Finite Group Databasebeta

← #46820256

Subgroups — #46820256

Order 34432.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#31771441C211yesmaximal, characteristic, complemented
2C2×#22575479C212yesmaximal, complemented
3#31770777C214yesmaximal, complemented
4#31770772C214yesmaximal, complemented
5#31770623C212yesmaximal, complemented
6#31770589C211yesmaximal, characteristic, complemented
7#535×C269C211yesmaximal, characteristic, complemented
8#22575501C2212yescomplemented
9#22575500—22no—
10#22575499—22no—
11#22575498—22no—
12#22575498C2212yescomplemented
13C2×#16864402—22no—
14C2×#16864402C2212yes—
15#41×D269—24no—
16#22575479—24no—
17#22575479C2218yescomplemented
18C2×#16864390—22no—
19C2×#16864387—22no—
20C2×#16864386C2212yescomplemented
21#22575381C2218yescomplemented
22#22575372C2212yescomplemented
23#22575370C2212yescomplemented
24C2×#41×C269C2211yescharacteristic
25C22×D4×C269C2211yescharacteristic, complemented
26#127×C269C2212yescomplemented
27#124×C269C2212yescomplemented
28#121×C269—22no—
29#120×C269—22no—
30#117×C269C2211yescharacteristic, complemented
31#16864403—24no—
32#16864402—24no—
33#16864402—24no—
34#16864402—22no—
35#16864402—24no—
36#16864402—22no—
37#16864402D414yescomplemented
38#16864402C2314yes—
39#16864401—28no—
40#16864401—24no—
41C2×Dic538—22no—
42C2×Dic538D412yescomplemented
43Q8×D269—24no—
44C2×D1076—22no—
45C2×D1076D412yescomplemented
46C2×C22⋊D269—24no—
47C2×C22⋊D269—22no—
48D4×D269—28no—
49D4×D269—24no—
50C2×C4×D269—24no—
51C2×C4×D269—22no—
52#16864390—28no—
53#16864387—28no—
54#16864386—22no—
55#16864386C2314yes—
56C2×#14524130—22no—
57C24×C269—21no—
58#41×C269—22no—
59#41×C269—22no—
60C2×Q8×C269C2311yescharacteristic
61C2×D4×C269—24no—
62C2×D4×C269—22no—
63C2×D4×C269—22no—
64C2×D4×C269—21no—
65C2×D4×C269D412yes—
66C2×D4×C269C2312yes—
67C2×D4×C269C2311yescharacteristic
68C22×C4×C269D412yes—
69C22×C4×C269C2311yescharacteristic
70C4⋊C4×C269—21no—
71C22⋊C4×C269—21no—
72C22⋊C4×C269—24no—
73C42×C269C2312yes—
74C22×D269—24no—
75C22×D269—22no—
76Dic538—24no—
77Dic538—22no—
78Dic538C2×D414yes—
79D1076—24no—
80D1076—22no—
81D1076C2×D414yes—
82C22⋊D269—28no—
83C22⋊D269—24no—
84C22⋊D269—24no—
85C2×Dic269—24no—
86C2×Dic269—22no—
87C4×D269—28no—
88C4×D269—24no—
89C4×D269—24no—
90#14524130—24no—
91C23×C269—22no—
92C23×C269—42no—
93C23×C269—82no—
94C23×C269C2×D412yes—
95C23×C269C2×D411yescharacteristic
96Q8×C269—22no—
97D4×C269—24no—
98D4×C269—22no—
99D4×C269—24no—
100D4×C269—41no—
101D4×C269—41no—
102D4×C269—84no—
103C2×C4×C269—22no—
104C2×C4×C269—22no—
105C2×C4×C269—22no—
106C2×C4×C269C2×D412yes—
107C2×C4×C269C2×D412yes—
108C2×C4×C269C2×D412yes—
109C2×C4×C269C2×D411yescharacteristic
110C2×C4×C269C2411yescharacteristic
111C22×C269—22no—
112C22×C269—41no—
113C22×C269—42no—
114C22×C269—44no—
115C22×C269—42no—
116C22×C269—84no—
117C22×C269#12012yes—
118C22×C269C22×D411yescharacteristic
119C2×D269—24no—
120C2×D269—82no—
121C2×D269—84no—
122C4×C269—22no—
123C4×C269—42no—
124C4×C269#12011yescharacteristic
125C4×C269#12011yescharacteristic
126C4×C269C22×D412yes—
127Dic269—24no—
128C2×C269—21no—
129C2×C269—22no—
130C2×C269—42no—
131C2×C269—82no—
132C2×C269C2×#12011yescharacteristic
133D269—44no—
134C269#302111yescharacteristic, complemented, Sylow
135#3021—2691nomaximal, Sylow
136#585—2691no—
137C2×AGL(1,ℤ/8)—2692no—
138#535D26911yescharacteristic, complemented
