Finite Group Databasebeta

← #30342

Subgroups — #30342

Order 256.

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

normal subgroups only

This group has 427 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#2860C211yesmaximal, characteristic, complemented
2#2858C211yesmaximal, characteristic, complemented
3#2065C212yesmaximal, complemented
4#2058C211yesmaximal, characteristic, complemented
5#2054C212yesmaximal, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic
9#1972C212yesmaximal, complemented
10#1936C212yesmaximal, complemented
11#1767C211yesmaximal, characteristic, complemented
12C22×D8C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#127C2211yescharacteristic
15C2×#126—21no—
16C2×#124—21no—
17C2×#124—21no—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#97C2212yes—
25#390—21no—
26#390—21no—
27#390C2211yescharacteristic
28#390C2212yes—
29#390C2212yes—
30#390C2211yescharacteristic
31#390C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#388—21no—
36#388—21no—
37#388—21no—
38C43C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×D4—21no—
45C22×D4—21no—
46C22×D4C2312yes—
47C22×D4C2311yescharacteristic
48C2×D8—24no—
49C2×D8—42no—
50C2×D8—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—42no—
54#127—42no—
55#126—44no—
56#124—44no—
57#124—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#106—42no—
84C4⋊C8—24no—
85#102—24no—
86#102—42no—
87#102—42no—
88#102—42no—
89#97—42no—
90#95—21no—
91#95—21no—
92#95—21no—
93C24—21no—
94C24—22no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—42no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125D8—48no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8#4114yes—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—44no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—44no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—22no—
156C42—21no—
157C42—22no—
158C42—21no—
159C42—41no—
160C23—42no—
161C23—41no—
162C23—42no—
163C23—41no—
164C23—42no—
165C23—41no—
166C23—41no—
167C23—42no—
168C23—42no—
169C23—41no—
170C23—41no—
171C23—42no—
172C23—41no—
173C23—42no—
174C23C22×D411yescharacteristic
175Q8—84no—
176D4—42no—
177D4—42no—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—84no—
183D4—84no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4AGL(1,ℤ/8)12yes—
212C2×C4AGL(1,ℤ/8)12yes—
213C2×C4AGL(1,ℤ/8)12yes—
214C2×C4C2×#4112yes—
215C2×C4H(5,𝔽2)11yescharacteristic
216C2×C4H(5,𝔽2)12yes—
217C2×C42−1+411yescharacteristic
218C8—24no—
219C8—42no—
220C22—81no—
221C22—82no—
222C22—82no—
223C22—81no—
224C22—81no—
225C22—82no—
226C22—82no—
227C22—81no—
228C22—82no—
229C22—81no—
230C22—82no—
231C22—81no—
232C22—81no—
233C22—82no—
234C22#53211yescharacteristic
235C22#53511yescharacteristic
236C22#53711yescharacteristic
237C22C2×AGL(1,ℤ/8)11yescharacteristic
238C22C2×AGL(1,ℤ/8)11yescharacteristic
239C22C2×AGL(1,ℤ/8)11yescharacteristic
240C22#57511yescharacteristic
241C4—22no—
242C4—22no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—81no—
248C4—81no—
249C4—82no—
250C4—81no—
251C4#53812yes—
252C4#54212yes—
253C2—162no—
254C2—161no—
255C2#266211yescharacteristic
256C2#318011yescharacteristic
257C2#318411yescharacteristic
258C2#321311yescharacteristic
259C2#321711yescharacteristic
260C2#325111yescharacteristic
261C2#325311yescharacteristic
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. #2860C2semidirect
2. #2858C2semidirect
3. #2065C2semidirect
4. #2058C2semidirect
5. #2054C2semidirect
6. #2041C2semidirect
7. #2023C2semidirect
9. #1972C2semidirect
10. #1936C2semidirect
11. #1767C2semidirect

Maximal subgroups

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

subgroupindex
#28602
#28582
#20652
#20582
#20542
#20412
#20232
#20112
#19722
#19362
#17672