Finite Group Databasebeta

← #32966

Subgroups — #32966

Order 256.

707 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 155 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×#518C211yesmaximal, characteristic
2C2×#478C212yesmaximal, complemented
3#2629C211yesmaximal, characteristic, complemented
4#1864C212yesmaximal, complemented
5#1862C212yesmaximal, complemented
6#1854C212yesmaximal, complemented
7#1802C212yesmaximal, complemented
8#1798C211yesmaximal, characteristic, complemented
9#1786C212yesmaximal
10C23×D4C2211yescharacteristic
11C2×#125C2211yescharacteristic, complemented
12#518—28no—
13C2×C4×D4—22no—
14C2×C4×D4C2212yes—
15C2×C4×D4C2212yes—
16C2×#117C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C22×C22⋊C4—22no—
19#478—28no—
20C2×#107C2212yescomplemented
21C2×#106C2212yescomplemented
22C2×C4⋊C8C2212yescomplemented
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2212yes—
28#398—22no—
29#396—22no—
30#389C2212yescomplemented
31#384C2211yescharacteristic, complemented
32#380C2212yes—
33#340C2212yescomplemented
34#340C2212yescomplemented
35#336C2211yescharacteristic, complemented
36#336C2211yescharacteristic, complemented
37C25—21no—
38C22×D4—24no—
39C22×D4—24no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4Q814yes—
43C22×D4C2312yes—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49#125—24no—
50C4×D4—216no—
51C4×D4—24no—
52C4×D4—24no—
53C4×D4—24no—
54C4×D4D418yes—
55#117—24no—
56#117—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D412yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—216no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4D412yes—
70C2×C42—21no—
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—24no—
74#106—24no—
75C4⋊C8—24no—
76#102—24no—
77#102—24no—
78#102—24no—
79#102—24no—
80C22⋊C8—24no—
81#95—22no—
82#95—22no—
83C24—24no—
84C24—24no—
85C24—22no—
86C24—24no—
87C24—82no—
88C24—82no—
89C24C2×D411yescharacteristic
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—216no—
93C2×D4—216no—
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4C2×D414yes—
99C2×D4C2×Q814yes—
100C2×D4#4114yes—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—28no—
105C22×C4—28no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—82no—
109C22×C4—82no—
110C22×C4—82no—
111C22×C4—82no—
112C22×C4C2×D412yes—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×Q812yes—
116C22×C4#4112yes—
117C22×C4C2411yescharacteristic
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—44no—
131C4⋊C4—82no—
132C4⋊C4—82no—
133C4⋊C4C2×D414yes—
134C22⋊C4—24no—
135C22⋊C4—28no—
136C22⋊C4—24no—
137C22⋊C4—28no—
138C22⋊C4—82no—
139C42—24no—
140C42—22no—
141C42C2×D414yes—
142C23—216no—
143C23—21no—
144C23—24no—
145C23—24no—
146C23—21no—
147C23—24no—
148C23—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—88no—
155C23—84no—
156C23—84no—
157C23—88no—
158C23C22⋊Q812yes—
159C23C22×D411yescharacteristic
160D4—216no—
161D4—24no—
162D4—24no—
163D4—24no—
164D4C22⋊Q818yes—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—88no—
184C2×C4—88no—
185C2×C4#12014yes—
186C2×C4C22⋊Q812yes—
187C2×C4C22×D412yes—
188C2×C4C22×Q811yescharacteristic
189C2×C4C2×#4112yes—
190C2×C4C2×#4111yescharacteristic
191C8—42no—
192C8—42no—
193C22—24no—
194C22—22no—
195C22—28no—
196C22—44no—
197C22—44no—
198C22—41no—
199C22—44no—
200C22—41no—
201C22—44no—
202C22—44no—
203C22—44no—
204C22—42no—
205C22—816no—
206C22C2×#12011yescharacteristic
207C22C2×C22⋊Q811yescharacteristic
208C22#52511yescharacteristic
209C22#57612yes—
210C22#57712yes—
211C4—22no—
212C4—24no—
213C4—42no—
214C4—44no—
215C4—82no—
216C4—82no—
217C4C2×C22⋊Q812yes—
218C4#52512yes—
219C2—42no—
220C2—48no—
221C2#284811yescharacteristic
222C2#301212yes—
223C2#308112yes—
224C2#312211yescharacteristic
225C2#312311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#478C2semidirect
3. #2629C2semidirect
4. #1864C2semidirect
5. #1862C2semidirect
6. #1854C2semidirect
7. #1802C2semidirect
8. #1798C2semidirect
11. C2×#125C22semidirect
20. C2×#107C22semidirect
21. C2×#106C22semidirect
22. C2×C4⋊C8C22semidirect
30. #389C22semidirect
31. #384C22semidirect
33. #340C22semidirect
34. #340C22semidirect
35. #336C22semidirect
36. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5182
C2×#4782
#26292
#18642
#18622
#18542
#18022
#17982
#17862