Finite Group Databasebeta

← #129927

Subgroups — #129927

Order 512.

704 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 126 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×#3170C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3C2×#1348C214yesmaximal
4#10820C211yesmaximal, characteristic
5C2×#533—21no—
6#3170—28no—
7C2×#500C411yescharacteristic, complemented
8C2×#494C411yescharacteristic, complemented
9C2×#487—21no—
10C2×#467—21no—
11C2×#467C2212yes—
12C2×C4×Q16—21no—
13C2×C4×Q16C412yescomplemented
14C22×C4×C8C2211yescharacteristic
15C2×C22⋊C16C2212yes—
16C2×#340—21no—
17C2×#340C2212yes—
18C4⋊#95—21no—
19#1374—21no—
20#1374—21no—
21#1348—28no—
22C22×Q16—21no—
23C22×Q16C2×C412yes—
24C23×C8D411yescharacteristic
25C23×C8C2311yescharacteristic
26#533—28no—
27#533—44no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×#125—41no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—41no—
33C2×C4×Q8—22no—
34C2×C4×Q8—41no—
35C2×#117—41no—
36C22×C4⋊C4—22no—
37C22×C42D411yescharacteristic
38C22×C16—22no—
39#500—24no—
40#494—24no—
41#487—28no—
42#467—28no—
43#467—24no—
44#467D418yes—
45C4×Q16—28no—
46C4×Q16—24no—
47C4×Q16—24no—
48C2×#107—21no—
49C2×#107C2×C412yes—
50C2×#103—22no—
51C2×#103—22no—
52C2×C4×C8—24no—
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57C2×#95—41no—
58C2×#95—41no—
59C22⋊C16—24no—
60#340—24no—
61#340—24no—
62C22×Q8—22no—
63C22×Q8—41no—
64C23×C4—42no—
65C23×C4D811yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4C2×D411yescharacteristic
68C2×Q16—24no—
69C2×Q16—22no—
70C2×Q16—42no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×Q16—42no—
74C2×Q16C22⋊C414yes—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22×C411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D414yes—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89#125—48no—
90C22⋊Q8—28no—
91C22⋊Q8—48no—
92C22⋊Q8—44no—
93C4×Q8—28no—
94C4×Q8—44no—
95C4×Q8—48no—
96#117—48no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—42no—
113C2×C42C22⋊C411yescharacteristic
114C2×C42C22⋊C411yescharacteristic
115C2×C42SD1611yescharacteristic
116C2×C42SD1611yescharacteristic
117C2×C16—42no—
118C2×C16—42no—
119C2×C16—42no—
120#107—24no—
121#107—22no—
122#107C22⋊C414yes—
123#103—28no—
124#103—28no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129#95—82no—
130#95—82no—
131C24C2×D811yescharacteristic
132C2×Q8—24no—
133C2×Q8—44no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—82no—
140C2×Q8—82no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—82no—
154C22×C4—82no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4#10211yescharacteristic
160C22×C4#10211yescharacteristic
161C22×C4#10211yescharacteristic
162C22×C4#10211yescharacteristic
163C22×C4SD3212yes—
164C22×C4Q3212yes—
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4#12012yes—
167C22×C4C2×SD1611yescharacteristic
168C22×C4AGL(1,ℤ/8)12yes—
169Q16—44no—
170Q16—44no—
171Q16—48no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—44no—
182C2×C8—44no—
183C2×C8#10412yes—
184C2×C8C2×C22⋊C412yes—
185C2×C8#12012yes—
186C4⋊C4—24no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—48no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—42no—
194C4⋊C4—48no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C22⋊C4—48no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—22no—
203C42—22no—
204C42—48no—
205C16—82no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23C2×#10211yescharacteristic
211C23#44712yes—
212C23C2×SD3211yescharacteristic
213C23C2×Q3211yescharacteristic
214C23#51012yes—
215Q8—44no—
216Q8—44no—
217Q8—88no—
218Q8—82no—
219Q8—82no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—84no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4—162no—
248C2×C4—162no—
249C2×C4—162no—
250C2×C4—162no—
251C2×C4#35812yes—
252C2×C4#35812yes—
253C2×C4#36012yes—
254C2×C4#37911yescharacteristic
255C2×C4#41011yescharacteristic
256C2×C4#41011yescharacteristic
257C2×C4#41811yescharacteristic
258C2×C4C2×#10411yescharacteristic
259C2×C4#42111yescharacteristic
260C2×C4#45012yes—
261C8—22no—
262C8—22no—
263C8—22no—
264C8—41no—
265C22—21no—
266C22—21no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22#150211yescharacteristic
274C22#151711yescharacteristic
275C22#178611yescharacteristic
276C22C2×#35811yescharacteristic
277C22#214111yescharacteristic
278C22C2×#36011yescharacteristic
279C22#214511yescharacteristic
280C22#218512yes—
281C22#218812yes—
282C4—22no—
283C4—22no—
284C4—22no—
285C4—22no—
286C4—41no—
287C4—84no—
288C4—84no—
289C4—164no—
290C4#150511yescharacteristic
291C4#151411yescharacteristic
292C4#178211yescharacteristic
293C4#178911yescharacteristic
294C2—22no—
295C2—22no—
296C2#833211yescharacteristic
297C2#1174311yescharacteristic
298C2#1174511yescharacteristic
299C2#1185711yescharacteristic
300C2#1185911yescharacteristic
301C2#1273211yescharacteristic
302C2#1273511yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#500C4semidirect
8. C2×#494C4semidirect
13. C2×C4×Q16C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
#128442
C2×#13482
#108202