Finite Group Databasebeta

← #46847854

Subgroups — #46847854

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×#22576187C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3#31774559C211yesmaximal, characteristic, complemented
4#31774532C211yesmaximal, characteristic, complemented
5#31774284C211yesmaximal, characteristic, complemented
6#31774142C211yesmaximal, characteristic, complemented
7#31774132C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic, complemented
10#31774041C211yesmaximal, characteristic, complemented
11#31774034C211yesmaximal, characteristic, complemented
12#31774032C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14#31773967C211yesmaximal, characteristic, complemented
15#460×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic, complemented
17C2×Q8×D271C2211yescharacteristic, complemented
18C2×D4×D271—21no—
19C2×D4×D271C2211yescharacteristic
20C2×D4×D271C2211yescharacteristic
21#22576187—24no—
22C2×#16864643C2211yescharacteristic
23C2×#16864642C2211yescharacteristic, complemented
24C2×#16864642C2211yescharacteristic, complemented
25C2×#16864641C2211yescharacteristic
26SD16×D271—24no—
27C2×D2168C2211yescharacteristic
28C2×#16864639C2211yescharacteristic, complemented
29C2×#16864639C2211yescharacteristic, complemented
30C2×#16864637C2211yescharacteristic, complemented
31C2×C8×D271C2211yescharacteristic, complemented
32#22576148—21no—
33#22576148C2211yescharacteristic, complemented
34#22576140—21no—
35#22576140C2211yescharacteristic, complemented
36#22576135—21no—
37#22576135C2211yescharacteristic, complemented
38#22576131—21no—
39#22576131C2211yescharacteristic
40#22576131C2211yescharacteristic
41#22576128—21no—
42#22576128—21no—
43C4×D1084—21no—
44C4×D1084C2211yescharacteristic
45C4×D1084C2211yescharacteristic
46#22576093C2211yescharacteristic, complemented
47#22576085C2211yescharacteristic, complemented
48#22576084C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53#22576071C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic, complemented
56#127×C271C2211yescharacteristic
57C4×Q8×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864655—22no—
64#16864655—41no—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—41no—
69C2×D1084—21no—
70C2×D1084—21no—
71C2×D1084D411yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×D1084C2311yescharacteristic
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271—21no—
78D4×D271—22no—
79D4×D271—22no—
80D4×D271—41no—
81D4×D271—41no—
82D4×D271—44no—
83C2×C4×D271—21no—
84C2×C4×D271—21no—
85C2×C4×D271—21no—
86C2×C4×D271C2311yescharacteristic
87C2×C4×D271C2311yescharacteristic
88#16864643—22no—
89#16864642—22no—
90#16864642—41no—
91#16864641D414yescomplemented
92D2168—22no—
93#16864639—22no—
94#16864639—41no—
95#16864637—22no—
96C2×C271⋊C8C2311yescharacteristic
97C2×C271⋊C8C2311yescharacteristic
98C8×D271—22no—
99C4⋊Dic271—21no—
100C4⋊Dic271D411yescharacteristic
101C4⋊Dic271C2311yescharacteristic
102C22⋊Dic271—21no—
103C22⋊Dic271—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4—21no—
109(C2×D271)⋊C4D411yescharacteristic
110(C2×D271)⋊C4D411yescharacteristic
111C2×Q8×C271C2311yescharacteristic
112C2×D4×C271—21no—
113C2×D4×C271—21no—
114C2×D4×C271C2311yescharacteristic
115C2×D4×C271C2311yescharacteristic
116SD16×C271—22no—
117SD16×C271—22no—
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C4⋊C4×C271—21no—
