Finite Group Databasebeta

← #24503

Subgroups — #24503

Order 256.

527 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 141 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×#448C211yesmaximal, characteristic
2C2×#431C211yesmaximal, characteristic, complemented
3#2854C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic, complemented
5#2066C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#1968C212yesmaximal, complemented
8#1933C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic
10#1853C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×SD16C2212yes—
14C2×#123—21no—
15C2×#121C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#448—24no—
19#431—24no—
20C2×#106C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×C4×C8C2212yes—
29#391—21no—
30#391—21no—
31#391C2212yes—
32#391C2212yes—
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#388—21no—
38#388—21no—
39#386—21no—
40#380C2211yescharacteristic
41C2×#95—21no—
42C2×#95—21no—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—24no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—42no—
50C22×Q8C2312yes—
51C22×D4—21no—
52C22×D4C2311yescharacteristic
53C23×C4—21no—
54C23×C4—21no—
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×SD16—24no—
59C2×SD16—24no—
60C2×SD16—42no—
61C22×C8C2312yes—
62C22×C8C2312yes—
63#123—44no—
64#121—22no—
65#121—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86#106—42no—
87C4⋊C8—24no—
88#103—24no—
89#103—24no—
90#103—42no—
91#102—42no—
92C22⋊C8—22no—
93C22⋊C8D414yes—
94C4×C8—24no—
95#95—22no—
96#95—22no—
97#95—22no—
98#95—22no—
99#95—21no—
100#95—21no—
101#95—21no—
102C24—21no—
103C24C2×D411yescharacteristic
104C2×Q8—24no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—48no—
111C2×Q8—48no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8#4114yes—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—24no—
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×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4SD1614yescomplemented
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D412yes—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4#4112yes—
148C22×C4C2411yescharacteristic
149SD16—48no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8C2×D414yes—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C42—22no—
172C42—22no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23#12112yes—
189C23C2×SD1612yes—
190C23#13512yes—
191C23C22×D411yescharacteristic
192Q8—24no—
193Q8—24no—
194Q8—48no—
195Q8—42no—
196Q8—42no—
197D4—42no—
198D4—42no—
199D4—84no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—28no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—44no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—84no—
232C2×C4#12112yes—
233C2×C4C2×SD1612yes—
234C2×C4C2×SD1612yes—
235C2×C4#13712yes—
236C2×C4C22×D411yescharacteristic
237C2×C4C2×#4112yes—
238C2×C4C2×#4111yescharacteristic
239C2×C4H(5,𝔽2)12yes—
240C8—24no—
241C8—42no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22#46514yes—
256C22C2×#12111yescharacteristic
257C22#52511yescharacteristic
258C22#53511yescharacteristic
259C22C22×SD1611yescharacteristic
260C22C2×#13511yescharacteristic
261C22C2×#13711yescharacteristic
262C22#57511yescharacteristic
263C4—22no—
264C4—22no—
265C4—22no—
266C4—22no—
267C4—44no—
268C4—41no—
269C4—81no—
270C4—81no—
271C4—82no—
272C4—81no—
273C4#54612yes—
274C4#55112yes—
275C2—24no—
276C2—161no—
277C2#241111yescharacteristic
278C2C2×#46511yescharacteristic
279C2#305611yescharacteristic
280C2#310311yescharacteristic
281C2#310811yescharacteristic
282C2#320611yescharacteristic
283C2#321111yescharacteristic
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
2. C2×#431C2semidirect
4. C2×#391C2semidirect
5. #2066C2semidirect
6. #2055C2semidirect
7. #1968C2semidirect
8. #1933C2semidirect
10. #1853C2semidirect
11. #1787C2semidirect
140. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4482
C2×#4312
#28542
C2×#3912
#20662
#20552
#19682
#19332
#18742
#18532
#17872