Finite Group Databasebeta

← #491345

Subgroups — #491345

Order 512.

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

normal subgroups only

This group has 377 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
1#33796C211yesmaximal, characteristic
2#33499C211yesmaximal, characteristic
3#19342C211yesmaximal, characteristic
4#19209C211yesmaximal, characteristic
5#12481C212yesmaximal, complemented
6#12477C212yesmaximal, complemented
7#12461C212yesmaximal, complemented
8#12458C212yesmaximal, complemented
9#12034C211yesmaximal, characteristic
10#11975C211yesmaximal, characteristic
11#11953C211yesmaximal, characteristic
12#3237—21no—
13#3230—21no—
14#3170C2211yescharacteristic
15C2×#500C2211yescharacteristic
16#2960C2211yescharacteristic
17#2252C2211yescharacteristic
18#2252C2211yescharacteristic
19#2241C2211yescharacteristic
20#2241C2211yescharacteristic
21#2236—22no—
22#2227—22no—
23#2183—22no—
24#2183C2212yes—
25Q8×C16C2212yes—
26#2164—22no—
27#2163C2211yescharacteristic
28#2162C2211yescharacteristic
29#1704—22no—
30#1696—22no—
31#1676C2212yes—
32#1676C2212yes—
33#1676C2212yes—
34#1670C2212yes—
35#1568C2211yescharacteristic
36#1564C2211yescharacteristic
37#1559—21no—
38#1548—21no—
39#1372C2212yes—
40#1371C2212yes—
41#1365C2212yes—
42#1365C2212yes—
43#1365C2212yes—
44#1362C2212yes—
45#533—21no—
46#533D411yescharacteristic
47C2×C4⋊Q8—21no—
48#501—41no—
49#500—21no—
50#500—21no—
51#500—21no—
52#500—41no—
53#500C2311yescharacteristic
54#500C2311yescharacteristic
55#500C2311yescharacteristic
56#494C2311yescharacteristic
57#491—41no—
58#487—21no—
59#487—41no—
60#485—42no—
61#484—42no—
62#477—22no—
63#477—22no—
64#477—22no—
65#475—22no—
66#467—21no—
67#467—21no—
68#446—21no—
69#446D412yes—
70C8×Q8—21no—
71C8×Q8C2312yes—
72C4×Q16—21no—
73C4×Q16—21no—
74#432—21no—
75#431—21no—
76C2×#107—21no—
77C2×#107—21no—
78#424—21no—
79#423D411yescharacteristic
80#405—21no—
81C2×C4×C8C2311yescharacteristic
82#366—22no—
83#366—22no—
84C4⋊C16—22no—
85C4⋊C16—22no—
86C4⋊C16D412yes—
87C4⋊C16C2312yes—
88#358—22no—
89#358—22no—
90C22⋊C16—21no—
91C22⋊C16D412yes—
92#346—22no—
93#346—21no—
94C4×C16—22no—
95C4×C16C2311yescharacteristic
96C4×C16C2311yescharacteristic
97#335C2312yes—
98#335C2312yes—
99#334—21no—
100#334—21no—
101#333—22no—
102#333—22no—
103#333—22no—
104#332—22no—
105C22×Q8—41no—
106C2×Q16—21no—
107C2×Q16—21no—
108C2×M16—21no—
109C22×C8C2×D411yescharacteristic
110C22×C8#4111yescharacteristic
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8D811yescharacteristic
119C4⋊Q8D811yescharacteristic
120#125—21no—
121#125—41no—
122C22⋊Q8—21no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8D812yes—
126C4×Q8—21no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8C2×D412yes—
130#117—21no—
131#117—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C42C2×D411yescharacteristic
135C2×C16—22no—
136C2×C16C2×D412yes—
137C2×C16C2×D412yes—
138#107—21no—
139#107—21no—
140#107—21no—
141#107—21no—
142#107—21no—
143#107—21no—
144#107—41no—
145#107—41no—
146#107—42no—
147#107—42no—
148#107—42no—
149#106—42no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8C2×D412yes—
156#103—42no—
157#103—42no—
158#103—42no—
159#103—42no—
160#103—41no—
161#103—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164#97—22no—
165#97—41no—
166C4×C8—22no—
167C4×C8—41no—
168C4×C8C2×D411yescharacteristic
169C4×C8C2×D411yescharacteristic
170C4×C8#4111yescharacteristic
171C4×C8C2411yescharacteristic
172C2×Q8—21no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—162no—
179C2×Q8C2×D812yes—
180C22×C4—81no—
181C22×C4#12111yescharacteristic
182C22×C4C2×D811yescharacteristic
183C22×C4#13511yescharacteristic
184Q16—42no—
185Q16—81no—
186M16—81no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8#12111yescharacteristic
193C2×C8#12111yescharacteristic
194C2×C8C22×D411yescharacteristic
195C2×C8C2×#4111yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—22no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4C2×D812yes—
214C22⋊C4—22no—
215C22⋊C4—81no—
216C42—22no—
217C42—81no—
218C42#12111yescharacteristic
219C42C2×D811yescharacteristic
220C42#13511yescharacteristic
221C42C22×D411yescharacteristic
222C16—22no—
223C16—22no—
224C16—42no—
225C23#46611yescharacteristic
226Q8—24no—
227Q8—81no—
228Q8—81no—
229Q8—81no—
230Q8—161no—
231Q8—161no—
232Q8—161no—
233Q8—162no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—162no—
245C2×C4#46611yescharacteristic
246C2×C4#46611yescharacteristic
247C2×C4#46611yescharacteristic
248C2×C4C2×#12111yescharacteristic
249C2×C4C22×D811yescharacteristic
250C2×C4C2×#13511yescharacteristic
251C8—21no—
252C8—21no—
253C8—21no—
254C8—42no—
255C8—82no—
256C8—82no—
257C8#57812yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22C2×#46611yescharacteristic
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—161no—
269C4—161no—
270C4—161no—
271C4—161no—
272C4#304411yescharacteristic
273C4#306911yescharacteristic
274C4#341712yes—
275C2—41no—
276C2#1940411yescharacteristic
277C2#3360811yescharacteristic
278C2#3361811yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radicalC23#466
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12481C2semidirect
6. #12477C2semidirect
7. #12461C2semidirect
8. #12458C2semidirect

Maximal subgroups

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

subgroupindex
#337962
#334992
#193422
#192092
#124812
#124772
#124612
#124582
#120342
#119752
#119532