Finite Group Databasebeta

← #46824212

Subgroups — #46824212

Order 34432.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D269C212yesmaximal, complemented
2#31771182C214yesmaximal, complemented
3#31771180C212yesmaximal, complemented
4#31771175C212yesmaximal, complemented
5#31771173C212yesmaximal, complemented
6#31771168C212yesmaximal, complemented
7#31771163C214yesmaximal, complemented
8#31771155C211yesmaximal, characteristic, complemented
9#31771151C212yesmaximal, complemented
10#31771114C218yesmaximal, complemented
11C2×#22575410C211yesmaximal, characteristic, complemented
12#532×C269C211yesmaximal, characteristic, complemented
13C2×#16864403C2212yes—
14C2×#16864402C2214yes—
15C2×#16864402C2212yescomplemented
16C2×#16864401C2212yes—
17C2×#16864401C2214yescomplemented
18#41×D269—216no—
19C2×Q8×D269C2212yescomplemented
20C2×D4×D269C2214yes—
21C2×D4×D269C2212yescomplemented
22C22×C4×D269C2211yescharacteristic
23C22×C4×D269C2214yescomplemented
24#22575453C2211yescharacteristic, complemented
25#22575451C2212yescomplemented
26#22575450C2211yescharacteristic, complemented
27#22575449C2212yescomplemented
28#22575442C2212yescomplemented
29#22575441C2212yescomplemented
30#22575440C2212yescomplemented
31#22575438C2218yescomplemented
32#22575435C2218yescomplemented
33#22575431C2218yescomplemented
34#22575430C2218yescomplemented
35#22575427C2214yescomplemented
36#22575426C2218yescomplemented
37#22575425C2218yescomplemented
38#22575425C2218yescomplemented
39#22575423C2218yescomplemented
40#22575421C2218yescomplemented
41C4×D1076C2214yescomplemented
42C4×C22⋊D269C2218yescomplemented
43#22575410C2212yescomplemented
44#22575410C2212yescomplemented
45#22575410C2214yescomplemented
46C2×#16864381C2212yescomplemented
47C2×#16864379C2212yes—
48C2×C4×Dic269C2211yescharacteristic, complemented
49C2×#41×C269C2212yescomplemented
50C4⋊Q8×C269C2211yescharacteristic, complemented
51#127×C269C2211yescharacteristic, complemented
52#124×C269C2212yescomplemented
53#121×C269C2214yescomplemented
54C4×D4×C269C2214yescomplemented
55C2×C42×C269C2211yescharacteristic, complemented
56C23×D269C2314yes—
57C23×D269C2311yescharacteristic
58#16864403—28no—
59#16864402—216no—
60#16864402—28no—
61#16864401—216no—
62#16864401—28no—
63C2×Dic538C2314yes—
64C2×Dic538C2312yes—
65Q8×D269—28no—
66C2×D1076C2314yes—
67C2×D1076C2312yes—
68C2×D1076C2314yes—
69C2×C22⋊D269C2314yes—
70C2×C22⋊D269C2314yes—
71C2×C22⋊D269C2314yes—
72C2×C22⋊D269C2318yes—
73D4×D269—216no—
74D4×D269—28no—
75C22×Dic269C2314yes—
76C22×Dic269C2312yes—
77C22×Dic269C2311yescharacteristic
78C2×C4×D269—216no—
79C2×C4×D269D418yescomplemented
80C2×C4×D269C2314yes—
81C2×C4×D269C2314yes—
82C2×C4×D269C2314yes—
83C2×C4×D269C2312yes—
84C2×C4×D269C2312yes—
85C4⋊Dic269C2314yes—
86#16864381C2314yes—
87#16864381C2314yes—
88#16864381C2318yes—
89C22⋊Dic269C2318yes—
90#16864379C2314yes—
91#16864379C2314yes—
92#16864379C2318yes—
93#16864379C2318yes—
94C4×Dic269C2312yes—
95C4×Dic269C2312yes—
96#41×C269—28no—
97C2×Q8×C269C2312yes—
98C2×D4×C269C2314yes—
99C2×D4×C269C2312yes—
100C2×D4×C269C2314yes—
101C22×C4×C269C2314yes—
102C22×C4×C269C2312yes—
103C22×C4×C269C2311yescharacteristic
104C4⋊C4×C269C2314yes—
105C22⋊C4×C269C2318yes—
106C42×C269C2312yes—
107C42×C269C2311yescharacteristic
108C42×C269C2311yescharacteristic
109C22×D269—28no—
