Finite Group Databasebeta

← #24118

Subgroups — #24118

Order 256.

609 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 147 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×#554C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic, complemented
4#2537C211yesmaximal, characteristic
5#2035C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1954C211yesmaximal, characteristic, complemented
9#1874C212yesmaximal, complemented
10#1853C212yesmaximal, complemented
11#1803C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—22no—
18C2×C22⋊Q8C2212yes—
19C2×#121—21no—
20C2×#121C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2212yes—
26#476—24no—
27C4×SD16—24no—
28C2×#106C2211yescharacteristic
29C2×#106C2212yes—
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2212yes—
37C2×C4×C8C2211yescharacteristic
38#394—22no—
39#384C2211yescharacteristic
40#380—21no—
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×D4—22no—
48C22×D4D411yescharacteristic
49C22×D4C2311yescharacteristic
50C23×C4—22no—
51C23×C4—21no—
52C23×C4C2312yes—
53C2×SD16—22no—
54C2×SD16—22no—
55C2×SD16—22no—
56C22×C8C2311yescharacteristic
57C22×C8C2312yes—
58C22×C8C2311yescharacteristic
59C4⋊Q8—22no—
60C4⋊Q8D414yescomplemented
61#125—28no—
62C22⋊Q8—216no—
63C22⋊Q8—24no—
64C22⋊Q8—24no—
65#121—24no—
66#121—42no—
67#121—44no—
68C4×Q8—22no—
69C4×Q8D414yescomplemented
70C4×D4—28no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4D412yes—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95#106—22no—
96#106—24no—
97C4⋊C8—22no—
98#103—22no—
99#103—42no—
100#102—22no—
101#102—24no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104C4×C8—22no—
105#95—22no—
106C24—21no—
107C24C2×D412yes—
108C2×Q8—24no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—81no—
115C2×Q8—81no—
116C2×Q8C2×D412yes—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4—82no—
127C2×D4SD1614yescomplemented
128C2×D4#4112yes—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—28no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—82no—
145C22×C4—82no—
146C22×C4C2×D412yes—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2411yescharacteristic
150SD16—28no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—21no—
159C2×C8#4112yes—
160C4⋊C4—28no—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—81no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D414yes—
175C22⋊C4—28no—
176C22⋊C4—24no—
177C22⋊C4—44no—
178C22⋊C4—44no—
179C22⋊C4—42no—
180C42—24no—
181C42C2×D412yes—
182C42C2×D412yes—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23C2×SD1614yes—
197C23C22×D411yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—84no—
203D4—22no—
204D4—22no—
205D4—28no—
206D4—41no—
207D4—41no—
208D4—88no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—88no—
234C2×C4—84no—
235C2×C4#12012yes—
236C2×C4#12012yes—
237C2×C4C2×SD1612yes—
238C2×C4#13712yes—
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4C2×#4111yescharacteristic
242C2×C4C2×#4111yescharacteristic
243C2×C4H(5,𝔽2)11yescharacteristic
244C2×C42−1+411yescharacteristic
245C8—22no—
246C8—41no—
247C8—42no—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22C2×#12011yescharacteristic
263C22#54611yescharacteristic
264C22#54711yescharacteristic
265C22C22×SD1611yescharacteristic
266C22C2×#13711yescharacteristic
267C22#57611yescharacteristic
268C22#57711yescharacteristic
269C4—22no—
270C4—42no—
271C4—42no—
272C4—42no—
273C4—44no—
274C4—81no—
275C4—82no—
276C4#44814yes—
277C4#52511yescharacteristic
278C4#53711yescharacteristic
279C4#54811yescharacteristic
280C4#55011yescharacteristic
281C2—44no—
282C2—82no—
283C2#240711yescharacteristic
284C2C2×#44811yescharacteristic
285C2#301211yescharacteristic
286C2#329511yescharacteristic
287C2#333111yescharacteristic
288C2#333511yescharacteristic
289C2#335711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#554C2semidirect
2. C2×#476C2semidirect
3. C2×C4×SD16C2semidirect
5. #2035C2semidirect
6. #2000C2semidirect
7. #1990C2semidirect
8. #1954C2semidirect
9. #1874C2semidirect
10. #1853C2semidirect
11. #1803C2semidirect
60. C4⋊Q8D4semidirect
69. C4×Q8D4semidirect
127. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4762
C2×C4×SD162
#25372
#20352
#20002
#19902
#19542
#18742
#18532
#18032