Finite Group Databasebeta

← #46847858

Subgroups — #46847858

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×#22576184C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3#31774555C211yesmaximal, characteristic, complemented
4#31774532C211yesmaximal, characteristic, complemented
5#31774140C211yesmaximal, characteristic, complemented
6#31774130C211yesmaximal, characteristic, complemented
7#31774109C211yesmaximal, characteristic, complemented
8#31774094C211yesmaximal, characteristic, complemented
9#31774046C211yesmaximal, characteristic, complemented
10#31774039C211yesmaximal, characteristic, complemented
11#31774034C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13#31773999C211yesmaximal, characteristic, complemented
14#31773970C211yesmaximal, characteristic, complemented
15#496×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic, complemented
17C2×Q8×D271C2211yescharacteristic, complemented
18C2×D4×D271—21no—
19C2×D4×D271C2211yescharacteristic
20C2×D4×D271C2211yescharacteristic
21#22576184—24no—
22C2×#16864643C2211yescharacteristic
23C2×#16864642C2211yescharacteristic, complemented
24C2×#16864642C2211yescharacteristic, complemented
25C2×#16864641C2211yescharacteristic
26SD16×D271—24no—
27C2×#16864639C2211yescharacteristic
28C2×#16864637C2211yescharacteristic, complemented
29C2×C8×D271C2211yescharacteristic, complemented
30#22576146—21no—
31#22576137—21no—
32#22576136C2211yescharacteristic, complemented
33#22576133—21no—
34#22576131—21no—
35#22576131C2211yescharacteristic
36#22576131C2211yescharacteristic
37#22576131C2211yescharacteristic, complemented
38#22576128—21no—
39#22576128—21no—
40C4×Dic542C2211yescharacteristic, complemented
41C4×D1084C2211yescharacteristic
42C4×D1084C2211yescharacteristic
43(C22⋊D271)⋊C4—21no—
44#22576094C2211yescharacteristic, complemented
45C4⋊(C271⋊C8)C2211yescharacteristic, complemented
46#22576085C2211yescharacteristic, complemented
47#22576084C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic
49#22576081C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53(C2×D271)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic, complemented
56C2×D8×C271C2211yescharacteristic, complemented
57C2×D8×C271C2211yescharacteristic, complemented
58#127×C271C2211yescharacteristic
59#124×C271C2211yescharacteristic, complemented
60#97×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864653—22no—
64#16864653—41no—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—41no—
69C2×D1084D411yescharacteristic
70C2×D1084C2311yescharacteristic
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77D4×D271—22no—
78D4×D271—22no—
79D4×D271—41no—
80D4×D271—41no—
81D4×D271—44no—
82C22×Dic271—21no—
83C2×C4×D271—21no—
84C2×C4×D271—21no—
85C2×C4×D271C2311yescharacteristic
86C2×C4×D271C2311yescharacteristic
87#16864643D414yescomplemented
88#16864642—22no—
89#16864642—22no—
90#16864641—22no—
91#16864639—22no—
92#16864637—22no—
93C2×C271⋊C8C2311yescharacteristic
94C2×C271⋊C8C2311yescharacteristic
95C8×D271—22no—
96C4⋊Dic271D411yescharacteristic
97C4⋊Dic271C2311yescharacteristic
98C4⋊Dic271C2311yescharacteristic
99Dic271⋊C4—21no—
100Dic271⋊C4—21no—
101C22⋊Dic271—21no—
102C22⋊Dic271—21no—
103C22⋊Dic271—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4D411yescharacteristic
107(C2×D271)⋊C4D411yescharacteristic
108C4×Dic271—21no—
109C2×Q8×C271C2311yescharacteristic
110C2×D4×C271—21no—
111C2×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C2×D4×C271C2311yescharacteristic
114C2×D4×C271C2311yescharacteristic
115SD16×C271—22no—
116SD16×C271—41no—
117D8×C271—22no—
118D8×C271—41no—
119C2×C8×C271C2311yescharacteristic
120C2×C8×C271C2311yescharacteristic
121C22⋊C4×C271—21no—
122C22⋊C4×C271—21no—
123C42×C271C2311yescharacteristic
