Finite Group Databasebeta

← #46820284

Subgroups — #46820284

Order 34432.

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

normal subgroups only

This group has 800 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
1C2×#22575498C211yesmaximal, characteristic, complemented
2#31771167C212yesmaximal, complemented
3#31771164C211yesmaximal, characteristic, complemented
4#31770796C214yesmaximal, complemented
5#31770793C214yesmaximal, complemented
6C2×#22575367C212yesmaximal, complemented
7C2×#99×C269C211yesmaximal, characteristic, complemented
8#22575498—24no—
9#22575498—22no—
10#22575498C2214yes—
11C2×#16864402—21no—
12C2×#16864401—22no—
13C22×C22⋊D269—21no—
14C22×C22⋊D269C2212yes—
15C22×C22⋊D269C2211yescharacteristic, complemented
16C2×D4×D269—22no—
17#22575437—22no—
18#22575437C2214yescomplemented
19#22575436—21no—
20#22575434C2212yescomplemented
21#22575427—24no—
22#22575425—22no—
23#22575423—22no—
24#22575423C2214yescomplemented
25#22575422—21no—
26#22575421C2212yescomplemented
27C2×C22⋊Dic269C2211yescharacteristic, complemented
28#22575367C2218yescomplemented
29C22×D4×C269C2211yescharacteristic
30C2×C22⋊C4×C269C2212yescomplemented
31#99×C269C2214yescomplemented
32C23×D269—21no—
33C23×D269D412yescomplemented
34C23×D269C2311yescharacteristic, complemented
35#16864402—24no—
36#16864402—24no—
37#16864401—28no—
38#16864401—28no—
39C2×Dic538—21no—
40C2×D1076—21no—
41C2×D1076—24no—
42C2×C22⋊D269—22no—
43C2×C22⋊D269—24no—
44C2×C22⋊D269—22no—
45C2×C22⋊D269—22no—
46C2×C22⋊D269—24no—
47C2×C22⋊D269—22no—
48C2×C22⋊D269—22no—
49C2×C22⋊D269—21no—
50C2×C22⋊D269—22no—
51C2×C22⋊D269—21no—
52C2×C22⋊D269—22no—
53C2×C22⋊D269—22no—
54C2×C22⋊D269D414yescomplemented
55C2×C22⋊D269C2314yes—
56D4×D269—28no—
57D4×D269—28no—
58C22×Dic269—21no—
59C22×Dic269—21no—
60C22×Dic269D412yescomplemented
61C22×Dic269C2311yescharacteristic, complemented
62C2×C4×D269—22no—
63C4⋊Dic269—24no—
64#16864381—22no—
65C22⋊Dic269—21no—
66C22⋊Dic269—22no—
67C22⋊Dic269C2312yescomplemented
68#16864379—24no—
69#16864379—24no—
70#16864379—22no—
71C24×C269D411yescharacteristic, complemented
72C24×C269C2311yescharacteristic
73C2×D4×C269—22no—
74C2×D4×C269—22no—
75C2×D4×C269D412yescomplemented
76C2×D4×C269C2312yes—
77C22×C4×C269—22no—
78C22×C4×C269D411yescharacteristic, complemented
79C22⋊C4×C269—22no—
80C22⋊C4×C269C2314yescomplemented
81C22×D269—22no—
82C22×D269—24no—
83C22×D269—22no—
84C22×D269—21no—
85C22×D269—22no—
86C22×D269—81no—
87C22×D269—82no—
88C22×D269—82no—
89C22×D269C2×D414yes—
90Dic538—24no—
91D1076—24no—
92D1076—84no—
93C22⋊D269—28no—
94C22⋊D269—24no—
95C22⋊D269—24no—
96C22⋊D269—24no—
97C22⋊D269—24no—
98C22⋊D269—41no—
99C22⋊D269—41no—
100C22⋊D269—42no—
101C22⋊D269—82no—
102C2×Dic269—24no—
