Finite Group Databasebeta

← #46820172

Subgroups — #46820172

Order 34432.

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

normal subgroups only

This group has 732 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
1C2×#22575501C211yesmaximal, characteristic, complemented
2C2×#22575481C211yesmaximal, characteristic, complemented
3C22×#16864389C211yesmaximal, characteristic, complemented
4#31771174C211yesmaximal, characteristic, complemented
5#31770753C214yesmaximal, complemented
6#31770745C214yesmaximal, complemented
7#31770599C211yesmaximal, characteristic, complemented
8#31770573C211yesmaximal, characteristic, complemented
9#419×C269C211yesmaximal, characteristic, complemented
10#22575501—24no—
11#22575501—24no—
12C2×#16864403—21no—
13C2×#16864403C2212yescomplemented
14C2×#16864402—22no—
15C2×#16864402C2211yescharacteristic
16C2×#16864402C2211yescharacteristic
17C2×Q8×D269—21no—
18C2×Q8×D269C2212yescomplemented
19C22×D1076C2211yescharacteristic
20#22575481—24no—
21C2×#16864391C2212yes—
22C2×#16864389—24no—
23C2×#16864389C2214yes—
24C2×D2152C2212yes—
25C22×#14524130C2211yescharacteristic
26#22575450—21no—
27#22575441—21no—
28#22575428C2214yes—
29#22575423—22no—
30#22575376C2212yescomplemented
31#22575373C2212yescomplemented
32#22575365C2214yescomplemented
33C22×Q8×C269C2211yescharacteristic
34C2×M16×C269C2211yescharacteristic
35#117×C269C2211yescharacteristic
36#103×C269C2214yescomplemented
37C23×D269—21no—
38#16864403—28no—
39#16864403—24no—
40#16864403—24no—
41#16864402—216no—
42#16864402—22no—
43#16864402—22no—
44#16864402—22no—
45#16864402D414yescomplemented
46C2×Dic538—22no—
47C2×Dic538D411yescharacteristic
48C2×Dic538C2311yescharacteristic
49Q8×D269—28no—
50Q8×D269—24no—
51Q8×D269—24no—
52C2×D1076—22no—
53C2×D1076—24no—
54C2×D1076—22no—
55C2×D1076—22no—
56C2×D1076—21no—
57C2×D1076D411yescharacteristic
58C2×D1076C2312yes—
59C2×D1076C2311yescharacteristic
60C2×C22⋊D269—21no—
61C2×C22⋊D269D412yes—
62C2×C4×D269—22no—
63C2×C4×D269—22no—
64C2×C4×D269—21no—
65C2×C4×D269C2312yes—
66#16864391—24no—
67#16864389—28no—
68D2152—24no—
69C2×#14524130—22no—
70C2×#14524130C2312yes—
71#16864379—24no—
72C2×Q8×C269—21no—
73C2×Q8×C269—22no—
74C2×Q8×C269D414yescomplemented
75C2×Q8×C269C2312yes—
76C22×C4×C269—21no—
77C22×C4×C269C2311yescharacteristic
78M16×C269—22no—
79C2×C8×C269C2312yes—
80C4⋊C4×C269C2312yes—
81C22⋊C4×C269—21no—
82C42×C269—21no—
83C22×D269—21no—
84C22×D269—22no—
85C22×D269—42no—
86C22×D269—82no—
87C22×D269C2×D412yes—
88Dic538—28no—
89Dic538—22no—
90Dic538—21no—
91Dic538C2×D412yes—
92D1076—28no—
93D1076—22no—
94D1076—21no—
95D1076—22no—
96D1076—22no—
97D1076—42no—
98D1076—84no—
99D1076C2×D412yes—
100C22⋊D269—24no—
101C22⋊D269—24no—
102C2×Dic269—21no—
103C2×Dic269C2×D412yes—
104C4×D269—28no—
105C4×D269—28no—
106C4×D269—24no—
107C4×D269—22no—
108C4×D269—22no—
109#14524130—22no—
110C23×C269C2×D411yescharacteristic
111Q8×C269—24no—
112Q8×C269—22no—
113Q8×C269C2×D414yes—
114C2×C4×C269—24no—
115C2×C4×C269—22no—
116C2×C4×C269—22no—
117C2×C4×C269C2×D411yescharacteristic
118C2×C4×C269C2×D412yes—
119C2×C4×C269C2×D412yes—
120C2×C4×C269C2411yescharacteristic
121C8×C269—22no—
122C22×C269—22no—
123C22×C269#12012yes—
124C22×C269C22×D411yescharacteristic
125C2×D269—22no—
126C2×D269—22no—
127C2×D269—24no—
128C2×D269—42no—
129C2×D269—42no—
130C2×D269—44no—
131C2×D269—84no—
132C4×C269—24no—
133C4×C269—42no—
134C4×C269#12012yes—
135C4×C269C22×D411yescharacteristic
136C4×C269C22×D411yescharacteristic
137Dic269—22no—
138Dic269—22no—
139C2×C269—22no—
140C2×C269C2×#12011yescharacteristic
141C2×C269#57712yes—
142D269—42no—
143D269—42no—
144D269—82no—