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—81no—
130C22×D271—81no—
131C22×D271C2×D411yescharacteristic
132C22×D271C2×D411yescharacteristic
133Dic542—21no—
134Dic542—42no—
135Dic542C2×D412yescomplemented
136D1084—22no—
137D1084—21no—
138D1084C2×D412yes—
139C22⋊D271—42no—
140C22⋊D271—42no—
141C22⋊D271—42no—
142C22⋊D271—42no—
143C22⋊D271—42no—
144C22⋊D271—42no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—42no—
149C2×Dic271C2×D411yescharacteristic
150C2×Dic271C2×D411yescharacteristic
151C4×D271—21no—
152C4×D271—21no—
153C4×D271—21no—
154C4×D271—21no—
155C4×D271—42no—
156C4×D271—42no—
157C271⋊C8—21no—
158C271⋊C8—21no—
159C23×C271—21no—
160C23×C271—21no—
161C23×C271—21no—
162Q8×C271—21no—
163Q8×C271—21no—
164D4×C271—21no—
165D4×C271—21no—
166D4×C271—21no—
167D4×C271—21no—
168D4×C271—21no—
169D4×C271—21no—
170D4×C271—42no—
171D4×C271—81no—
172C2×C4×C271—21no—
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2411yescharacteristic
176C8×C271—21no—
177C8×C271C2×D412yescomplemented
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271C22×D411yescharacteristic
188C2×D271—21no—
189C2×D271—21no—
190C2×D271—21no—
191C2×D271—22no—
192C2×D271—21no—
193C2×D271—21no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—84no—
197C2×D271—84no—
198C2×D271AGL(1,ℤ/8)12yescomplemented
199C4×C271—21no—
200C4×C271—21no—
201C4×C271—41no—
202C4×C271C22×D411yescharacteristic
203C4×C271H(5,𝔽2)11yescharacteristic
204Dic271—21no—
205Dic271—21no—
206Dic271—22no—
207Dic271—41no—
208Dic271—41no—
209C2×C271—81no—
210C2×C271—81no—
211C2×C271—81no—
212C2×C271D4211yescharacteristic
213C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C271#57711yescharacteristic
215D271—22no—
216D271—41no—
217D271—42no—
218C271#326511yescharacteristic, complemented, Sylow
219#3265—2711nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2711no—
221C22×SD16—2711no—
222#548—2711no—
223D42—2711no—
224#496D27111yescharacteristic, complemented
225#468—2711no—
226#466—2711no—
227#463—2711no—
228#460—2711no—
229#450—2711no—
230#449—2711no—
231#448—2711no—
232#447—2711no—
233#440—2711no—
234#435—2711no—
235C2×#41—2711no—
236C22×Q8—2711no—
237C22×D4—2711no—
238C22×D4—2711no—
239C22×D4—5421no—
240AGL(1,ℤ/8)—5424no—
241C2×SD16—2711no—
242C2×SD16—2711no—
243C2×SD16—2711no—
244C2×SD16—2711no—
245C2×SD16—5424no—
246C2×SD16C2×D27111yescharacteristic, complemented
247C2×SD16C2×D27111yescharacteristic, complemented
248C2×D8—2711no—
249C2×D8C2×D27111yescharacteristic, complemented
250C2×D8C2×D27111yescharacteristic, complemented
251C2×M16—2711no—
252C22×C8—2711no—
253#127C2×D27111yescharacteristic
254#124—5421no—
255#124C2×D27111yescharacteristic, complemented
256C22⋊Q8—2711no—
257C22⋊Q8—5421no—
258#121—2711no—
259#121—2711no—
260#121—2711no—
261#121—5421no—
262#121—5421no—
263#120—5421no—
264#120—5421no—
265C4×Q8—2711no—
266C4×D4—2711no—
267C4×D4—2711no—
268C4×D4—5421no—
269#107—2711no—
270C4⋊C8—2711no—
271#103—2711no—
272#103—2711no—
273#102—2711no—
274#102—2711no—
275#102—2711no—
276#102—2711no—
277C22⋊C8—2711no—
278C22⋊C8—2711no—
279#97C2×D27111yescharacteristic, complemented
280C24—5421no—
281C24—5421no—
282#41—5422no—
283#41—10841no—
284C2×Q8—2711no—
285C2×Q8—5421no—
286C2×Q8—5422no—
287C2×Q8—10841no—
288C2×Q8C22×D27111yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2711no—
292C2×D4—2711no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5422no—
296C2×D4—5422no—
297C2×D4—5421no—
298C2×D4—5421no—
299C2×D4—5421no—
300C2×D4—5421no—
301C2×D4—10841no—
302C2×D4—10841no—
303C2×D4—10844no—
304C2×D4C22×D27111yescharacteristic
305C2×D4C22×D27111yescharacteristic
306C2×D4C22×D27111yescharacteristic
307C22×C4—2711no—
308C22×C4—2711no—
309C22×C4—5421no—
310C22×C4—5421no—
311C22×C4—5421no—
