Finite Group Databasebeta

← #29368

Subgroups — #29368

Order 256.

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

normal subgroups only

This group has 411 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
1C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic
5#2043C211yesmaximal, characteristic, complemented
6#2013C211yesmaximal, characteristic
7#1927C212yesmaximal, complemented
8#1880C212yesmaximal
9#1877C212yesmaximal
10#1865C212yesmaximal, complemented
11#1839C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2212yes—
15C2×C4×D4C2211yescharacteristic
16C2×#117—21no—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22#391C2212yes—
23#390C2211yescharacteristic
24#390C2211yescharacteristic
25#389—21no—
26#389—21no—
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#388—21no—
30#388—21no—
31#384C2211yescharacteristic
32C4×C4⋊C4—21no—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C2×SD16—24no—
47C2×SD16—24no—
48C2×SD16—42no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—24no—
52C4×Q8—42no—
53C4×Q8—42no—
54C4×D4—22no—
55C4×D4—22no—
56#117—44no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—21no—
74C2×C42—22no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#106—42no—
83#103—24no—
84#103—24no—
85#103—42no—
86#102—42no—
87#95—21no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95—21no—
92C24—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8#4114yes—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4#4112yes—
103C22×C4—24no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118SD16—48no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—44no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4#4112yes—
138C22⋊C4—24no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C42—22no—
144C42—21no—
145C42—21no—
146C42—22no—
147C42—22no—
148C42—41no—
149C42—41no—
150C42—44no—
151C42—42no—
152C42SD1614yescomplemented
153C23—22no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23C22×D411yescharacteristic
161Q8—24no—
162Q8—24no—
163Q8—42no—
164Q8—42no—
165D4—24no—
166D4—42no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—48no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4C2×SD1612yes—
198C2×C4C2×SD1612yes—
199C2×C4C2×SD1612yes—
200C2×C4#13712yes—
201C2×C4C2×#4111yescharacteristic
202C2×C4C2×#4112yes—
203C2×C4C2×#4111yescharacteristic
204C2×C4H(5,𝔽2)12yes—
205C8—42no—
206C8—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22#52511yescharacteristic
215C22#52511yescharacteristic
216C22#53511yescharacteristic
217C22C22×SD1611yescharacteristic
218C22C2×#13711yescharacteristic
219C22#57511yescharacteristic
220C22#57511yescharacteristic
221C4—22no—
222C4—22no—
223C4—22no—
224C4—22no—
225C4—42no—
226C4—44no—
227C4—41no—
228C4—41no—
229C4—81no—
230C4—81no—
231C4—82no—
232C4#55112yes—
233C4#55312yes—
234C2—82no—
235C2#259811yescharacteristic
236C2#310111yescharacteristic
237C2#310311yescharacteristic
238C2#311611yescharacteristic
239C2#311711yescharacteristic
240C2#320611yescharacteristic
241C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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. C4×C4⋊Q8C2semidirect
3. #2100C2semidirect
5. #2043C2semidirect
7. #1927C2semidirect
10. #1865C2semidirect
11. #1839C2semidirect
152. C42SD16semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊Q82
#22932
#21002
#20942
#20432
#20132
#19272
#18802
#18772
#18652
#18392