Finite Group Databasebeta

← #24598

Subgroups — #24598

Order 256.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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#2502C211yesmaximal, characteristic
4#2346C211yesmaximal, characteristic
5#2083C212yesmaximal, complemented
6#2068C212yesmaximal, complemented
7#1948C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1894C211yesmaximal, characteristic
10#1851C211yesmaximal, characteristic
11#1823C211yesmaximal, 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#398—21no—
28#398—21no—
29#391C2212yes—
30#389C2212yes—
31#388—22no—
32#385—22no—
33#384C2212yes—
34#382—21no—
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⋊C4—22no—
61C2×C4⋊C4—22no—
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⋊C4—21no—
70C2×C22⋊C4D412yes—
71C2×C42—21no—
72C2×C42—22no—
73C2×C42C2312yes—
74#107—24no—
75#107—22no—
76#106—24no—
77#106—22no—
78C4⋊C8—24no—
79#103—24no—
80#103—22no—
81#103—22no—
82C22⋊C8—22no—
83#95—21no—
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—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—42no—
104C22×C4—81no—
105C22×C4—81no—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4#4112yes—
111C22×C4C2411yescharacteristic
112M16—28no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8#4114yes—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—82no—
131C4⋊C4—82no—
132C4⋊C4—82no—
133C4⋊C4C2×D412yes—
134C4⋊C4C2×D412yes—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—42no—
138C22⋊C4—81no—
139C42—24no—
140C42—22no—
141C42—22no—
142C42—44no—
143C23—22no—
144C23—22no—
145C23—21no—
146C23—21no—
147C23#12112yes—
148C23C22×D411yescharacteristic
149Q8—41no—
150Q8—41no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—84no—
178C2×C4#12112yes—
179C2×C4#13714yes—
180C2×C4C22×D411yescharacteristic
181C2×C4C2×#4112yes—
182C2×C4C2×#4111yescharacteristic
183C2×C42−1+412yes—
184C8—24no—
185C8—41no—
186C8—41no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22C2×#12111yescharacteristic
193C22#54712yes—
194C22C2×#13712yes—
195C22#57611yescharacteristic
196C22#57711yescharacteristic
197C4—24no—
198C4—22no—
199C4—42no—
200C4—42no—
201C4—42no—
202C4—82no—
203C4—81no—
204C4—81no—
205C4—82no—
206C4#52511yescharacteristic
207C4#53611yescharacteristic
208C4#55212yes—
209C2—42no—
210C2#241311yescharacteristic
211C2#303911yescharacteristic
212C2#304511yescharacteristic
213C2#334412yes—
214C2#335712yes—
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. #2083C2semidirect
6. #2068C2semidirect
7. #1948C2semidirect
8. #1927C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4282
#25022
#23462
#20832
#20682
#19482
#19272
#18942
#18512
#18232