Finite Group Databasebeta

← #29800

Subgroups — #29800

Order 256.

497 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 123 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×#463C212yesmaximal, complemented
2#2858C211yesmaximal, characteristic, complemented
3#2293C211yesmaximal, characteristic, complemented
4#2043C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#2012C212yesmaximal, complemented
9#2011C211yesmaximal, characteristic
10#1983C212yesmaximal, complemented
11#1831C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#124C2212yes—
17C2×#124C2212yes—
18C2×C4×D4C2211yescharacteristic
19C2×#117—21no—
20#463—28no—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28#390—21no—
29#390—21no—
30#390—21no—
31#390—21no—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#388—21no—
38C4×C4⋊C4C2211yescharacteristic
39C4×C22⋊C4—21no—
40C43C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43#336C2211yescharacteristic
44C22×Q8C2312yes—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C2×SD16—24no—
51C2×SD16—42no—
52C2×SD16—42no—
53C2×D8—24no—
54C2×D8—42no—
55C2×D8—42no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—42no—
59#127—42no—
60#124—24no—
61#124—24no—
62#124—42no—
63#124—42no—
64C4×D4—22no—
65C4×D4D414yescomplemented
66#117—44no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C42—21no—
84C2×C42D411yescharacteristic
85C2×C42D411yescharacteristic
86C2×C42D411yescharacteristic
87C2×C42D411yescharacteristic
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91C4⋊C8—24no—
92#103—24no—
93#103—42no—
94#103—42no—
95#102—24no—
96#102—42no—
97#102—42no—
98#95—21no—
99#95—21no—
100#95—21no—
101C24—21no—
102C24—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×D4—22no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4C2×D412yes—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2411yescharacteristic
139SD16—48no—
140D8—48no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4C2×D412yes—
155C22⋊C4—24no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C42—22no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42—41no—
169C42—41no—
170C42—42no—
171C42C2×D412yes—
172C42#4112yes—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23C22×D411yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—42no—
191Q8—42no—
192D4—24no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—84no—
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—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—48no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4#12112yes—
228C2×C4#12112yes—
229C2×C4#13512yes—
230C2×C4AGL(1,ℤ/8)12yes—
231C2×C4C22×D411yescharacteristic
232C2×C4C2×#4111yescharacteristic
233C2×C4H(5,𝔽2)12yes—
234C2×C4H(5,𝔽2)12yes—
235C8—42no—
236C8—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—82no—
245C22—82no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22—82no—
250C22—82no—
251C22C2×#12111yescharacteristic
252C22#53511yescharacteristic
253C22#53511yescharacteristic
254C22C2×#13511yescharacteristic
255C22C2×AGL(1,ℤ/8)11yescharacteristic
256C22#57511yescharacteristic
257C22#57511yescharacteristic
258C4—22no—
259C4—22no—
260C4—22no—
261C4—22no—
262C4—41no—
263C4—41no—
264C4—42no—
265C4—82no—
266C4—81no—
267C4—81no—
268C4#53512yes—
269C4#54012yes—
270C2—82no—
271C2—162no—
272C2#261911yescharacteristic
273C2#303511yescharacteristic
274C2#303611yescharacteristic
275C2#320711yescharacteristic
276C2#321111yescharacteristic
277C2#321311yescharacteristic
278C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#463C2semidirect
2. #2858C2semidirect
3. #2293C2semidirect
4. #2043C2semidirect
5. #2042C2semidirect
6. #2041C2semidirect
7. #2013C2semidirect
8. #2012C2semidirect
10. #1983C2semidirect
11. #1831C2semidirect
65. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
#28582
#22932
#20432
#20422
#20412
#20132
#20122
#20112
#19832
#18312