Finite Group Databasebeta

← #548×D271 #46849826

Subgroups — #548×D271 #46849826

Order 34688.

1620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D271C211yesmaximal, characteristic, complemented
2C22×Q8×D271C211yesmaximal, characteristic, complemented
3#31774559C211yesmaximal, characteristic, complemented
4#31774558C211yesmaximal, characteristic, complemented
5#31774557C213yesmaximal, complemented
6#31774556C213yesmaximal, complemented
7#31774555C213yesmaximal, complemented
8#31774554C211yesmaximal, characteristic, complemented
9#31774553C213yesmaximal, complemented
10#124×D271C213yesmaximal, complemented
11C22⋊Q8×D271C213yesmaximal, complemented
12#121×D271C213yesmaximal, complemented
13C4×Q8×D271C211yesmaximal, characteristic, complemented
14C4×D4×D271C213yesmaximal, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic, complemented
17C2×#16864654C2213yescomplemented
18C2×#16864653C2213yescomplemented
19#41×D271—28no—
20C22×Dic542C2213yescomplemented
21C2×Q8×D271—28no—
22C2×Q8×D271C2213yes—
23C2×Q8×D271C2211yescharacteristic
24C2×Q8×D271C2211yescharacteristic, complemented
25C2×Q8×D271C2213yescomplemented
26C2×D4×D271C2213yes—
27C2×D4×D271C2213yes—
28C22×C4×D271C2213yes—
29C22×C4×D271C2213yes—
30#22576149C2213yescomplemented
31#22576148C2213yescomplemented
32#22576147C2213yescomplemented
33#22576146C2216yescomplemented
34#22576146C2216yescomplemented
35#22576140C2213yescomplemented
36#22576139C2213yescomplemented
37#22576138C2216yescomplemented
38#22576137C2216yescomplemented
39#22576136C2213yescomplemented
40#22576135C2213yescomplemented
41#22576134C2213yescomplemented
42#22576133C2216yescomplemented
43#22576132C2216yescomplemented
44#22576131C2213yescomplemented
45Q8×Dic271C2211yescharacteristic, complemented
46C4×Dic542C2213yescomplemented
47#22576125C2213yescomplemented
48C4×D1084C2213yescomplemented
49D4×Dic271C2213yescomplemented
50#22576122C2213yescomplemented
51C4×C22⋊D271C2213yescomplemented
52C4×C22⋊D271C2213yescomplemented
53(C22⋊D271)⋊C4C2213yescomplemented
54(C22⋊D271)⋊C4C2213yescomplemented
55C4⋊C4×D271C2213yes—
56C4⋊C4×D271C2213yes—
57C22⋊C4×D271C2211yescharacteristic
58C22⋊C4×D271C2213yes—
59C22⋊C4×D271C2216yes—
60C42×D271C2213yes—
61C2×#41×C271C2211yescharacteristic, complemented
62C22×Q8×C271C2211yescharacteristic, complemented
63#124×C271C2213yescomplemented
64C22⋊Q8×C271C2213yescomplemented
65#121×C271C2213yescomplemented
66C4×Q8×C271C2211yescharacteristic, complemented
67C4×D4×C271C2213yescomplemented
68C23×D271C2313yes—
69C23×D271C2311yescharacteristic
70#16864655—24no—
71#16864654—212no—
72#16864653—212no—
73C2×Dic542—212no—
74C2×Dic542C2316yes—
75C2×Dic542C2313yes—
76C2×Dic542C2313yes—
77Q8×D271—212no—
78Q8×D271—212no—
79Q8×D271—24no—
80Q8×D271D418yescomplemented
81C2×D1084C2313yes—
82C2×D1084C2313yes—
83C2×C22⋊D271C2313yes—
84C2×C22⋊D271C2313yes—
85C2×C22⋊D271C2316yes—
86D4×D271—212no—
87D4×D271—212no—
88C22×Dic271C2313yes—
89C22×Dic271C2313yes—
90C2×C4×D271—212no—
91C2×C4×D271—212no—
92C2×C4×D271C2313yes—
93C2×C4×D271C2313yes—
94C2×C4×D271C2313yes—
