Finite Group Databasebeta

← #46820113

Subgroups — #46820113

Order 34432.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×D4×D269C211yesmaximal, characteristic, complemented
2C2×#22575480C212yesmaximal, complemented
3#31771118C211yesmaximal, characteristic, complemented
4#31770755C211yesmaximal, characteristic, complemented
5#31770754C212yesmaximal, complemented
6#31770741C211yesmaximal, characteristic, complemented
7#31770740C212yesmaximal, complemented
8#31770739C211yesmaximal, characteristic, complemented
9#31770608C212yesmaximal, complemented
10#31770567C211yesmaximal, characteristic, complemented
11#450×C269C211yesmaximal, characteristic, complemented
12C24×D269—21no—
13C2×#16864403C2211yescharacteristic, complemented
14C22×D1076C2211yescharacteristic
15C22×C22⋊D269—21no—
16C2×D4×D269—28no—
17C2×D4×D269—21no—
18C2×D4×D269—21no—
19C2×D4×D269C2212yes—
20C2×D4×D269C2212yes—
21C22×C4×D269C2211yescharacteristic
22#22575480—28no—
23C2×#16864389C2212yescomplemented
24C2×D2152C2212yescomplemented
25C2×#16864387C2212yescomplemented
26C2×#16864385C2212yescomplemented
27#22575438—21no—
28#22575429C2211yescharacteristic, complemented
29#22575428C2211yescharacteristic, complemented
30#22575427C2211yescharacteristic, complemented
31#22575423—21no—
32#22575415—21no—
33#22575413—21no—
34#22575411C2211yescharacteristic, complemented
35#22575375C2212yescomplemented
36#22575374C2212yescomplemented
37#22575373C2212yescomplemented
38C22⋊#14524130C2211yescharacteristic, complemented
39#22575365C2212yescomplemented
40C22×D4×C269C2211yescharacteristic
41C2×SD16×C269C2212yescomplemented
42C22⋊Q8×C269C2211yescharacteristic, complemented
43#102×C269C2212yescomplemented
44C22⋊C8×C269C2211yescharacteristic, complemented
45C23×D269—28no—
46C23×D269—22no—
47C23×D269—22no—
48C23×D269—21no—
49C23×D269—44no—
50C23×D269—44no—
51C23×D269D411yescharacteristic
52#16864403—24no—
53C2×D1076—24no—
54C2×D1076—21no—
55C2×D1076—21no—
56C2×D1076—21no—
57C2×D1076C2312yes—
58C2×D1076C2311yescharacteristic, complemented
59C2×C22⋊D269—28no—
60C2×C22⋊D269—22no—
61C2×C22⋊D269—22no—
62C2×C22⋊D269—21no—
63D4×D269—28no—
64D4×D269—28no—
65D4×D269—24no—
66D4×D269—24no—
67D4×D269—24no—
68D4×D269D418yescomplemented
69C22×Dic269D411yescharacteristic
70C2×C4×D269—24no—
71C2×C4×D269—21no—
72C2×C4×D269D411yescharacteristic
73C2×C4×D269D411yescharacteristic
74C2×C4×D269C2311yescharacteristic
75C2×C4×D269C2311yescharacteristic
76#16864389—24no—
77D2152—24no—
78#16864387—24no—
79#16864385—24no—
80C2×#14524130C2312yescomplemented
81C4⋊Dic269C2311yescharacteristic, complemented
82#16864381—21no—
83#16864379—21no—
84#16864379—21no—
85#16864379—21no—
86#16864379—21no—
87C4×Dic269—21no—
88C24×C269—21no—
89C2×Q8×C269C2311yescharacteristic, complemented
90C2×D4×C269—24no—
91C2×D4×C269—21no—
92C2×D4×C269C2312yes—
93C22×C4×C269C2311yescharacteristic
94SD16×C269—24no—
95C2×C8×C269C2312yescomplemented
96C4⋊C4×C269—21no—
97C4⋊C4×C269C2311yescharacteristic, complemented
98C22⋊C4×C269—21no—
99C22×D269—22no—
100C22×D269—22no—
101C22×D269—22no—
102C22×D269—28no—
103C22×D269—24no—
104C22×D269—22no—
105C22×D269—28no—
106C22×D269—21no—
107C22×D269—432no—
108C22×D269—44no—
109C22×D269—44no—
110C22×D269—44no—
111C22×D269—42no—
112C22×D269—42no—
113C22×D269—44no—
114C22×D269C2×D411yescharacteristic