121C4⋊C4×C271C2311yescharacteristic
122C42×C271—21no—
123C42×C271C2311yescharacteristic
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—42no—
130C22×D271—81no—
131C22×D271—81no—
132C22×D271C2×D411yescharacteristic
133C22×D271C2×D411yescharacteristic
134Dic542—21no—
135Dic542—21no—
136Dic542—42no—
137D1084—22no—
138D1084—21no—
139D1084—21no—
140D1084—21no—
141D1084—42no—
142D1084—42no—
143D1084C2×D412yes—
144C22⋊D271—42no—
145C22⋊D271—42no—
146C22⋊D271—42no—
147C22⋊D271—42no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271C2×D411yescharacteristic
151C2×Dic271C2×D411yescharacteristic
152C4×D271—21no—
153C4×D271—21no—
154C4×D271—21no—
155C4×D271—21no—
156C4×D271—42no—
157C4×D271—42no—
158C4×D271—42no—
159C271⋊C8—21no—
160C271⋊C8C2×D412yescomplemented
161C23×C271—21no—
162C23×C271—21no—
163Q8×C271—21no—
164Q8×C271C2×D412yescomplemented
165D4×C271—21no—
166D4×C271—21no—
167D4×C271—21no—
168D4×C271—21no—
169D4×C271—42no—
170D4×C271—81no—
171C2×C4×C271—21no—
172C2×C4×C271—21no—
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2411yescharacteristic
176C8×C271—21no—
177C8×C271—21no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271C22×D411yescharacteristic
185C2×D271—21no—
186C2×D271—21no—
187C2×D271—21no—
188C2×D271—22no—
189C2×D271—21no—
190C2×D271—21no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—84no—
197C2×D271—84no—
198C2×D271AGL(1,ℤ/8)12yescomplemented
199C4×C271—22no—
200C4×C271—21no—
201C4×C271—21no—
202C4×C271—41no—
203C4×C271C22×D411yescharacteristic
204C4×C271H(5,𝔽2)11yescharacteristic
205Dic271—21no—
206Dic271—21no—
207Dic271—41no—
208Dic271—41no—
209C2×C271—81no—
210C2×C271—81no—
211C2×C271D4211yescharacteristic
212C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C271#57711yescharacteristic
214D271—22no—
215D271—41no—
216D271—42no—
217D271—81no—
218C271#326511yescharacteristic, complemented, Sylow
219#3265—2711nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2711no—
221C22×SD16—2711no—
222#548—2711no—
223D42—2711no—
224#496—2711no—
225#468—2711no—
226#466—2711no—
227#463—2711no—
228#460D27111yescharacteristic, complemented
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×D8—2711no—
250C2×D8—2711no—
251C2×M16—2711no—
252C22×C8—2711no—
253#127C2×D27111yescharacteristic
254#124—2711no—
255#124—5421no—
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×Q8C2×D27111yescharacteristic, complemented
266C4×D4—2711no—
267C4×D4—2711no—
268C4×D4—5421no—
269#107—2711no—
270C4⋊C8C2×D27111yescharacteristic, complemented
271#103—2711no—
272#103—2711no—
273#102—2711no—
274#102—2711no—
275#102C2×D27111yescharacteristic, complemented
276#102C2×D27111yescharacteristic, complemented
277C22⋊C8—2711no—
278C22⋊C8—2711no—
279#97—2711no—
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—2711no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4—5422no—
299C2×D4—5422no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—10841no—
303C2×D4—10841no—
304C2×D4—10844no—
305C2×D4C22×D27111yescharacteristic
306C2×D4C22×D27111yescharacteristic
307C22×C4—2711no—
308C22×C4—2711no—
309C22×C4—5421no—
310C22×C4—5421no—
311C22×C4—5421no—
312SD16—22no—
313SD16—22no—
314SD16—2714no—
315SD16—5422no—
316SD16—5422no—
317SD16—10841no—
318D8—5422no—
319D8—5422no—
320D8—10841no—
321M16—5422no—
322C2×C8—2711no—
323C2×C8—2711no—
324C2×C8—5422no—
325C2×C8C22×D27111yescharacteristic
