Finite Group Databasebeta

← #25497

Subgroups — #25497

Order 256.

597 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 149 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×#533C211yesmaximal, characteristic, complemented
2C2×#465C212yesmaximal
3#2854C211yesmaximal, characteristic
4#2058C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#1892C212yesmaximal, complemented
9#1880C212yesmaximal, complemented
10#1865C212yesmaximal, complemented
11C2×#336C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C23×C8C2212yes—
14#533—28no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—21no—
18C2×#123—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2212yes—
21C2×#121C2211yescharacteristic
22C2×C4×Q8—21no—
23C2×C4×Q8C2212yes—
24C2×#117—21no—
25C22×C42C2211yescharacteristic
26#465—28no—
27C2×#106C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#102C2212yes—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#388—21no—
38#386—21no—
39#380C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2212yes—
44#336C2211yescharacteristic
45C22×Q8—21no—
46C22×Q8C2312yes—
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16—42no—
55C22×C8—24no—
56C22×C8—24no—
57C22×C8—42no—
58C22×C8C2312yes—
59C22×C8C2312yes—
60C4⋊Q8—24no—
61C4⋊Q8—24no—
62#125—28no—
63#123—44no—
64C22⋊Q8—28no—
65C22⋊Q8—24no—
66C22⋊Q8D418yes—
67#121—24no—
68C4×Q8—28no—
69C4×Q8—24no—
70C4×Q8—24no—
71#117—28no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C42—22no—
91C2×C42—24no—
92C2×C42D411yescharacteristic
93C2×C42D411yescharacteristic
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#106—24no—
97#103—24no—
98#103—24no—
99#103—42no—
100#102—24no—
101#95—21no—
102#95—21no—
103#95—21no—
104C24—21no—
105C24C2×D411yescharacteristic
106C2×Q8—24no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8C2×D414yes—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4SD1614yescomplemented
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D414yes—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143SD16—48no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—48no—
150C2×C8#4114yes—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—28no—
154C4⋊C4—24no—
155C4⋊C4—28no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4C2×D414yes—
164C22⋊C4—28no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C42—28no—
172C42—22no—
173C42—22no—
174C42—22no—
175C42—22no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23#12012yes—
188C23C2×SD1612yes—
189C23#13712yes—
190C23C22×D411yescharacteristic
191Q8—24no—
192Q8—24no—
193Q8—48no—
194Q8—42no—
195Q8—42no—
196D4—42no—
197D4—42no—
198D4—84no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—28no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—21no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4#12012yes—
230C2×C4C2×SD1612yes—
231C2×C4C2×SD1612yes—
232C2×C4#13512yes—
233C2×C4C22×D412yes—
234C2×C4C2×#4112yes—
235C2×C4H(5,𝔽2)11yescharacteristic
236C2×C42−1+411yescharacteristic
237C8—24no—
238C8—42no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22#44814yes—
253C22C2×#12011yescharacteristic
254C22#53211yescharacteristic
255C22#53711yescharacteristic
256C22C22×SD1611yescharacteristic
257C22C2×#13511yescharacteristic
258C22C2×#13711yescharacteristic
259C22#57511yescharacteristic
260C4—22no—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—44no—
265C4—41no—
266C4—44no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4#54612yes—
271C4#54712yes—
272C2—24no—
273C2—161no—
274C2#243011yescharacteristic
275C2C2×#44811yescharacteristic
276C2#300211yescharacteristic
277C2#316711yescharacteristic
278C2#318111yescharacteristic
279C2#324611yescharacteristic
280C2#324811yescharacteristic
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×#533C2semidirect
4. #2058C2semidirect
5. #2043C2semidirect
6. #2023C2semidirect
7. #2013C2semidirect
8. #1892C2semidirect
9. #1880C2semidirect
10. #1865C2semidirect
11. C2×#336C2semidirect
135. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4652
#28542
#20582
#20432
#20232
#20132
#18922
#18802
#18652
C2×#3362