4—41no—
67C2×D4—42no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—81no—
71C2×D4—81no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86SD16—82no—
87C2×C8—41no—
88C2×C8—41no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—41no—
92C4⋊C4—42no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—81no—
96C22⋊C4—41no—
97C22⋊C4—42no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—81no—
104C22⋊C4—81no—
105C22⋊C4—81no—
106C22⋊C4—81no—
107C22⋊C4—81no—
108C42—81no—
109C23—21no—
110C23—21no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23#12011yescharacteristic
121C23#13511yescharacteristic
122C23#13711yescharacteristic
123Q8—81no—
124Q8—81no—
125D4—42no—
126D4—82no—
127D4—82no—
128D4—81no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
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—42no—
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—81no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—81no—
157C8—81no—
158C8—81no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22#45211yescharacteristic
176C22#45511yescharacteristic
177C22B(4,𝔽2)11yescharacteristic
178C4—42no—
179C4—41no—
180C4—41no—
181C4—81no—
182C4—81no—
183C4—81no—
184C4—81no—
185C4—81no—
186C4—81no—
187C4—161no—
188C2—21no—
189C2—21no—
190C2—41no—
191C2—41no—
192C2—161no—
193C2#160611yescharacteristic
194C2#219811yescharacteristic
195C2#220011yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#392C22
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
2. #2030C2semidirect
3. #1614C2semidirect
5. #1352C2semidirect
7. #1317C2semidirect

Maximal subgroups

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

subgroupindex
#27522
#20302
#16142
#15962
#13522
#13432
#13172