Finite Group Databasebeta

← #29648

Subgroups — #29648

Order 256.

474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 127 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×#544C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic
3C2×#483C211yesmaximal, characteristic
4#2625C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2058C212yesmaximal, complemented
7#2026C212yesmaximal, complemented
8#2024C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic
10#1824C211yesmaximal, characteristic
11#1818C211yesmaximal, characteristic
12#544—24no—
13#544—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2212yes—
16C2×#125—21no—
17C2×#125C2212yes—
18C2×C22⋊Q8—22no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#485—24no—
24#483—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2212yes—
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#400—21no—
34#398—21no—
35#397—22no—
36#391C2212yes—
37#389C2212yes—
38#380C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—22no—
46C22×Q8C2311yescharacteristic
47C23×C4—21no—
48C23×C4C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C4⋊Q8—28no—
53C4⋊Q8—24no—
54C4⋊Q8—24no—
55#125—28no—
56#125—24no—
57#125—24no—
58C22⋊Q8—216no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61#117—22no—
62#117—22no—
63#117—22no—
64#117D414yes—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4D411yescharacteristic
74C2×C4⋊C4D414yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4D412yes—
84C2×C42—21no—
85C2×C42C2312yes—
86#107—22no—
87#106—24no—
88#106—22no—
89#103—24no—
90#103—22no—
91#103—22no—
92C22⋊C8—22no—
93C22⋊C8—22no—
94#95—21no—
95#95—22no—
96#95—22no—
97#95—22no—
98C24C2×D411yescharacteristic
99C2×Q8—28no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—84no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—82no—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4C2411yescharacteristic
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C4⋊C4—28no—
129C4⋊C4—28no—
130C4⋊C4—28no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—82no—
143C4⋊C4C2×D414yes—
144C4⋊C4C2×D412yes—
145C4⋊C4C2×D412yes—
146C22⋊C4—24no—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C22⋊C4—81no—
150C42—24no—
151C42—22no—
152C42—22no—
153C23—22no—
154C23—21no—
155C23—22no—
156C23—21no—
157C23#12012yes—
158C23C22×D411yescharacteristic
159Q8—42no—
160Q8—41no—
161Q8—41no—
162Q8—88no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—84no—
188C2×C4#12012yes—
189C2×C4C22×D411yescharacteristic
190C2×C4C22×D411yescharacteristic
191C2×C42−1+412yes—
192C2×C42−1+412yes—
193C8—42no—
194C8—41no—
195C8—41no—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22C2×#12011yescharacteristic
202C22#53612yes—
203C22#57712yes—
204C22#57711yescharacteristic
205C22#57811yescharacteristic
206C4—22no—
207C4—42no—
208C4—44no—
209C4—42no—
210C4—42no—
211C4—42no—
212C4—81no—
213C4—82no—
214C4—81no—
215C4#53611yescharacteristic
216C4#53612yes—
217C4#53611yescharacteristic
218C2—42no—
219C2#261411yescharacteristic
220C2#300911yescharacteristic
221C2#301311yescharacteristic
222C2#324212yes—
223C2#324412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2059C2semidirect
6. #2058C2semidirect
7. #2026C2semidirect
8. #2024C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4852
C2×#4832
#26252
#20592
#20582
#20262
#20242
#18252
#18242
#18182