Finite Group Databasebeta

← #141168

Subgroups — #141168

Order 512.

447 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 90 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#33172C211yesmaximal, characteristic
2#12788C211yesmaximal, characteristic
3#11712C212yesmaximal
4#11706C212yesmaximal
5#10818C211yesmaximal, characteristic
6C2×#494C411yescharacteristic, complemented
7C2×#467—21no—
8C2×#467C2212yes—
9C22×C4×C8C2211yescharacteristic
10#2857—21no—
11C2×#364—21no—
12#2112C2212yes—
13#1969—21no—
14#1969C412yescomplemented
15#1956—21no—
16#1845C411yescharacteristic, complemented
17C2×#341C2211yescharacteristic
18#1735C2211yescharacteristic
19C2×#339—21no—
20#1729—21no—
21#1376—21no—
22#1350—24no—
23#1348—24no—
24C22×Q16—21no—
25C22×Q16C2×C412yes—
26C22×M16—21no—
27C23×C8D411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×C4⋊Q8—21no—
30C2×C22⋊Q8—22no—
31C2×#117—21no—
32C22×C42D411yescharacteristic
33C2×M32—22no—
34#494—42no—
35#467—22no—
36#467—22no—
37#467—22no—
38#467—22no—
39#467—42no—
40#467—42no—
41C2×#107—21no—
42C2×#107C2×C412yes—
43C2×#103—22no—
44C2×#103—22no—
45C2×C4×C8—42no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#391—22no—
51#389—42no—
52#384—21no—
53#380—42no—
54#364—42no—
55C22⋊C16—22no—
56C22⋊C16—22no—
57#341—21no—
58#341D412yes—
59#340—21no—
60#340D412yes—
61#339—42no—
62#336—42no—
63C22×Q8—22no—
64C23×C4—21no—
65C23×C4C2×D411yescharacteristic
66C2×Q16—22no—
67C2×Q16—22no—
68C2×Q16—42no—
69C2×Q16—42no—
70C2×Q16—42no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×M16—22no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8D812yes—
87C22×C8SD1612yes—
88C22×C8C22×C411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92C22×C8#4111yescharacteristic
93C22×C8#4111yescharacteristic
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C22⋊Q8—42no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C22⋊Q8—42no—
100#117—41no—
101#117—41no—
102#117—41no—
103#117—41no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—42no—
111C2×C42—22no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C22⋊C411yescharacteristic
116C2×C42C22⋊C411yescharacteristic
117M32—42no—
118M32—42no—
119C2×C16—42no—
120#107—22no—
121#107—22no—
122#107—42no—
123#103—44no—
124#103—42no—
125#103—42no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—41no—
129C4×C8—41no—
130#95—44no—
131C24AGL(1,ℤ/8)11yescharacteristic
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4#12111yescharacteristic
152C22×C4#12111yescharacteristic
153C22×C4C2×D811yescharacteristic
154C22×C4C2×SD1611yescharacteristic
155C22×C4AGL(1,ℤ/8)11yescharacteristic
156Q16—82no—
157Q16—82no—
158Q16—84no—
159M16—42no—
160M16—42no—
161M16—81no—
162M16—81no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—81no—
179C2×C8—81no—
180C2×C8#10212yes—
181C2×C8#10212yes—
182C2×C8C4×D411yescharacteristic
183C2×C8C4×D411yescharacteristic
184C2×C8#12411yescharacteristic
185C2×C8#12711yescharacteristic
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C22⋊C4—82no—
197C22⋊C4—82no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—42no—
202C42—81no—
203C42—81no—
204C42#9912yes—
205C16—82no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23#41811yescharacteristic
211C23#45911yescharacteristic
212C23#46011yescharacteristic
213Q8—82no—
214Q8—82no—
215Q8—82no—
216Q8—82no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—48no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—82no—
235C2×C4—84no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4#39011yescharacteristic
243C2×C4C2×#9911yescharacteristic
244C2×C4#41011yescharacteristic
245C2×C4C2×#10211yescharacteristic
246C2×C4#42111yescharacteristic
247C2×C4#42111yescharacteristic
248C2×C4#45911yescharacteristic
249C2×C4#46011yescharacteristic
250C8—21no—
251C8—21no—
252C8—21no—
253C8—21no—
254C8—22no—
255C8—22no—
256C8—82no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22#151011yescharacteristic
267C22#151411yescharacteristic
268C22#198211yescharacteristic
269C22#221412yes—
270C4—24no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—162no—
275C4—162no—
276C4—162no—
277C4#151711yescharacteristic
278C4#151911yescharacteristic
279C4#196311yescharacteristic
280C4#196411yescharacteristic
281C2—21no—
282C2—22no—
283C2—21no—
284C2—41no—
285C2#971511yescharacteristic
286C2#1325911yescharacteristic
287C2#1326211yescharacteristic
subgroupG/H
center, socleC22#1982
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ 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
6. C2×#494C4semidirect
14. #1969C4semidirect
16. #1845C4semidirect

Maximal subgroups

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

subgroupindex
#331722
#127882
#117122
#117062
#108182