Finite Group Databasebeta

← #29668

Subgroups — #29668

Order 256.

579 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 133 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×#464C212yesmaximal, complemented
3#2850C211yesmaximal, characteristic
4#2024C211yesmaximal, characteristic, complemented
5#2023C211yesmaximal, characteristic, complemented
6#2016C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#2001C212yesmaximal, complemented
9#2000C212yesmaximal
10#1805C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14#533—28no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—21no—
18C2×#124C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2212yes—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×Q8—21no—
24C2×C4×Q8C2212yes—
25C2×#117—21no—
26C22×C42C2211yescharacteristic
27#464—28no—
28C2×C4⋊C8C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×C22⋊C8C2212yes—
34#391C2211yescharacteristic
35#390C2211yescharacteristic
36#388—21no—
37#388—21no—
38#386—21no—
39#386—21no—
40#382C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×Q8C2312yes—
48C22×D4—21no—
49C22×D4C2311yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C2×Q16—24no—
54C2×Q16—42no—
55C2×Q16—42no—
56C2×SD16—24no—
57C2×SD16—42no—
58C2×SD16—42no—
59C22×C8C2312yes—
60C22×C8C2312yes—
61C4⋊Q8—24no—
62C4⋊Q8—24no—
63#125—28no—
64#124—24no—
65C22⋊Q8—28no—
66C22⋊Q8—24no—
67C22⋊Q8—24no—
68C22⋊Q8—42no—
69#121—42no—
70C4×Q8—28no—
71C4×Q8—24no—
72C4×Q8D418yescomplemented
73#117—28no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C42—22no—
91C2×C42—24no—
92C2×C42D411yescharacteristic
93C2×C42D411yescharacteristic
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96C4⋊C8—24no—
97#103—24no—
98#103—42no—
99#103—42no—
100#102—24no—
101C22⋊C8—42no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106C24—21no—
107C24C2×D411yescharacteristic
108C2×Q8—24no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8C2×D414yes—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—42no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2411yescharacteristic
149Q16—48no—
150SD16—48no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—28no—
160C4⋊C4—24no—
161C4⋊C4—28no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4C2×D414yes—
167C22⋊C4—28no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C42—22no—
175C42—22no—
176C42—28no—
177C42—22no—
178C42C2×D414yes—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23#13712yes—
193C23C22×D411yescharacteristic
194Q8—24no—
195Q8—24no—
196Q8—48no—
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201D4—42no—
202D4—42no—
203D4—84no—
204C2×C4—28no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4#12012yes—
235C2×C4#12012yes—
236C2×C4#13512yes—
237C2×C4#13712yes—
238C2×C4C22×D412yes—
239C2×C4H(5,𝔽2)11yescharacteristic
240C2×C4H(5,𝔽2)12yes—
241C2×C42−1+411yescharacteristic
242C8—42no—
243C8—42no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22C2×#12011yescharacteristic
259C22#53511yescharacteristic
260C22#53711yescharacteristic
261C22C2×#13511yescharacteristic
262C22C2×#13711yescharacteristic
263C22C2×#13711yescharacteristic
264C22#57511yescharacteristic
265C4—22no—
266C4—22no—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—44no—
271C4—44no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4#53512yes—
276C4#53712yes—
277C2—42no—
278C2—161no—
279C2#261511yescharacteristic
280C2#300211yescharacteristic
281C2#300411yescharacteristic
282C2#321011yescharacteristic
283C2#321511yescharacteristic
284C2#324811yescharacteristic
285C2#325211yescharacteristic
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×#464C2semidirect
4. #2024C2semidirect
5. #2023C2semidirect
6. #2016C2semidirect
7. #2013C2semidirect
8. #2001C2semidirect
10. #1805C2semidirect
11. #1797C2semidirect
72. 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×#4642
#28502
#20242
#20232
#20162
#20132
#20012
#20002
#18052
#17972