103C2×Dic269—22no—
104C2×Dic269—22no—
105C2×Dic269—22no—
106C2×Dic269—22no—
107C2×Dic269—21no—
108C2×Dic269—81no—
109C2×Dic269C2×D414yes—
110C4×D269—28no—
111C23×C269—21no—
112C23×C269—21no—
113C23×C269—81no—
114C23×C269—81no—
115C23×C269C2×D412yes—
116C23×C269C2×D412yes—
117C23×C269C2×D412yes—
118C23×C269C2×D411yescharacteristic
119C23×C269C2×D411yescharacteristic
120C23×C269C2411yescharacteristic
121D4×C269—24no—
122D4×C269—24no—
123C2×C4×C269—21no—
124C2×C4×C269—21no—
125C2×C4×C269—24no—
126C2×C4×C269—82no—
127C2×C4×C269C2×D412yes—
128C22×C269—22no—
129C22×C269—42no—
130C22×C269—41no—
131C22×C269—41no—
132C22×C269—42no—
133C22×C269—41no—
134C22×C269—42no—
135C22×C269—42no—
136C22×C269—81no—
137C22×C269#12011yescharacteristic
138C22×C269#12011yescharacteristic
139C22×C269#12012yes—
140C22×C269C22×D412yes—
141C22×C269C22×D411yescharacteristic
142C2×D269—24no—
143C2×D269—44no—
144C2×D269—44no—
145C2×D269—42no—
146C2×D269—44no—
147C2×D269—41no—
148C2×D269—82no—
149C2×D269—84no—
150C4×C269—22no—
151C4×C269—44no—
152Dic269—24no—
153Dic269—42no—
154Dic269—41no—
155C2×C269—21no—
156C2×C269—22no—
157C2×C269—21no—
158C2×C269—22no—
159C2×C269—21no—
160C2×C269—42no—
161C2×C269C2×#12011yescharacteristic
162D269—44no—
163D269—81no—
164C269#302611yescharacteristic, complemented, Sylow
165#3026—2691nomaximal, Sylow
166C2×H(5,𝔽2)—2691no—
167#534—2691no—
168#534—2692no—
169#458—2694no—
170B(4,𝔽2)—2694no—
171C2×#99—2692no—
172C2×#99D26911yescharacteristic, complemented
173H(5,𝔽2)—2694no—
174H(5,𝔽2)—5384no—
175H(5,𝔽2)—5382no—
176C2×#41—5382no—
177C2×#41—5381no—
178C22×D4—2692no—
179C22×D4—2691no—
180C22×D4—5382no—
181C22×D4—5381no—
182C22×D4C2×D26911yescharacteristic
183#123—2692no—
184#123—2694no—
185#123—5382no—
186#123—5381no—
187#121—5384no—
188#121—5382no—
189#120—2692no—
190#120—2694no—
191#120—5382no—
192#120—5381no—
193C2×C22⋊C4—2691no—
194C2×C22⋊C4C2×D26912yescomplemented
195#99—2698no—
196#99C2×D26914yescomplemented
197C24—2692no—
198C24—2691no—
199C24—5381no—
200C24D107611yescharacteristic, complemented
201C24C22×D26911yescharacteristic
202#41—5384no—
203#41—5388no—
204#41—5388no—
205#41—5384no—
206C2×Q8—5381no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—2694no—
210C2×D4—2694no—
211C2×D4—5381no—
212C2×D4—5384no—
213C2×D4—5388no—
214C2×D4—5388no—
215C2×D4—5382no—
216C2×D4—5384no—
217C2×D4—5382no—
218C2×D4—5382no—
219C2×D4—5382no—
220C2×D4—5382no—
221C2×D4—5384no—
222C2×D4—5382no—
223C2×D4—5382no—
224C2×D4—5381no—
225C2×D4—5381no—
226C2×D4—5382no—
227C2×D4D107612yescomplemented
228C2×D4C22×D26912yes—
229C22×C4—22no—
230C22×C4—2692no—
231C22×C4—2691no—
232C22×C4—5382no—
233C22×C4—5381no—
234C22×C4—5381no—
235C22×C4D107611yescharacteristic, complemented
