Finite Group Databasebeta

← #46847851

Subgroups — #46847851

Order 34688.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576188C211yesmaximal, characteristic, complemented
2C2×D8×D271C211yesmaximal, characteristic, complemented
3#31774544C211yesmaximal, characteristic, complemented
4#31774531C211yesmaximal, characteristic, complemented
5#31774285C211yesmaximal, characteristic, complemented
6#31774127C211yesmaximal, characteristic, complemented
7#31774111C211yesmaximal, characteristic, complemented
8#31774091C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic, complemented
10#31774040C211yesmaximal, characteristic, complemented
11#31774032C211yesmaximal, characteristic, complemented
12#31774030C211yesmaximal, characteristic, complemented
13#31774008C211yesmaximal, characteristic, complemented
14#31773966C211yesmaximal, characteristic, complemented
15#469×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic, complemented
17C22×D1084C2211yescharacteristic, complemented
18C2×D4×D271—21no—
19C2×D4×D271C2211yescharacteristic
20C2×D4×D271C2211yescharacteristic, complemented
21#22576188—24no—
22C2×#16864643C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic, complemented
24C2×#16864642C2211yescharacteristic, complemented
25C2×D2168C2211yescharacteristic
26C2×D2168C2211yescharacteristic, complemented
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic, complemented
29C2×#16864639C2211yescharacteristic, complemented
30D8×D271—24no—
31C2×C8×D271C2211yescharacteristic, complemented
32#22576156C2211yescharacteristic, complemented
33#22576147C2211yescharacteristic, complemented
34#22576142—21no—
35#22576140C2211yescharacteristic, complemented
36#22576135C2211yescharacteristic
37#22576133—21no—
38#22576131—21no—
39#22576130—21no—
40#22576128—21no—
41#22576128—21no—
42#22576122C2211yescharacteristic, complemented
43#22576122C2211yescharacteristic, complemented
44C4×C22⋊D271C2211yescharacteristic
45C22⋊C4×D271—21no—
46Dic271⋊C8C2211yescharacteristic, complemented
47#22576083C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic, complemented
51C22⋊(C271⋊C8)C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53#22576070C2211yescharacteristic, complemented
54C2×D8×C271C2211yescharacteristic, complemented
55C2×M16×C271C2211yescharacteristic, complemented
56#121×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic, complemented
58#106×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63C23×D271—21no—
64#16864654—22no—
65#16864654—41no—
66C2×Dic542C2311yescharacteristic
67C2×D1084—22no—
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084—41no—
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×D1084C2311yescharacteristic
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271—21no—
78C2×C22⋊D271—21no—
79C2×C22⋊D271D411yescharacteristic
80D4×D271—22no—
81D4×D271—22no—
82D4×D271—41no—
83D4×D271—41no—
84D4×D271—44no—
85C2×C4×D271—21no—
86C2×C4×D271—21no—
87C2×C4×D271—21no—
88C2×C4×D271C2311yescharacteristic
89#16864643—22no—
90#16864643—22no—
91#16864642—41no—
92D2168—22no—
93D2168D414yescomplemented
94#16864639—22no—
95#16864639—22no—
96#16864639—41no—
97C2×C271⋊C8C2311yescharacteristic
98C2×C271⋊C8C2311yescharacteristic
99C8×D271—22no—
100C4⋊Dic271—21no—
101Dic271⋊C4D411yescharacteristic
102C22⋊Dic271—21no—
103C22⋊Dic271—21no—
104C22⋊Dic271D411yescharacteristic
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4—21no—
109(C2×D271)⋊C4D411yescharacteristic
110C4×Dic271C2311yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271—21no—
113C2×D4×C271C2311yescharacteristic
114C2×D4×C271C2311yescharacteristic
115C22×C4×C271C2311yescharacteristic
116D8×C271—22no—
117M16×C271—22no—
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C4⋊C4×C271C2311yescharacteristic
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C22⋊C4×C271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—42no—
130C22×D271—42no—
131C22×D271—42no—
132C22×D271—42no—
133C22×D271—81no—