139#454—2694no—
140#453—2694no—
141#421—2692no—
142#413—2691no—
1432−1+4—2692no—
1442−1+4—5382no—
145H(5,𝔽2)—2692no—
146H(5,𝔽2)—5382no—
147H(5,𝔽2)—5382no—
148C2×#41—2692no—
149C2×#41—5384no—
150C2×#41—5382no—
151C2×#41C2×D26911yescharacteristic
152C22×D4C2×D26911yescharacteristic, complemented
153AGL(1,ℤ/8)—2698no—
154AGL(1,ℤ/8)—5384no—
155C2×SD16—5382no—
156C2×D8—5382no—
157C2×M16—2692no—
158#127C2×D26912yescomplemented
159#124C2×D26912yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D26911yescharacteristic, complemented
163#104—2698no—
164#101—2692no—
165#100—2692no—
166C24—21no—
167#41—22no—
168#41—22no—
169#41—2694no—
170#41—2694no—
171#41—5384no—
172#41—5384no—
173#41—5384no—
174#41—5388no—
175#41—5384no—
176#41—5382no—
177#41—5382no—
178#41—5384no—
179C2×Q8—2692no—
180C2×Q8—5382no—
181C2×Q8—5384no—
182C2×Q8C22×D26911yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—2692no—
188C2×D4—5382no—
189C2×D4—5384no—
190C2×D4—5388no—
191C2×D4—5382no—
192C2×D4—5384no—
193C2×D4C22⋊D26912yes—
194C2×D4C22×D26912yes—
195C2×D4C22×D26911yescharacteristic
196C22×C4—5384no—
197C22×C4—5382no—
198C22×C4C22⋊D26912yes—
199C22×C4C22×D26911yescharacteristic
200SD16—5388no—
201D8—5388no—
202M16—2694no—
203M16—5382no—
204C2×C8—5382no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D26912yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—5384no—
213C23—5382no—
214C23C2×C22⋊D26912yes—
215C23C2×C22⋊D26911yescharacteristic
216Q8—22no—
217Q8—2694no—
218Q8—5384no—
219Q8—5382no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—2694no—
227D4—5384no—
228D4—5388no—
229D4—5384no—
230D4—5384no—
231D4—5382no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—5388no—
236C2×C4—5384no—
237C2×C4—5382no—
238C2×C4—5384no—
239C2×C4—5384no—
240C2×C4D4×D26912yes—
241C2×C4D4×D26912yes—
242C2×C4C2×C22⋊D26912yes—
243C2×C4C2×C22⋊D26911yescharacteristic
244C2×C4C23×D26911yescharacteristic
245C8—5384no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—5384no—
253C22—21522no—
254C22—21524no—
255C22#2257542112yes—
256C22C22×C22⋊D26911yescharacteristic
257C4—22no—
258C4—42no—
259C4—5384no—
260C4#2257542111yescharacteristic
261C4#2257542111yescharacteristic
262C4C2×D4×D26912yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—10764no—
268C2C2×#2257542111yescharacteristic
subgroupG/H
centerC2C2×#22575421
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#535×C269C2
socleC2×C269C2×#120
hypercenter#535D269
hypocenterC269#3021
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. C2×#22575479C2semidirect
3. #31770777C2semidirect
4. #31770772C2semidirect
5. #31770623C2semidirect
6. #31770589C2semidirect
7. #535×C269C2semidirect
8. #22575501C22semidirect
12. #22575498C22semidirect
17. #22575479C22semidirect
20. C2×#16864386C22semidirect
21. #22575381C22semidirect
22. #22575372C22semidirect
23. #22575370C22semidirect
25. C22×D4×C269C22semidirect
26. #127×C269C22semidirect
27. #124×C269C22semidirect
30. #117×C269C22semidirect
37. #16864402D4semidirect
42. C2×Dic538D4semidirect
45. C2×D1076D4semidirect
134. C269#3021semidirect
138. #535D269semidirect
152. C22×D4C2×D269semidirect
158. #127C2×D269semidirect
159. #124C2×D269semidirect
162. #117C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
C2×#225754792
#317707772
#317707722
#317706232
#317705892
#535×C2692
#3021269