Finite Group Databasebeta

← #25639

Subgroups — #25639

Order 256.

599 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 131 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×#559C211yesmaximal, characteristic, complemented
2C2×#496C211yesmaximal, characteristic
3C2×#464C211yesmaximal, characteristic, complemented
4#2592C211yesmaximal, characteristic, complemented
5#2015C212yesmaximal, complemented
6#2012C212yesmaximal, complemented
7#1985C211yesmaximal, characteristic, complemented
8#1982C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C22×SD16C2212yes—
14C22×SD16C2211yescharacteristic
15C22×D8C2212yes—
16#559—24no—
17#559—24no—
18C2×#127C2211yescharacteristic
19C2×#124—21no—
20C2×#124C2211yescharacteristic
21C2×#124C2212yes—
22C2×#123—22no—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2212yes—
27#496—24no—
28#464—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#97C2211yescharacteristic
36#397—21no—
37#393—21no—
38#391C2212yes—
39#390C2212yes—
40#388—22no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8D411yescharacteristic
46C22×Q8C2311yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4—21no—
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C22×D4C2312yes—
52C23×C4—22no—
53C2×Q16—22no—
54C2×Q16—42no—
55C2×SD16—22no—
56C2×SD16—24no—
57C2×SD16—24no—
58C2×SD16—42no—
59C2×SD16—42no—
60C2×D8—24no—
61C2×D8—24no—
62C2×D8—42no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2312yes—
66#127—22no—
67#127D414yes—
68#124—28no—
69#124—24no—
70#124—24no—
71#124—22no—
72#124—22no—
73#123—216no—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8D414yes—
78C4×D4—28no—
79C4×D4—24no—
80C4×D4—24no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D412yes—
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C2312yes—
96C2×C42C2311yescharacteristic
97C4⋊C8—22no—
98#103—24no—
99#103—22no—
100#102—22no—
101#102—42no—
102#102—42no—
103#97—22no—
104#95—21no—
105#95—22no—
106#95—22no—
107C24—22no—
108C24—21no—
109C2×Q8—22no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—42no—
118C2×Q8C2×D412yes—
119C2×D4—24no—
120C2×D4—24no—
121C2×D4—22no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—44no—
131C2×D4—81no—
132C2×D4—81no—
133C2×D4C2×D414yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—28no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2411yescharacteristic
146Q16—44no—
147SD16—48no—
148SD16—44no—
149D8—48no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8#4114yes—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—24no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4C2×D412yes—
164C22⋊C4—28no—
165C22⋊C4—28no—
166C22⋊C4—28no—
167C22⋊C4—44no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—82no—
171C42—24no—
172C42—22no—
173C42—22no—
174C42—81no—
175C42C2×D412yes—
176C42C2×D412yes—
177C23—24no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—41no—
185C23—42no—
186C23—42no—
187C23—41no—
188C23—41no—
189C23C22×D411yescharacteristic
190Q8—22no—
191Q8—22no—
192Q8—44no—
193Q8—42no—
194Q8—41no—
195Q8—41no—
196Q8—42no—
197D4—24no—
198D4—24no—
199D4—48no—
200D4—42no—
201D4—41no—
202D4—42no—
203D4—41no—
204D4—42no—
205D4—84no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—88no—
226C2×C4#12012yes—
227C2×C4#12012yes—
228C2×C4C22×D411yescharacteristic
229C2×C4C22×D411yescharacteristic
230C2×C4C2×#4112yes—
231C2×C4H(5,𝔽2)12yes—
232C8—24no—
233C8—41no—
234C8—41no—
235C22—44no—
236C22—44no—
237C22—44no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22—82no—
243C22—82no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22C2×#12011yescharacteristic
248C22#55012yes—
249C22#57512yes—
250C22#57611yescharacteristic
251C22#57711yescharacteristic
252C4—22no—
253C4—22no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—82no—
259C4—81no—
260C4#53211yescharacteristic
261C4#53511yescharacteristic
262C4#54612yes—
263C2—84no—
264C2—161no—
265C2#243111yescharacteristic
266C2#300511yescharacteristic
267C2#301211yescharacteristic
268C2#340112yes—
269C2#340312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#559C2semidirect
3. C2×#464C2semidirect
4. #2592C2semidirect
5. #2015C2semidirect
6. #2012C2semidirect
7. #1985C2semidirect
8. #1982C2semidirect
9. #1879C2semidirect
10. #1877C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4962
C2×#4642
#25922
#20152
#20122
#19852
#19822
#18792
#18772
#17632