95C2×C4×D271C2313yes—
96C2×C4×D271C2311yescharacteristic
97C2×C4×D271C2313yes—
98C2×C4×D271C2313yes—
99C2×C4×D271C2311yescharacteristic
100C2×C4×D271C2313yes—
101C4⋊Dic271C2313yes—
102C4⋊Dic271C2313yes—
103Dic271⋊C4C2316yes—
104Dic271⋊C4C2313yes—
105Dic271⋊C4C2313yes—
106C22⋊Dic271C2316yes—
107C22⋊Dic271C2313yes—
108C22⋊Dic271C2311yescharacteristic
109(C2×D271)⋊C4C2316yes—
110(C2×D271)⋊C4C2316yes—
111(C2×D271)⋊C4C2313yes—
112(C2×D271)⋊C4C2313yes—
113(C2×D271)⋊C4C2311yescharacteristic
114(C2×D271)⋊C4C2311yescharacteristic
115C4×Dic271C2313yes—
116C4×Dic271C2313yes—
117C4×Dic271C2313yes—
118#41×C271—24no—
119C2×Q8×C271—24no—
120C2×Q8×C271C2313yes—
121C2×Q8×C271C2311yescharacteristic
122C2×D4×C271C2313yes—
123C2×D4×C271C2313yes—
124C22×C4×C271C2313yes—
125C22×C4×C271C2313yes—
126C4⋊C4×C271C2313yes—
127C4⋊C4×C271C2313yes—
128C22⋊C4×C271C2316yes—
129C22⋊C4×C271C2313yes—
130C22⋊C4×C271C2311yescharacteristic
131C42×C271C2313yes—
132C22×D271—22no—
133C22×D271—22no—
134C22×D271—26no—
135C22×D271—26no—
136C22×D271—26no—
137C22×D271#4114yescomplemented
138C22×D271C2413yes—
139C22×D271C2411yescharacteristic
140C22×D271C2411yescharacteristic
141Dic542—26no—
142Dic542—212no—
143Dic542C2×D4112yescomplemented
144D1084—26no—
145D1084—26no—
146C22⋊D271—26no—
147C22⋊D271—26no—
148C22⋊D271—212no—
149C2×Dic271—26no—
150C2×Dic271—26no—
151C2×Dic271C2411yescharacteristic
152C2×Dic271C2413yes—
153C2×Dic271C2413yes—
154C2×Dic271C2411yescharacteristic
155C2×Dic271C2413yes—
156C4×D271—22no—
157C4×D271—26no—
158C4×D271—26no—
159C4×D271—22no—
160C4×D271—26no—
161C4×D271—26no—
162C4×D271—26no—
163C4×D271—26no—
164C4×D271C2×D4112yes—
165C23×C271C2413yes—
166C23×C271C2411yescharacteristic
167Q8×C271—26no—
168Q8×C271C2×D414yescomplemented
169D4×C271—26no—
170D4×C271—26no—
171C2×C4×C271—26no—
172C2×C4×C271—26no—
173C2×C4×C271C2411yescharacteristic
174C2×C4×C271C2413yes—
175C2×C4×C271C2413yes—
176C2×C4×C271C2411yescharacteristic
177C2×C4×C271C2413yes—
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—23no—
181C22×C271—23no—
182C22×C271—23no—
183C22×C271C2×#4112yescomplemented
184C22×C271C2511yescharacteristic
185C2×D271—21no—
186C2×D271—21no—
187C2×D271—28no—
188C2×D271—23no—
189C2×D271—23no—
190C2×D271—23no—
191C2×D271—412no—
192C2×D271C22×D412yes—
193C2×D271C2×#4112yes—
194C2×D271C2×#4112yescomplemented
195C2×D2712−1+412yes—
196C4×C271—21no—
197C4×C271—23no—
198C4×C271—23no—
199C4×C271—21no—
200C4×C271C22×D416yes—
201Dic271—21no—
202Dic271—23no—
203Dic271—23no—
204Dic271—21no—
205Dic271C22×D416yes—
206C2×C271—22no—
207C2×C271—43no—
208C2×C271C23×D411yescharacteristic
209C2×C271C22×#4111yescharacteristic
210C2×C271C2×2−1+411yescharacteristic
211D271—22no—
212D271—43no—
213D271#54814yesdirect product factor
214C271C2×#54811yescharacteristic, complemented, Sylow
215C2×#548—2711nomaximal, Sylow
216C22×#41—2711no—
217C23×Q8—2711no—
218#548—2711no—
219#548—2711no—
220#548—2713no—
221#548—2713no—
222#548—2713no—
223#548—2711no—
224#548—2713no—
225#548D27111yescharacteristic, direct product factor
226C2×#124—2713no—
