Finite Group Databasebeta

← #12214

Subgroups — #12214

Order 256.

289 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 37 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#2437C211yesmaximal, characteristic
2#2029C211yesmaximal, characteristic
3#1614C211yesmaximal, characteristic, complemented
4#1602C211yesmaximal, characteristic, complemented
5#1352C211yesmaximal, characteristic, complemented
6#1351C211yesmaximal, characteristic, complemented
7#1316C211yesmaximal, characteristic
8#543C2211yescharacteristic
9C2×C22⋊Q8—21no—
10C22×C22⋊C4—21no—
11#484—21no—
12#451—21no—
13#399C2211yescharacteristic
14#396—21no—
15#393C2211yescharacteristic
16#385—21no—
17#385—21no—
18#380—21no—
19#379—21no—
20#379C2211yescharacteristic
21#342—21no—
22#342C2211yescharacteristic
23#328—21no—
24#328—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C25D411yescharacteristic
28C22×Q8D411yescharacteristic
29C23×C4—21no—
30C23×C4—21no—
31C2×Q16—41no—
32C22⋊Q8—21no—
33C22⋊Q8—21no—
34C22⋊Q8—44no—
35C22⋊Q8—41no—
36C22⋊Q8—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4D411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—44no—
51C2×C22⋊C4—41no—
52C2×C22⋊C4—41no—
53C2×C22⋊C4D411yescharacteristic
54C2×C22⋊C4C2311yescharacteristic
55#107—41no—
56#103—41no—
57#103—41no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#95—21no—
61#95—21no—
62#95D411yescharacteristic
63#95D411yescharacteristic
64C24—21no—
65C24—21no—
66C24—42no—
67C24—41no—
68C24—42no—
69C24—41no—
70C24Q1612yescomplemented
71C24C2×D411yescharacteristic
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—42no—
75C2×Q8—41no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—41no—
88C22×C4—81no—
89C22×C4—81no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92Q16—82no—
93C2×C8—41no—
94C2×C8—41no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—41no—
98C4⋊C4—42no—
99C4⋊C4—42no—
100C4⋊C4—42no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—81no—
104C22⋊C4—24no—
105C22⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—82no—
117C22⋊C4—82no—
118C23—22no—
119C23—22no—
120C23—22no—
121C23—21no—
122C23—21no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—44no—
127C23—42no—
128C23—41no—
129C23—44no—
130C23—41no—
131C23—41no—
132C23—42no—
133C23—42no—
134C23—81no—
135C23—84no—
136C23—81no—
137C23—81no—
138C23#12011yescharacteristic
139C23C2×Q1611yescharacteristic
140C23#13711yescharacteristic
141Q8—42no—
142Q8—81no—
143Q8—81no—
144Q8—81no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—84no—
167C2×C4—84no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C8—81no—
174C8—81no—
175C22—21no—
176C22—21no—
177C22—24no—
178C22—21no—
179C22—21no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—44no—
186C22—42no—
187C22—41no—
188C22—42no—
189C22—41no—
190C22—42no—
191C22—81no—
192C22—82no—
193C22—82no—
194C22—82no—
195C22#45111yescharacteristic
196C22#45511yescharacteristic
197C22B(4,𝔽2)11yescharacteristic
198C4—42no—
199C4—41no—
200C4—42no—
201C4—41no—
202C4—81no—
203C4—81no—
204C4—81no—
205C4—81no—
206C4—81no—
207C4—161no—
208C2—21no—
209C2—21no—
210C2—41no—
211C2—44no—
212C2—41no—
213C2#160211yescharacteristic
214C2#219611yescharacteristic
215C2#219711yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radicalC25D4
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
3. #1614C2semidirect
4. #1602C2semidirect
5. #1352C2semidirect
6. #1351C2semidirect
70. C24Q16semidirect

Maximal subgroups

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

subgroupindex
#24372
#20292
#16142
#16022
#13522
#13512
#13162