Finite Group Databasebeta

← #46820167

Subgroups — #46820167

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×#22575479C211yesmaximal, characteristic, complemented
3#31771172C211yesmaximal, characteristic, complemented
4#31770749C214yesmaximal, complemented
5#31770742C214yesmaximal, complemented
6#31770614C211yesmaximal, characteristic, complemented
7#31770602C211yesmaximal, characteristic, complemented
8#31770571C211yesmaximal, characteristic, complemented
9C22×SD16×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
19#22575479—24no—
20C2×#16864390C2212yes—
21C2×#16864387C2212yes—
22C2×#16864386C2211yescharacteristic
23#22575449—21no—
24#22575440—21no—
25#22575424C2214yes—
26#22575422—22no—
27#22575412C2211yescharacteristic
28#22575378C2212yescomplemented
29#22575377C2214yescomplemented
30#22575375C2212yescomplemented
31#22575365C2214yescomplemented
32C22×Q8×C269C2211yescharacteristic
33C22×D4×C269C2211yescharacteristic
34C2×SD16×C269—24no—
35C2×SD16×C269C2214yes—
36C22×C8×C269C2211yescharacteristic
37#16864403—28no—
38#16864403—24no—
39#16864403—24no—
40#16864402—216no—
41#16864402—22no—
42#16864402—22no—
43#16864402—22no—
44#16864402D414yescomplemented
45C2×Dic538—22no—
46C2×Dic538D411yescharacteristic
47C2×Dic538C2311yescharacteristic
48Q8×D269—28no—
49Q8×D269—24no—
50Q8×D269—24no—
51C2×D1076—22no—
52C2×D1076D411yescharacteristic
53C2×D1076C2311yescharacteristic
54C2×C22⋊D269—24no—
55C2×C22⋊D269—21no—
56C2×C22⋊D269D412yes—
57C2×C4×D269—22no—
58C2×C4×D269—22no—
59C2×C4×D269—21no—
60C2×C4×D269C2312yes—
61#16864390—24no—
62#16864387—24no—
63#16864386—22no—
64C2×#14524130C2312yes—
65C4⋊Dic269C2312yes—
66C22⋊Dic269—24no—
67C22⋊Dic269—21no—
68C4×Dic269—21no—
69C24×C269—21no—
70C2×Q8×C269—21no—
71C2×Q8×C269—22no—
72C2×Q8×C269D414yescomplemented
73C2×Q8×C269C2312yes—
74C2×D4×C269—22no—
75C2×D4×C269—22no—
76C2×D4×C269—21no—
77C2×D4×C269C2312yes—
78C22×C4×C269—21no—
79C22×C4×C269C2311yescharacteristic
80SD16×C269—28no—
81C2×C8×C269—22no—
82C2×C8×C269C2312yes—
83C22×D269—21no—
84C22×D269C2×D412yes—
85Dic538—28no—
86Dic538—22no—
87Dic538—21no—
88Dic538C2×D412yes—
89D1076—28no—
90D1076—22no—
91D1076—21no—
92D1076C2×D412yes—
93C22⋊D269—24no—
94C22⋊D269—24no—
95C22⋊D269—84no—
96C2×Dic269—21no—
97C2×Dic269—22no—
98C2×Dic269C2×D412yes—
99C4×D269—28no—
100C4×D269—28no—
101C4×D269—24no—
102C4×D269—22no—
103C4×D269—22no—
104#14524130—22no—
105C23×C269—22no—
106C23×C269—42no—
107C23×C269—82no—
108C23×C269C2×D411yescharacteristic
109Q8×C269—24no—
110Q8×C269—22no—
111Q8×C269C2×D414yes—
112D4×C269—22no—
113D4×C269—22no—
114D4×C269—42no—
115C2×C4×C269—24no—
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—42no—
124C22×C269—44no—
125C22×C269—84no—
126C22×C269#12012yes—
127C22×C269C22×D411yescharacteristic
128C2×D269—22no—
129C2×D269—22no—
130C2×D269—24no—
131C2×D269—42no—
132C4×C269—24no—
133C4×C269#12012yes—
134C4×C269C22×D411yescharacteristic
135C4×C269C22×D411yescharacteristic
136Dic269—22no—
137Dic269—22no—
138Dic269—42no—
139C2×C269—22no—
140C2×C269—82no—
141C2×C269C2×#12011yescharacteristic
142C2×C269#57712yes—
143D269—42no—
144D269—42no—
145C269#301011yescharacteristic, complemented, Sylow