134C22×D271—81no—
135C22×D271—81no—
136C22×D271C2×D411yescharacteristic
137Dic542—21no—
138Dic542—21no—
139D1084—21no—
140D1084—21no—
141D1084—21no—
142D1084—21no—
143D1084—42no—
144D1084—42no—
145D1084C2×D412yes—
146D1084C2×D412yescomplemented
147C22⋊D271—22no—
148C22⋊D271—42no—
149C22⋊D271—42no—
150C22⋊D271—42no—
151C22⋊D271—42no—
152C22⋊D271—81no—
153C2×Dic271—21no—
154C2×Dic271C2×D411yescharacteristic
155C2×Dic271C2×D411yescharacteristic
156C2×Dic271C2×D411yescharacteristic
157C4×D271—21no—
158C4×D271—21no—
159C4×D271—42no—
160C4×D271—42no—
161C4×D271—81no—
162C271⋊C8—21no—
163C271⋊C8—21no—
164C23×C271—21no—
165C23×C271—21no—
166C23×C271C2×D411yescharacteristic
167D4×C271—21no—
168D4×C271—21no—
169D4×C271—21no—
170D4×C271—21no—
171D4×C271—42no—
172D4×C271—81no—
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—21no—
177C2×C4×C271C2×D411yescharacteristic
178C2×C4×C271C2411yescharacteristic
179C8×C271—21no—
180C8×C271C2×D412yescomplemented
181C22×C271—21no—
182C22×C271—21no—
183C22×C271—21no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—41no—
188C22×C271—41no—
189C22×C271—41no—
190C22×C271C22×D411yescharacteristic
191C2×D271—21no—
192C2×D271—21no—
193C2×D271—22no—
194C2×D271—22no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—41no—
198C2×D271—41no—
199C2×D271—41no—
200C2×D271—41no—
201C2×D271—41no—
202C2×D271—84no—
203C2×D271—84no—
204C2×D271—84no—
205C2×D271AGL(1,ℤ/8)12yescomplemented
206C4×C271—21no—
207C4×C271—41no—
208C4×C271—41no—
209C4×C271C22×D411yescharacteristic
210C4×C271H(5,𝔽2)11yescharacteristic
211Dic271—21no—
212Dic271—21no—
213Dic271—21no—
214Dic271—41no—
215C2×C271—41no—
216C2×C271—81no—
217C2×C271—81no—
218C2×C271D4211yescharacteristic
219C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C271#57611yescharacteristic
221D271—22no—
222D271—42no—
223D271—42no—
224D271—81no—
225C271#326411yescharacteristic, complemented, Sylow
226#3264—2711nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2711no—
228C22×D8—2711no—
229#546—2711no—
230D42—2711no—
231#496—2711no—
232#469D27111yescharacteristic, complemented
233#465—2711no—
234#463—2711no—
235#459—2711no—
236#450—2711no—
237#449—2711no—
238#447—2711no—
239#447—2711no—
240#442—2711no—
241#435—2711no—
242C2×#41—2711no—
243C22×D4—2711no—
244C22×D4—2711no—
245C22×D4—2711no—
246C22×D4—5421no—
247AGL(1,ℤ/8)—5424no—
248C2×SD16—2711no—
249C2×SD16—2711no—
250C2×SD16—2711no—
251C2×D8—2711no—
252C2×D8—2711no—
253C2×D8—2711no—
254C2×D8—2711no—
255C2×D8—2711no—
256C2×D8—5424no—
257C2×D8C2×D27111yescharacteristic, complemented
258C2×M16C2×D27111yescharacteristic, complemented
259C22×C8—2711no—
260#127—2711no—
261#124—2711no—
262#123—5421no—
263C22⋊Q8—2711no—
264#121—2711no—
265#121—5421no—
266#121—5421no—
267#121C2×D27111yescharacteristic, complemented
268#121C2×D27111yescharacteristic, complemented
269#120—5421no—
270#120—5421no—
271#120—5421no—
272C4×D4—2711no—
273C4×D4—2711no—
274C4×D4—2711no—
275C2×C22⋊C4—5421no—
276#106C2×D27111yescharacteristic, complemented
277C4⋊C8—2711no—
278#103—2711no—
279#102—2711no—
280#102—2711no—
281#102—2711no—
282#102C2×D27111yescharacteristic, complemented
283#102C2×D27111yescharacteristic, complemented
284C22⋊C8—2711no—
285C22⋊C8—2711no—
286#97—2711no—
287C24—5421no—
288C24—5421no—
289C24—5421no—
290#41—5422no—
291#41—10841no—
292C2×Q8—2711no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2711no—
296C2×D4—2711no—
297C2×D4—2711no—
298C2×D4—2711no—
299C2×D4—5422no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—5421no—
303C2×D4—5421no—
304C2×D4—5421no—
305C2×D4—5422no—
306C2×D4—5422no—
307C2×D4—5421no—
308C2×D4—5421no—
309C2×D4—10841no—
310C2×D4—10841no—
311C2×D4—10841no—
312C2×D4—10844no—
313C2×D4C22×D27111yescharacteristic
314C2×D4C22×D27111yescharacteristic
315C22×C4—2711no—
316C22×C4—5421no—
317C22×C4—5421no—
318C22×C4—5421no—
319C22×C4C22×D27111yescharacteristic
320SD16—5422no—
321SD16—5422no—
322SD16—10841no—
323D8—22no—
324D8—2714no—
