Finite Group Databasebeta

← #29719

Subgroups — #29719

Order 256.

519 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 129 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×#451C211yesmaximal, characteristic, complemented
2C2×#448C211yesmaximal, characteristic
3#2850C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic, complemented
5#2074C212yesmaximal, complemented
6#2056C212yesmaximal, complemented
7#2055C212yesmaximal, complemented
8#2015C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic
10#1872C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C42C2211yescharacteristic
20#451—24no—
21#448—24no—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#391—21no—
30#391—21no—
31#391C2212yes—
32#391C2211yescharacteristic
33#391C2212yes—
34#390C2211yescharacteristic
35#388—21no—
36#388—21no—
37#386—21no—
38#386—21no—
39#382C2211yescharacteristic
40C2×#95—21no—
41C2×#95—21no—
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×Q8—42no—
49C22×Q8D414yes—
50C22×Q8C2311yescharacteristic
51C22×Q8C2312yes—
52C22×D4—21no—
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4C2311yescharacteristic
59C2×Q16—24no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×SD16—24no—
63C2×SD16—42no—
64C2×SD16—42no—
65C22×C8C2312yes—
66C22×C8C2312yes—
67#124—42no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70#121—22no—
71#121—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C42—42no—
85C2×C42D411yescharacteristic
86C2×C42D411yescharacteristic
87C2×C42C2311yescharacteristic
88C2×C42C2311yescharacteristic
89C4⋊C8—42no—
90#103—24no—
91#103—24no—
92#103—42no—
93#102—42no—
94C22⋊C8—22no—
95C22⋊C8—22no—
96#95—22no—
97#95—22no—
98#95—22no—
99#95—22no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104C24—21no—
105C24C2×D411yescharacteristic
106C2×Q8—24no—
107C2×Q8—22no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—48no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—48no—
120C2×Q8C2×D414yes—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—42no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D412yes—
152C22×C4C2×D411yescharacteristic
153C22×C4#4112yes—
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—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C42—22no—
178C42—22no—
179C42—41no—
180C42—41no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23#12112yes—
193C23#13512yes—
194C23#13712yes—
195C23C22×D411yescharacteristic
196Q8—24no—
197Q8—24no—
198Q8—48no—
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—42no—
203D4—42no—
204D4—42no—
205D4—84no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—21no—
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—44no—
223C2×C4—44no—
224C2×C4—48no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—84no—
240C2×C4#12112yes—
241C2×C4#13712yes—
242C2×C4C22×D411yescharacteristic
243C2×C4C2×#4111yescharacteristic
244C2×C4H(5,𝔽2)12yes—
245C2×C4H(5,𝔽2)12yes—
246C8—42no—
247C8—42no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22#46414yes—
262C22C2×#12111yescharacteristic
263C22#53411yescharacteristic
264C22#53511yescharacteristic
265C22C2×#13511yescharacteristic
266C22C2×#13711yescharacteristic
267C22C2×#13711yescharacteristic
268C22#57511yescharacteristic
269C4—22no—
270C4—22no—
271C4—44no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—81no—
276C4—82no—
277C4—81no—
278C4—81no—
279C4#53412yes—
280C4#53812yes—
281C2—24no—
282C2—161no—
283C2#261711yescharacteristic
284C2C2×#46411yescharacteristic
285C2#303711yescharacteristic
286C2#319011yescharacteristic
287C2#319611yescharacteristic
288C2#321111yescharacteristic
289C2#321811yescharacteristic
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×#451C2semidirect
4. C2×#391C2semidirect
5. #2074C2semidirect
6. #2056C2semidirect
7. #2055C2semidirect
8. #2015C2semidirect
10. #1872C2semidirect
11. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
C2×#4482
#28502
C2×#3912
#20742
#20562
#20552
#20152
#18742
#18722
#17972