326C2×C8C22×D27111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2711no—
329C4⋊C4—2711no—
330C4⋊C4—5421no—
331C4⋊C4C22×D27111yescharacteristic
332C22⋊C4—2711no—
333C22⋊C4—2711no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C22⋊C4—5421no—
339C22⋊C4—5421no—
340C22⋊C4—5421no—
341C42—21no—
342C42C22×D27111yescharacteristic
343C23—21no—
344C23—21no—
345C23—2711no—
346C23—2711no—
347C23—5421no—
348C23—5421no—
349C23—10842no—
350C23—10842no—
351C23—10842no—
352C23—10842no—
353C23—21681no—
354C23—21681no—
355Q8—21no—
356Q8—5421no—
357Q8—5421no—
358Q8—10842no—
359Q8D4×D27112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—81no—
366D4—2712no—
367D4—5422no—
368D4—5421no—
369D4—5421no—
370D4—5421no—
371D4—10842no—
372D4—10842no—
373D4—10842no—
374D4—10842no—
375D4—10842no—
376D4—10842no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—2711no—
380C2×C4—2711no—
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—21no—
395C8—2712no—
396C8—5421no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2712no—
404C22—5421no—
405C22—5421no—
406C22—5421no—
407C22—5422no—
408C22—5421no—
409C22—5421no—
410C22—10841no—
411C22—10841no—
412C22—10841no—
413C22—10841no—
414C22—10841no—
415C22—21684no—
416C22—21684no—
417C22C2×D4×D27111yescharacteristic
418C4—22no—
419C4—21no—
420C4—21no—
421C4—41no—
422C4—5421no—
423C4—5421no—
424C4—10841no—
425C4—10841no—
426C4C2×D4×D27111yescharacteristic
427C4#2257620611yescharacteristic
428C2—81no—
429C2—81no—
430C2—5422no—
431C2—10841no—
432C2—10842no—
433C2—21681no—
434C2#3177453111yescharacteristic
435C2AGL(1,ℤ/8)×D27111yescharacteristic
436C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#460×C271C2
socleC22×C271C22×D4
hypercenter#460D271
hypocenterC271#3265
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×SD16×D271C2semidirect
3. #31774559C2semidirect
4. #31774532C2semidirect
5. #31774284C2semidirect
6. #31774142C2semidirect
7. #31774132C2semidirect
8. #31774113C2semidirect
9. #31774049C2semidirect
10. #31774041C2semidirect
11. #31774034C2semidirect
12. #31774032C2semidirect
13. #31773995C2semidirect
14. #31773967C2semidirect
15. #460×C271C2semidirect
16. C2×#16864655C22semidirect
17. C2×Q8×D271C22semidirect
23. C2×#16864642C22semidirect
24. C2×#16864642C22semidirect
28. C2×#16864639C22semidirect
29. C2×#16864639C22semidirect
30. C2×#16864637C22semidirect
31. C2×C8×D271C22semidirect
33. #22576148C22semidirect
35. #22576140C22semidirect
37. #22576135C22semidirect
46. #22576093C22semidirect
47. #22576085C22semidirect
48. #22576084C22semidirect
49. #22576082C22semidirect
51. (C2×D271)⋊C8C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. #22576071C22semidirect
54. C2×SD16×C271C22semidirect
55. C2×SD16×C271C22semidirect
57. C4×Q8×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #102×C271C22semidirect
60. #102×C271C22semidirect
91. #16864641D4semidirect
160. C271⋊C8C2×D4semidirect
164. Q8×C271C2×D4semidirect
198. C2×D271AGL(1,ℤ/8)semidirect
218. C271#3265semidirect
228. #460D271semidirect
246. C2×SD16C2×D271semidirect
247. C2×SD16C2×D271semidirect
265. C4×Q8C2×D271semidirect
270. C4⋊C8C2×D271semidirect
275. #102C2×D271semidirect
276. #102C2×D271semidirect
359. Q8D4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×SD16×D2712
#317745592
#317745322
#317742842
#317741422
#317741322
#317741132
#317740492
#317740412
#317740342
#317740322
#317739952
#317739672
#460×C2712
#3265271