Finite Group Databasebeta

← #30045

Subgroups — #30045

Order 256.

464 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#2858C211yesmaximal, characteristic, complemented
2C4×#124C211yesmaximal, characteristic, complemented
3#2073C212yesmaximal, complemented
4#2024C211yesmaximal, characteristic, complemented
5#2023C211yesmaximal, characteristic, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2012C212yesmaximal, complemented
8#2011C211yesmaximal, characteristic
9#1880C212yesmaximal, complemented
10#1879C212yesmaximal, complemented
11#1839C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#124—21no—
17C2×#124—21no—
18C2×#124C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25#391C2211yescharacteristic
26#390—21no—
27#390—21no—
28#390C2212yes—
29#390C2212yes—
30#390C2211yescharacteristic
31#388—21no—
32#388—21no—
33#382C2211yescharacteristic
34C4×C22⋊C4—21no—
35C4×C22⋊C4—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2212yes—
40#336C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
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—22no—
61#124—22no—
62#124—44no—
63#124—44no—
64C4×Q8—42no—
65C4×D4—42no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C42—21no—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42D411yescharacteristic
85C2×C42C2311yescharacteristic
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88#103—42no—
89#102—42no—
90#102—42no—
91#102—42no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96C24—21no—
97C24—22no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8#4112yes—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4#4112yes—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2411yescharacteristic
135SD16—48no—
136D8—48no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C22⋊C4—24no—
150C22⋊C4—24no—
151C22⋊C4—42no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C22⋊C4—44no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—44no—
158C42—22no—
159C42—22no—
160C42—22no—
161C42—21no—
162C42—21no—
163C42—22no—
164C42—22no—
165C42—41no—
166C42—42no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23C22×D411yescharacteristic
182Q8—24no—
183Q8—42no—
184Q8—84no—
185D4—24no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—42no—
190D4—42no—
191D4—84no—
192D4—84no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—84no—
221C2×C4#13512yes—
222C2×C4AGL(1,ℤ/8)12yes—
223C2×C4AGL(1,ℤ/8)12yes—
224C2×C4C2×#4111yescharacteristic
225C2×C4C2×#4111yescharacteristic
226C2×C4H(5,𝔽2)12yes—
227C2×C4H(5,𝔽2)12yes—
228C8—42no—
229C8—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—82no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22—82no—
243C22—82no—
244C22#52511yescharacteristic
245C22#53511yescharacteristic
246C22#53511yescharacteristic
247C22C2×#13511yescharacteristic
248C22C2×AGL(1,ℤ/8)11yescharacteristic
249C22C2×AGL(1,ℤ/8)11yescharacteristic
250C22#57511yescharacteristic
251C4—22no—
252C4—22no—
253C4—22no—
254C4—41no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4—82no—
261C4#53812yes—
262C4#53812yes—
263C2—82no—
264C2—162no—
265C2#263111yescharacteristic
266C2#310611yescharacteristic
267C2#311511yescharacteristic
268C2#320711yescharacteristic
269C2#321111yescharacteristic
270C2#321311yescharacteristic
271C2#321711yescharacteristic
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. #2858C2semidirect
2. C4×#124C2semidirect
3. #2073C2semidirect
4. #2024C2semidirect
5. #2023C2semidirect
6. #2014C2semidirect
7. #2012C2semidirect
9. #1880C2semidirect
10. #1879C2semidirect
11. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28582
C4×#1242
#20732
#20242
#20232
#20142
#20122
#20112
#18802
#18792
#18392