Finite Group Databasebeta

← #23935

Subgroups — #23935

Order 256.

638 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 137 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×#452C212yesmaximal
3#2297C211yesmaximal, characteristic
4#2024C211yesmaximal, characteristic, complemented
5#2023C211yesmaximal, characteristic, complemented
6#2016C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#1880C212yesmaximal, complemented
9#1878C212yesmaximal, complemented
10#1873C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15#533—28no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2212yes—
21C2×C4×Q8—21no—
22C2×C4×Q8C2212yes—
23C2×#117—21no—
24C2×#117C2211yescharacteristic
25C22×C42C2211yescharacteristic
26#452—28no—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#102C2212yes—
31C2×C22⋊C8C2212yes—
32#391C2211yescharacteristic
33#391C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#386—21no—
37#386—21no—
38C4×C22⋊C4—21no—
39C4×C22⋊C4—21no—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—24no—
48C2×#41—42no—
49C22×Q8—21no—
50C22×Q8C2312yes—
51C22×Q8C2311yescharacteristic
52C22×D4—21no—
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×Q16—24no—
59C2×Q16—24no—
60C2×Q16—42no—
61C2×SD16—24no—
62C2×SD16—24no—
63C2×SD16—42no—
64C22×C8C2312yes—
65C22×C8C2312yes—
66C4⋊Q8—24no—
67C4⋊Q8—24no—
68#125—28no—
69C22⋊Q8—28no—
70C22⋊Q8—24no—
71C22⋊Q8D418yes—
72C4×Q8—28no—
73C4×Q8—24no—
74C4×Q8—24no—
75#117—28no—
76#117—42no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C42—22no—
92C2×C42—24no—
93C2×C42—21no—
94C2×C42D411yescharacteristic
95C2×C42D411yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#103—24no—
99#103—42no—
100#103—42no—
101#102—24no—
102C22⋊C8—24no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107C24—21no—
108C24C2×D411yescharacteristic
109#41—48no—
110#41—48no—
111C2×Q8—24no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8C2×D414yes—
123C2×Q8#4112yes—
124C2×D4—24no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—22no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4#4112yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D414yes—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
155Q16—48no—
156SD16—48no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—28no—
166C4⋊C4—24no—
167C4⋊C4—28no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4C2×D414yes—
172C22⋊C4—28no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—28no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42—22no—
183C42—42no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23#12012yes—
198C23C22×D411yescharacteristic
199Q8—24no—
200Q8—24no—
201Q8—24no—
202Q8—48no—
203Q8—42no—
204Q8—42no—
205Q8—42no—
206D4—24no—
207D4—48no—
208D4—42no—
209C2×C4—28no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4#12012yes—
241C2×C4#13512yes—
242C2×C4#13712yes—
243C2×C4#13712yes—
244C2×C4C22×D412yes—
245C2×C4C2×#4111yescharacteristic
246C2×C4C2×#4111yescharacteristic
247C2×C4H(5,𝔽2)12yes—
248C8—42no—
249C8—42no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22C2×#12011yescharacteristic
265C22#52511yescharacteristic
266C22#53511yescharacteristic
267C22C2×#13511yescharacteristic
268C22C2×#13711yescharacteristic
269C22C2×#13711yescharacteristic
270C22#57511yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—44no—
277C4—41no—
278C4—44no—
279C4—81no—
280C4—81no—
281C4#54612yes—
282C4#54612yes—
283C2—42no—
284C2—82no—
285C2#240411yescharacteristic
286C2#300211yescharacteristic
287C2#300411yescharacteristic
288C2#310711yescharacteristic
289C2#311611yescharacteristic
290C2#321011yescharacteristic
291C2#321611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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. #2024C2semidirect
5. #2023C2semidirect
6. #2016C2semidirect
7. #2013C2semidirect
8. #1880C2semidirect
9. #1878C2semidirect
10. #1873C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4522
#22972
#20242
#20232
#20162
#20132
#18802
#18782
#18732
#18172