236C4⋊C4—5384no—
237C4⋊C4—5382no—
238C22⋊C4—22no—
239C22⋊C4—2692no—
240C22⋊C4—5384no—
241C22⋊C4—5384no—
242C22⋊C4—5381no—
243C22⋊C4—5382no—
244C22⋊C4—5382no—
245C22⋊C4C22×D26914yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—2694no—
251C23—5384no—
252C23—5382no—
253C23—5382no—
254C23—5382no—
255C23—5381no—
256C23—21521no—
257C23—21522no—
258C23—21522no—
259C23D4×D26912yes—
260C23D4×D26912yes—
261C23C2×D107612yes—
262C23C2×D107611yescharacteristic
263C23C2×D107611yescharacteristic
264C23C23×D26911yescharacteristic
265Q8—5384no—
266D4—24no—
267D4—24no—
268D4—5384no—
269D4—5388no—
270D4—5384no—
271D4—5384no—
272D4—5384no—
273D4—5384no—
274D4—10761no—
275D4—10761no—
276D4—10762no—
277D4—21524no—
278D4—21522no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—2694no—
284C2×C4—5388no—
285C2×C4—5384no—
286C2×C4—5382no—
287C2×C4—5382no—
288C2×C4—5382no—
289C2×C4—5382no—
290C2×C4—5381no—
291C2×C4—21521no—
292C2×C4C2×D107612yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—5384no—
303C22—10764no—
304C22—10764no—
305C22—10764no—
306C22—10762no—
307C22—10761no—
308C22—21522no—
309C22—21524no—
310C22#2257542311yescharacteristic
311C22#2257542311yescharacteristic
312C22#2257542312yes—
313C22C2×D4×D26912yes—
314C22C22×D107611yescharacteristic
315C4—22no—
316C4—44no—
317C4—5384no—
318C4—10762no—
319C4—10761no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—10764no—
327C2—21521no—
328C2C2×#2257542311yescharacteristic
subgroupG/H
centerC2C2×#22575423
commutator, cosocleC23×C269C24
FrattiniC23C23×D269
FittingC2×#99×C269C2
socleC2×C269C2×#120
hypercenterC2×#99D269
hypocenterC269#3026
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#22575498C2semidirect
2. #31771167C2semidirect
3. #31771164C2semidirect
4. #31770796C2semidirect
5. #31770793C2semidirect
6. C2×#22575367C2semidirect
7. C2×#99×C269C2semidirect
15. C22×C22⋊D269C22semidirect
18. #22575437C22semidirect
20. #22575434C22semidirect
24. #22575423C22semidirect
26. #22575421C22semidirect
27. C2×C22⋊Dic269C22semidirect
28. #22575367C22semidirect
30. C2×C22⋊C4×C269C22semidirect
31. #99×C269C22semidirect
33. C23×D269D4semidirect
34. C23×D269C23semidirect
54. C2×C22⋊D269D4semidirect
60. C22×Dic269D4semidirect
61. C22×Dic269C23semidirect
67. C22⋊Dic269C23semidirect
71. C24×C269D4semidirect
75. C2×D4×C269D4semidirect
78. C22×C4×C269D4semidirect
80. C22⋊C4×C269C23semidirect
164. C269#3026semidirect
172. C2×#99D269semidirect
194. C2×C22⋊C4C2×D269semidirect
196. #99C2×D269semidirect
200. C24D1076semidirect
227. C2×D4D1076semidirect
235. C22×C4D1076semidirect
245. C22⋊C4C22×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754982
#317711672
#317711642
#317707962
#317707932
C2×#225753672
C2×#99×C2692
#3026269