Finite Group Databasebeta

← #46847961

Subgroups — #46847961

Order 34688.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576188C211yesmaximal, characteristic, complemented
2C2×#22576187C211yesmaximal, characteristic, complemented
3#31774549C211yesmaximal, characteristic, complemented
4#31774531C211yesmaximal, characteristic, complemented
5#31774268C211yesmaximal, characteristic, complemented
6#31774144C211yesmaximal, characteristic, complemented
7#31774101C211yesmaximal, characteristic, complemented
8#31774091C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic, complemented
10#31774047C211yesmaximal, characteristic, complemented
11#31774045C211yesmaximal, characteristic, complemented
12#31774040C211yesmaximal, characteristic, complemented
13#31773979C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15#468×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic, complemented
17C2×#16864654C2211yescharacteristic, complemented
18C22×D1084C2211yescharacteristic
19C2×D4×D271—21no—
20C2×D4×D271C2211yescharacteristic
21#22576188—24no—
22#22576187—24no—
23C2×#16864643C2211yescharacteristic
24C2×#16864643C2211yescharacteristic, complemented
25C2×#16864642C2211yescharacteristic, complemented
26C2×#16864641C2211yescharacteristic
27C2×#16864641C2211yescharacteristic, complemented
28C2×D2168C2211yescharacteristic
29C2×D2168C2211yescharacteristic, complemented
30C2×#16864639C2211yescharacteristic, complemented
31C2×#16864637C2211yescharacteristic, complemented
32#22576158C2211yescharacteristic, complemented
33#22576156C2211yescharacteristic
34#22576145—21no—
35#22576142—21no—
36#22576140C2211yescharacteristic, complemented
37#22576135C2211yescharacteristic
38#22576133—21no—
39#22576133—21no—
40#22576130—21no—
41#22576128—21no—
42#22576122C2211yescharacteristic
43#22576122C2211yescharacteristic, complemented
44C4×C22⋊D271C2211yescharacteristic
45C4⋊C4×D271—21no—
46Dic271⋊C8C2211yescharacteristic, complemented
47#22576083C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic, complemented
51C22⋊(C271⋊C8)C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53C8×Dic271C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×M16×C271C2211yescharacteristic, complemented
56C22⋊Q8×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#107×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864655—22no—
64#16864655—41no—
65#16864654—22no—
66#16864654—41no—
67C2×Dic542C2311yescharacteristic
68C2×D1084—22no—
69C2×D1084—21no—
70C2×D1084—21no—
71C2×D1084—21no—
72C2×D1084—41no—
73C2×D1084C2311yescharacteristic
74C2×D1084C2311yescharacteristic
75C2×D1084C2311yescharacteristic
76C2×C22⋊D271—21no—
77C2×C22⋊D271—21no—
78C2×C22⋊D271—21no—
79C2×C22⋊D271—21no—
80C2×C22⋊D271D411yescharacteristic
81D4×D271—22no—
82D4×D271—41no—
83D4×D271—44no—
84C2×C4×D271—21no—
85C2×C4×D271—21no—
86C2×C4×D271—21no—
87C2×C4×D271—21no—
88C2×C4×D271C2311yescharacteristic
89#16864643—22no—
90#16864643—22no—
91#16864642—41no—
92#16864641—22no—
93#16864641—41no—
94D2168—22no—
95D2168D414yescomplemented
96#16864639—22no—
97#16864637—22no—
98C2×C271⋊C8C2311yescharacteristic
99C2×C271⋊C8C2311yescharacteristic
100C4⋊Dic271—21no—
101Dic271⋊C4—21no—
102Dic271⋊C4—21no—
103Dic271⋊C4D411yescharacteristic
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×Q8×C271C2311yescharacteristic
112C2×D4×C271—21no—
113C2×D4×C271C2311yescharacteristic
114C22×C4×C271C2311yescharacteristic
115SD16×C271—22no—
116M16×C271—22no—
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C4⋊C4×C271—21no—
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—81no—
132C22×D271—81no—
133C22×D271C2×D411yescharacteristic
134Dic542—21no—
135Dic542—21no—
136D1084—21no—
137D1084—21no—
