Finite Group Databasebeta

← #46822680

Subgroups — #46822680

Order 34432.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#31771416C212yesmaximal, complemented
2#31771415C212yesmaximal, complemented
3#31770977C211yesmaximal, characteristic, complemented
4#31770867C212yesmaximal, complemented
5#31770866C212yesmaximal, complemented
6#31770789C212yesmaximal, complemented
7#31770772C211yesmaximal, characteristic, complemented
8#31770719C212yesmaximal, complemented
9#453×C269C211yesmaximal, characteristic, complemented
10#22575500C2212yes—
11#22575498C2211yescharacteristic, complemented
12#22575486C2212yescomplemented
13#22575483—22no—
14#22575483C2212yescomplemented
15#22575482—22no—
16#22575479—22no—
17#22575479C2212yes—
18#22575477—22no—
19#22575473—22no—
20#22575473C2212yes—
21#22575472—22no—
22C2×#16864390—22no—
23C2×#16864390C2212yescomplemented
24C2×#16864390C2212yescomplemented
25SD16×D269—22no—
26#22575461C2212yescomplemented
27#22575460C2212yescomplemented
28#22575453C2211yescharacteristic, complemented
29#22575388C2212yescomplemented
30#22575381C2212yescomplemented
31#22575371C2212yescomplemented
32#22575370C2211yescharacteristic, complemented
33C4×#14524130C2211yescharacteristic, complemented
34H(5,𝔽2)×C269C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C269C2212yes—
36#127×C269C2211yescharacteristic
37#104×C269C2212yescomplemented
38#100×C269C2211yescharacteristic, complemented
39#16864402—22no—
40#16864402C2312yes—
41#16864401—22no—
42#16864401—22no—
43#16864401—42no—
44C2×Dic538—22no—
45C2×Dic538C2312yes—
46Q8×D269—22no—
47C2×C22⋊D269—41no—
48D4×D269—22no—
49Dic1076—22no—
50#16864392—22no—
51#16864392D412yescomplemented
52#16864391—22no—
53#16864390—22no—
54#16864390—22no—
55#16864390—22no—
56#16864390—22no—
57#16864390—22no—
58#16864390—22no—
59#16864390D412yescomplemented
60#16864389D412yescomplemented
61#16864387—22no—
62#16864387D412yescomplemented
63#16864386—22no—
64#16864386C2312yes—
65#16864385—22no—
66C2×#14524130—22no—
67C2×#14524130C2312yes—
68C8×D269—22no—
69C4⋊Dic269—41no—
70#41×C269—22no—
71#41×C269C2312yes—
72C2×D4×C269—22no—
73C2×D4×C269—41no—
74C2×D4×C269—41no—
75C2×D4×C269C2311yescharacteristic
76C2×D4×C269C2311yescharacteristic
77SD16×C269—22no—
78D8×C269—22no—
79M16×C269C2312yes—
80C42×C269C2311yescharacteristic
81C22×D269—41no—
82Dic538—22no—
83Dic538—22no—
84Dic538—42no—
85Dic538C2×D412yes—
86D1076C2×D412yes—
87C22⋊D269—22no—
88C22⋊D269—42no—
89C22⋊D269—42no—
90C2×Dic269—22no—
91C2×Dic269—41no—
92C2×Dic269—42no—
93C4×D269—22no—
94C4×D269—42no—
95#14524130—22no—
96#14524130C2×D412yes—
97#14524130C2×D412yes—
98C23×C269—21no—
99C23×C269—21no—
100C23×C269—41no—
101Q8×C269C2×D412yes—
102D4×C269—22no—
103D4×C269—22no—
104D4×C269—22no—
105D4×C269—42no—
106D4×C269—82no—
107D4×C269C2×D412yes—
108C2×C4×C269—21no—
109C2×C4×C269—22no—
110C2×C4×C269—41no—
111C2×C4×C269C2411yescharacteristic
112C8×C269—22no—
113C22×C269—22no—
114C22×C269—41no—
115C22×C269—41no—
116C22×C269—81no—
117C22×C269—81no—
118C22×C269—161no—
119C22×C269H(5,𝔽2)11yescharacteristic
120C2×D269—22no—
121C2×D269—161no—
122C4×C269—22no—
123C4×C269—22no—
124C4×C269C22×D412yes—
125Dic269—22no—
126Dic269—42no—
127C2×C269—21no—
128C2×C269—42no—
129C2×C269—81no—
130C2×C269—81no—
131C2×C269D4211yescharacteristic
132D269—42no—
133C269#329111yescharacteristic, complemented, Sylow
134#3291—2691nomaximal, Sylow
135#577—2692no—
136#577—2692no—
137#492—2691no—
138#471—2692no—
