Finite Group Databasebeta

← #29955

Subgroups — #29955

Order 256.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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
2#2297C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2068C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10#1797C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal
12C22×#41C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×#117C2211yescharacteristic
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2212yes—
25#391—21no—
26#391C2211yescharacteristic
27#391C2211yescharacteristic
28#390C2211yescharacteristic
29#390C2211yescharacteristic
30#389—21no—
31#389C2212yes—
32#386—21no—
33#386—21no—
34#384C2211yescharacteristic
35#380—21no—
36#380C2212yes—
37C4×C22⋊C4—21no—
38C4×C22⋊C4—21no—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C2×#41—22no—
44C2×#41—42no—
45C2×#41—42no—
46C2×#41Q814yescomplemented
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4D411yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—42no—
58C22⋊Q8—44no—
59#117—42no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42—42no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—42no—
81#106—42no—
82C4⋊C8—42no—
83#103—42no—
84#103—42no—
85#102—42no—
86#102—42no—
87C22⋊C8—24no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95—22no—
94C24—21no—
95C24C2×D411yescharacteristic
96#41—48no—
97#41—48no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8C2×Q812yes—
105C2×D4—24no—
106C2×D4—21no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4C2×Q812yes—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—24no—
116C22×C4—22no—
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—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×Q812yes—
134C22×C4#4112yes—
135C22×C4C2411yescharacteristic
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—44no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—44no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C42—22no—
159C42—22no—
160C42—41no—
161C42—41no—
162C42—42no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23C22⋊Q812yes—
177C23C22×D411yescharacteristic
178Q8—24no—
179Q8—42no—
180Q8—84no—
181D4—24no—
182D4—48no—
183D4—42no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—48no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4C22⋊Q812yes—
216C2×C4#13512yes—
217C2×C4AGL(1,ℤ/8)12yes—
218C2×C4#13712yes—
219C2×C4C22×Q811yescharacteristic
220C2×C4C2×#4111yescharacteristic
221C2×C4H(5,𝔽2)12yes—
222C2×C4H(5,𝔽2)12yes—
223C8—42no—
224C8—42no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22C2×C22⋊Q811yescharacteristic
240C22#53411yescharacteristic
241C22#53511yescharacteristic
242C22C2×#13511yescharacteristic
243C22C2×AGL(1,ℤ/8)11yescharacteristic
244C22C2×#13711yescharacteristic
245C22#57511yescharacteristic
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—42no—
251C4—41no—
252C4—82no—
253C4—81no—
254C4—82no—
255C4—81no—
256C4#53812yes—
257C4#54312yes—
258C2—42no—
259C2—82no—
260C2#262611yescharacteristic
261C2#307611yescharacteristic
262C2#307911yescharacteristic
263C2#319011yescharacteristic
264C2#319411yescharacteristic
265C2#320711yescharacteristic
266C2#321611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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. #2096C2semidirect
4. #2093C2semidirect
5. #2068C2semidirect
6. #2065C2semidirect
7. #2024C2semidirect
8. #2023C2semidirect
9. #1854C2semidirect
10. #1797C2semidirect
46. C2×#41Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
#22972
#20962
#20932
#20682
#20652
#20242
#20232
#18542
#17972
#17882