Finite Group Databasebeta

← #25603

Subgroups — #25603

Order 256.

586 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 135 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×#497C211yesmaximal, characteristic, complemented
3C2×#489C211yesmaximal, characteristic, complemented
4#2850C211yesmaximal, characteristic
5#2001C212yesmaximal, complemented
6#2000C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1878C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14#533—24no—
15#533—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—21no—
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8C2212yes—
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8—21no—
25C2×C4×Q8C2212yes—
26C2×#117—21no—
27C22×C42C2211yescharacteristic
28#497—24no—
29#489—24no—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×C22⋊C8C2212yes—
35C2×#97C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#388—21no—
40#388—21no—
41#386—21no—
42#386—21no—
43#382C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×Q8C2312yes—
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C2×Q16—24no—
56C2×Q16—24no—
57C2×Q16—42no—
58C2×SD16—24no—
59C2×SD16—24no—
60C2×SD16—42no—
61C22×C8C2312yes—
62C22×C8C2312yes—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65C4⋊Q8D414yescomplemented
66C4⋊Q8D414yescomplemented
67#125—28no—
68#124—22no—
69#124—22no—
70C22⋊Q8—28no—
71C22⋊Q8—24no—
72C22⋊Q8—24no—
73C22⋊Q8—42no—
74#121—42no—
75C4×Q8—28no—
76C4×Q8—24no—
77C4×Q8—24no—
78#117—28no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42D411yescharacteristic
99C2×C42D411yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#103—24no—
103#103—42no—
104#103—42no—
105#102—24no—
106C22⋊C8—24no—
107#97—22no—
108#97—22no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113C24—21no—
114C24C2×D411yescharacteristic
115C2×Q8—24no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8C2×D414yes—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—42no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
156Q16—48no—
157SD16—48no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8#4114yes—
164C4⋊C4—28no—
165C4⋊C4—28no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4C2×D414yes—
174C22⋊C4—28no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—28no—
182C42—22no—
183C42—22no—
184C42—21no—
185C42—21no—
186C42C2×D412yes—
187C42C2×D412yes—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23#13712yes—
202C23C22×D411yescharacteristic
203Q8—24no—
204Q8—24no—
205Q8—48no—
206Q8—42no—
207Q8—42no—
208Q8—42no—
209Q8—42no—
210D4—42no—
211D4—42no—
212D4—84no—
213C2×C4—28no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4#12012yes—
245C2×C4#12012yes—
246C2×C4#13712yes—
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C2×C4C2×#4112yes—
250C2×C4H(5,𝔽2)12yes—
251C8—24no—
252C8—42no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22C2×#12011yescharacteristic
268C22#53211yescharacteristic
269C22#53511yescharacteristic
270C22C2×#13711yescharacteristic
271C22C2×#13711yescharacteristic
272C22#57511yescharacteristic
273C22#57511yescharacteristic
274C4—22no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—44no—
279C4—41no—
280C4—44no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4#54612yes—
285C4#55012yes—
286C2—42no—
287C2—161no—
288C2#243111yescharacteristic
289C2#300411yescharacteristic
290C2#300511yescharacteristic
291C2#317211yescharacteristic
292C2#318111yescharacteristic
293C2#321511yescharacteristic
294C2#321611yescharacteristic
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
2. C2×#497C2semidirect
3. C2×#489C2semidirect
5. #2001C2semidirect
7. #1993C2semidirect
8. #1960C2semidirect
9. #1880C2semidirect
10. #1878C2semidirect
11. #1749C2semidirect
65. C4⋊Q8D4semidirect
66. C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4972
C2×#4892
#28502
#20012
#20002
#19932
#19602
#18802
#18782
#17492