312SD16—22no—
313SD16—41no—
314SD16—2714no—
315SD16—5422no—
316SD16—5422no—
317SD16—5422no—
318D8—22no—
319D8—41no—
320D8—5422no—
321M16—5422no—
322C2×C8—2711no—
323C2×C8—2711no—
324C2×C8—5422no—
325C2×C8C22×D27111yescharacteristic
326C2×C8C22×D27111yescharacteristic
327C4⋊C4—2711no—
328C4⋊C4—2711no—
329C4⋊C4—2711no—
330C4⋊C4—5421no—
331C4⋊C4—5421no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—2711no—
335C22⋊C4—2711no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C22⋊C4—5421no—
339C22⋊C4—5421no—
340C22⋊C4—5421no—
341C42—5421no—
342C42C22×D27111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—2711no—
347C23—2711no—
348C23—5421no—
349C23—10842no—
350C23—10842no—
351C23—10842no—
352C23—10842no—
353C23—21681no—
354C23—21681no—
355Q8—21no—
356Q8—21no—
357Q8—2712no—
358Q8—5421no—
359Q8—10842no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—2712no—
369D4—5422no—
370D4—5421no—
371D4—10842no—
372D4—10842no—
373D4—10842no—
374D4—10842no—
375D4—10842no—
376D4—10842no—
377C2×C4—21no—
378C2×C4—2711no—
379C2×C4—2711no—
380C2×C4—5421no—
381C2×C4—5421no—
382C2×C4—5421no—
383C2×C4—5421no—
384C2×C4—5421no—
385C2×C4—5421no—
386C2×C4—5421no—
387C2×C4—10842no—
388C2×C4—10842no—
389C2×C4—10842no—
390C2×C4D4×D27111yescharacteristic
391C2×C4D4×D27111yescharacteristic
392C2×C4C23×D27111yescharacteristic
393C8—21no—
394C8—5421no—
395C8—5421no—
396C8D4×D27112yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2712no—
407C22—5421no—
408C22—5421no—
409C22—5421no—
410C22—5422no—
411C22—5421no—
412C22—5421no—
413C22—10841no—
414C22—10841no—
415C22—21684no—
416C22—21684no—
417C22C2×D4×D27111yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—5421no—
422C4—5421no—
423C4—5422no—
424C4—10841no—
425C4—10841no—
426C4C2×D4×D27111yescharacteristic
427C4#2257620511yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—5422no—
432C2—10841no—
433C2—10842no—
434C2#3177453211yescharacteristic
435C2AGL(1,ℤ/8)×D27111yescharacteristic
436C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#496×C271C2
socleC22×C271C22×D4
hypercenter#496D271
hypocenterC271#3265
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. C2×SD16×D271C2semidirect
3. #31774555C2semidirect
4. #31774532C2semidirect
5. #31774140C2semidirect
6. #31774130C2semidirect
7. #31774109C2semidirect
8. #31774094C2semidirect
9. #31774046C2semidirect
10. #31774039C2semidirect
11. #31774034C2semidirect
12. #31774029C2semidirect
13. #31773999C2semidirect
14. #31773970C2semidirect
15. #496×C271C2semidirect
16. C2×#16864653C22semidirect
17. C2×Q8×D271C22semidirect
23. C2×#16864642C22semidirect
24. C2×#16864642C22semidirect
28. C2×#16864637C22semidirect
29. C2×C8×D271C22semidirect
32. #22576136C22semidirect
37. #22576131C22semidirect
40. C4×Dic542C22semidirect
44. #22576094C22semidirect
45. C4⋊(C271⋊C8)C22semidirect
46. #22576085C22semidirect
47. #22576084C22semidirect
49. #22576081C22semidirect
50. #22576081C22semidirect
51. #22576081C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. (C2×D271)⋊C8C22semidirect
54. C2×SD16×C271C22semidirect
55. C2×SD16×C271C22semidirect
56. C2×D8×C271C22semidirect
57. C2×D8×C271C22semidirect
59. #124×C271C22semidirect
60. #97×C271C22semidirect
87. #16864643D4semidirect
135. Dic542C2×D4semidirect
177. C8×C271C2×D4semidirect
198. C2×D271AGL(1,ℤ/8)semidirect
218. C271#3265semidirect
224. #496D271semidirect
246. C2×SD16C2×D271semidirect
247. C2×SD16C2×D271semidirect
249. C2×D8C2×D271semidirect
250. C2×D8C2×D271semidirect
255. #124C2×D271semidirect
279. #97C2×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×#225761842
C2×SD16×D2712
#317745552
#317745322
#317741402
#317741302
#317741092
#317740942
#317740462
#317740392
#317740342
#317740292
#317739992
#317739702
#496×C2712
#3265271