Finite Group Databasebeta

← #29790

Subgroups — #29790

Order 256.

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

normal subgroups only

This group has 363 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#3095C211yesmaximal, characteristic, complemented
2#3094C211yesmaximal, characteristic
3C2×#465C211yesmaximal, characteristic
4C2×#424C211yesmaximal, characteristic, complemented
5#2090C212yesmaximal, complemented
6#2064C212yesmaximal, complemented
7#2063C212yesmaximal, complemented
8#2046C212yesmaximal, complemented
9#1901C211yesmaximal, characteristic, complemented
10#1900C211yesmaximal, characteristic
11#1810C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C2×#117C2211yescharacteristic
18#485—22no—
19#484—21no—
20#483—21no—
21#482—22no—
22#481—21no—
23#480—21no—
24#465—22no—
25#465—22no—
26C2×#108C2212yes—
27#428C2212yes—
28#428C2212yes—
29C2×#106C2211yescharacteristic
30#424—22no—
31#424C2212yes—
32#424C2212yes—
33C2×C4⋊C8—22no—
34#419C2212yes—
35#418C2212yes—
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#101C2212yes—
39C2×#100C2212yes—
40#407C2211yescharacteristic
41#407C2211yescharacteristic
42#343C2212yes—
43#343C2212yes—
44#341C2212yes—
45C22×Q8C2311yescharacteristic
46C22×D4—21no—
47C22×D4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C2×SD16—22no—
50C2×SD16—42no—
51C2×M16—21no—
52C2×M16—42no—
53C2×M16C2312yes—
54C2×M16C2312yes—
55C2×M16C2312yes—
56C22×C8—21no—
57C22×C8—42no—
58C22×C8D412yes—
59C22×C8D412yes—
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C22⋊Q8—21no—
65C22⋊Q8—21no—
66#121—21no—
67#121—21no—
68#121—21no—
69#121—21no—
70#117—21no—
71#117—21no—
72#117D412yes—
73#117C2312yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—21no—
83#108—42no—
84#107—22no—
85#107—22no—
86#106—22no—
87#106—22no—
88#106—22no—
89C4⋊C8—24no—
90C4⋊C8—24no—
91#103—22no—
92#103—22no—
93#103—22no—
94#102—22no—
95#102—22no—
96#102—22no—
97#101—42no—
98#100—42no—
99C22⋊C8—21no—
100C22⋊C8—21no—
101C22⋊C8—21no—
102C22⋊C8—21no—
103C24—21no—
104C24#4111yescharacteristic
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4—82no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4#4111yescharacteristic
128C22×C4C2411yescharacteristic
129SD16—44no—
130M16—22no—
131M16—22no—
132M16—22no—
133M16—22no—
134M16—22no—
135M16—22no—
136M16—42no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8C2×D412yes—
146C2×C8C2×D412yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4C2×D412yes—
156C4⋊C4C2×D412yes—
157C22⋊C4—22no—
158C22⋊C4—22no—
159C22⋊C4—41no—
160C22⋊C4—41no—
161C22⋊C4—41no—
162C22⋊C4—41no—
163C42—22no—
164C42—22no—
165C23—22no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—42no—
170C23—81no—
171C23—81no—
172C23#12111yescharacteristic
173C23#12111yescharacteristic
174C23C2×#4111yescharacteristic
175Q8—41no—
176Q8—41no—
177Q8—42no—
178D4—41no—
179D4—41no—
180D4—42no—
181D4—82no—
182D4—82no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4#12111yescharacteristic
201C2×C4#12111yescharacteristic
202C2×C4C22×D411yescharacteristic
203C2×C4C22×D411yescharacteristic
204C2×C4H(5,𝔽2)12yes—
205C2×C42−1+412yes—
206C8—22no—
207C8—22no—
208C8—42no—
209C8—42no—
210C8—42no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—44no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—162no—
223C22#47112yes—
224C22C2×#12111yescharacteristic
225C22#53912yes—
226C4—21no—
227C4—21no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4—81no—
232C4—81no—
233C4#53411yescharacteristic
234C4#53611yescharacteristic
235C4#53712yes—
236C2—22no—
237C2—22no—
238C2—41no—
239C2—161no—
240C2#261811yescharacteristic
241C2C2×#47111yescharacteristic
242C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3095C2semidirect
4. C2×#424C2semidirect
5. #2090C2semidirect
6. #2064C2semidirect
7. #2063C2semidirect
8. #2046C2semidirect
9. #1901C2semidirect
11. #1810C2semidirect

Maximal subgroups

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

subgroupindex
#30952
#30942
C2×#4652
C2×#4242
#20902
#20642
#20632
#20462
#19012
#19002
#18102