Finite Group Databasebeta

← #491336

Subgroups — #491336

Order 512.

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

normal subgroups only

This group has 395 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#33792C211yesmaximal, characteristic
2#33500C211yesmaximal, characteristic, complemented
3#19344C211yesmaximal, characteristic
4#19209C211yesmaximal, characteristic, complemented
5#12480C212yesmaximal, complemented
6#12476C212yesmaximal, complemented
7#12461C212yesmaximal, complemented
8#12458C212yesmaximal, complemented
9#12036C211yesmaximal, characteristic, complemented
10#11972C211yesmaximal, characteristic, complemented
11#11948C211yesmaximal, characteristic
12#3254—21no—
13#3230—21no—
14#3169C2211yescharacteristic
15C2×#500C2211yescharacteristic
16#2960C2211yescharacteristic
17#2252C2212yes—
18#2240C2212yes—
19#2232—22no—
20#2224—21no—
21#2224—21no—
22#2183—21no—
23#2183C2212yes—
24Q8×C16—21no—
25Q8×C16C2212yes—
26#2164—22no—
27#2162C2211yescharacteristic
28#2162C2211yescharacteristic
29#1698—22no—
30#1696—22no—
31#1676C2212yes—
32#1676C2212yes—
33#1676C2212yes—
34#1668C2212yes—
35#1567C2211yescharacteristic
36#1564C2211yescharacteristic
37#1560—21no—
38#1548—21no—
39#1372C2212yes—
40#1371C2212yes—
41#1365C2211yescharacteristic
42#1365C2212yes—
43#1365C2211yescharacteristic
44#1365C2212yes—
45#1361C2211yescharacteristic
46#1361C2211yescharacteristic
47#533D411yescharacteristic
48#532—21no—
49C2×C4⋊Q8—21no—
50#500—21no—
51#500—21no—
52#500—21no—
53#500—41no—
54#500C2312yes—
55#500C2311yescharacteristic
56#493C2311yescharacteristic
57#487—41no—
58#486—21no—
59#484—42no—
60#477—22no—
61#477—22no—
62#477—22no—
63#474—41no—
64#468—42no—
65#466—21no—
66#466—21no—
67#461—41no—
68#460—22no—
69#446—21no—
70#446D412yes—
71C8×Q8—21no—
72C8×Q8C2312yes—
73C4×D8—21no—
74C4×D8—21no—
75#431—21no—
76C2×#107—21no—
77C2×#107—21no—
78#424—21no—
79#423—21no—
80#423D411yescharacteristic
81#405—21no—
82C2×C4×C8C2311yescharacteristic
83#366—22no—
84#366—22no—
85C4⋊C16—22no—
86C4⋊C16—22no—
87C4⋊C16D411yescharacteristic
88C4⋊C16D411yescharacteristic
89C4⋊C16C2311yescharacteristic
90C4⋊C16C2311yescharacteristic
91#357—22no—
92#357—22no—
93C22⋊C16—21no—
94C22⋊C16D412yes—
95#346—22no—
96C4×C16—22no—
97C4×C16—21no—
98C4×C16C2312yes—
99#335—21no—
100#335C2311yescharacteristic
101#335C2311yescharacteristic
102#335C2312yes—
103#334—21no—
104#333—22no—
105#333—22no—
106#333—22no—
107#331—22no—
108C2×#41—41no—
109C22×Q8—41no—
110C2×SD16—42no—
111C2×D8—21no—
112C2×D8—21no—
113C2×M16—21no—
114C2×M16—41no—
115C22×C8C2×D411yescharacteristic
116C22×C8#4111yescharacteristic
117C4⋊Q8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8D811yescharacteristic
123C4⋊Q8D811yescharacteristic
124#127—21no—
125#125—21no—
126#124—41no—
127C22⋊Q8—21no—
128C22⋊Q8D812yes—
129#121—41no—
130#121—41no—
131C4×Q8—21no—
132C4×Q8C2×D412yes—
133C4×D4—41no—
134C4×D4—41no—
135#117—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C42C2×D411yescharacteristic
139C2×C16—21no—
140C2×C16—21no—
141C2×C16C2×D412yes—
142C2×C16C2×D412yes—
143#107—22no—
144#107—21no—
145#107—22no—
146#107—21no—
147#107—41no—
148#107—41no—
149#107—42no—
150#107—42no—
151#107—42no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—21no—
155C4⋊C8—22no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—41no—
159C4⋊C8C2×D412yes—
160#103—42no—
161#103—42no—
162#103—42no—
163#102—41no—
164#102—41no—
165#102—42no—
166C22⋊C8—41no—
167#97—22no—
168C4×C8—22no—
169C4×C8—41no—
170C4×C8C2×D411yescharacteristic
171C4×C8C2×D411yescharacteristic
172C4×C8#4111yescharacteristic
173C4×C8C2411yescharacteristic
174#41—162no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—81no—
179C2×Q8—162no—
180C2×Q8C2×D812yes—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4#12111yescharacteristic
188C22×C4C2×D811yescharacteristic
189C22×C4#13511yescharacteristic
190SD16—162no—
191D8—41no—
192D8—41no—
193D8—81no—
194M16—81no—
195M16—161no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8#12111yescharacteristic
204C2×C8#12111yescharacteristic
205C2×C8C22×D411yescharacteristic
206C2×C8C2×#4111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4C2×D812yes—
219C22⋊C4—22no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C42—22no—
223C42#12111yescharacteristic
224C42C2×D811yescharacteristic
225C42#13511yescharacteristic
226C42C22×D411yescharacteristic
227C16—22no—
228C16—22no—
229C16—41no—
230C16—41no—
231C23—81no—
232C23#46611yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—42no—
236Q8—162no—
237Q8—161no—
238Q8—162no—
239D4—81no—
240D4—81no—
241D4—81no—
242D4—81no—
243D4—161no—
244D4—162no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—162no—
255C2×C4—162no—
256C2×C4#46611yescharacteristic
257C2×C4#46611yescharacteristic
258C2×C4#46611yescharacteristic
259C2×C4C2×#12111yescharacteristic
260C2×C4C22×D811yescharacteristic
261C2×C4C2×#13511yescharacteristic
262C8—21no—
263C8—21no—
264C8—21no—
265C8—42no—
266C8—81no—
267C8—82no—
268C8—81no—
269C8#57811yescharacteristic
270C8#57811yescharacteristic
271C22—21no—
272C22—21no—
273C22—21no—
274C22—161no—
275C22—161no—
276C22—161no—
277C22C2×#46611yescharacteristic
278C4—21no—
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—161no—
284C4—162no—
285C4#222112yes—
286C4#304411yescharacteristic
287C4#306911yescharacteristic
288C4#341611yescharacteristic
289C4#341711yescharacteristic
290C2—41no—
291C2—321no—
292C2#1940411yescharacteristic
293C2#3361311yescharacteristic
294C2C2×#222111yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#3169C22
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
2. #33500C2semidirect
4. #19209C2semidirect
5. #12480C2semidirect
6. #12476C2semidirect
7. #12461C2semidirect
8. #12458C2semidirect
9. #12036C2semidirect
10. #11972C2semidirect

Maximal subgroups

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

subgroupindex
#337922
#335002
#193442
#192092
#124802
#124762
#124612
#124582
#120362
#119722
#119482