Finite Group Databasebeta

← #16507

Subgroups — #16507

Order 256.

819 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 819 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C22×C4⋊Q8C211yesmaximal, characteristic
2C2×#398C212yesmaximal
3C2×#395C212yesmaximal
4C2×#387C211yesmaximal, characteristic
5#1446C212yesmaximal
6#1445C212yesmaximal
7#1444C212yesmaximal
8#1442C212yesmaximal
9#1434C211yesmaximal, characteristic
10C23×Q8C2212yes—
11C2×C4⋊Q8—28no—
12C2×C4⋊Q8—24no—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2214yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2212yes—
18C22×C42C2211yescharacteristic
19#398—28no—
20#395—28no—
21#387—24no—
22C2×#95C2212yes—
23C2×#95C2214yes—
24C2×#95C2212yes—
25C2×#95C2214yes—
26C2×#95C2211yescharacteristic
27C2×#95C2211yescharacteristic
28C2×#95C2211yescharacteristic
29C2×#95C2211yescharacteristic
30C2×#95C2214yes—
31C2×#95C2212yes—
32C22×Q8—28no—
33C22×Q8—24no—
34C22×Q8—24no—
35C22×Q8—24no—
36C22×Q8—44no—
37C23×C4C2311yescharacteristic
38C23×C4C2311yescharacteristic
39C23×C4C2311yescharacteristic
40C23×C4C2314yes—
41C23×C4C2311yescharacteristic
42C23×C4C2314yes—
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C4⋊Q8—232no—
47C2×C4⋊C4—28no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—28no—
55C2×C4⋊C4—28no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—44no—
58C2×C4⋊C4D418yes—
59C2×C4⋊C4D414yes—
60C2×C42—24no—
61C2×C42D414yes—
62#95—22no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—24no—
67#95—28no—
68#95—24no—
69#95—28no—
70#95—44no—
71#95Q818yescomplemented
72C24C2411yescharacteristic
73C2×Q8—216no—
74C2×Q8—416no—
75C2×Q8—416no—
76C2×Q8—48no—
77C2×Q8—48no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—24no—
86C22×C4—24no—
87C22×C4—21no—
88C22×C4—24no—
89C22×C4—22no—
90C22×C4—24no—
91C22×C4—24no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—24no—
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4C2×D412yes—
113C22×C4C2×D418yes—
114C22×C4C2×D412yes—
115C22×C4C2×D414yes—
116C22×C4C2×D412yes—
117C22×C4C2×D412yes—
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4C2×Q818yes—
121C22×C4C2×Q814yes—
122C22×C4#4112yes—
123C4⋊C4—216no—
124C4⋊C4—216no—
125C4⋊C4—28no—
126C4⋊C4—416no—
127C4⋊C4—44no—
128C42—28no—
129C23C22×D411yescharacteristic
130C23C22×D412yes—
131C23C22×D411yescharacteristic
132C23C22×Q812yes—
133C23C2×#4111yescharacteristic
134C23H(5,𝔽2)12yes—
135C232−1+412yes—
136C232−1+412yes—
137C232−1+412yes—
138Q8—432no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—28no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—28no—
149C2×C4—28no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—28no—
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—28no—
157C2×C4—28no—
158C2×C4—28no—
159C2×C4—41no—
160C2×C4—44no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—42no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4#12014yes—
182C2×C4#12114yes—
183C2×C4C4⋊Q814yes—
184C22C2×#12011yescharacteristic
185C22C2×#12111yescharacteristic
186C22C2×C4⋊Q811yescharacteristic
187C22#53511yescharacteristic
188C22#53611yescharacteristic
189C22#53611yescharacteristic
190C22#53611yescharacteristic
191C22#53614yes—
192C22#53612yes—
193C22#54112yes—
194C22#54414yes—
195C22D4212yes—
196C22#54612yes—
197C22D4×Q812yes—
198C22D4×Q814yes—
199C22#55712yes—
200C22#56314yes—
201C4—24no—
202C4—24no—
203C4—24no—
204C4—42no—
205C4—42no—
206C4—48no—
207C4—48no—
208C4—81no—
209C4—81no—
210C2#240811yescharacteristic
211C2#243411yescharacteristic
212C2#254611yescharacteristic
213C2#261411yescharacteristic
214C2#262111yescharacteristic
215C2#266011yescharacteristic
216C2#267714yes—
217C2#271314yes—
218C2#282111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
71. #95Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#3982
C2×#3952
C2×#3872
#14462
#14452
#14442
#14422
#14342