Finite Group Databasebeta

← #219609

Subgroups — #219609

Order 512.

600 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 162 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#22282C211yesmaximal, characteristic, complemented
2#22275C211yesmaximal, characteristic, complemented
3#19660C211yesmaximal, characteristic
4#9777C211yesmaximal, characteristic, complemented
5#9760C211yesmaximal, characteristic, complemented
6#9300C212yesmaximal, complemented
7#9299C212yesmaximal, complemented
8#9293C212yesmaximal, complemented
9#9291C212yesmaximal, complemented
10#8792C211yesmaximal, characteristic, complemented
11#8142C211yesmaximal, characteristic, complemented
12#3226C2212yes—
13#3205C2212yes—
14#3147C2211yescharacteristic
15C2×#492C2211yescharacteristic
16#2932C2212yes—
17#2328C2211yescharacteristic
18#2327C2211yescharacteristic
19#1963C2212yes—
20#1961C2211yescharacteristic
21#1960C2211yescharacteristic
22#1880—22no—
23#1877—22no—
24#1876—22no—
25#1876—22no—
26#1837C2211yescharacteristic
27#1835C2211yescharacteristic
28#1746C2211yescharacteristic
29#1722C412yescomplemented
30#1721C412yescomplemented
31#1706C414yescomplemented
32#1546C2212yes—
33#1545C2212yes—
34#1517C2212yes—
35#1517C2212yes—
36#1452C2211yescharacteristic
37#1450C2211yescharacteristic
38#1286C2212yes—
39#1286C2212yes—
40#1275C2212yes—
41#1274C2212yes—
42C2×#135—22no—
43C22×SD16—22no—
44C2×#131—22no—
45#532C2312yes—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#127C2311yescharacteristic
48C2×C4×Q8—21no—
49C2×C4×D4—21no—
50#500C2×C414yescomplemented
51#495—22no—
52#494C2×C411yescharacteristic, complemented
53#493C2×C414yes—
54#493C2×C411yescharacteristic, complemented
55#492—22no—
56#492C2×C412yes—
57#492C2×C412yes—
58#492C2×C412yescomplemented
59#486C2×C414yes—
60#486C2×C414yescomplemented
61#480—42no—
62#474—42no—
63#459—42no—
64#447—42no—
65#435—42no—
66#435—42no—
67#431C2312yes—
68#431C2312yes—
69#423C2312yes—
70C2×#104—22no—
71C2×#104—22no—
72C2×#103—22no—
73C2×#102—22no—
74#408—22no—
75C2×#97C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—21no—
78#390—21no—
79#387—42no—
80#386—42no—
81#382C2311yescharacteristic
82#382C2311yescharacteristic
83#376C2311yescharacteristic
84#341—22no—
85#340—22no—
86#336D412yes—
87#336D412yes—
88#332—42no—
89#331—42no—
90#327—42no—
91#327—42no—
92#322—22no—
93#322C2×C412yescomplemented
94C82C2×C412yescomplemented
95C2×#41—22no—
96C22×Q8—21no—
97C22×D4—21no—
98C22×D4—22no—
99C23×C4—41no—
100#135—82no—
101#135—82no—
102C2×Q16—22no—
103C2×SD16—24no—
104C2×SD16—24no—
105C2×SD16—22no—
106C2×SD16—22no—
107C2×SD16—82no—
108C2×SD16—82no—
109C2×D8—22no—
110C2×D8—44no—
111#131—82no—
112#131—82no—
113C2×M16—22no—
114C2×M16—42no—
115C22×C8—42no—
116C22×C8C2×D412yes—
117C22×C8C2×D412yes—
118C4⋊Q8—21no—
119C4⋊Q8—22no—
120C4⋊Q8C22×C412yes—
121C4⋊Q8C22×C412yes—
122#127—21no—
123#127—22no—
124#127C22×C412yes—
125#127C22×C412yes—
126#124—22no—
127#121—22no—
128#121—22no—
129C4×Q8—82no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—82no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C42—41no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2411yescharacteristic
143#107—44no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148#104—82no—
149#104—82no—
150#103—82no—
151#102—44no—
152#102—44no—
153#102—82no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—44no—
157#97—22no—
158C4×C8—22no—
159C4×C8C22×C414yes—
160C4×C8C22×C411yescharacteristic
161C4×C8C22×C411yescharacteristic
162C4×C8#4112yes—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—42no—
167C24—41no—
168#41—42no—
169#41—42no—
170#41—42no—
171#41—42no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—42no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—162no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—82no—
195C22×C4—162no—
196C22×C4C22×D411yescharacteristic
197C22×C4H(5,𝔽2)12yes—
198Q16—82no—
199SD16—84no—
200SD16—84no—
201SD16—82no—
202SD16—82no—
203D8—82no—
204D8—164no—
205M16—42no—
206M16—42no—
207M16—84no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—84no—
217C2×C8C4×D412yes—
218C2×C8C4×D412yes—
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—82no—
229C22⋊C4—162no—
230C42—22no—
231C42—22no—
232C42—82no—
233C42C23×C411yescharacteristic
234C42C2×#4111yescharacteristic
235C42H(5,𝔽2)12yes—
236C23—42no—
237C23—42no—
238C23—82no—
239C23—82no—
240C23—81no—
241C23—81no—
242C23#53511yescharacteristic
243Q8—44no—
244Q8—42no—
245Q8—42no—
246Q8—82no—
247D4—42no—
248D4—42no—
249D4—44no—
250D4—82no—
251D4—84no—
252D4—82no—
253D4—82no—
254D4—82no—
255D4—82no—
256D4—162no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—164no—
273C2×C4C2×C4×D411yescharacteristic
274C2×C4#51812yes—
275C2×C4#53511yescharacteristic
276C2×C4#54012yes—
277C2×C4#57612yes—
278C2×C4#57612yes—
279C8—22no—
280C8—22no—
281C8—44no—
282C8—42no—
283C8—82no—
284C8—82no—
285C22—22no—
286C22—82no—
287C22—82no—
288C22—84no—
289C22—82no—
290C22—82no—
291C22—161no—
292C22—161no—
293C22—162no—
294C22#232811yescharacteristic
295C22#322011yescharacteristic
296C22#322011yescharacteristic
297C4—22no—
298C4—22no—
299C4—81no—
300C4—81no—
301C4—82no—
302C4—82no—
303C4—161no—
304C4—161no—
305C4#294812yes—
306C4#322012yes—
307C2—21no—
308C2—21no—
309C2—162no—
310C2—162no—
311C2#2227611yescharacteristic
312C2#2228412yes—
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#19660C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22282C2semidirect
2. #22275C2semidirect
4. #9777C2semidirect
5. #9760C2semidirect
6. #9300C2semidirect
7. #9299C2semidirect
8. #9293C2semidirect
9. #9291C2semidirect
10. #8792C2semidirect
11. #8142C2semidirect
29. #1722C4semidirect
30. #1721C4semidirect
31. #1706C4semidirect
50. #500C2×C4semidirect
52. #494C2×C4semidirect
54. #493C2×C4semidirect
58. #492C2×C4semidirect
60. #486C2×C4semidirect
93. #322C2×C4semidirect
94. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222822
#222752
#196602
#97772
#97602
#93002
#92992
#92932
#92912
#87922
#81422