110C22×D269—42no—
111C22×D269—48no—
112C22×D269C2×D414yes—
113C22×D269C2414yes—
114C22×D269C2412yes—
115Dic538—24no—
116Dic538—28no—
117D1076—28no—
118D1076—24no—
119D1076—44no—
120C22⋊D269—28no—
121C22⋊D269—28no—
122C22⋊D269—28no—
123C22⋊D269—48no—
124C2×Dic269—28no—
125C2×Dic269—42no—
126C2×Dic269C2×D414yes—
127C2×Dic269C2414yes—
128C2×Dic269C2414yes—
129C2×Dic269C2412yes—
130C4×D269—28no—
131C4×D269—28no—
132C4×D269—28no—
133C4×D269C2×D418yes—
134C4×D269C2×D418yes—
135C23×C269C2414yes—
136C23×C269C2411yescharacteristic
137Q8×C269—24no—
138D4×C269—28no—
139D4×C269—24no—
140D4×C269—44no—
141C2×C4×C269—28no—
142C2×C4×C269—42no—
143C2×C4×C269C2×D414yes—
144C2×C4×C269C2412yes—
145C2×C4×C269C2412yes—
146C2×C4×C269C2414yes—
147C2×C4×C269C2411yescharacteristic
148C2×C4×C269C2411yescharacteristic
149C22×C269—22no—
150C22×C269—24no—
151C22×C269—28no—
152C22×C269C22×D412yes—
153C22×C269C2511yescharacteristic
154C2×D269—24no—
155C2×D269—24no—
156C2×D269—28no—
157C2×D269—44no—
158C2×D269—416no—
159C2×D269C22×D414yes—
160C4×C269—24no—
161C4×C269—22no—
162C4×C269—22no—
163C4×C269C22×D412yes—
164C4×C269C22×D412yes—
165Dic269—24no—
166Dic269—24no—
167Dic269C22×D414yes—
168C2×C269—22no—
169C2×C269—44no—
170C2×C269C23×D411yescharacteristic
171C2×C269#58512yes—
172D269—24no—
173D269—44no—
174C269#344911yescharacteristic, complemented, Sylow
175#3449—2691nomaximal, Sylow
176C22×#41—2692no—
177#537—2694no—
178#536—2692no—
179#536—2692no—
180#535—2692no—
181#535—2692no—
182#534—2694no—
183#532—2692no—
184#532—2691no—
185#532D26911yescharacteristic, complemented
186#525—2698no—
187C2×#117—2691no—
188C2×#41—2692no—
189C2×#41—2692no—
190C2×#41—2694no—
191C2×#41—2694no—
192C2×#41—2692no—
193C2×#41—53816no—
194C2×#41C2×D26912yescomplemented
195C22×Q8—2692no—
196C22×D4—2694no—
197C22×D4—2692no—
198C23×C4—2691no—
199C23×C4—2694no—
200C4⋊Q8—2692no—
201C4⋊Q8—2691no—
202C4⋊Q8C2×D26911yescharacteristic, complemented
203#127—2692no—
204#127—2691no—
205#127C2×D26911yescharacteristic, complemented
206#124—2692no—
207#124—2692no—
208#124—2692no—
209#124C2×D26912yescomplemented
210#123—2698no—
211#123—2698no—
212C22⋊Q8—2698no—
213C22⋊Q8—2698no—
214#121—2698no—
215#121—2698no—
216#121—2694no—
217#121—2698no—
218#121C2×D26914yescomplemented
219#120—2698no—
220#120—2698no—
221C4×D4—2698no—
222C4×D4—2694no—
223C4×D4C2×D26914yescomplemented
224#117—2694no—
225#117—2692no—
226#117—2692no—
227C2×C4⋊C4—2692no—
228C2×C22⋊C4—2692no—
229C2×C42—2691no—
230C2×C42C2×D26911yescharacteristic, complemented
231C24—2691no—
232C24—2694no—
233#41—28no—
234#41—53816no—
235#41—53816no—
236#41—5388no—
237#41—5388no—
238#41—5388no—
239C2×Q8—2694no—
240C2×Q8—2692no—
241C2×Q8—5388no—
242C2×Q8C22×D26912yes—
243C2×D4—2694no—
244C2×D4—2694no—
245C2×D4—2694no—
246C2×D4—2698no—
247C2×D4—2694no—
248C2×D4—2692no—
249C2×D4—2694no—
250C2×D4—5388no—
251C2×D4—53816no—
252C2×D4C22×D26914yes—
253C2×D4C22×D26912yes—
254C2×D4C22×D26914yes—
255C22×C4—2691no—
256C22×C4—2698no—
257C22×C4—2692no—
258C22×C4—2692no—
259C22×C4—2694no—
260C22×C4—2692no—
261C22×C4—2694no—
262C22×C4—2694no—
263C22×C4—2694no—
264C22×C4—53816no—
265C22×C4C22×D26914yes—
266C22×C4C22×D26912yes—
267C22×C4C22×D26911yescharacteristic
268C4⋊C4—2694no—
269C4⋊C4—2694no—
270C4⋊C4—2698no—
271C4⋊C4—2694no—
272C4⋊C4C22×D26914yes—
273C22⋊C4—2698no—
