Finite Group Databasebeta

← #491335

Subgroups — #491335

Order 512.

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

normal subgroups only

This group has 390 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
2#33500C211yesmaximal, characteristic
3#19345C211yesmaximal, characteristic
4#19208C211yesmaximal, characteristic
5#12482C212yesmaximal, complemented
6#12478C212yesmaximal, complemented
7#12460C212yesmaximal, complemented
8#12457C212yesmaximal, complemented
9#12036C211yesmaximal, characteristic
10#11971C211yesmaximal, characteristic
11#11950C211yesmaximal, characteristic
12#3255—21no—
13#3209—21no—
14#3170C2211yescharacteristic
15C2×#500C2211yescharacteristic
16#2960C2211yescharacteristic
17#2252C2212yes—
18#2241C2212yes—
19#2236—22no—
20#2227—21no—
21#2227—21no—
22#2183—21no—
23#2183C2212yes—
24Q8×C16—21no—
25Q8×C16C2212yes—
26#2164—22no—
27#2162C2211yescharacteristic
28#2162C2211yescharacteristic
29#1700—22no—
30#1694—22no—
31#1674C2212yes—
32#1674C2212yes—
33#1674C2212yes—
34#1670C2212yes—
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
47#533—21no—
48#533D411yescharacteristic
49C2×C4⋊Q8—21no—
50#500—21no—
51#500—21no—
52#500—21no—
53#500C2312yes—
54#500C2311yescharacteristic
55#494—41no—
56#494C2311yescharacteristic
57#487—21no—
58#487—41no—
59#475—22no—
60#475—41no—
61#470—42no—
62#467—21no—
63#467—21no—
64#462—22no—
65#462—22no—
66#462—22no—
67#462—41no—
68#451—42no—
69#446—21no—
70#446D412yes—
71C8×Q8—21no—
72C8×Q8C2312yes—
73C4×Q16—21no—
74C4×Q16—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#358—22no—
92#358—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#332—22no—
105#332—22no—
106#332—22no—
107#332—22no—
108C22×Q8—41no—
109C2×Q16—21no—
110C2×Q16—21no—
111C2×Q16—42no—
112C2×Q16—42no—
113C2×Q16—42no—
114C2×M16—21no—
115C2×M16—41no—
116C22×C8C2×D411yescharacteristic
117C22×C8#4111yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8D811yescharacteristic
125C4⋊Q8D811yescharacteristic
126#125—21no—
127#125—41no—
128C22⋊Q8—21no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8D812yes—
132C4×Q8—21no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8C2×D412yes—
136#117—21no—
137#117—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C42C2×D411yescharacteristic
141C2×C16—21no—
142C2×C16—21no—
143C2×C16C2×D412yes—
144C2×C16C2×D412yes—
145#107—22no—
146#107—21no—
147#107—22no—
148#107—21no—
149#107—41no—
150#107—41no—
151#106—42no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—22no—
158C4⋊C8—41no—
159C4⋊C8C2×D412yes—
160#103—42no—
161#103—41no—
162#103—41no—
163#103—42no—
164#103—42no—
165#103—42no—
166C22⋊C8—41no—
167#97—22no—
168C4×C8—22no—
169C4×C8—41no—
170C4×C8C2×D411yescharacteristic
171C4×C8C2×D411yescharacteristic
172C4×C8#4111yescharacteristic
173C4×C8C2411yescharacteristic
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—81no—
179C2×Q8—161no—
180C2×Q8—161no—
181C2×Q8C2×D812yes—
182C22×C4—81no—
183C22×C4#12111yescharacteristic
184C22×C4C2×D811yescharacteristic
185C22×C4#13511yescharacteristic
186Q16—41no—
187Q16—41no—
188Q16—81no—
189Q16—162no—
190Q16—162no—
191Q16—162no—
192M16—81no—
193M16—161no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8#12111yescharacteristic
202C2×C8#12111yescharacteristic
203C2×C8C22×D411yescharacteristic
204C2×C8C2×#4111yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4C2×D812yes—
221C22⋊C4—22no—
222C22⋊C4—81no—
223C42—22no—
224C42—81no—
225C42#12111yescharacteristic
226C42C2×D811yescharacteristic
227C42#13511yescharacteristic
228C42C22×D411yescharacteristic
229C16—22no—
230C16—22no—
231C16—41no—
232C16—41no—
233C23#46611yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—42no—
237Q8—81no—
238Q8—82no—
239Q8—82no—
240Q8—81no—
241Q8—81no—
242Q8—81no—
243Q8—162no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—162no—
255C2×C4#46611yescharacteristic
256C2×C4#46611yescharacteristic
257C2×C4#46611yescharacteristic
258C2×C4C2×#12111yescharacteristic
259C2×C4C22×D811yescharacteristic
260C2×C4C2×#13511yescharacteristic
261C8—21no—
262C8—21no—
263C8—21no—
264C8—42no—
265C8—82no—
266C8—81no—
267C8—81no—
268C8#57811yescharacteristic
269C8#57811yescharacteristic
270C22—21no—
271C22—21no—
272C22—21no—
273C22C2×#46611yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—162no—
280C4—161no—
281C4—161no—
282C4#222212yes—
283C4#304411yescharacteristic
284C4#306911yescharacteristic
285C4#341511yescharacteristic
286C4#341611yescharacteristic
287C2—41no—
288C2#1940411yescharacteristic
289C2#3361211yescharacteristic
290C2C2×#222211yescharacteristic
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. #12482C2semidirect
6. #12478C2semidirect
7. #12460C2semidirect
8. #12457C2semidirect

Maximal subgroups

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

subgroupindex
#337962
#335002
#193452
#192082
#124822
#124782
#124602
#124572
#120362
#119712
#119502