Finite Group Databasebeta

← #219620

Subgroups — #219620

Order 512.

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

normal subgroups only

This group has 601 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#22276C211yesmaximal, characteristic, complemented
2#22236C211yesmaximal, characteristic, complemented
3#19698C211yesmaximal, characteristic
4#9773C211yesmaximal, characteristic, complemented
5#9757C211yesmaximal, characteristic, complemented
6#9299C212yesmaximal, complemented
7#9295C212yesmaximal, complemented
8#9294C212yesmaximal, complemented
9#9293C212yesmaximal, complemented
10#8794C211yesmaximal, characteristic, complemented
11#8142C211yesmaximal, characteristic, complemented
12#3208C2212yes—
13#3205C2212yes—
14C2×#493C2211yescharacteristic
15#3131C2211yescharacteristic
16#2961C2211yescharacteristic
17#2932C2211yescharacteristic
18#2328C2212yes—
19#1995C2211yescharacteristic
20#1963C2211yescharacteristic
21#1961C2212yes—
22#1929—22no—
23#1879—22no—
24#1879—22no—
25#1866—22no—
26#1848C2211yescharacteristic
27#1835C2211yescharacteristic
28#1746C2211yescharacteristic
29#1721C414yescomplemented
30#1717C414yescomplemented
31#1545C2212yes—
32#1545C2212yes—
33#1518C2212yes—
34#1517C2212yes—
35#1452C2211yescharacteristic
36#1450C2211yescharacteristic
37#1286C2212yes—
38#1286C2212yes—
39#1275C2212yes—
40#1274C2212yes—
41C22×D8—22no—
42C2×#131—21no—
43#533C2311yescharacteristic
44#532C2311yescharacteristic
45C2×#127C2312yes—
46C2×C4×D4—22no—
47#493—21no—
48#493—21no—
49#493C2×C414yes—
50#493C2×C414yescomplemented
51#492C2×C414yescomplemented
52#488—21no—
53#488—21no—
54#487C2×C412yescomplemented
55#486C2×C412yescomplemented
56#486C2×C414yescomplemented
57#486C2×C414yescomplemented
58#460—42no—
59#459—42no—
60#450—42no—
61#447—42no—
62#446—42no—
63#435—42no—
64#431C2312yes—
65#431C2312yes—
66C2×#108—22no—
67#424—21no—
68#423C2311yescharacteristic
69#423C2311yescharacteristic
70C2×#104—22no—
71#416—22no—
72C2×#102—22no—
73C2×#102—22no—
74#408—21no—
75#405—21no—
76C2×#97C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#390—22no—
79#386—42no—
80#386—42no—
81#382C2312yes—
82#376C2311yescharacteristic
83#340—22no—
84#339—22no—
85#336D412yes—
86#336D412yes—
87#331—42no—
88#331—42no—
89#328—42no—
90#327—42no—
91#322—21no—
92#322C2×C414yescomplemented
93C82—21no—
94C2×#41—21no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—22no—
98C23×C4—42no—
99C2×SD16—44no—
100C2×D8—24no—
101C2×D8—24no—
102C2×D8—44no—
103C2×D8—82no—
104C2×D8—82no—
105#131—82no—
106C2×M16—21no—
107C2×M16—21no—
108C2×M16—41no—
109C22×C8—41no—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D412yes—
113C4⋊Q8C22×C411yescharacteristic
114C4⋊Q8C22×C412yes—
115#127—22no—
116#127—22no—
117#127—22no—
118#127C22×C414yes—
119#127C22×C411yescharacteristic
120#125—21no—
121#124—21no—
122C22⋊Q8—22no—
123#121—22no—
124C4×Q8—22no—
125C4×D4—22no—
126C4×D4—82no—
127C4×D4—82no—
128#117—21no—
129C2×C4⋊C4—22no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C42C2×D412yes—
134C2×C42C2411yescharacteristic
135#108—82no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—42no—
141#104—82no—
142#103—22no—
143#103—22no—
144#102—22no—
145#102—22no—
146#102—44no—
147#102—44no—
148#102—82no—
149#102—82no—
150C22⋊C8—22no—
151C22⋊C8—22no—
152C22⋊C8—42no—
153#97—22no—
154#97—41no—
155C4×C8—22no—
156C4×C8—41no—
157C4×C8C22×C414yes—
158C4×C8C22×C412yes—
159C4×C8#4111yescharacteristic
160C4×C8#4111yescharacteristic
161#95—22no—
162#95—22no—
163#95—22no—
164C24—42no—
165#41—42no—
166#41—42no—
167C2×Q8—21no—
168C2×Q8—22no—
169C2×D4—21no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—162no—
183C2×D4—162no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—84no—
188C22×C4—162no—
189C22×C4—162no—
190C22×C4C22×D411yescharacteristic
191C22×C4H(5,𝔽2)12yes—
192SD16—164no—
193D8—84no—
194D8—84no—
195D8—164no—
196M16—42no—
197M16—42no—
198M16—82no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—82no—
210C2×C8C4×D412yes—
211C2×C8C4×D412yes—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—84no—
221C22⋊C4—162no—
222C22⋊C4—162no—
223C42—22no—
224C42—22no—
225C42—41no—
226C42C23×C411yescharacteristic
227C42C2×#4111yescharacteristic
228C42H(5,𝔽2)12yes—
229C23—41no—
230C23—44no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23—82no—
236C23—82no—
237C23#53511yescharacteristic
238Q8—42no—
239Q8—82no—
240D4—44no—
241D4—44no—
242D4—42no—
243D4—82no—
244D4—84no—
245D4—84no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—162no—
250D4—162no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—168no—
263C2×C4C2×C4×D411yescharacteristic
264C2×C4#51812yes—
265C2×C4#53511yescharacteristic
266C2×C4#54012yes—
267C2×C4#57612yes—
268C2×C4#57712yes—
269C8—22no—
270C8—22no—
271C8—44no—
272C8—41no—
273C8—41no—
274C8—82no—
275C8—81no—
276C8—81no—
277C22—22no—
278C22—81no—
279C22—84no—
280C22—84no—
281C22—84no—
282C22—82no—
283C22—162no—
284C22—162no—
285C22—162no—
286C22—162no—
287C22#232811yescharacteristic
288C22#322412yes—
289C4—22no—
290C4—22no—
291C4—82no—
292C4—81no—
293C4—82no—
294C4—162no—
295C4#294612yes—
296C4#322011yescharacteristic
297C4#322111yescharacteristic
298C2—22no—
299C2—161no—
300C2—164no—
301C2#2227911yescharacteristic
302C2#2228612yes—
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#19698C2
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. #22276C2semidirect
2. #22236C2semidirect
4. #9773C2semidirect
5. #9757C2semidirect
6. #9299C2semidirect
7. #9295C2semidirect
8. #9294C2semidirect
9. #9293C2semidirect
10. #8794C2semidirect
11. #8142C2semidirect
29. #1721C4semidirect
30. #1717C4semidirect
50. #493C2×C4semidirect
51. #492C2×C4semidirect
54. #487C2×C4semidirect
55. #486C2×C4semidirect
56. #486C2×C4semidirect
57. #486C2×C4semidirect
92. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222762
#222362
#196982
#97732
#97572
#92992
#92952
#92942
#92932
#87942
#81422