Finite Group Databasebeta

← #27939

Subgroups — #27939

Order 256.

482 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#483C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic
3C2×#465C211yesmaximal, characteristic
4C22×#107C211yesmaximal, characteristic, complemented
5#2503C211yesmaximal, characteristic, complemented
6#2501C211yesmaximal, characteristic
7#2095C212yesmaximal, complemented
8#2092C212yesmaximal, complemented
9#1973C212yesmaximal, complemented
10#1847C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×#123—22no—
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19#483—24no—
20#480—24no—
21#465—24no—
22C2×#107—24no—
23C2×#107C2212yes—
24C2×#107C2212yes—
25C2×#107C2212yes—
26C2×#106C2212yes—
27C2×#106C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#102C2212yes—
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2212yes—
34#400—22no—
35#399—22no—
36#391C2212yes—
37#390C2212yes—
38#384C2212yes—
39#382C2212yes—
40#381—22no—
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4C2311yescharacteristic
49C2×SD16—22no—
50C2×SD16—24no—
51C22×C8—24no—
52C22×C8D414yes—
53C22×C8C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56#123—48no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59#121—22no—
60#121—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4D412yes—
77C2×C42C2312yes—
78#107—24no—
79#107—24no—
80#107—24no—
81#106—22no—
82#106—24no—
83#103—22no—
84#103—24no—
85#102—22no—
86#102—24no—
87C22⋊C8D414yes—
88#97—24no—
89#95—24no—
90#95—24no—
91#95D412yes—
92C24—21no—
93C24C2×D411yescharacteristic
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—42no—
103C2×D4—44no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4C2×D414yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4C2×D412yes—
119C22×C4C2411yescharacteristic
120SD16—44no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—28no—
125C2×C8C2×D414yes—
126C2×C8C2×D412yes—
127C2×C8C2×D412yes—
128C2×C8#4114yes—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—48no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C22⋊C4—24no—
142C22⋊C4—48no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C42—22no—
146C42—22no—
147C23—22no—
148C23—22no—
149C23—41no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23#12712yes—
154C23AGL(1,ℤ/8)14yes—
155C23C22×D411yescharacteristic
156Q8—41no—
157Q8—41no—
158Q8—42no—
159D4—41no—
160D4—41no—
161D4—42no—
162D4—84no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—84no—
181C2×C4—84no—
182C2×C4#12712yes—
183C2×C4C22×D411yescharacteristic
184C2×C4C22×D411yescharacteristic
185C2×C4C2×#4112yes—
186C2×C42−1+412yes—
187C8—22no—
188C8—24no—
189C8#12314yescomplemented
190C22—28no—
191C22—22no—
192C22—22no—
193C22—84no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22#49614yes—
198C22C2×#12711yescharacteristic
199C22#54712yes—
200C22C2×AGL(1,ℤ/8)12yes—
201C22#57812yes—
202C4—22no—
203C4—44no—
204C4—82no—
205C4—82no—
206C4—81no—
207C4—81no—
208C4—81no—
209C4C2×#12311yescharacteristic
210C4#53611yescharacteristic
211C4#54712yes—
212C2—24no—
213C2—161no—
214C2#250111yescharacteristic
215C2C2×#49611yescharacteristic
216C2#315611yescharacteristic
217C2#334314yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#483C2semidirect
4. C22×#107C2semidirect
5. #2503C2semidirect
7. #2095C2semidirect
8. #2092C2semidirect
9. #1973C2semidirect
10. #1847C2semidirect
11. #1752C2semidirect
189. C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4802
C2×#4652
C22×#1072
#25032
#25012
#20952
#20922
#19732
#18472
#17522