Finite Group Databasebeta

← #27933

Subgroups — #27933

Order 256.

441 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 143 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×#484C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic
3C2×#467C211yesmaximal, characteristic
4C22×#106C211yesmaximal, characteristic
5#2503C211yesmaximal, characteristic
6#2502C211yesmaximal, characteristic
7#2097C212yesmaximal, complemented
8#2096C212yesmaximal, complemented
9#1971C212yesmaximal, complemented
10#1847C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#484—24no—
19#483—24no—
20#467—24no—
21C2×#107C2212yes—
22C2×#107C2211yescharacteristic
23C2×#106—24no—
24C2×#106C2212yes—
25C2×#106C2212yes—
26C2×#106C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2212yes—
33#400—22no—
34#398—22no—
35#391C2212yes—
36#391C2212yes—
37#385—22no—
38#384C2212yes—
39#382C2212yes—
40#381—22no—
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×Q8C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4C2311yescharacteristic
48C2×Q16—22no—
49C2×Q16—24no—
50C22×C8—24no—
51C22×C8D414yes—
52C22×C8C2311yescharacteristic
53C22×C8C2312yes—
54C22×C8C2311yescharacteristic
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4D412yes—
75C2×C42C2312yes—
76#107—22no—
77#107—24no—
78#106—24no—
79#106—24no—
80#106—24no—
81#103—22no—
82#103—22no—
83#103—24no—
84#103—24no—
85C22⋊C8D414yes—
86#97—24no—
87#95—24no—
88#95—24no—
89#95—24no—
90#95D412yes—
91C24C2×D411yescharacteristic
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—44no—
111C22×C4—82no—
112C22×C4C2×D414yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D412yes—
115C22×C4C2×D412yes—
116C22×C4C2411yescharacteristic
117Q16—44no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—28no—
122C2×C8C2×D414yes—
123C2×C8C2×D412yes—
124C2×C8C2×D412yes—
125C2×C8#4114yes—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—84no—
139C22⋊C4—24no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C42—22no—
143C42—22no—
144C23—22no—
145C23—22no—
146C23#12712yes—
147C23#13714yes—
148C23C22×D411yescharacteristic
149Q8—41no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—42no—
154Q8—42no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—44no—
173C2×C4—42no—
174C2×C4—84no—
175C2×C4—84no—
176C2×C4#12712yes—
177C2×C4C22×D411yescharacteristic
178C2×C4C22×D411yescharacteristic
179C2×C4C2×#4112yes—
180C2×C42−1+412yes—
181C8—22no—
182C8—24no—
183C8#12314yes—
184C22—28no—
185C22—22no—
186C22—22no—
187C22#49714yes—
188C22C2×#12711yescharacteristic
189C22#54712yes—
190C22C2×#13712yes—
191C22#57712yes—
192C4—22no—
193C4—44no—
194C4—82no—
195C4—81no—
196C4—81no—
197C4—82no—
198C4—81no—
199C4—81no—
200C4C2×#12311yescharacteristic
201C4#53611yescharacteristic
202C4#54712yes—
203C2—24no—
204C2#250111yescharacteristic
205C2C2×#49711yescharacteristic
206C2#315411yescharacteristic
207C2#334414yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #2097C2semidirect
8. #2096C2semidirect
9. #1971C2semidirect
10. #1847C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4832
C2×#4672
C22×#1062
#25032
#25022
#20972
#20962
#19712
#18472
#17522