Finite Group Databasebeta

← #30115

Subgroups — #30115

Order 256.

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

normal subgroups only

This group has 467 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#2856C211yesmaximal, characteristic
2C4×C22⋊Q8C211yesmaximal, characteristic, complemented
3#2100C211yesmaximal, characteristic, complemented
4#2095C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2033C212yesmaximal, complemented
7#2014C211yesmaximal, characteristic, complemented
8#2000C212yesmaximal, complemented
9#1925C212yesmaximal, complemented
10#1880C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×#127C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×#121—21no—
16C2×#121—21no—
17C2×#121C2212yes—
18C2×C4×Q8C2211yescharacteristic
19C22×C42C2211yescharacteristic
20C2×#107C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×C22⋊C8C2212yes—
26#391C2211yescharacteristic
27#390—21no—
28#390—21no—
29#390C2212yes—
30#389C2211yescharacteristic
31#387—21no—
32#385—21no—
33#384C2211yescharacteristic
34#380C2211yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—22no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C2×SD16—24no—
52C2×SD16—42no—
53C2×SD16—42no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56#127—42no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59#121—42no—
60#121—42no—
61#121—44no—
62#121—44no—
63C4×Q8—42no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—42no—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85#107—42no—
86#103—42no—
87#102—24no—
88#102—42no—
89#102—42no—
90C22⋊C8—42no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96C24—22no—
97C24C2×D411yescharacteristic
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8#4112yes—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—44no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—42no—
116C2×D4—42no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138SD16—48no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C4⋊C4—24no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4#4112yes—
156C22⋊C4—24no—
157C22⋊C4—44no—
158C22⋊C4—44no—
159C22⋊C4—42no—
160C22⋊C4—44no—
161C42—22no—
162C42—22no—
163C42—41no—
164C42—41no—
165C42—42no—
166C42—42no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23AGL(1,ℤ/8)12yes—
181C23C22×D411yescharacteristic
182Q8—24no—
183Q8—42no—
184D4—42no—
185D4—42no—
186D4—42no—
187D4—42no—
188D4—84no—
189D4—84no—
190D4—88no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4#13512yes—
223C2×C4AGL(1,ℤ/8)12yes—
224C2×C4#13712yes—
225C2×C4C2×#4111yescharacteristic
226C2×C4C2×#4111yescharacteristic
227C2×C4H(5,𝔽2)12yes—
228C2×C4H(5,𝔽2)12yes—
229C8—42no—
230C8—42no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22—82no—
243C22—82no—
244C22—82no—
245C22#52511yescharacteristic
246C22#53411yescharacteristic
247C22#53511yescharacteristic
248C22C2×#13511yescharacteristic
249C22C2×AGL(1,ℤ/8)11yescharacteristic
250C22C2×AGL(1,ℤ/8)11yescharacteristic
251C22C2×#13711yescharacteristic
252C4—22no—
253C4—22no—
254C4—22no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4—41no—
259C4—81no—
260C4—81no—
261C4—82no—
262C4#53812yes—
263C4#54012yes—
264C2—42no—
265C2—162no—
266C2#263511yescharacteristic
267C2#310611yescharacteristic
268C2#311211yescharacteristic
269C2#318811yescharacteristic
270C2#319311yescharacteristic
271C2#321011yescharacteristic
272C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2856C2
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
2. C4×C22⋊Q8C2semidirect
3. #2100C2semidirect
4. #2095C2semidirect
5. #2041C2semidirect
6. #2033C2semidirect
7. #2014C2semidirect
8. #2000C2semidirect
9. #1925C2semidirect
10. #1880C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28562
C4×C22⋊Q82
#21002
#20952
#20412
#20332
#20142
#20002
#19252
#18802
#18172