115C22×D269C2×D411yescharacteristic
116D1076—24no—
117D1076—22no—
118D1076—22no—
119D1076—42no—
120D1076—42no—
121D1076C2×D414yes—
122C22⋊D269—28no—
123C22⋊D269—28no—
124C22⋊D269—42no—
125C2×Dic269—22no—
126C2×Dic269—21no—
127C2×Dic269C2×D411yescharacteristic
128C2×Dic269C2×D411yescharacteristic
129C4×D269—22no—
130C4×D269—22no—
131C4×D269—22no—
132C4×D269—42no—
133C4×D269C2×D414yes—
134#14524130—22no—
135C23×C269—24no—
136C23×C269—22no—
137C23×C269—42no—
138C23×C269—42no—
139C23×C269C2×D411yescharacteristic
140Q8×C269—22no—
141D4×C269—24no—
142D4×C269—22no—
143D4×C269C2×D414yes—
144C2×C4×C269—22no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2411yescharacteristic
149C8×C269—22no—
150C22×C269—21no—
151C22×C269—21no—
152C22×C269—24no—
153C22×C269—48no—
154C22×C269—42no—
155C22×C269—42no—
156C22×C269AGL(1,ℤ/8)12yescomplemented
157C22×C269C22×D411yescharacteristic
158C2×D269—28no—
159C2×D269—21no—
160C2×D269—21no—
161C2×D269—21no—
162C2×D269—21no—
163C2×D269—24no—
164C2×D269—416no—
165C2×D269—416no—
166C2×D269—48no—
167C2×D269—42no—
168C2×D269—42no—
169C2×D269—41no—
170C2×D269—41no—
171C2×D269—41no—
172C2×D269#12012yes—
173C2×D269AGL(1,ℤ/8)12yescomplemented
174C4×C269—21no—
175C4×C269—41no—
176C4×C269—41no—
177C4×C269C22×D412yes—
178Dic269—21no—
179Dic269—41no—
180Dic269#12012yes—
181C2×C269—22no—
182C2×C269—44no—
183C2×C269C2×#12011yescharacteristic
184C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
186D269—22no—
187D269—22no—
188D269—44no—
189D269—81no—
190C269#300311yescharacteristic, complemented, Sylow
191#3003—2691nomaximal, Sylow
192C23×D4—2691no—
193C2×AGL(1,ℤ/8)—2692no—
194#525—2691no—
195#450—2692no—
196#450—2691no—
197#450D26911yescharacteristic, complemented
198#447—2691no—
199#447—2692no—
200#447—2691no—
201#418—2692no—
202#407—2691no—
203C25—5381no—
204C2×#41—2691no—
205C22×D4—2692no—
206C22×D4—2692no—
207C22×D4—2691no—
208C22×D4—5381no—
209C22×D4—5381no—
210C22×D4—5381no—
211C22×D4—5388no—
212C22×D4C2×D26911yescharacteristic
213C23×C4—2691no—
214AGL(1,ℤ/8)—5388no—
215C2×SD16—2692no—
216C2×SD16C2×D26912yescomplemented
217C2×D8—2692no—
218C2×D8—2692no—
219C2×M16—2692no—
220#123—5381no—
221C22⋊Q8—2691no—
222C22⋊Q8C2×D26911yescharacteristic, complemented
223#121—2691no—
224#121—2691no—
225#120—5381no—
226C4×D4—5381no—
227C4×D4—5381no—
228#117—2691no—
229#102—2692no—
230#102—2692no—
231#102—2692no—
232#102C2×D26912yescomplemented
233C22⋊C8—2691no—
234C22⋊C8—2692no—
235C22⋊C8C2×D26911yescharacteristic, complemented
236C24—21no—
237C24—2691no—
238C24—5382no—
239C24—5382no—
240C24—5388no—
241C24—5381no—
242C24—10764no—
243C24—10764no—
244#41—5384no—
245C2×Q8C22×D26911yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—2698no—
249C2×D4—2692no—
250C2×D4—2691no—
251C2×D4—5384no—
252C2×D4—5381no—
253C2×D4—5381no—
254C2×D4—5384no—
255C2×D4—5382no—
256C2×D4—5382no—
257C2×D4—5384no—
258C2×D4—5388no—
259C2×D4—5388no—
260C2×D4—5388no—
261C2×D4—5384no—
262C2×D4—5381no—
263C2×D4—5381no—
264C2×D4C22×D26912yes—
265C22×C4—2691no—
266C22×C4—2691no—
267C22×C4—2691no—
268C22×C4—2691no—
269C22×C4—2691no—
270C22×C4—5384no—
271C22×C4—5381no—
272C22×C4C22×D26911yescharacteristic
273SD16—24no—
274SD16—5384no—
275D8—5384no—
276D8—5384no—
277M16—5384no—
278C2×C8—2692no—
279C2×C8C22×D26912yescomplemented
280C4⋊C4—21no—
281C4⋊C4—2691no—
