Finite Group Databasebeta

← #32953

Subgroups — #32953

Order 256.

882 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C22×C4×D4C211yesmaximal, characteristic
2C2×#474C212yesmaximal, complemented
3#2857C211yesmaximal, characteristic, complemented
4#1862C214yesmaximal, complemented
5#1852C212yesmaximal, complemented
6#1802C212yesmaximal, complemented
7#1797C211yesmaximal, characteristic, complemented
8#1786C212yesmaximal
9C23×D4C2211yescharacteristic
10C24×C4—21no—
11C2×C4⋊Q8C2211yescharacteristic, complemented
12C2×C22⋊Q8—21no—
13C2×C4×D4—28no—
14C2×C4×D4—21no—
15C2×C4×D4—21no—
16C2×C4×D4C2212yes—
17C2×C4×D4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4—21no—
20C22×C42C2211yescharacteristic
21#474—28no—
22C2×#107C2214yescomplemented
23C2×C4⋊C8C2212yescomplemented
24C2×#102C2214yes—
25C2×#102C2214yes—
26C2×C22⋊C8C2212yes—
27#391—21no—
28#389—21no—
29#389C2212yescomplemented
30#384C2211yescharacteristic, complemented
31#380—21no—
32#380C2212yes—
33#340C2214yescomplemented
34#336C2212yescomplemented
35C25—21no—
36C22×Q8—21no—
37C22×D4—24no—
38C22×D4—24no—
39C22×D4—21no—
40C22×D4—22no—
41C22×D4Q814yescomplemented
42C22×D4C2312yes—
43C23×C4—28no—
44C23×C4—21no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—44no—
48C23×C4—44no—
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C22×C8C2314yes—
53C4⋊Q8—24no—
54C22⋊Q8—44no—
55C4×D4—28no—
56C4×D4—28no—
57C4×D4—24no—
58C4×D4—24no—
59C4×D4—24no—
60C4×D4D418yes—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—28no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—24no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—28no—
80C4⋊C8—24no—
81#102—28no—
82#102—28no—
83C22⋊C8—24no—
84#95—22no—
85#95—22no—
86C24—24no—
87C24—24no—
88C24—24no—
89C24—22no—
90C24—42no—
91C24—42no—
92C24—42no—
93C24—42no—
94C24C2×D411yescharacteristic
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×D4—24no—
99C2×D4—24no—
100C2×D4—216no—
101C2×D4—216no—
102C2×D4—24no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4C2×D414yes—
107C2×D4C2×Q814yes—
108C2×D4#4114yes—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—28no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—41no—
122C22×C4—44no—
123C22×C4—432no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4D814yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×Q812yes—
136C22×C4#4112yes—
137C22×C4C2411yescharacteristic
138C2×C8—24no—
139C2×C8—24no—
140C2×C8—24no—
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4C2×D414yes—
150C22⋊C4—28no—
151C22⋊C4—28no—
152C22⋊C4—44no—
153C42—22no—
154C42—22no—
155C42—22no—
156C42C2×D414yes—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—216no—
162C23—24no—
163C23—24no—
164C23—24no—
165C23—48no—
166C23—48no—
167C23—48no—
168C23—48no—
169C23—48no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—48no—
175C23C22⋊Q812yes—
176C23C2×D812yes—
177C23#13712yes—
178C23C22×D411yescharacteristic
179Q8—84no—
180D4—216no—
181D4—24no—
182D4—24no—
183D4—24no—
184D4C22⋊Q818yescomplemented
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—28no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—44no—
199C2×C4—416no—
200C2×C4—48no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—416no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4#12012yes—
211C2×C4#12012yes—
212C2×C4C22⋊Q812yes—
213C2×C4C2×D812yes—
214C2×C4C2×D812yes—
215C2×C4AGL(1,ℤ/8)12yes—
216C2×C4C22×D412yes—
217C2×C4C22×Q811yescharacteristic
218C2×C4C2×#4112yes—
219C2×C4C2×#4111yescharacteristic
220C8—44no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—28no—
228C22—432no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—44no—
234C22—44no—
235C22#47414yescomplemented
236C22C2×#12011yescharacteristic
237C22C2×C22⋊Q811yescharacteristic
238C22#52511yescharacteristic
239C22C22×D811yescharacteristic
240C22C2×AGL(1,ℤ/8)11yescharacteristic
241C22C2×#13711yescharacteristic
242C22#57511yescharacteristic
243C4—22no—
244C4—22no—
245C4—22no—
246C4—44no—
247C4—41no—
248C4—82no—
249C4—82no—
250C4#44714yes—
251C4C2×C22⋊Q812yes—
252C4#52512yes—
253C2—24no—
254C2—48no—
255C2#284811yescharacteristic
256C2C2×#44711yescharacteristic
257C2#300711yescharacteristic
258C2C2×#47411yescharacteristic
259C2#307711yescharacteristic
260C2#310011yescharacteristic
261C2#311011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#474C2semidirect
3. #2857C2semidirect
4. #1862C2semidirect
5. #1852C2semidirect
6. #1802C2semidirect
7. #1797C2semidirect
11. C2×C4⋊Q8C22semidirect
22. C2×#107C22semidirect
23. C2×C4⋊C8C22semidirect
29. #389C22semidirect
30. #384C22semidirect
33. #340C22semidirect
34. #336C22semidirect
41. C22×D4Q8semidirect
184. D4C22⋊Q8semidirect
235. C22#474semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#4742
#28572
#18622
#18522
#18022
#17972
#17862