138D1084—21no—
139D1084—21no—
140D1084—42no—
141D1084—42no—
142D1084—42no—
143D1084C2×D412yes—
144D1084C2×D412yescomplemented
145C22⋊D271—22no—
146C22⋊D271—42no—
147C22⋊D271—42no—
148C22⋊D271—42no—
149C22⋊D271—81no—
150C2×Dic271—21no—
151C2×Dic271C2×D411yescharacteristic
152C2×Dic271C2×D411yescharacteristic
153C2×Dic271C2×D411yescharacteristic
154C4×D271—21no—
155C4×D271—21no—
156C4×D271—42no—
157C4×D271—42no—
158C4×D271—42no—
159C4×D271—42no—
160C271⋊C8—21no—
161C271⋊C8—21no—
162C23×C271—21no—
163C23×C271C2×D411yescharacteristic
164Q8×C271—21no—
165Q8×C271—21no—
166D4×C271—21no—
167D4×C271—21no—
168D4×C271—42no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—21no—
173C2×C4×C271—21no—
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2411yescharacteristic
176C8×C271—21no—
177C8×C271C2×D412yescomplemented
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—21no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271C22×D411yescharacteristic
185C2×D271—21no—
186C2×D271—22no—
187C2×D271—22no—
188C2×D271—21no—
189C2×D271—21no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—84no—
198C2×D271—84no—
199C4×C271—21no—
200C4×C271—41no—
201C4×C271—41no—
202C4×C271—41no—
203C4×C271C22×D411yescharacteristic
204C4×C271H(5,𝔽2)11yescharacteristic
205Dic271—21no—
206Dic271—21no—
207Dic271—41no—
208Dic271AGL(1,ℤ/8)12yescomplemented
209C2×C271—41no—
210C2×C271—81no—
211C2×C271D4211yescharacteristic
212C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C271#57711yescharacteristic
214D271—41no—
215D271—42no—
216D271—42no—
217D271—81no—
218C271#327211yescharacteristic, complemented, Sylow
219#3272—2711nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2711no—
221C2×AGL(1,ℤ/8)—2711no—
222#547—2711no—
223D42—2711no—
224#492—2711no—
225#468—2711no—
226#468D27111yescharacteristic, complemented
227#461—2711no—
228#459—2711no—
229#450—2711no—
230#450—2711no—
231#449—2711no—
232#449—2711no—
233C4×D8—2711no—
234#436—2711no—
235C2×#41—2711no—
236C2×#41—2711no—
237C22×D4—2711no—
238C22×D4—2711no—
239C22×D4—5421no—
240AGL(1,ℤ/8)—5424no—
241AGL(1,ℤ/8)—5424no—
242C2×SD16—2711no—
243C2×SD16—2711no—
244C2×SD16—2711no—
245C2×SD16—2711no—
246C2×SD16—2711no—
247C2×SD16C2×D27111yescharacteristic, complemented
248C2×D8—2711no—
249C2×D8—2711no—
250C2×D8—2711no—
251C2×M16—2711no—
252C2×M16C2×D27111yescharacteristic, complemented
253C4⋊Q8—2711no—
254#127—2711no—
255#123—5421no—
256#123—5421no—
257C22⋊Q8—2711no—
258C22⋊Q8C2×D27111yescharacteristic, complemented
259#121—2711no—
260#121—5421no—
261#121—5421no—
262#121C2×D27111yescharacteristic
263#120—5421no—
264#120—5421no—
265C4×D4—2711no—
266C4×D4—2711no—
267C4×D4—2711no—
268C2×C4⋊C4—5421no—
269#107C2×D27111yescharacteristic, complemented
270C4⋊C8—2711no—
271#103—2711no—
272#103C2×D27111yescharacteristic, complemented
273#102—2711no—
274#102—2711no—
275#102—2711no—
276#102C2×D27111yescharacteristic, complemented
277C22⋊C8—2711no—
278C22⋊C8—2711no—
279C4×C8—2711no—
280C24—5421no—
281C24—5421no—
282#41—5422no—
283#41—5422no—
284#41—10841no—
285#41—10841no—
286C2×Q8—2711no—
287C2×Q8C22×D27111yescharacteristic
288C2×D4—21no—
289C2×D4—2711no—
290C2×D4—2711no—
291C2×D4—2711no—
292C2×D4—2711no—
293C2×D4—5422no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4—5421no—
299C2×D4—5422no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—10841no—
303C2×D4—10841no—
304C2×D4—10844no—
305C2×D4C22×D27111yescharacteristic
306C22×C4—2711no—
307C22×C4—5421no—
308C22×C4—5421no—
309C22×C4—5421no—
310C22×C4—5421no—
311C22×C4C22×D27111yescharacteristic
312SD16—22no—
313SD16—5422no—
314SD16—5422no—
315SD16—5422no—
316SD16—10841no—
317SD16—10841no—
318D8—2714no—
319D8—5422no—
320D8—5422no—
321M16—22no—
