Finite Group Databasebeta

← #46820125

Subgroups — #46820125

Order 34432.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#16864402C211yesmaximal, characteristic, complemented
2C2×#22575483C211yesmaximal, characteristic, complemented
3C2×#22575478C211yesmaximal, characteristic, complemented
4#31770765C214yesmaximal, complemented
5#31770748C214yesmaximal, complemented
6#31770613C211yesmaximal, characteristic, complemented
7#31770609C211yesmaximal, characteristic, complemented
8#31770569C211yesmaximal, characteristic, complemented
9#525×C269C211yesmaximal, characteristic, complemented
10C2×#16864402—24no—
11C2×#16864402—24no—
12C2×#16864402—22no—
13C2×#16864402C2214yes—
14C22×Dic538C2211yescharacteristic
15C22×D1076C2211yescharacteristic
16C22×C22⋊D269—21no—
17C22×C4×D269—21no—
18#22575483—24no—
19#22575478—24no—
20C2×#16864392C2212yescomplemented
21C2×#16864390C2212yescomplemented
22C2×#16864389C2212yescomplemented
23C2×#16864387C2212yescomplemented
24C2×#16864386C2211yescharacteristic
25C2×#16864386C2211yescharacteristic
26#22575376C2214yescomplemented
27#22575373C2214yescomplemented
28C22⋊#14524130C2214yescomplemented
29C2×#41×C269C2211yescharacteristic
30C23×C4×C269C2211yescharacteristic
31#123×C269—21no—
32C22⋊Q8×C269C2212yescomplemented
33#121×C269C2212yescomplemented
34#120×C269—21no—
35C4×D4×C269—22no—
36#117×C269C2211yescharacteristic
37C23×D269—21no—
38#16864402—216no—
39#16864402—28no—
40#16864402—28no—
41C2×Dic538—22no—
42C2×Dic538—21no—
43C2×Dic538D414yescomplemented
44C2×Dic538C2312yes—
45C2×D1076—22no—
46C2×D1076—21no—
47C2×D1076D414yescomplemented
48C2×D1076C2312yes—
49C2×C22⋊D269—28no—
50C2×C22⋊D269—24no—
51C22×Dic269—21no—
52C2×C4×D269—28no—
53C2×C4×D269—24no—
54#16864392—24no—
55#16864390—24no—
56#16864389—24no—
57#16864387—24no—
58#16864386—22no—
59#16864386—22no—
60C2×#14524130C2312yes—
61C2×#14524130C2312yes—
62C24×C269D411yescharacteristic
63#41×C269—22no—
64#41×C269—22no—
65C2×Q8×C269C2311yescharacteristic
66C2×D4×C269—22no—
67C2×D4×C269—21no—
68C2×D4×C269C2311yescharacteristic
69C22×C4×C269—22no—
70C22×C4×C269—22no—
71C22×C4×C269—21no—
72C22×C4×C269D411yescharacteristic
73C22×C4×C269D412yes—
74C22×C4×C269C2312yes—
75C22×C4×C269C2311yescharacteristic
76C4⋊C4×C269—22no—
77C4⋊C4×C269C2312yes—
78C22⋊C4×C269—21no—
79C22⋊C4×C269—22no—
80C22⋊C4×C269—22no—
81C42×C269—21no—
82C22×D269—24no—
83C22×D269—22no—
84C22×D269—44no—
85Dic538—24no—
86Dic538—22no—
87Dic538C2×D414yes—
88D1076—24no—
89D1076—22no—
90D1076C2×D414yes—
91C22⋊D269—216no—
92C2×Dic269—24no—
93C2×Dic269—22no—
94C4×D269—216no—
95#14524130—22no—
96#14524130—22no—
97C23×C269—22no—
98C23×C269—21no—
99C23×C269—42no—
100C23×C269C2×D412yes—
101C23×C269C2×D411yescharacteristic
102Q8×C269—21no—
103Q8×C269—21no—
104D4×C269—21no—
105D4×C269—21no—
106D4×C269—44no—
107D4×C269—42no—
108C2×C4×C269—22no—
109C2×C4×C269—24no—
110C2×C4×C269—22no—
111C2×C4×C269—22no—
112C2×C4×C269—22no—
113C2×C4×C269—21no—
114C2×C4×C269—42no—
115C2×C4×C269C2×D412yes—
116C2×C4×C269C2×D412yes—
117C2×C4×C269C2×D412yes—
118C2×C4×C269C2×D411yescharacteristic
119C2×C4×C269C2411yescharacteristic
120C22×C269—22no—
121C22×C269—22no—
122C22×C269—22no—
123C22×C269—22no—
124C22×C269—44no—
125C22×C269—41no—
126C22×C269—42no—
127C22×C269#12012yes—
128C22×C269C22×D411yescharacteristic
129C2×D269—24no—
130C2×D269—48no—
131C2×D269—44no—
132C4×C269—22no—
133C4×C269—41no—
134C4×C269—42no—
135C4×C269#12012yes—
136C4×C269C22×D411yescharacteristic
137C4×C269C22×D411yescharacteristic
138Dic269—24no—
139C2×C269—22no—
140C2×C269—42no—
141C2×C269—81no—
142C2×C269C2×#12011yescharacteristic
143C2×C269#57512yes—
144D269—44no—