282C4⋊C4—5381no—
283C4⋊C4C22×D26911yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—5381no—
286C22⋊C4—5381no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C42—5381no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—2691no—
295C23—2691no—
296C23—5382no—
297C23—5382no—
298C23—5388no—
299C23—5384no—
300C23—5388no—
301C23—5382no—
302C23—5382no—
303C23—5381no—
304C23—10762no—
305C23—10764no—
306C23—107632no—
307C23—10764no—
308C23—10764no—
309C23—10764no—
310C23—10762no—
311C23D4×D26911yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—2694no—
316D4—5384no—
317D4—5388no—
318D4—5388no—
319D4—5382no—
320D4—5382no—
321D4—10762no—
322D4—10762no—
323D4—10762no—
324D4D4×D26914yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—2691no—
329C2×C4—2694no—
330C2×C4—2691no—
331C2×C4—5382no—
332C2×C4—5382no—
333C2×C4—5382no—
334C2×C4—5382no—
335C2×C4—5381no—
336C2×C4—10762no—
337C2×C4D4×D26911yescharacteristic
338C2×C4C23×D26911yescharacteristic
339C8—22no—
340C8—5382no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—2692no—
348C22—2692no—
349C22—5388no—
350C22—5381no—
351C22—5381no—
352C22—5381no—
353C22—5381no—
354C22—5384no—
355C22—10768no—
356C22—107616no—
357C22—107616no—
358C22—10762no—
359C22—10762no—
360C22—10761no—
361C22—10761no—
362C22—10761no—
363C22#2257548012yescomplemented
364C22C2×D4×D26911yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—2692no—
369C4—5381no—
370C4—10761no—
371C4C2×D4×D26912yes—
372C2—22no—
373C2—44no—
374C2—5382no—
375C2—5382no—
376C2—10764no—
377C2—21521no—
378C2#120×D26911yescharacteristic
379C2AGL(1,ℤ/8)×D26911yescharacteristic
380C2C2×#2257548011yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#450×C269C2
socleC22×C269C22×D4
hypercenter#450D269
hypocenterC269#3003
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D269C2semidirect
2. C2×#22575480C2semidirect
3. #31771118C2semidirect
4. #31770755C2semidirect
5. #31770754C2semidirect
6. #31770741C2semidirect
7. #31770740C2semidirect
8. #31770739C2semidirect
9. #31770608C2semidirect
10. #31770567C2semidirect
11. #450×C269C2semidirect
13. C2×#16864403C22semidirect
23. C2×#16864389C22semidirect
24. C2×D2152C22semidirect
25. C2×#16864387C22semidirect
26. C2×#16864385C22semidirect
28. #22575429C22semidirect
29. #22575428C22semidirect
30. #22575427C22semidirect
34. #22575411C22semidirect
35. #22575375C22semidirect
36. #22575374C22semidirect
37. #22575373C22semidirect
38. C22⋊#14524130C22semidirect
39. #22575365C22semidirect
41. C2×SD16×C269C22semidirect
42. C22⋊Q8×C269C22semidirect
43. #102×C269C22semidirect
44. C22⋊C8×C269C22semidirect
58. C2×D1076C23semidirect
68. D4×D269D4semidirect
80. C2×#14524130C23semidirect
81. C4⋊Dic269C23semidirect
89. C2×Q8×C269C23semidirect
95. C2×C8×C269C23semidirect
97. C4⋊C4×C269C23semidirect
156. C22×C269AGL(1,ℤ/8)semidirect
173. C2×D269AGL(1,ℤ/8)semidirect
190. C269#3003semidirect
197. #450D269semidirect
216. C2×SD16C2×D269semidirect
222. C22⋊Q8C2×D269semidirect
232. #102C2×D269semidirect
235. C22⋊C8C2×D269semidirect
245. C2×Q8C22×D269semidirect
279. C2×C8C22×D269semidirect
283. C4⋊C4C22×D269semidirect
363. C22#22575480semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2692
C2×#225754802
#317711182
#317707552
#317707542
#317707412
#317707402
#317707392
#317706082
#317705672
#450×C2692
#3003269