Finite Group Databasebeta

← #491271

Subgroups — #491271

Order 512.

418 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 128 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, complemented
2#33493C211yesmaximal, characteristic
3#19333C211yesmaximal, characteristic
4#19192C211yesmaximal, characteristic
5#12474C212yesmaximal, complemented
6#12470C212yesmaximal, complemented
7#12454C212yesmaximal, complemented
8#12451C212yesmaximal, complemented
9#12036C211yesmaximal, characteristic, complemented
10#11971C211yesmaximal, characteristic, complemented
11#11950C211yesmaximal, characteristic, complemented
12#3254—21no—
13#3206—21no—
14#3170C2211yescharacteristic
15C2×#500C2211yescharacteristic
16#2931C2211yescharacteristic
17#2252C2212yes—
18#2241C2212yes—
19#2236—22no—
20#2227—21no—
21#2227—21no—
22#2169—21no—
23#2169C2212yes—
24D4×C16—21no—
25D4×C16C2212yes—
26#2162C2211yescharacteristic
27#2162C2211yescharacteristic
28#2113—22no—
29#1699—22no—
30#1693—22no—
31#1673C2212yes—
32#1673C2212yes—
33#1673C2212yes—
34#1669C2212yes—
35#1567C2211yescharacteristic
36#1564C2211yescharacteristic
37#1561—21no—
38#1549—21no—
39#1372C2212yes—
40#1371C2212yes—
41#1364C2211yescharacteristic
42#1364C2212yes—
43#1364C2211yescharacteristic
44#1364C2212yes—
45#1362C2211yescharacteristic
46#1362C2211yescharacteristic
47C2×#131—21no—
48#533—21no—
49#532D411yescharacteristic
50C2×C4⋊Q8—21no—
51C2×M32—22no—
52C22×C16—22no—
53#500—21no—
54#500—21no—
55#500—21no—
56#500C2312yes—
57#500C2311yescharacteristic
58#494C2311yescharacteristic
59#492—41no—
60#487—21no—
61#487—41no—
62#477—41no—
63#474—22no—
64#468—42no—
65#467—21no—
66#467—21no—
67#461—22no—
68#461—22no—
69#461—22no—
70#460—41no—
71#448—42no—
72C4×Q16—21no—
73C4×Q16—21no—
74#436—21no—
75#436D412yes—
76C8×D4—21no—
77C8×D4C2312yes—
78#431—21no—
79C2×#107—21no—
80C2×#107—21no—
81#423—21no—
82#423D411yescharacteristic
83#408—21no—
84C2×C4×C8C2311yescharacteristic
85#366—22no—
86#366—22no—
87C4⋊C16D411yescharacteristic
88C4⋊C16D411yescharacteristic
89C4⋊C16C2311yescharacteristic
90C4⋊C16C2311yescharacteristic
91#358—22no—
92#358—22no—
93C22⋊C16—21no—
94C22⋊C16—22no—
95C22⋊C16—22no—
96C22⋊C16D412yes—
97C4×C16—21no—
98C4×C16C2312yes—
99#335—21no—
100#335C2311yescharacteristic
101#335C2311yescharacteristic
102#335C2312yes—
103#334—21no—
104#332—22no—
105#332—22no—
106#332—22no—
107#332—22no—
108C2×#41—21no—
109C22×Q8—41no—
110C2×Q16—21no—
111C2×Q16—21no—
112C2×SD16—42no—
113C2×SD16—42no—
114C2×SD16—42no—
115#131—81no—
116#131—81no—
117C2×M16—21no—
118C2×M16—22no—
119C2×M16—41no—
120C22×C8—22no—
121C22×C8C2×D411yescharacteristic
122C22×C8#4111yescharacteristic
123C4⋊Q8—22no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8D811yescharacteristic
130#127D811yescharacteristic
131#125—41no—
132#124—21no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135#121—21no—
136#121D812yes—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×D4—21no—
140C4×D4C2×D412yes—
141#117—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C42C2×D411yescharacteristic
145M32—82no—
146C2×C16—21no—
147C2×C16—21no—
148C2×C16—82no—
149C2×C16C2×D412yes—
150C2×C16C2×D412yes—
151#107—22no—
152#107—21no—
153#107—22no—
154#107—21no—
155#107—41no—
156#107—41no—
157#107—42no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—22no—
162C4⋊C8—41no—
163C4⋊C8C2×D412yes—
164#103—41no—
165#103—41no—
166#103—42no—
167#103—42no—
168#103—42no—
169#102—42no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—41no—
173C4×C8—41no—
174C4×C8C2×D411yescharacteristic
175C4×C8C2×D411yescharacteristic
176C4×C8#4111yescharacteristic
177C4×C8C2411yescharacteristic
178#41—81no—
179#41—81no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—81no—
185C2×Q8—161no—
186C2×Q8—161no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4C2×D812yes—
191C22×C4—22no—
192C22×C4—81no—
193C22×C4#12111yescharacteristic
194C22×C4C2×D811yescharacteristic
195C22×C4#13511yescharacteristic
196Q16—41no—
197Q16—41no—
198Q16—81no—
199SD16—162no—
200SD16—162no—
201SD16—162no—
202M16—82no—
203M16—81no—
204M16—161no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8#12111yescharacteristic
215C2×C8#12111yescharacteristic
216C2×C8C22×D411yescharacteristic
217C2×C8C2×#4111yescharacteristic
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4C2×D812yes—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—81no—
233C42—81no—
234C42#12111yescharacteristic
235C42C2×D811yescharacteristic
236C42#13511yescharacteristic
237C42C22×D411yescharacteristic
238C16—22no—
239C16—22no—
240C16—41no—
241C16—41no—
242C23—22no—
243C23#46611yescharacteristic
244Q8—81no—
245Q8—81no—
246Q8—82no—
247Q8—82no—
248Q8—81no—
249Q8—81no—
250Q8—81no—
251Q8—162no—
252D4—22no—
253D4—22no—
254D4—42no—
255D4—81no—
256D4—82no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—162no—
269C2×C4#46611yescharacteristic
270C2×C4#46611yescharacteristic
271C2×C4#46611yescharacteristic
272C2×C4C2×#12111yescharacteristic
273C2×C4C22×D811yescharacteristic
274C2×C4C2×#13511yescharacteristic
275C8—21no—
276C8—21no—
277C8—21no—
278C8—42no—
279C8—82no—
280C8—81no—
281C8—81no—
282C8#57711yescharacteristic
283C8#57711yescharacteristic
284C22—21no—
285C22—21no—
286C22—21no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22C2×#46611yescharacteristic
291C4—21no—
292C4—21no—
293C4—21no—
294C4—42no—
295C4—162no—
296C4—161no—
297C4—161no—
298C4#222212yes—
299C4#304211yescharacteristic
300C4#306911yescharacteristic
301C4#341511yescharacteristic
302C4#341611yescharacteristic
303C2—41no—
304C2—82no—
305C2#1939211yescharacteristic
306C2#3361211yescharacteristic
307C2C2×#222211yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#532D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33796C2semidirect
5. #12474C2semidirect
6. #12470C2semidirect
7. #12454C2semidirect
8. #12451C2semidirect
9. #12036C2semidirect
10. #11971C2semidirect
11. #11950C2semidirect

Maximal subgroups

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

subgroupindex
#337962
#334932
#193332
#191922
#124742
#124702
#124542
#124512
#120362
#119712
#119502