Finite Group Databasebeta

← #30113

Subgroups — #30113

Order 256.

439 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 115 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#2857C211yesmaximal, characteristic, complemented
2C4×#121C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2036C212yesmaximal, complemented
7#2013C211yesmaximal, characteristic, complemented
8#2000C212yesmaximal
9#1928C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2212yes—
15C2×#121C2211yescharacteristic
16C2×C4×D4—21no—
17C2×C4×D4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19C2×#107C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2212yes—
25#391C2212yes—
26#390C2211yescharacteristic
27#389—21no—
28#389—21no—
29#389C2211yescharacteristic
30#388—21no—
31#386—21no—
32#384C2211yescharacteristic
33#380—21no—
34#380—21no—
35#380C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2312yes—
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C2×SD16—24no—
51C2×SD16—42no—
52C2×SD16—42no—
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—42no—
56C22⋊Q8—42no—
57C22⋊Q8—42no—
58#121—22no—
59#121—22no—
60C4×D4—42no—
61C4×D4—44no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C42—21no—
79C2×C42—42no—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42C2311yescharacteristic
83C2×C42C2311yescharacteristic
84#107—42no—
85#103—24no—
86#103—42no—
87#103—42no—
88#102—42no—
89C22⋊C8—42no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—22no—
94#95—22no—
95C24—21no—
96C24C2×D411yescharacteristic
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×D4—24no—
104C2×D4—21no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4#4112yes—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—41no—
124C22×C4—41no—
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×C4C2411yescharacteristic
134SD16—48no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—44no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4#4112yes—
153C22⋊C4—24no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—44no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—41no—
162C42—41no—
163C42—42no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23#13712yes—
178C23C22×D411yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8—42no—
183D4—24no—
184D4—48no—
185D4—42no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—84no—
218C2×C4#13512yes—
219C2×C4AGL(1,ℤ/8)12yes—
220C2×C4#13712yes—
221C2×C4C2×#4111yescharacteristic
222C2×C4C2×#4111yescharacteristic
223C2×C4H(5,𝔽2)12yes—
224C2×C4H(5,𝔽2)12yes—
225C8—42no—
226C8—42no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22#52511yescharacteristic
242C22#53411yescharacteristic
243C22#53511yescharacteristic
244C22C2×#13511yescharacteristic
245C22C2×AGL(1,ℤ/8)11yescharacteristic
246C22C2×#13711yescharacteristic
247C22C2×#13711yescharacteristic
248C4—22no—
249C4—22no—
250C4—22no—
251C4—42no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4—81no—
256C4—82no—
257C4—81no—
258C4#53812yes—
259C4#54012yes—
260C2—42no—
261C2—82no—
262C2#263511yescharacteristic
263C2#310711yescharacteristic
264C2#311011yescharacteristic
265C2#319011yescharacteristic
266C2#319311yescharacteristic
267C2#320711yescharacteristic
268C2#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. #2857C2semidirect
3. #2099C2semidirect
4. #2093C2semidirect
5. #2043C2semidirect
6. #2036C2semidirect
7. #2013C2semidirect
9. #1928C2semidirect
10. #1877C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28572
C4×#1212
#20992
#20932
#20432
#20362
#20132
#20002
#19282
#18772
#18172