145C269#301011yescharacteristic, complemented, Sylow
146#3010—2691nomaximal, Sylow
147C2×2−1+4—2691no—
148C2×AGL(1,ℤ/8)—2691no—
149C22×SD16—2691no—
150#535—2691no—
151#449—2694no—
152#448—2694no—
153#419D26911yescharacteristic, complemented
154#416—2691no—
155#408—2691no—
1562−1+4—5384no—
1572−1+4—5384no—
158C2×#41—2691no—
159C2×#41—2692no—
160C2×#41—2691no—
161C2×#41—5382no—
162C2×#41—5381no—
163C22×Q8—2692no—
164C22×Q8—5381no—
165C22×Q8C2×D26911yescharacteristic
166C22×D4—2691no—
167AGL(1,ℤ/8)—5384no—
168C2×SD16—2694no—
169C2×SD16—2692no—
170C2×SD16—5384no—
171C2×D8—2692no—
172C2×M16C2×D26911yescharacteristic
173C22×C8—2691no—
174#127—5381no—
175#124—5381no—
176#121—2694no—
177#120—5382no—
178#117C2×D26911yescharacteristic
179#103—2692no—
180#103C2×D26914yescomplemented
181#102—2692no—
182C22⋊C8—2694no—
183C24—5381no—
184#41—2694no—
185#41—53816no—
186#41—5384no—
187#41—5388no—
188#41—5384no—
189#41—5382no—
190#41—5382no—
191#41—5382no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—2691no—
195C2×Q8—2691no—
196C2×Q8—5382no—
197C2×Q8—5384no—
198C2×Q8—5388no—
199C2×Q8—5384no—
200C2×Q8D107614yescomplemented
201C2×Q8C22×D26912yes—
202C2×D4—2692no—
203C2×D4—2691no—
204C2×D4—2691no—
205C2×D4—2692no—
206C2×D4—5382no—
207C2×D4—5384no—
208C2×D4—5381no—
209C2×D4—5382no—
210C2×D4—5381no—
211C2×D4—5382no—
212C22×C4—21no—
213C22×C4—2692no—
214C22×C4—5382no—
215C22×C4—5382no—
216C22×C4—5381no—
217C22×C4C22×D26911yescharacteristic
218SD16—5384no—
219SD16—5388no—
220D8—5384no—
221M16—22no—
222C2×C8—2692no—
223C2×C8—5382no—
224C2×C8C22×D26912yes—
225C4⋊C4C22×D26912yes—
226C22⋊C4—21no—
227C22⋊C4—5384no—
228C42—21no—
229C23—2692no—
230C23—5381no—
231C23—5382no—
232C23—10762no—
233C23—21522no—
234C23D4×D26911yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—2692no—
238Q8—5388no—
239Q8—5382no—
240Q8—5381no—
241Q8C2×D107614yes—
242D4—2692no—
243D4—5388no—
244D4—5384no—
245D4—5384no—
246D4—5382no—
247D4—5382no—
248D4—5381no—
249D4—5382no—
250D4—10762no—
251D4—21524no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—2692no—
256C2×C4—5388no—
257C2×C4—5388no—
258C2×C4—5381no—
259C2×C4—5384no—
260C2×C4—5382no—
261C2×C4—5382no—
262C2×C4D4×D26911yescharacteristic
263C2×C4D4×D26912yes—
264C2×C4C2×D107612yes—
265C2×C4C23×D26911yescharacteristic
266C8—22no—
267C8—5382no—
268C22—22no—
269C22—5384no—
270C22—5382no—
271C22—5382no—
272C22—10762no—
273C22—10764no—
274C22—10762no—
275C22—21524no—
276C22#2257542312yes—
277C22C2×D4×D26911yescharacteristic
278C4—24no—
279C4—42no—
280C4—5382no—
281C4—5382no—
282C4#2257542312yes—
283C4C2×D4×D26911yescharacteristic
284C4C22×D107611yescharacteristic
285C2—22no—
286C2—10762no—
287C2—10762no—
288C2—21522no—
289C2C2×#2257542311yescharacteristic
290C2#3177141412yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#419×C269C2
socleC22×C269C22×D4
hypercenter#419D269
hypocenterC269#3010
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#22575501C2semidirect
2. C2×#22575481C2semidirect
3. C22×#16864389C2semidirect
4. #31771174C2semidirect
5. #31770753C2semidirect
6. #31770745C2semidirect
7. #31770599C2semidirect
8. #31770573C2semidirect
9. #419×C269C2semidirect
13. C2×#16864403C22semidirect
18. C2×Q8×D269C22semidirect
30. #22575376C22semidirect
31. #22575373C22semidirect
32. #22575365C22semidirect
36. #103×C269C22semidirect
45. #16864402D4semidirect
74. C2×Q8×C269D4semidirect
145. C269#3010semidirect
153. #419D269semidirect
180. #103C2×D269semidirect
200. C2×Q8D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225755012
C2×#225754812
C22×#168643892
#317711742
#317707532
#317707452
#317705992
#317705732
#419×C2692
#3010269