Finite Group Databasebeta

← #32811

Subgroups — #32811

Order 256.

371 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 109 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
2#2861C211yesmaximal, characteristic
3#2056C212yesmaximal, complemented
4#2055C212yesmaximal
5#2026C212yesmaximal, complemented
6#2024C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic
8#2016C211yesmaximal, characteristic, complemented
9#2015C212yesmaximal
10#2013C211yesmaximal, characteristic
11#1831C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23#391—21no—
24#391—21no—
25#391C2212yes—
26#391C2211yescharacteristic
27#391C2211yescharacteristic
28#391C2212yes—
29#390C2211yescharacteristic
30#390C2211yescharacteristic
31#388—21no—
32#388—21no—
33#388—21no—
34#388—21no—
35#382—21no—
36#382—21no—
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2312yes—
46C22×D4C2311yescharacteristic
47C2×Q16—42no—
48C2×Q16—42no—
49C2×Q16—42no—
50C2×SD16—42no—
51C2×SD16—42no—
52C2×SD16—42no—
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—42no—
56C4⋊Q8—42no—
57#125—42no—
58#124—42no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80C4⋊C8—42no—
81#103—42no—
82#103—42no—
83#103—42no—
84#102—42no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93C24—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2411yescharacteristic
123Q16—48no—
124SD16—48no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—21no—
150C42—22no—
151C42—22no—
152C42—21no—
153C42—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23C22×D411yescharacteristic
162Q8—42no—
163Q8—42no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168Q8—84no—
169D4—42no—
170D4—42no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
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—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4#13512yes—
202C2×C4#13712yes—
203C2×C4#13712yes—
204C2×C4H(5,𝔽2)11yescharacteristic
205C2×C4H(5,𝔽2)12yes—
206C2×C4H(5,𝔽2)12yes—
207C2×C42−1+411yescharacteristic
208C8—42no—
209C8—42no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22#53511yescharacteristic
218C22#53511yescharacteristic
219C22#53711yescharacteristic
220C22C2×#13511yescharacteristic
221C22C2×#13711yescharacteristic
222C22C2×#13711yescharacteristic
223C22#57511yescharacteristic
224C4—22no—
225C4—22no—
226C4—22no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—81no—
231C4—82no—
232C4—81no—
233C4—82no—
234C4—81no—
235C4H(3,𝔽4)12yes—
236C4#56212yes—
237C2—161no—
238C2#281911yescharacteristic
239C2#321011yescharacteristic
240C2#321111yescharacteristic
241C2#321611yescharacteristic
242C2#321811yescharacteristic
243C2#324811yescharacteristic
244C2#325211yescharacteristic
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. #2056C2semidirect
5. #2026C2semidirect
6. #2024C2semidirect
8. #2016C2semidirect
11. #1831C2semidirect

Maximal subgroups

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

subgroupindex
#28622
#28612
#20562
#20552
#20262
#20242
#20232
#20162
#20152
#20132
#18312