Finite Group Databasebeta

← #138456

Subgroups — #138456

Order 512.

414 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 86 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
1#22703C211yesmaximal, characteristic
2#19259C211yesmaximal, characteristic
3#11857C211yesmaximal, characteristic, complemented
4#11745C211yesmaximal, characteristic, complemented
5#11631C211yesmaximal, characteristic
6#7299C211yesmaximal, characteristic, complemented
7#7222C211yesmaximal, characteristic, complemented
8#3170C2211yescharacteristic
9#3110—21no—
10C2×C8×D4C2211yescharacteristic
11D4×C4⋊C4—21no—
12#2163C2211yescharacteristic
13#2162C2211yescharacteristic
14#1948—21no—
15#1853—22no—
16#1834C2211yescharacteristic
17#1807—21no—
18C4×#106C2211yescharacteristic
19C2×#340—21no—
20#1679—22no—
21#1673C412yescomplemented
22#1669C412yescomplemented
23#1568C2211yescharacteristic
24#1552—21no—
25#1348—22no—
26#1348—21no—
27#1348—21no—
28D4⋊C16—22no—
29C23×C8—21no—
30#533—21no—
31C2×C4×D4—41no—
32C2×C4×D4D411yescharacteristic
33C22×C4⋊C4—22no—
34#500C2×C411yescharacteristic, complemented
35#494C2×C411yescharacteristic, complemented
36#487—21no—
37#484—41no—
38#474—22no—
39#467—21no—
40#467—21no—
41#461—22no—
42#450—41no—
43#440—42no—
44C4×Q16—21no—
45C4×Q16—21no—
46C8×D4—21no—
47C8×D4—21no—
48C8×D4C2×C412yes—
49C8×D4D412yes—
50#432—21no—
51C2×#106—21no—
52C2×#106—22no—
53C2×C4⋊C8D411yescharacteristic
54C2×C22⋊C8—21no—
55C2×C4×C8C2311yescharacteristic
56#389—41no—
57#385—42no—
58#384—21no—
59#380—22no—
60C4×C4⋊C4—21no—
61C4⋊C16—21no—
62C4⋊C16—21no—
63C22⋊C16—21no—
64C22⋊C16—21no—
65#346—21no—
66C4×C16—21no—
67#340—22no—
68#340—21no—
69#340D411yescharacteristic
70#340D412yes—
71#340D411yescharacteristic
72#335C2×C412yescomplemented
73#334—21no—
74#332—22no—
75#332—22no—
76C22×D4D811yescharacteristic
77C23×C4—21no—
78C23×C4—42no—
79C2×Q16—21no—
80C2×Q16—21no—
81C2×SD16—42no—
82C22×C8—21no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—82no—
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90#125—41no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—84no—
98C4×D4C22⋊C412yes—
99C4×D4SD1612yes—
100#117—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—82no—
109C2×C4⋊C4D811yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—42no—
112C2×C42—41no—
113C2×C42C2×D411yescharacteristic
114C2×C16—41no—
115C2×C16—41no—
116#107—21no—
117#107—21no—
118#107—42no—
119#106—42no—
120#106—82no—
121C4⋊C8—22no—
122C4⋊C8—21no—
123C4⋊C8C22⋊C412yes—
124#103—41no—
125#103—41no—
126#103—42no—
127#102—42no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131#97—41no—
132C4×C8C22×C411yescharacteristic
133C4×C8C2×D411yescharacteristic
134C4×C8#4111yescharacteristic
135C4×C8#4111yescharacteristic
136#95—42no—
137#95—41no—
138C24—21no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×D4—24no—
142C2×D4—21no—
143C2×D4#10212yes—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—84no—
153C22×C4—84no—
154C22×C4—162no—
155C22×C4#12011yescharacteristic
156C22×C4C2×D811yescharacteristic
157C22×C4AGL(1,ℤ/8)11yescharacteristic
158Q16—81no—
159Q16—81no—
160SD16—84no—
161C2×C8—22no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—82no—
167C2×C8—82no—
168C2×C8C4×D411yescharacteristic
169C2×C8C4×D411yescharacteristic
170C2×C8#12111yescharacteristic
171C2×C8#12111yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—82no—
184C4⋊C4—84no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—161no—
188C4⋊C4#10212yes—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—84no—
192C22⋊C4—81no—
193C42—82no—
194C42—81no—
195C42C2×C22⋊C411yescharacteristic
196C42#12311yescharacteristic
197C42C2×SD1611yescharacteristic
198C42#13711yescharacteristic
199C16—81no—
200C16—81no—
201C23—24no—
202C23—22no—
203C23—42no—
204C23—41no—
205C23—41no—
206C23#44711yescharacteristic
207Q8—82no—
208Q8—81no—
209Q8—81no—
210D4—24no—
211D4—22no—
212D4#36014yes—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—81no—
234C2×C4—168no—
235C2×C4#38611yescharacteristic
236C2×C4C2×#10211yescharacteristic
237C2×C4#41711yescharacteristic
238C2×C4#44811yescharacteristic
239C2×C4#48011yescharacteristic
240C2×C4#48111yescharacteristic
241C8—21no—
242C8—21no—
243C8—21no—
244C8—21no—
245C8—42no—
246C8—82no—
247C22—21no—
248C22—21no—
249C22—24no—
250C22—44no—
251C22—44no—
252C22—42no—
253C22—42no—
254C22#187411yescharacteristic
255C22#219411yescharacteristic
256C22#219511yescharacteristic
257C4—21no—
258C4—21no—
259C4—21no—
260C4—21no—
261C4—42no—
262C4—82no—
263C4—161no—
264C4—162no—
265C4—161no—
266C4—161no—
267C4#190411yescharacteristic
268C4#190611yescharacteristic
269C4C2×#36012yes—
270C2—21no—
271C2—21no—
272C2—44no—
273C2#917911yescharacteristic
274C2#1299711yescharacteristic
275C2#1300111yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11857C2semidirect
4. #11745C2semidirect
6. #7299C2semidirect
7. #7222C2semidirect
21. #1673C4semidirect
22. #1669C4semidirect
34. #500C2×C4semidirect
35. #494C2×C4semidirect
72. #335C2×C4semidirect

Maximal subgroups

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

subgroupindex
#227032
#192592
#118572
#117452
#116312
#72992
#72222