227C2×C22⋊Q8—2713no—
228C2×#121—2713no—
229C2×C4×Q8—2711no—
230C2×C4×D4—2713no—
231C2×#41—2711no—
232C2×#41—2713no—
233C2×#41—2713no—
234C2×#41—5428no—
235C2×#41C2×D27111yescharacteristic, complemented
236C22×Q8—2711no—
237C22×Q8—2713no—
238C22×Q8—2713no—
239C22×Q8—2711no—
240C22×Q8—2713no—
241C22×Q8—5428no—
242C22×Q8C2×D27111yescharacteristic, complemented
243C22×D4—2713no—
244C22×D4—2713no—
245C23×C4—2713no—
246C23×C4—2713no—
247#124—2713no—
248#124—2713no—
249#124—2716no—
250#124—2716no—
251#124—2713no—
252#124C2×D27113yescomplemented
253C22⋊Q8—2713no—
254C22⋊Q8—2713no—
255C22⋊Q8—2716no—
256C22⋊Q8—2713no—
257C22⋊Q8—2716no—
258C22⋊Q8C2×D27113yescomplemented
259#121—2713no—
260#121—2716no—
261#121—2713no—
262#121—2713no—
263#121—2716no—
264#121C2×D27113yescomplemented
265C4×Q8—2711no—
266C4×Q8—2713no—
267C4×Q8—2713no—
268C4×Q8C2×D27111yescharacteristic, complemented
269C4×D4—2713no—
270C4×D4—2713no—
271C4×D4—2713no—
272C4×D4—2713no—
273C4×D4—2713no—
274C4×D4—2713no—
275C4×D4—2713no—
276C4×D4C2×D27113yescomplemented
277C2×C4⋊C4—2713no—
278C2×C4⋊C4—2713no—
279C2×C22⋊C4—2711no—
280C2×C22⋊C4—2713no—
281C2×C22⋊C4—2716no—
282C2×C42—2713no—
283C24—2711no—
284C24—2713no—
285#41—24no—
286#41—54212no—
287#41—54212no—
288#41—5424no—
289C2×Q8—24no—
290C2×Q8—2718no—
291C2×Q8—2713no—
292C2×Q8—2716no—
293C2×Q8—2713no—
294C2×Q8—54212no—
295C2×Q8—54212no—
296C2×Q8—54212no—
297C2×Q8—5424no—
298C2×Q8C22×D27113yes—
299C2×Q8C22×D27111yescharacteristic
300C2×D4—2713no—
301C2×D4—2713no—
302C2×D4—2716no—
303C2×D4—2713no—
304C2×D4—2713no—
305C2×D4—54212no—
306C2×D4—54212no—
307C2×D4C22×D27113yes—
308C2×D4C22×D27113yes—
309C22×C4—2713no—
310C22×C4—2713no—
311C22×C4—2713no—
312C22×C4—2711no—
313C22×C4—2713no—
314C22×C4—2713no—
315C22×C4—2711no—
316C22×C4—2713no—
317C22×C4—2713no—
318C22×C4—2713no—
319C22×C4—2713no—
320C22×C4—54212no—
321C22×C4—54212no—
322C22×C4C22×D27113yes—
323C22×C4C22×D27113yes—
324C4⋊C4—2713no—
325C4⋊C4—2713no—
326C4⋊C4—2713no—
327C4⋊C4—2716no—
328C4⋊C4—2713no—
329C4⋊C4C22×D27113yes—
330C4⋊C4C22×D27113yes—
331C22⋊C4—2711no—
332C22⋊C4—2713no—
333C22⋊C4—2716no—
334C22⋊C4—2711no—
335C22⋊C4—2711no—
336C22⋊C4—2713no—
337C22⋊C4—2713no—
338C22⋊C4—2716no—
339C22⋊C4—2716no—
340C22⋊C4C22×D27111yescharacteristic
341C22⋊C4C22×D27113yes—
342C22⋊C4C22×D27116yes—
343C42—2713no—
344C42—2713no—
345C42—2713no—
346C42C22×D27113yes—
347C23—2711no—
348C23—2714no—
349C23—2711no—
350C23—2713no—
351C23—5422no—
352C23—5422no—
353C23—5426no—
354C23—5426no—
355C23—5426no—
356C23C23×D27113yes—
357C23C23×D27111yescharacteristic
358Q8—26no—
359Q8—27112no—
360Q8—54212no—
361Q8—5426no—
362Q8D4×D27114yescomplemented
363D4—26no—
364D4—26no—
365D4—5426no—
366D4—54212no—
367D4—5426no—
368D4—5426no—
369D4—5426no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—2713no—
373C2×C4—27112no—
374C2×C4—2711no—
375C2×C4—2713no—
376C2×C4—2713no—
377C2×C4—2711no—
378C2×C4—5426no—
379C2×C4—5426no—
380C2×C4—5422no—
381C2×C4—5426no—
382C2×C4—5426no—
383C2×C4—5422no—
384C2×C4—5426no—
385C2×C4—5426no—
386C2×C4—5426no—
387C2×C4—5426no—
