Finite Group Databasebeta

← #25726

Subgroups — #25726

Order 256.

780 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 163 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, complemented
2C2×#497C212yesmaximal, complemented
3C2×#451C212yesmaximal, complemented
4C2×#448C212yesmaximal
5#2850C211yesmaximal, characteristic
6#2016C212yesmaximal, complemented
7#2013C212yesmaximal, complemented
8#1960C212yesmaximal, complemented
9#1749C211yesmaximal, characteristic, complemented
10C23×Q8C2212yes—
11C22×Q16C2214yes—
12C22×SD16C2214yes—
13C2×C4⋊Q8—28no—
14C2×C4⋊Q8—22no—
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2212yes—
17C2×#124C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C22×C4⋊C4—22no—
21C22×C42C2211yescharacteristic
22#497—28no—
23#451—28no—
24#448—28no—
25C2×#103C2214yes—
26C2×#103C2214yes—
27C2×C22⋊C8C2212yes—
28C2×#97C2212yes—
29#391C2211yescharacteristic
30#390C2211yescharacteristic
31#388—22no—
32#386—22no—
33#382C2211yescharacteristic
34#336C2212yes—
35C22×Q8—28no—
36C22×Q8—22no—
37C22×Q8—24no—
38C22×Q8—24no—
39C22×Q8—44no—
40C22×Q8C2311yescharacteristic
41C22×Q8C2314yes—
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4—22no—
45C23×C4D412yes—
46C23×C4C2311yescharacteristic
47C2×Q16—28no—
48C2×Q16—28no—
49C2×Q16—44no—
50C2×SD16—28no—
51C2×SD16—28no—
52C2×SD16—44no—
53C22×C8C2314yes—
54C4⋊Q8—216no—
55C4⋊Q8—28no—
56C4⋊Q8—24no—
57C4⋊Q8D418yescomplemented
58#124—24no—
59C22⋊Q8—24no—
60#121—24no—
61C2×C4⋊C4—216no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—24no—
72C2×C42D412yes—
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#103—28no—
76#103—28no—
77C22⋊C8D418yes—
78#97—24no—
79#95—22no—
80#95—22no—
81C24—21no—
82C24C2×D411yescharacteristic
83C2×Q8—28no—
84C2×Q8—24no—
85C2×Q8—24no—
86C2×Q8—24no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—24no—
90C2×Q8—24no—
91C2×Q8—22no—
92C2×Q8—416no—
93C2×Q8—416no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8C2×D418yes—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—42no—
101C2×D4—44no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—28no—
108C22×C4—24no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D414yes—
117C22×C4C2×D412yes—
118C22×C4C2411yescharacteristic
119Q16—416no—
120SD16—416no—
121C2×C8—24no—
122C2×C8—24no—
123C2×C8C2×D418yes—
124C4⋊C4—216no—
125C4⋊C4—216no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—44no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C42—24no—
134C42—22no—
135C42C2×D414yes—
136C23—21no—
137C23—21no—
138C23—22no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—42no—
144C23#12712yes—
145C23#13714yes—
146C23C22×D411yescharacteristic
147Q8—28no—
148Q8—28no—
149Q8—416no—
150Q8—44no—
151Q8—44no—
152Q8—44no—
153D4—44no—
154D4—84no—
155C2×C4—216no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—28no—
163C2×C4—28no—
164C2×C4—416no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4#12014yes—
175C2×C4#12712yes—
176C2×C4#13714yes—
177C2×C4C22×D412yes—
178C2×C4C22×D412yes—
179C2×C4H(5,𝔽2)12yes—
180C8—28no—
181C22—22no—
182C22—22no—
183C22—24no—
184C22—24no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—82no—
189C22—82no—
190C22#49714yescomplemented
191C22C2×#12011yescharacteristic
192C22C2×#12711yescharacteristic
193C22#53511yescharacteristic
194C22C2×#13712yes—
195C22C2×#13712yes—
196C4—22no—
197C4—24no—
198C4—24no—
199C4—48no—
200C4—81no—
201C4—81no—
202C4—81no—
203C4D4214yes—
204C2—24no—
205C2—161no—
206C2#243311yescharacteristic
207C2#300412yes—
208C2C2×#49711yescharacteristic
209C2#315211yescharacteristic
210C2#321812yes—
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. C22×C4⋊Q8C2semidirect
2. C2×#497C2semidirect
3. C2×#451C2semidirect
6. #2016C2semidirect
7. #2013C2semidirect
8. #1960C2semidirect
9. #1749C2semidirect
57. C4⋊Q8D4semidirect
190. C22#497semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#4972
C2×#4512
C2×#4482
#28502
#20162
#20132
#19602
#17492