322M16—5422no—
323C2×C8—2711no—
324C2×C8—2711no—
325C2×C8C22×D27111yescharacteristic
326C2×C8C22×D27111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2711no—
329C4⋊C4—5421no—
330C4⋊C4—5421no—
331C4⋊C4—5421no—
332C4⋊C4C22×D27111yescharacteristic
333C4⋊C4C22×D27111yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—2711no—
337C22⋊C4—2711no—
338C22⋊C4—5421no—
339C22⋊C4—5421no—
340C22⋊C4—5421no—
341C22⋊C4—5421no—
342C42—2711no—
343C23—21no—
344C23—2711no—
345C23—5421no—
346C23—5421no—
347C23—5421no—
348C23—10842no—
349C23—10842no—
350C23—10842no—
351C23—10842no—
352C23—21681no—
353C23—21681no—
354C23D4×D27111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—5421no—
358Q8—5421no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—2712no—
363D4—2712no—
364D4—5422no—
365D4—5421no—
366D4—5421no—
367D4—5421no—
368D4—5421no—
369D4—10842no—
370D4—10842no—
371D4—10842no—
372D4—10842no—
373D4—10842no—
374D4—10842no—
375D4—21681no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—2711no—
382C2×C4—2711no—
383C2×C4—2711no—
384C2×C4—5421no—
385C2×C4—5421no—
386C2×C4—5421no—
387C2×C4—10842no—
388C2×C4—10842no—
389C2×C4—10842no—
390C2×C4—10842no—
391C2×C4D4×D27111yescharacteristic
392C2×C4C23×D27111yescharacteristic
393C8—21no—
394C8—5421no—
395C8—5421no—
396C8D4×D27112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—5421no—
404C22—5422no—
405C22—5422no—
406C22—5421no—
407C22—5421no—
408C22—10841no—
409C22—10841no—
410C22—10841no—
411C22—10841no—
412C22—10841no—
413C22—10841no—
414C22—10841no—
415C22—21684no—
416C22—21684no—
417C22C2×D4×D27111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—2712no—
423C4—5421no—
424C4—5421no—
425C4—10841no—
426C4C2×D4×D27111yescharacteristic
427C4#2257620611yescharacteristic
428C2—41no—
429C2—81no—
430C2—10841no—
431C2—10842no—
432C2—10842no—
433C2—21681no—
434C2#3177453111yescharacteristic
435C2AGL(1,ℤ/8)×D27111yescharacteristic
436C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#468×C271C2
socleC22×C271C22×D4
hypercenter#468D271
hypocenterC271#3272
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C2×#22576187C2semidirect
3. #31774549C2semidirect
4. #31774531C2semidirect
5. #31774268C2semidirect
6. #31774144C2semidirect
7. #31774101C2semidirect
8. #31774091C2semidirect
9. #31774049C2semidirect
10. #31774047C2semidirect
11. #31774045C2semidirect
12. #31774040C2semidirect
13. #31773979C2semidirect
14. #31773973C2semidirect
15. #468×C271C2semidirect
16. C2×#16864655C22semidirect
17. C2×#16864654C22semidirect
24. C2×#16864643C22semidirect
25. C2×#16864642C22semidirect
27. C2×#16864641C22semidirect
29. C2×D2168C22semidirect
30. C2×#16864639C22semidirect
31. C2×#16864637C22semidirect
32. #22576158C22semidirect
36. #22576140C22semidirect
43. #22576122C22semidirect
46. Dic271⋊C8C22semidirect
47. #22576083C22semidirect
49. #22576080C22semidirect
50. #22576080C22semidirect
51. C22⋊(C271⋊C8)C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. C8×Dic271C22semidirect
54. C2×SD16×C271C22semidirect
55. C2×M16×C271C22semidirect
56. C22⋊Q8×C271C22semidirect
58. #107×C271C22semidirect
59. #103×C271C22semidirect
60. #102×C271C22semidirect
95. D2168D4semidirect
144. D1084C2×D4semidirect
177. C8×C271C2×D4semidirect
208. Dic271AGL(1,ℤ/8)semidirect
218. C271#3272semidirect
226. #468D271semidirect
247. C2×SD16C2×D271semidirect
252. C2×M16C2×D271semidirect
258. C22⋊Q8C2×D271semidirect
269. #107C2×D271semidirect
272. #103C2×D271semidirect
276. #102C2×D271semidirect
396. 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×#225761872
#317745492
#317745312
#317742682
#317741442
#317741012
#317740912
#317740492
#317740472
#317740452
#317740402
#317739792
#317739732
#468×C2712
#3272271