388C2×C4C23×D27111yescharacteristic
389C2×C4C23×D27113yes—
390C2×C4C23×D27113yes—
391C2×C4C23×D27111yescharacteristic
392C2×C4C23×D27113yes—
393C22—21no—
394C22—21no—
395C22—23no—
396C22—23no—
397C22—23no—
398C22—2712no—
399C22—2712no—
400C22—2712no—
401C22—2712no—
402C22—5428no—
403C22—5421no—
404C22—5421no—
405C22—5423no—
406C22—5423no—
407C22—5423no—
408C22—108412no—
409C22#41×D27112yescomplemented
410C22C24×D27111yescharacteristic
411C4—23no—
412C4—21no—
413C4—21no—
414C4—23no—
415C4—2716no—
416C4—5423no—
417C4—5421no—
418C4—5421no—
419C4—5423no—
420C4C2×D4×D27116yes—
421C2—22no—
422C2—43no—
423C2—2714no—
424C2—5422no—
425C2—10843no—
426C2C22×D4×D27111yescharacteristic
427C2C2×#41×D27111yescharacteristic
428C22−1+4×D27111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#548×C271C2
hypercenter#548D271
hypocenterC271C2×#548
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#41×D271C2semidirect
2. C22×Q8×D271C2semidirect
3. #31774559C2semidirect
4. #31774558C2semidirect
5. #31774557C2semidirect
6. #31774556C2semidirect
7. #31774555C2semidirect
8. #31774554C2semidirect
9. #31774553C2semidirect
10. #124×D271C2semidirect
11. C22⋊Q8×D271C2semidirect
12. #121×D271C2semidirect
13. C4×Q8×D271C2semidirect
14. C4×D4×D271C2semidirect
15. #548×C271C2semidirect
16. C2×#16864655C22semidirect
17. C2×#16864654C22semidirect
18. C2×#16864653C22semidirect
20. C22×Dic542C22semidirect
24. C2×Q8×D271C22semidirect
25. C2×Q8×D271C22semidirect
30. #22576149C22semidirect
31. #22576148C22semidirect
32. #22576147C22semidirect
33. #22576146C22semidirect
34. #22576146C22semidirect
35. #22576140C22semidirect
36. #22576139C22semidirect
37. #22576138C22semidirect
38. #22576137C22semidirect
39. #22576136C22semidirect
40. #22576135C22semidirect
41. #22576134C22semidirect
42. #22576133C22semidirect
43. #22576132C22semidirect
44. #22576131C22semidirect
45. Q8×Dic271C22semidirect
46. C4×Dic542C22semidirect
47. #22576125C22semidirect
48. C4×D1084C22semidirect
49. D4×Dic271C22semidirect
50. #22576122C22semidirect
51. C4×C22⋊D271C22semidirect
52. C4×C22⋊D271C22semidirect
53. (C22⋊D271)⋊C4C22semidirect
54. (C22⋊D271)⋊C4C22semidirect
61. C2×#41×C271C22semidirect
62. C22×Q8×C271C22semidirect
63. #124×C271C22semidirect
64. C22⋊Q8×C271C22semidirect
65. #121×C271C22semidirect
66. C4×Q8×C271C22semidirect
67. C4×D4×C271C22semidirect
80. Q8×D271D4semidirect
137. C22×D271#41semidirect
143. Dic542C2×D4semidirect
168. Q8×C271C2×D4semidirect
183. C22×C271C2×#41semidirect
194. C2×D271C2×#41semidirect
213. D271#548direct (N × H)
214. C271C2×#548semidirect
225. #548D271direct (N × H)
235. C2×#41C2×D271semidirect
242. C22×Q8C2×D271semidirect
252. #124C2×D271semidirect
258. C22⋊Q8C2×D271semidirect
264. #121C2×D271semidirect
268. C4×Q8C2×D271semidirect
276. C4×D4C2×D271semidirect
362. Q8D4×D271semidirect
409. C22#41×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2712
C22×Q8×D2712
#317745592
#317745582
#317745572
#317745562
#317745552
#317745542
#317745532
#124×D2712
C22⋊Q8×D2712
#121×D2712
C4×Q8×D2712
C4×D4×D2712
#548×C2712
C2×#548271