Finite Group Databasebeta

← #491274

Subgroups — #491274

Order 512.

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

normal subgroups only

This group has 412 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#33795C211yesmaximal, characteristic, complemented
2#33493C211yesmaximal, characteristic
3#19333C211yesmaximal, characteristic
4#19193C211yesmaximal, characteristic
5#12473C212yesmaximal, complemented
6#12469C212yesmaximal, complemented
7#12456C212yesmaximal, complemented
8#12453C212yesmaximal, complemented
9#12036C211yesmaximal, characteristic, complemented
10#11972C211yesmaximal, characteristic, complemented
11#11950C211yesmaximal, characteristic, complemented
12#3254—21no—
13#3227—21no—
14#3170C2211yescharacteristic
15C2×#500C2211yescharacteristic
16#2931C2211yescharacteristic
17#2254C2212yes—
18#2241C2212yes—
19#2235—22no—
20#2225—21no—
21#2225—21no—
22#2169—21no—
23#2169C2212yes—
24D4×C16—21no—
25D4×C16C2212yes—
26#2162C2211yescharacteristic
27#2162C2211yescharacteristic
28#2113—22no—
29#1699—22no—
30#1695—22no—
31#1675C2212yes—
32#1675C2212yes—
33#1673C2212yes—
34#1669C2212yes—
35#1567C2211yescharacteristic
36#1564C2211yescharacteristic
37#1561—21no—
38#1548—21no—
39#1372C2212yes—
40#1371C2212yes—
41#1365C2212yes—
42#1365C2212yes—
43#1364C2211yescharacteristic
44#1364C2211yescharacteristic
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#500C2311yescharacteristic
57#500C2312yes—
58#498—41no—
59#494C2311yescharacteristic
60#487—21no—
61#486—41no—
62#481—42no—
63#477—41no—
64#476—22no—
65#476—22no—
66#474—22no—
67#468—42no—
68#467—21no—
69#467—21no—
70#461—22no—
71#460—41no—
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#367—22no—
86#367—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#333—22no—
105#333—22no—
106#332—22no—
107#332—22no—
108C2×#41—21no—
109C22×Q8—41no—
110C2×Q16—21no—
111C2×Q16—21no—
112C2×SD16—42no—
113#131—81no—
114#131—81no—
115C2×M16—21no—
116C2×M16—22no—
117C2×M16—41no—
118C22×C8—22no—
119C22×C8C2×D411yescharacteristic
120C22×C8#4111yescharacteristic
121C4⋊Q8—22no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126C4⋊Q8—41no—
127C4⋊Q8D811yescharacteristic
128#127D811yescharacteristic
129#125—41no—
130#124—21no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133#121—21no—
134#121D812yes—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×D4—21no—
138C4×D4C2×D412yes—
139#117—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C42C2×D411yescharacteristic
143M32—82no—
144C2×C16—21no—
145C2×C16—21no—
146C2×C16—82no—
147C2×C16C2×D412yes—
148C2×C16C2×D412yes—
149#107—22no—
150#107—21no—
151#107—22no—
152#107—21no—
153#107—41no—
154#107—41no—
155#107—42no—
156#106—42no—
157#106—42no—
158C4⋊C8—21no—
159C4⋊C8—22no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—41no—
163C4⋊C8C2×D412yes—
164#103—41no—
165#103—41no—
166#103—42no—
167#102—42no—
168#102—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—162no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4C2×D812yes—
190C22×C4—22no—
191C22×C4—81no—
192C22×C4#12111yescharacteristic
193C22×C4C2×D811yescharacteristic
194C22×C4#13511yescharacteristic
195Q16—41no—
196Q16—41no—
197Q16—81no—
198SD16—162no—
199M16—82no—
200M16—81no—
201M16—161no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8#12111yescharacteristic
212C2×C8#12111yescharacteristic
213C2×C8C22×D411yescharacteristic
214C2×C8C2×#4111yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4C2×D812yes—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—81no—
230C42—81no—
231C42#12111yescharacteristic
232C42C2×D811yescharacteristic
233C42#13511yescharacteristic
234C42C22×D411yescharacteristic
235C16—22no—
236C16—22no—
237C16—41no—
238C16—41no—
239C23—22no—
240C23#46611yescharacteristic
241Q8—81no—
242Q8—81no—
243Q8—81no—
244Q8—81no—
245Q8—81no—
246Q8—162no—
247Q8—162no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—81no—
252D4—82no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—162no—
265C2×C4#46611yescharacteristic
266C2×C4#46611yescharacteristic
267C2×C4#46611yescharacteristic
268C2×C4C2×#12111yescharacteristic
269C2×C4C22×D811yescharacteristic
270C2×C4C2×#13511yescharacteristic
271C8—21no—
272C8—21no—
273C8—21no—
274C8—42no—
275C8—81no—
276C8—82no—
277C8—81no—
278C8#57711yescharacteristic
279C8#57711yescharacteristic
280C22—21no—
281C22—21no—
282C22—21no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22C2×#46611yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—161no—
292C4—162no—
293C4—161no—
294C4#222312yes—
295C4#304211yescharacteristic
296C4#306911yescharacteristic
297C4#341611yescharacteristic
298C4#341711yescharacteristic
299C2—41no—
300C2—82no—
301C2#1939211yescharacteristic
302C2#3361311yescharacteristic
303C2C2×#222311yescharacteristic
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. #33795C2semidirect
5. #12473C2semidirect
6. #12469C2semidirect
7. #12456C2semidirect
8. #12453C2semidirect
9. #12036C2semidirect
10. #11972C2semidirect
11. #11950C2semidirect

Maximal subgroups

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

subgroupindex
#337952
#334932
#193332
#191932
#124732
#124692
#124562
#124532
#120362
#119722
#119502