325D8—5422no—
326D8—5422no—
327D8—5422no—
328D8—10841no—
329M16—22no—
330C2×C8—2711no—
331C2×C8—2711no—
332C2×C8—5422no—
333C2×C8C22×D27111yescharacteristic
334C2×C8C22×D27111yescharacteristic
335C4⋊C4—2711no—
336C4⋊C4—5421no—
337C4⋊C4C22×D27111yescharacteristic
338C4⋊C4C22×D27111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2711no—
342C22⋊C4—2711no—
343C22⋊C4—5421no—
344C22⋊C4—5421no—
345C22⋊C4—5421no—
346C22⋊C4—5421no—
347C22⋊C4—5421no—
348C22⋊C4—5421no—
349C42—2711no—
350C23—21no—
351C23—21no—
352C23—2711no—
353C23—5421no—
354C23—5421no—
355C23—5421no—
356C23—10842no—
357C23—10842no—
358C23—10842no—
359C23—10842no—
360C23—10842no—
361C23—10842no—
362C23—21681no—
363C23—21681no—
364C23—21681no—
365C23D4×D27111yescharacteristic
366Q8—5421no—
367Q8—5421no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—2712no—
375D4—2712no—
376D4—5422no—
377D4—5421no—
378D4—5421no—
379D4—5421no—
380D4—5421no—
381D4—10842no—
382D4—10842no—
383D4—10842no—
384D4—10842no—
385D4—10842no—
386D4—10842no—
387D4—21681no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—2711no—
393C2×C4—2711no—
394C2×C4—2711no—
395C2×C4—5421no—
396C2×C4—5421no—
397C2×C4—5421no—
398C2×C4—10842no—
399C2×C4—10842no—
400C2×C4—21681no—
401C2×C4D4×D27111yescharacteristic
402C2×C4C23×D27111yescharacteristic
403C8—21no—
404C8—5421no—
405C8—5421no—
406C8D4×D27112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2712no—
417C22—5421no—
418C22—5421no—
419C22—5422no—
420C22—5422no—
421C22—10841no—
422C22—10841no—
423C22—10841no—
424C22—10841no—
425C22—10841no—
426C22—10841no—
427C22—10841no—
428C22—21684no—
429C22—21684no—
430C22—21684no—
431C22C2×D4×D27111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—5421no—
436C4—5421no—
437C4—5421no—
438C4—10841no—
439C4C2×D4×D27111yescharacteristic
440C4#2257620611yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—5422no—
445C2—10842no—
446C2—10842no—
447C2—21681no—
448C2#3177453111yescharacteristic
449C2AGL(1,ℤ/8)×D27111yescharacteristic
450C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#469×C271C2
socleC22×C271C22×D4
hypercenter#469D271
hypocenterC271#3264
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C2×D8×D271C2semidirect
3. #31774544C2semidirect
4. #31774531C2semidirect
5. #31774285C2semidirect
6. #31774127C2semidirect
7. #31774111C2semidirect
8. #31774091C2semidirect
9. #31774049C2semidirect
10. #31774040C2semidirect
11. #31774032C2semidirect
12. #31774030C2semidirect
13. #31774008C2semidirect
14. #31773966C2semidirect
15. #469×C271C2semidirect
16. C2×#16864654C22semidirect
17. C22×D1084C22semidirect
20. C2×D4×D271C22semidirect
22. C2×#16864643C22semidirect
23. C2×#16864643C22semidirect
24. C2×#16864642C22semidirect
26. C2×D2168C22semidirect
28. C2×#16864639C22semidirect
29. C2×#16864639C22semidirect
31. C2×C8×D271C22semidirect
32. #22576156C22semidirect
33. #22576147C22semidirect
35. #22576140C22semidirect
42. #22576122C22semidirect
43. #22576122C22semidirect
46. Dic271⋊C8C22semidirect
47. #22576083C22semidirect
48. #22576080C22semidirect
49. #22576080C22semidirect
50. #22576080C22semidirect
51. C22⋊(C271⋊C8)C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. #22576070C22semidirect
54. C2×D8×C271C22semidirect
55. C2×M16×C271C22semidirect
56. #121×C271C22semidirect
57. #121×C271C22semidirect
58. #106×C271C22semidirect
59. #102×C271C22semidirect
60. #102×C271C22semidirect
93. D2168D4semidirect
146. D1084C2×D4semidirect
180. C8×C271C2×D4semidirect
205. C2×D271AGL(1,ℤ/8)semidirect
225. C271#3264semidirect
232. #469D271semidirect
257. C2×D8C2×D271semidirect
258. C2×M16C2×D271semidirect
267. #121C2×D271semidirect
268. #121C2×D271semidirect
276. #106C2×D271semidirect
282. #102C2×D271semidirect
283. #102C2×D271semidirect
406. C8D4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
C2×D8×D2712
#317745442
#317745312
#317742852
#317741272
#317741112
#317740912
#317740492
#317740402
#317740322
#317740302
#317740082
#317739662
#469×C2712
#3264271