Finite Group Databasebeta

← #25941

Subgroups — #25941

Order 256.

511 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 135 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×#557C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic
3C2×#464C211yesmaximal, characteristic
4#2592C211yesmaximal, characteristic
5#2026C212yesmaximal, complemented
6#2025C212yesmaximal
7#1987C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1878C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14#557—24no—
15#557—24no—
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—22no—
19C2×#125C2212yes—
20C2×#124C2211yescharacteristic
21C2×C4×Q8—21no—
22C2×C4×Q8C2212yes—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25#489—24no—
26#464—24no—
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2211yescharacteristic
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#397—21no—
35#393—21no—
36#391C2212yes—
37#390C2212yes—
38#388—22no—
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8D411yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×Q8C2311yescharacteristic
49C22×D4C2311yescharacteristic
50C2×Q16—22no—
51C2×Q16—24no—
52C2×SD16—22no—
53C2×SD16—24no—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22×C8C2312yes—
57C4⋊Q8—28no—
58C4⋊Q8—22no—
59C4⋊Q8D414yes—
60#125—216no—
61#125—24no—
62#125—24no—
63#124—22no—
64#124—22no—
65C4×Q8—28no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8D414yes—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D412yes—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C42—22no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42C2312yes—
90C2×C42C2311yescharacteristic
91C4⋊C8—22no—
92#103—24no—
93#103—22no—
94#103—24no—
95#102—22no—
96#102—24no—
97#97—22no—
98#95—21no—
99#95—22no—
100#95—22no—
101C24—21no—
102C2×Q8—22no—
103C2×Q8—24no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—42no—
111C2×Q8—44no—
112C2×Q8C2×D412yes—
113C2×Q8#4114yes—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128Q16—44no—
129SD16—44no—
130C2×C8—22no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C4⋊C4—28no—
138C4⋊C4—28no—
139C4⋊C4—28no—
140C4⋊C4—24no—
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—24no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4C2×D414yes—
149C4⋊C4C2×D412yes—
150C22⋊C4—44no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—82no—
154C42—28no—
155C42—24no—
156C42—22no—
157C42—22no—
158C42—81no—
159C42C2×D412yes—
160C42C2×D412yes—
161C23—41no—
162C23—42no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23C22×D411yescharacteristic
167Q8—22no—
168Q8—24no—
169Q8—44no—
170Q8—48no—
171Q8—41no—
172Q8—41no—
173Q8—42no—
174Q8#12414yes—
175D4—41no—
176D4—41no—
177D4—42no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—44no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4#12012yes—
200C2×C4#12012yes—
201C2×C4C22×D411yescharacteristic
202C2×C4C22×D411yescharacteristic
203C2×C4C2×#4112yes—
204C2×C42−1+412yes—
205C8—41no—
206C8—41no—
207C8—42no—
208C22—82no—
209C22—82no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22C2×#12011yescharacteristic
214C22#54812yes—
215C22#57512yes—
216C22#57711yescharacteristic
217C22#57811yescharacteristic
218C4—24no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—44no—
223C4—42no—
224C4—42no—
225C4—82no—
226C4—81no—
227C4C2×#12411yescharacteristic
228C4#53611yescharacteristic
229C4#54812yes—
230C2—161no—
231C2#243611yescharacteristic
232C2#300511yescharacteristic
233C2#301311yescharacteristic
234C2#337112yes—
235C2#337712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#557C2semidirect
5. #2026C2semidirect
7. #1987C2semidirect
8. #1985C2semidirect
10. #1878C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4892
C2×#4642
#25922
#20262
#20252
#19872
#19852
#18802
#18782
#17632