Finite Group Databasebeta

← #24579

Subgroups — #24579

Order 256.

399 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 131 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
2C22×#106C211yesmaximal, characteristic
3#2503C211yesmaximal, characteristic
4#2345C211yesmaximal, characteristic
5#2085C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#1949C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1892C211yesmaximal, characteristic
10#1850C211yesmaximal, characteristic
11#1823C211yesmaximal, characteristic
12C23×C8C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16#484—24no—
17C2×#107C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#106—24no—
20C2×#106C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27#400—22no—
28#391C2212yes—
29#389C2212yes—
30#387—22no—
31#382C2212yes—
32#381—21no—
33#381—21no—
34#380—21no—
35#380C2211yescharacteristic
36C4×C4⋊C4C2212yes—
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4C2311yescharacteristic
47C22×C8—22no—
48C22×C8—24no—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4D414yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4D411yescharacteristic
70C2×C22⋊C4D411yescharacteristic
71C2×C42—22no—
72C2×C42C2312yes—
73#107—22no—
74#107—22no—
75#106—24no—
76#106—24no—
77C4⋊C8—24no—
78#103—24no—
79#103—22no—
80#103—22no—
81C22⋊C8—22no—
82#95—22no—
83#95—22no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95D412yes—
90C24C2×D411yescharacteristic
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—21no—
94C2×Q8—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—24no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—82no—
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D412yes—
112C22×C4#4112yes—
113C22×C4C2411yescharacteristic
114C2×C8—28no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8#4114yes—
122C4⋊C4—24no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—82no—
134C4⋊C4C2×D412yes—
135C4⋊C4C2×D412yes—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—22no—
141C42—22no—
142C42—82no—
143C23—22no—
144C23—22no—
145C23#12112yes—
146C23#13512yes—
147C23#13712yes—
148C23C22×D411yescharacteristic
149Q8—41no—
150Q8—41no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—84no—
177C2×C4—84no—
178C2×C4#12112yes—
179C2×C4C22×D411yescharacteristic
180C2×C4C2×#4112yes—
181C2×C4C2×#4111yescharacteristic
182C2×C42−1+412yes—
183C8—24no—
184C8—41no—
185C8—41no—
186C22—24no—
187C22—24no—
188C22—22no—
189C22—22no—
190C22#46414yes—
191C22C2×#12111yescharacteristic
192C22#54712yes—
193C22C2×#13511yescharacteristic
194C22C2×#13711yescharacteristic
195C22#57712yes—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4—81no—
203C4—82no—
204C4—81no—
205C4#52511yescharacteristic
206C4#53611yescharacteristic
207C4#55212yes—
208C2—24no—
209C2#241311yescharacteristic
210C2C2×#46411yescharacteristic
211C2#304211yescharacteristic
212C2#333712yes—
213C2#334412yes—
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
5. #2085C2semidirect
6. #2066C2semidirect
7. #1949C2semidirect
8. #1927C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C22×#1062
#25032
#23452
#20852
#20662
#19492
#19272
#18922
#18502
#18232