Finite Group Databasebeta

← #24595

Subgroups — #24595

Order 256.

382 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 123 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×#485C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic
3#2503C211yesmaximal, characteristic
4#2346C211yesmaximal, characteristic
5#2085C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#1949C212yesmaximal, complemented
8#1928C212yesmaximal, complemented
9#1894C211yesmaximal, characteristic
10#1851C211yesmaximal, characteristic
11#1824C211yesmaximal, characteristic
12C22×M16C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16#485—24no—
17#428—24no—
18C2×#107C2211yescharacteristic
19C2×#107C2212yes—
20C2×#106C2211yescharacteristic
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27#400—21no—
28#400—21no—
29#391C2212yes—
30#389C2212yes—
31#388—22no—
32#382—21no—
33#382C2212yes—
34#381—22no—
35#380C2211yescharacteristic
36C4×C4⋊C4C2212yes—
37C4×C22⋊C4C2211yescharacteristic
38#376—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4—21no—
45C23×C4C2311yescharacteristic
46C2×M16—22no—
47C2×M16—24no—
48C22×C8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C22×C8C2312yes—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53#117—22no—
54#117D414yes—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4D412yes—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42C2312yes—
72#107—24no—
73#107—22no—
74#106—24no—
75#106—22no—
76C4⋊C8—24no—
77#103—24no—
78#103—22no—
79#103—22no—
80C22⋊C8—22no—
81#95—21no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95D411yescharacteristic
89#95D411yescharacteristic
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—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—44no—
104C22×C4—42no—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D412yes—
109C22×C4#4112yes—
110C22×C4C2411yescharacteristic
111M16—28no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8#4114yes—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—82no—
130C4⋊C4C2×D412yes—
131C4⋊C4C2×D412yes—
132C22⋊C4—24no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—81no—
136C42—24no—
137C42—22no—
138C42—22no—
139C42—44no—
140C23—22no—
141C23—22no—
142C23—21no—
143C23—21no—
144C23#12112yes—
145C23C22×D411yescharacteristic
146Q8—41no—
147Q8—41no—
148Q8—42no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
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—41no—
173C2×C4—41no—
174C2×C4—84no—
175C2×C4#12112yes—
176C2×C4C22×D411yescharacteristic
177C2×C4C2×#4112yes—
178C2×C4C2×#4111yescharacteristic
179C2×C42−1+412yes—
180C8—24no—
181C8—41no—
182C8—41no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—42no—
187C22—42no—
188C22C2×#12111yescharacteristic
189C22#54712yes—
190C22#57512yes—
191C22#57711yescharacteristic
192C22#57811yescharacteristic
193C4—22no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—82no—
199C4—81no—
200C4—82no—
201C4—81no—
202C4#52511yescharacteristic
203C4#53611yescharacteristic
204C4#55212yes—
205C2—42no—
206C2#241311yescharacteristic
207C2#304011yescharacteristic
208C2#304611yescharacteristic
209C2#334812yes—
210C2#335212yes—
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. #1928C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4282
#25032
#23462
#20852
#20662
#19492
#19282
#18942
#18512
#18242