Finite Group Databasebeta

← #28345

Subgroups — #28345

Order 256.

521 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#483C211yesmaximal, characteristic
2C2×#467C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic
4C22×#103C211yesmaximal, characteristic
5#2503C211yesmaximal, characteristic
6#2434C211yesmaximal, characteristic
7#2097C212yesmaximal, complemented
8#2016C212yesmaximal, complemented
9#1971C212yesmaximal, complemented
10#1960C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×Q16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×C4⋊Q8C2212yes—
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20#483—24no—
21#467—24no—
22#451—24no—
23C2×#107C2211yescharacteristic
24C2×#106C2212yes—
25C2×#103—24no—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2212yes—
34#400—22no—
35#395—22no—
36#391C2212yes—
37#391C2212yes—
38#385—22no—
39#382C2212yes—
40#381—22no—
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—24no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×Q8C2312yes—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C2×Q16—24no—
55C2×Q16—24no—
56C2×Q16—22no—
57C2×Q16—42no—
58C2×Q16—42no—
59C22×C8—42no—
60C22×C8D414yes—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2312yes—
64C4⋊Q8—24no—
65C4⋊Q8—42no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4D412yes—
82C2×C42C2312yes—
83#107—22no—
84#106—24no—
85#103—24no—
86#103—24no—
87#103—24no—
88#103—22no—
89#103—22no—
90#103—24no—
91C22⋊C8D414yes—
92#97—42no—
93#95—24no—
94#95—24no—
95#95—24no—
96#95D412yes—
97C24C2×D411yescharacteristic
98C2×Q8—22no—
99C2×Q8—28no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—44no—
113C2×Q8—48no—
114C2×Q8—44no—
115C2×Q8—42no—
116C2×Q8#4114yes—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—44no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—44no—
129C22×C4C2×D414yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D412yes—
132C22×C4C2×D412yes—
133C22×C4C2411yescharacteristic
134Q16—48no—
135Q16—44no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—44no—
141C2×C8C2×D414yes—
142C2×C8C2×D412yes—
143C2×C8C2×D412yes—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—48no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C22⋊C4—24no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C42—22no—
158C42—22no—
159C23—22no—
160C23—22no—
161C23#12712yes—
162C23#13714yes—
163C23C22×D411yescharacteristic
164Q8—24no—
165Q8—24no—
166Q8—48no—
167Q8—42no—
168Q8—42no—
169Q8—41no—
170Q8—41no—
171Q8—41no—
172Q8—41no—
173Q8—42no—
174Q8—42no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—48no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—84no—
197C2×C4#12712yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4112yes—
201C2×C4H(5,𝔽2)12yes—
202C8—22no—
203C8—22no—
204C8—24no—
205C22—22no—
206C22—22no—
207C22—28no—
208C22#49714yes—
209C22C2×#12711yescharacteristic
210C22#54612yes—
211C22C2×#13712yes—
212C22#57812yes—
213C4—22no—
214C4—44no—
215C4—44no—
216C4—82no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4—81no—
221C4#52511yescharacteristic
222C4#53411yescharacteristic
223C4#54612yes—
224C2—24no—
225C2#253911yescharacteristic
226C2C2×#49711yescharacteristic
227C2#315611yescharacteristic
228C2#331214yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #2097C2semidirect
8. #2016C2semidirect
9. #1971C2semidirect
10. #1960C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4672
C2×#4512
C22×#1032
#25032
#24342
#20972
#20162
#19712
#19602
#17522