139#471—2692no—
140#456—2692no—
141#453—2691no—
142#453D26911yescharacteristic, complemented
143#443—2692no—
1442−1+4—2692no—
145H(5,𝔽2)—2691no—
146H(5,𝔽2)C2×D26911yescharacteristic, complemented
147#137—2692no—
148#137—2692no—
149#137—5382no—
150#137—5382no—
151AGL(1,ℤ/8)—2692no—
152AGL(1,ℤ/8)—5382no—
153AGL(1,ℤ/8)—5382no—
154AGL(1,ℤ/8)C2×D26912yes—
155#135—2692no—
156#135—5382no—
157#135—5382no—
158C2×SD16—2692no—
159C2×SD16—2692no—
160C2×SD16—5382no—
161C2×SD16—5382no—
162#131—2692no—
163#131—2692no—
164C4⋊Q8—2691no—
165#127C2×D26911yescharacteristic
166#108—2692no—
167#104—2692no—
168#104C2×D26912yescomplemented
169#101—2692no—
170#100—2691no—
171#100C2×D26911yescharacteristic, complemented
172C4×C8—2691no—
173#41—22no—
174#41—2692no—
175#41—5382no—
176#41—5382no—
177#41—5382no—
178#41—10762no—
179#41C22×D26912yes—
180C2×Q8—2692no—
181C2×Q8—5382no—
182C2×Q8—5382no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—5382no—
187C2×D4—10761no—
188C2×D4C22×D26911yescharacteristic
189C2×D4C22×D26911yescharacteristic
190Q16—2692no—
191Q16—5382no—
192Q16—5382no—
193SD16—22no—
194SD16—2692no—
195SD16—2692no—
196SD16—5382no—
197SD16—5382no—
198SD16—5382no—
199SD16—5382no—
200SD16—5382no—
201SD16—5382no—
202SD16—5382no—
203D8—22no—
204D8—2692no—
205D8—5382no—
206M16—2692no—
207M16—5382no—
208M16—5382no—
209M16C22×D26912yes—
210C2×C8—2692no—
211C2×C8—5382no—
212C2×C8—5382no—
213C4⋊C4—10761no—
214C42C22×D26911yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—10761no—
219Q8—2692no—
220Q8—5382no—
221Q8—5382no—
222Q8—10762no—
223Q8D4×D26912yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—2692no—
230D4—5382no—
231D4—10762no—
232D4—10762no—
233D4D4×D26912yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—5382no—
238C2×C4—5382no—
239C2×C4—10762no—
240C2×C4—10761no—
241C2×C4—10762no—
242C2×C4C23×D26911yescharacteristic
243C8—22no—
244C8—2692no—
245C8—2692no—
246C8—5382no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—5382no—
254C22—43041no—
255C22#2257549811yescharacteristic
256C4—22no—
257C4—22no—
258C4—5382no—
259C4—10762no—
260C4C2×D4×D26912yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—10762no—
266C2#3177123711yescharacteristic
subgroupG/H
centerC2#31771237
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#453×C269C2
socleC2×C269D42
hypercenter#453D269
hypocenterC269#3291
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #31771416C2semidirect
2. #31771415C2semidirect
3. #31770977C2semidirect
4. #31770867C2semidirect
5. #31770866C2semidirect
6. #31770789C2semidirect
7. #31770772C2semidirect
8. #31770719C2semidirect
9. #453×C269C2semidirect
11. #22575498C22semidirect
12. #22575486C22semidirect
14. #22575483C22semidirect
23. C2×#16864390C22semidirect
24. C2×#16864390C22semidirect
26. #22575461C22semidirect
27. #22575460C22semidirect
28. #22575453C22semidirect
29. #22575388C22semidirect
30. #22575381C22semidirect
31. #22575371C22semidirect
32. #22575370C22semidirect
33. C4×#14524130C22semidirect
34. H(5,𝔽2)×C269C22semidirect
37. #104×C269C22semidirect
38. #100×C269C22semidirect
51. #16864392D4semidirect
59. #16864390D4semidirect
60. #16864389D4semidirect
62. #16864387D4semidirect
133. C269#3291semidirect
142. #453D269semidirect
146. H(5,𝔽2)C2×D269semidirect
168. #104C2×D269semidirect
171. #100C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714162
#317714152
#317709772
#317708672
#317708662
#317707892
#317707722
#317707192
#453×C2692
#3291269