Finite Group Databasebeta

← #30046

Subgroups — #30046

Order 256.

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

normal subgroups only

This group has 402 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#2862C211yesmaximal, characteristic, complemented
2C4×#124C211yesmaximal, characteristic
3#2074C212yesmaximal, complemented
4#2024C211yesmaximal, characteristic, complemented
5#2023C211yesmaximal, characteristic
6#2016C211yesmaximal, characteristic, complemented
7#2015C212yesmaximal
8#2013C211yesmaximal, characteristic
9#1878C212yesmaximal, complemented
10#1877C212yesmaximal
11#1839C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23#391—21no—
24#391—21no—
25#391C2212yes—
26#391C2212yes—
27#391C2211yescharacteristic
28#390C2211yescharacteristic
29#388—21no—
30#388—21no—
31#382—21no—
32#382—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×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×Q16—24no—
49C2×Q16—42no—
50C2×Q16—42no—
51C2×SD16—24no—
52C2×SD16—42no—
53C2×SD16—42no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—42no—
57C4⋊Q8—42no—
58#124—22no—
59#124—22no—
60C4×Q8—42no—
61C4×D4—42no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#103—42no—
83#103—42no—
84#103—42no—
85#102—42no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94C24—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8#4112yes—
108C2×D4—21no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4#4112yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2411yescharacteristic
127Q16—48no—
128SD16—48no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C42—22no—
149C42—22no—
150C42—22no—
151C42—21no—
152C42—21no—
153C42—22no—
154C42—22no—
155C42—41no—
156C42—42no—
157C23—22no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23C22×D411yescharacteristic
165Q8—24no—
166Q8—42no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170Q8—42no—
171Q8—84no—
172D4—24no—
173D4—42no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—84no—
205C2×C4#13512yes—
206C2×C4#13712yes—
207C2×C4#13712yes—
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4H(5,𝔽2)12yes—
211C2×C4H(5,𝔽2)12yes—
212C8—42no—
213C8—42no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22#52511yescharacteristic
222C22#53511yescharacteristic
223C22#53511yescharacteristic
224C22C2×#13511yescharacteristic
225C22C2×#13711yescharacteristic
226C22C2×#13711yescharacteristic
227C22#57511yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—41no—
232C4—41no—
233C4—42no—
234C4—41no—
235C4—82no—
236C4—81no—
237C4—82no—
238C4—81no—
239C4#53812yes—
240C4#53812yes—
241C2—82no—
242C2#263111yescharacteristic
243C2#310711yescharacteristic
244C2#311611yescharacteristic
245C2#321011yescharacteristic
246C2#321111yescharacteristic
247C2#321511yescharacteristic
248C2#321811yescharacteristic
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. #2862C2semidirect
3. #2074C2semidirect
4. #2024C2semidirect
6. #2016C2semidirect
9. #1878C2semidirect
11. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28622
C4×#1242
#20742
#20242
#20232
#20162
#20152
#20132
#18782
#18772
#18392