Finite Group Databasebeta

← #18310

Subgroups — #18310

Order 256.

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

normal subgroups only

This group has 454 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
1#3458C211yesmaximal, characteristic
2#3229C212yesmaximal, complemented
3#2621C211yesmaximal, characteristic
4#1611C214yesmaximal, complemented
5#1608C214yesmaximal
6#1518C212yesmaximal, complemented
7#1458C211yesmaximal, characteristic
8#544—22no—
9#544—22no—
10#544—21no—
11#544C2212yes—
12#543—21no—
13#543C2212yes—
14#533—24no—
15#533—21no—
16#533C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×#117—21no—
20C2×#117C2211yescharacteristic
21#498—22no—
22#487—22no—
23#483C2218yes—
24#475—24no—
25#431C2212yes—
26#398—21no—
27#398C2212yes—
28#397—21no—
29#397C2212yes—
30#391—21no—
31#382C2211yescharacteristic
32#328C2218yes—
33#324C2212yes—
34#323C2212yes—
35C22×Q8—21no—
36C23×C4D411yescharacteristic
37C4⋊Q8—24no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—24no—
41C4⋊Q8—22no—
42C4⋊Q8—22no—
43C4⋊Q8D412yes—
44C4⋊Q8D412yes—
45#125—24no—
46#125—22no—
47#125—24no—
48#125—22no—
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51C22⋊Q8—24no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55C22⋊Q8—24no—
56C22⋊Q8—44no—
57C22⋊Q8C2314yes—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×Q8D414yes—
62#117—22no—
63#117—22no—
64#117—22no—
65#117—22no—
66#117—22no—
67#117—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4D411yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4D411yescharacteristic
79C2×C22⋊C4C2311yescharacteristic
80C2×C42—22no—
81C2×C42D411yescharacteristic
82C2×C42C2311yescharacteristic
83#106—28no—
84C4⋊C8—22no—
85#103—28no—
86C22⋊C8C2314yes—
87C4×C8—22no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95C2312yes—
92C24C2×D411yescharacteristic
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—41no—
97C2×Q8—41no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8C2×D414yes—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4SD1614yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2411yescharacteristic
115C2×C8—24no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—44no—
133C4⋊C4C2×D414yes—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—44no—
139C42—24no—
140C42—24no—
141C42—21no—
142C42—21no—
143C42C2×D412yes—
144C42C2×D412yes—
145C23—21no—
146C23—21no—
147C23—41no—
148C23—41no—
149C23C2×SD1612yes—
150C23C22×D411yescharacteristic
151Q8—44no—
152Q8—82no—
153Q8—82no—
154Q8—82no—
155Q8—82no—
156Q8—82no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4#12012yes—
178C2×C4#12011yescharacteristic
179C2×C4#12011yescharacteristic
180C2×C4C2×SD1612yes—
181C2×C4C2×SD1612yes—
182C2×C4#13712yes—
183C2×C4C22×D412yes—
184C8—44no—
185C22—21no—
186C22—21no—
187C22—22no—
188C22—42no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22C2×#12011yescharacteristic
193C22C22×SD1611yescharacteristic
194C22C2×#13711yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—41no—
199C4—44no—
200C4—41no—
201C4—44no—
202C4—82no—
203C4—82no—
204C4#44812yes—
205C4#44812yes—
206C2—21no—
207C2—21no—
208C2—42no—
209C2C2×#44811yescharacteristic
210C2#302211yescharacteristic
211C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3229C2semidirect
4. #1611C2semidirect
6. #1518C2semidirect

Maximal subgroups

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

subgroupindex
#34582
#32292
#26212
#16112
#16082
#15182
#14582