146#3010—2691nomaximal, Sylow
147C2×2−1+4—2691no—
148C2×AGL(1,ℤ/8)—2691no—
149C22×SD16D26911yescharacteristic, complemented
150#535—2691no—
151#449—2694no—
152#448—2694no—
153#419—2691no—
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×D4C2×D26911yescharacteristic
167AGL(1,ℤ/8)—5384no—
168C2×SD16—24no—
169C2×SD16—2692no—
170C2×SD16C2×D26914yes—
171C2×D8—2692no—
172C2×M16—2691no—
173C22×C8C2×D26911yescharacteristic
174#127—5381no—
175#124—5381no—
176#121—2694no—
177#120—5382no—
178#117—2691no—
179#103—2694no—
180#103—2692no—
181#102—2692no—
182C22⋊C8—2694no—
183C24—21no—
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×Q8C22⋊D26914yescomplemented
201C2×Q8C22×D26912yes—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—2692no—
206C2×D4—2691no—
207C2×D4—2691no—
208C2×D4—5382no—
209C2×D4—5381no—
210C2×D4—5384no—
211C2×D4C22×D26912yes—
212C22×C4—21no—
213C22×C4—2692no—
214C22×C4—5382no—
215C22×C4—5382no—
216C22×C4—5381no—
217C22×C4C22×D26911yescharacteristic
218SD16—28no—
219SD16—5384no—
220D8—5384no—
221M16—5382no—
222C2×C8—22no—
223C2×C8—2692no—
224C2×C8C22×D26912yes—
225C4⋊C4—2692no—
226C22⋊C4—5384no—
227C22⋊C4—5381no—
228C42—5381no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—2692no—
233C23—5381no—
234C23D4×D26911yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—2692no—
238Q8—5388no—
239Q8—5382no—
240Q8—5381no—
241Q8C2×C22⋊D26914yes—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—2692no—
246D4—5388no—
247D4—5384no—
248D4—5384no—
249D4—5382no—
250D4—5381no—
251D4—21524no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—2692no—
255C2×C4—5388no—
256C2×C4—5388no—
257C2×C4—5381no—
258C2×C4—5384no—
259C2×C4—5382no—
260C2×C4—5382no—
261C2×C4—5382no—
262C2×C4D4×D26911yescharacteristic
263C2×C4D4×D26912yes—
264C2×C4C2×C22⋊D26912yes—
265C2×C4C23×D26911yescharacteristic
266C8—22no—
267C8—5382no—
268C22—22no—
269C22—44no—
270C22—42no—
271C22—84no—
272C22—5384no—
273C22—5382no—
274C22—5382no—
275C22—10762no—
276C22#2257542112yes—
277C22C2×D4×D26911yescharacteristic
278C4—24no—
279C4—5382no—
280C4—5382no—
281C4—10762no—
282C4#2257542112yes—
283C4C2×D4×D26911yescharacteristic
284C4C22×C22⋊D26911yescharacteristic
285C2—22no—
286C2—82no—
287C2—10762no—
288C2—10762no—
289C2C2×#2257542111yescharacteristic
290C2#3177141112yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC22×SD16×C269C2
socleC22×C269C22×D4
hypercenterC22×SD16D269
hypocenterC269#3010
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22575501C2semidirect
2. C2×#22575479C2semidirect
3. #31771172C2semidirect
4. #31770749C2semidirect
5. #31770742C2semidirect
6. #31770614C2semidirect
7. #31770602C2semidirect
8. #31770571C2semidirect
9. C22×SD16×C269C2semidirect
13. C2×#16864403C22semidirect
18. C2×Q8×D269C22semidirect
28. #22575378C22semidirect
29. #22575377C22semidirect
30. #22575375C22semidirect
31. #22575365C22semidirect
44. #16864402D4semidirect
72. C2×Q8×C269D4semidirect
145. C269#3010semidirect
149. C22×SD16D269semidirect
200. C2×Q8C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225755012
C2×#225754792
#317711722
#317707492
#317707422
#317706142
#317706022
#317705712
C22×SD16×C2692
#3010269