145C269#300511yescharacteristic, complemented, Sylow
146#3005—2691nomaximal, Sylow
147C22×#41—2691no—
148C2×#137—2691no—
149C2×AGL(1,ℤ/8)—2691no—
150#525D26911yescharacteristic, complemented
151#452—2694no—
152#449—2694no—
153#419—2691no—
154#418—2691no—
155#407—2691no—
156C2×#41—2694no—
157C2×#41—5384no—
158C2×#41—5384no—
159C2×#41—5382no—
160C2×#41C2×D26911yescharacteristic
161C22×Q8—2691no—
162C22×D4—2691no—
163C22×D4—5381no—
164C23×C4—5381no—
165C23×C4C2×D26911yescharacteristic
166#137—5384no—
167AGL(1,ℤ/8)—5384no—
168C2×Q16—2692no—
169C2×SD16—2692no—
170C2×SD16—2692no—
171C2×D8—2692no—
172C2×M16—2691no—
173C2×M16—2691no—
174#123—21no—
175C22⋊Q8C2×D26912yescomplemented
176#121C2×D26912yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D26911yescharacteristic
180#103—2694no—
181#102—2694no—
182C22⋊C8—2694no—
183C24—5381no—
184C24C22⋊D26911yescharacteristic
185#41—22no—
186#41—22no—
187#41—5388no—
188#41—5388no—
189#41—53816no—
190C2×Q8—2694no—
191C2×Q8—2692no—
192C2×Q8—5382no—
193C2×Q8—5381no—
194C2×Q8C22×D26911yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—2694no—
198C2×D4—2692no—
199C2×D4—5388no—
200C2×D4—5384no—
201C2×D4—5382no—
202C2×D4—5381no—
203C2×D4C22×D26911yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—5388no—
208C22×C4—5381no—
209C22×C4—5384no—
210C22×C4C22⋊D26912yes—
211C22×C4C22⋊D26911yescharacteristic
212C22×C4C22×D26912yes—
213C22×C4C22×D26911yescharacteristic
214Q16—5384no—
215SD16—5384no—
216SD16—5384no—
217D8—5384no—
218M16—5382no—
219M16—5382no—
220C2×C8—2692no—
221C2×C8—2692no—
222C4⋊C4—22no—
223C4⋊C4C22×D26912yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—5384no—
232C23—5382no—
233C23—10764no—
234C23C2×C22⋊D26912yes—
235C23C2×C22⋊D26911yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—2694no—
239Q8—5384no—
240Q8—5382no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—2694no—
246D4—53816no—
247D4—5384no—
248D4—5382no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—53816no—
257C2×C4—5384no—
258C2×C4—5382no—
259C2×C4D4×D26912yes—
260C2×C4D4×D26912yes—
261C2×C4C2×C22⋊D26912yes—
262C2×C4C2×C22⋊D26911yescharacteristic
263C2×C4C23×D26911yescharacteristic
264C8—5382no—
265C8—5382no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—42no—
272C22—41no—
273C22—5384no—
274C22—10768no—
275C22—10764no—
276C22#2257542112yes—
277C22C22×C22⋊D26911yescharacteristic
278C4—22no—
279C4—42no—
280C4—41no—
281C4—5384no—
282C4#2257542112yes—
283C4C2×D4×D26911yescharacteristic
284C4C2×D4×D26911yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—10764no—
289C2C2×#2257542111yescharacteristic
290C2#3177140312yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#525×C269C2
socleC22×C269C22×D4
hypercenter#525D269
hypocenterC269#3005
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#16864402C2semidirect
2. C2×#22575483C2semidirect
3. C2×#22575478C2semidirect
4. #31770765C2semidirect
5. #31770748C2semidirect
6. #31770613C2semidirect
7. #31770609C2semidirect
8. #31770569C2semidirect
9. #525×C269C2semidirect
20. C2×#16864392C22semidirect
21. C2×#16864390C22semidirect
22. C2×#16864389C22semidirect
23. C2×#16864387C22semidirect
26. #22575376C22semidirect
27. #22575373C22semidirect
28. C22⋊#14524130C22semidirect
32. C22⋊Q8×C269C22semidirect
33. #121×C269C22semidirect
43. C2×Dic538D4semidirect
47. C2×D1076D4semidirect
145. C269#3005semidirect
150. #525D269semidirect
175. C22⋊Q8C2×D269semidirect
176. #121C2×D269semidirect

Maximal subgroups

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

subgroupindex
C22×#168644022
C2×#225754832
C2×#225754782
#317707652
#317707482
#317706132
#317706092
#317705692
#525×C2692
#3005269