274C22⋊C4—2694no—
275C22⋊C4—2694no—
276C22⋊C4—2698no—
277C22⋊C4—2698no—
278C22⋊C4C22×D26918yes—
279C42—2692no—
280C42—2692no—
281C42C22×D26911yescharacteristic
282C42C22×D26911yescharacteristic
283C42C22×D26912yes—
284C23—2692no—
285C23—2694no—
286C23—2694no—
287C23—5388no—
288C23—10762no—
289C23—10768no—
290C23C23×D26914yes—
291C23C23×D26911yescharacteristic
292Q8—24no—
293Q8—5388no—
294Q8—5384no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—5388no—
299D4—5388no—
300D4—5388no—
301D4—5388no—
302D4—5384no—
303D4—10764no—
304D4—10768no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—2694no—
308C2×C4—2692no—
309C2×C4—2698no—
310C2×C4—2698no—
311C2×C4—2694no—
312C2×C4—2694no—
313C2×C4—5388no—
314C2×C4—5388no—
315C2×C4—5388no—
316C2×C4—5388no—
317C2×C4—10762no—
318C2×C4D4×D26914yes—
319C2×C4C23×D26912yes—
320C2×C4C23×D26912yes—
321C2×C4C23×D26914yes—
322C2×C4C23×D26911yescharacteristic
323C2×C4C23×D26911yescharacteristic
324C22—22no—
325C22—24no—
326C22—28no—
327C22—2694no—
328C22—5384no—
329C22—5384no—
330C22—5388no—
331C22—10764no—
332C22—107616no—
333C22C2×D4×D26912yes—
334C22C24×D26911yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—2694no—
339C4—5384no—
340C4—5384no—
341C4C2×D4×D26912yes—
342C4C2×D4×D26912yes—
343C2—22no—
344C2—44no—
345C2—5384no—
346C2—10764no—
347C2C22×D4×D26911yescharacteristic
348C2#3177144112yes—
subgroupG/H
centerC2×C4C23×D269
commutator, socle, cosocleC22×C269C25
FrattiniC22C24×D269
Fitting#532×C269C2
hypercenter#532D269
hypocenterC269#3449
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D269C2semidirect
2. #31771182C2semidirect
3. #31771180C2semidirect
4. #31771175C2semidirect
5. #31771173C2semidirect
6. #31771168C2semidirect
7. #31771163C2semidirect
8. #31771155C2semidirect
9. #31771151C2semidirect
10. #31771114C2semidirect
11. C2×#22575410C2semidirect
12. #532×C269C2semidirect
15. C2×#16864402C22semidirect
17. C2×#16864401C22semidirect
19. C2×Q8×D269C22semidirect
21. C2×D4×D269C22semidirect
23. C22×C4×D269C22semidirect
24. #22575453C22semidirect
25. #22575451C22semidirect
26. #22575450C22semidirect
27. #22575449C22semidirect
28. #22575442C22semidirect
29. #22575441C22semidirect
30. #22575440C22semidirect
31. #22575438C22semidirect
32. #22575435C22semidirect
33. #22575431C22semidirect
34. #22575430C22semidirect
35. #22575427C22semidirect
36. #22575426C22semidirect
37. #22575425C22semidirect
38. #22575425C22semidirect
39. #22575423C22semidirect
40. #22575421C22semidirect
41. C4×D1076C22semidirect
42. C4×C22⋊D269C22semidirect
43. #22575410C22semidirect
44. #22575410C22semidirect
45. #22575410C22semidirect
46. C2×#16864381C22semidirect
48. C2×C4×Dic269C22semidirect
49. C2×#41×C269C22semidirect
50. C4⋊Q8×C269C22semidirect
51. #127×C269C22semidirect
52. #124×C269C22semidirect
53. #121×C269C22semidirect
54. C4×D4×C269C22semidirect
55. C2×C42×C269C22semidirect
79. C2×C4×D269D4semidirect
174. C269#3449semidirect
185. #532D269semidirect
194. C2×#41C2×D269semidirect
202. C4⋊Q8C2×D269semidirect
205. #127C2×D269semidirect
209. #124C2×D269semidirect
218. #121C2×D269semidirect
223. C4×D4C2×D269semidirect
230. C2×C42C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2692
#317711822
#317711802
#317711752
#317711732
#317711682
#317711632
#317711552
#317711512
#317711142
C2×#225754102
#532×C2692
#3449269