Finite Group Databasebeta

← #28315

Subgroups — #28315

Order 256.

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

normal subgroups only

This group has 554 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×#484C211yesmaximal, characteristic, complemented
2C2×#481C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic
5#2850C211yesmaximal, characteristic
6C2×#391C211yesmaximal, characteristic, complemented
7#2097C212yesmaximal, complemented
8#2015C212yesmaximal, complemented
9#1973C212yesmaximal, complemented
10#1971C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C42C2211yescharacteristic
20#484—24no—
21#481—24no—
22#451—24no—
23#448—24no—
24C2×#107C2212yes—
25C2×#106C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2212yes—
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2212yes—
33#391—22no—
34#391C2211yescharacteristic
35#391C2212yes—
36#391C2212yes—
37#390C2211yescharacteristic
38#388—22no—
39#386—22no—
40#382C2211yescharacteristic
41C2×#95—22no—
42#336C2212yes—
43C22×Q8—24no—
44C22×Q8—21no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×Q8C2311yescharacteristic
49C22×Q8C2312yes—
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4D412yes—
55C23×C4C2311yescharacteristic
56C2×Q16—24no—
57C2×Q16—24no—
58C2×Q16—42no—
59C2×SD16—24no—
60C2×SD16—24no—
61C2×SD16—42no—
62C22×C8C2312yes—
63C22×C8C2312yes—
64#124—42no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67#121—22no—
68#121—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C42—42no—
81C2×C42D412yes—
82C2×C42C2311yescharacteristic
83C2×C42C2311yescharacteristic
84#107—24no—
85#106—24no—
86#103—24no—
87#103—24no—
88#103—24no—
89#102—24no—
90C22⋊C8D414yes—
91C22⋊C8D414yes—
92#97—42no—
93#95—24no—
94#95—24no—
95#95—22no—
96#95—22no—
97C24—21no—
98C24C2×D411yescharacteristic
99C2×Q8—24no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—48no—
110C2×Q8—48no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8#4114yes—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—42no—
119C2×D4—44no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4C2×D412yes—
140C22×C4C2411yescharacteristic
141Q16—48no—
142SD16—48no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8C2×D414yes—
148C2×C8C2×D414yes—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C22⋊C4—44no—
158C22⋊C4—44no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C42—22no—
162C42—22no—
163C42—42no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—42no—
172C23#12712yes—
173C23#13714yes—
174C23C22×D411yescharacteristic
175Q8—24no—
176Q8—24no—
177Q8—48no—
178Q8—42no—
179Q8—42no—
180Q8—42no—
181Q8—42no—
182D4—42no—
183D4—42no—
184D4—84no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—48no—
197C2×C4—48no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—84no—
210C2×C4#12712yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4112yes—
214C2×C4H(5,𝔽2)12yes—
215C8—24no—
216C8—24no—
217C22—24no—
218C22—22no—
219C22—24no—
220C22—22no—
221C22—82no—
222C22—81no—
223C22—81no—
224C22—82no—
225C22—81no—
226C22#49714yes—
227C22#52511yescharacteristic
228C22C2×#12711yescharacteristic
229C22#53411yescharacteristic
230C22C2×#13712yes—
231C22#57512yes—
232C4—22no—
233C4—42no—
234C4—42no—
235C4—44no—
236C4—81no—
237C4—81no—
238C4—82no—
239C4—81no—
240C4#54612yes—
241C4#54612yes—
242C2—24no—
243C2—161no—
244C2#253911yescharacteristic
245C2#311712yes—
246C2C2×#49711yescharacteristic
247C2#315311yescharacteristic
248C2#319612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
3. C2×#451C2semidirect
6. C2×#391C2semidirect
7. #2097C2semidirect
8. #2015C2semidirect
9. #1973C2semidirect
10. #1971C2semidirect
11. #1749C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4812
C2×#4512
C2×#4482
#28502
C2×#3912
#20972
#20152
#19732
#19712
#17492