Finite Group Databasebeta

← #27912

Subgroups — #27912

Order 256.

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

normal subgroups only

This group has 474 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×#484C211yesmaximal, characteristic, complemented
2C2×#481C211yesmaximal, characteristic
3C2×#483C211yesmaximal, characteristic, complemented
4C2×#480C211yesmaximal, characteristic
5#2850C211yesmaximal, characteristic
6C2×#382C211yesmaximal, characteristic, complemented
7#2097C212yesmaximal, complemented
8#2092C212yesmaximal, complemented
9#1958C212yesmaximal, complemented
10#1846C212yesmaximal, complemented
11C4×C22⋊C8C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#484—24no—
19#483—24no—
20#481—24no—
21#480—24no—
22C2×#107C2212yes—
23C2×#107C2212yes—
24C2×#106C2212yes—
25C2×#106C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C4×C8C2212yes—
33#391C2211yescharacteristic
34#390C2211yescharacteristic
35#388—22no—
36#386—22no—
37#382—22no—
38#382C2211yescharacteristic
39#382C2212yes—
40#382C2212yes—
41C2×#95—22no—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4D412yes—
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52#124—24no—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55#121—22no—
56#121—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—24no—
73C2×C42D412yes—
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—24no—
77#107—24no—
78#106—24no—
79#106—24no—
80#103—24no—
81#103—24no—
82#102—24no—
83#102—24no—
84C22⋊C8D414yes—
85C22⋊C8D414yes—
86C4×C8—24no—
87#95—24no—
88#95—24no—
89#95—22no—
90#95—22no—
91C24—21no—
92C24C2×D411yescharacteristic
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—44no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—42no—
113C22×C4—44no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—44no—
117C22×C4C2×D412yes—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2411yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8C2×D414yes—
128C2×C8C2×D414yes—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C22⋊C4—44no—
141C22⋊C4—42no—
142C22⋊C4—44no—
143C22⋊C4—42no—
144C42—24no—
145C42—22no—
146C42#4114yes—
147C23—21no—
148C23—21no—
149C23—22no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23#12712yes—
156C23#13514yes—
157C23C22×D411yescharacteristic
158Q8—42no—
159Q8—42no—
160D4—42no—
161D4—42no—
162D4—84no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—48no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—84no—
185C2×C4—84no—
186C2×C4#12314yes—
187C2×C4#12712yes—
188C2×C4C22×D411yescharacteristic
189C2×C4C22×D411yescharacteristic
190C2×C4C2×#4112yes—
191C2×C42−1+412yes—
192C8—24no—
193C8—24no—
194C22—24no—
195C22—22no—
196C22—24no—
197C22—22no—
198C22—82no—
199C22—81no—
200C22—81no—
201C22—82no—
202C22—81no—
203C22#49514yes—
204C22C2×#12311yescharacteristic
205C22C2×#12711yescharacteristic
206C22#53611yescharacteristic
207C22C2×#13512yes—
208C22#57512yes—
209C4—24no—
210C4—22no—
211C4—42no—
212C4—82no—
213C4—81no—
214C4—81no—
215C4—82no—
216C4—81no—
217C4#54712yes—
218C4#54712yes—
219C2—24no—
220C2—161no—
221C2#250111yescharacteristic
222C2#309112yes—
223C2C2×#49511yescharacteristic
224C2#315311yescharacteristic
225C2#323212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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. C2×#484C2semidirect
3. C2×#483C2semidirect
6. C2×#382C2semidirect
7. #2097C2semidirect
8. #2092C2semidirect
9. #1958C2semidirect
10. #1846C2semidirect
11. C4×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4812
C2×#4832
C2×#4802
#28502
C2×#3822
#20972
#20922
#19582
#18462
C4×C22⋊C82