Finite Group Databasebeta

← #46847864

Subgroups — #46847864

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
2C22×#16864643C211yesmaximal, characteristic, complemented
3#31774556C211yesmaximal, characteristic, complemented
4D4×D1084C211yesmaximal, characteristic, complemented
5#31774286C211yesmaximal, characteristic, complemented
6#31774143C211yesmaximal, characteristic, complemented
7#31774133C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774098C211yesmaximal, characteristic, complemented
10#31774050C211yesmaximal, characteristic, complemented
11#31774044C211yesmaximal, characteristic, complemented
12#31774038C211yesmaximal, characteristic, complemented
13#31774033C211yesmaximal, characteristic, complemented
14#31773999C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic, complemented
17C22×Dic542C2211yescharacteristic, complemented
18C22×D1084C2211yescharacteristic
19C22×D1084C2211yescharacteristic
20C2×D4×D271—21no—
21#22576188—24no—
22C2×#16864643—24no—
23C2×#16864643C2211yescharacteristic
24C2×#16864643C2211yescharacteristic
25C2×#16864643C2211yescharacteristic, complemented
26C2×#16864643C2211yescharacteristic, complemented
27C2×#16864643C2211yescharacteristic, complemented
28C2×#16864643C2211yescharacteristic, complemented
29C2×D2168C2211yescharacteristic
30C2×D2168C2211yescharacteristic, complemented
31C2×D2168C2211yescharacteristic, complemented
32#22576157C2211yescharacteristic
33#22576148C2211yescharacteristic, complemented
34#22576146—21no—
35#22576139C2211yescharacteristic, complemented
36#22576138—21no—
37#22576135—21no—
38#22576134C2211yescharacteristic
39#22576134C2211yescharacteristic
40#22576134C2211yescharacteristic, complemented
41#22576132—21no—
42#22576130—21no—
43#22576130—21no—
44C4×Dic542C2211yescharacteristic, complemented
45C4×D1084C2211yescharacteristic
46(C22⋊D271)⋊C4—21no—
47#22576094C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576085C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic
51#22576082C2211yescharacteristic, complemented
52#22576082C2211yescharacteristic, complemented
53#22576082C2211yescharacteristic, complemented
54C2×M16×C271C2211yescharacteristic, complemented
55C22×C8×C271C2211yescharacteristic, complemented
56C4×D4×C271C2211yescharacteristic
57C4⋊C8×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic, complemented
59C22⋊C8×C271C2211yescharacteristic, complemented
60#97×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864654—22no—
64#16864654—41no—
65C2×Dic542—22no—
66C2×Dic542—21no—
67C2×Dic542—41no—
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2311yescharacteristic
70C2×D1084—22no—
71C2×D1084—22no—
72C2×D1084—21no—
73C2×D1084—21no—
74C2×D1084—21no—
75C2×D1084—21no—
76C2×D1084—41no—
77C2×D1084—41no—
78C2×D1084C2311yescharacteristic
79C2×D1084C2311yescharacteristic
80C2×D1084C2311yescharacteristic
81C2×D1084C2311yescharacteristic
82C2×C22⋊D271—21no—
83C2×C22⋊D271—21no—
84C2×C22⋊D271—21no—
85C2×C22⋊D271—21no—
86D4×D271—44no—
87C22×Dic271—21no—
88C2×C4×D271—21no—
89C2×C4×D271—21no—
90#16864643—22no—
91#16864643—22no—
92#16864643—22no—
93#16864643—22no—
94#16864643—41no—
95#16864643D414yescomplemented
96D2168—22no—
97D2168—22no—
98D2168—41no—
99C4⋊Dic271C2311yescharacteristic
100C4⋊Dic271C2311yescharacteristic
101Dic271⋊C4—21no—
102Dic271⋊C4—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)⋊C4—21no—
110C4×Dic271—21no—
111C2×D4×C271D411yescharacteristic
112C22×C4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114M16×C271—22no—
115C2×C8×C271—22no—
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C4⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
122C22⋊C4×C271D411yescharacteristic
123C42×C271C2311yescharacteristic
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—42no—
129C22×D271—42no—
130C22×D271—42no—
131C22×D271—42no—
132C22×D271—81no—
133C22×D271—81no—
134Dic542—21no—
135Dic542—21no—
136Dic542—21no—
137Dic542—42no—
138Dic542C2×D412yescomplemented
139D1084—21no—
140D1084—21no—
141D1084—21no—
142D1084—21no—
143D1084—21no—
144D1084—21no—
145D1084—21no—
146D1084—42no—
147D1084—42no—
148D1084—42no—
149D1084—81no—
150D1084C2×D412yes—
151C22⋊D271—42no—
152C22⋊D271—42no—
153C22⋊D271—42no—
154C22⋊D271—42no—
155C2×Dic271—21no—
156C2×Dic271—21no—
157C2×Dic271—21no—
158C2×Dic271—21no—
159C2×Dic271—42no—
160C4×D271—42no—
161C4×D271—42no—
162C23×C271C2×D411yescharacteristic
163C23×C271C2×D411yescharacteristic
164D4×C271—22no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271C2×D411yescharacteristic
170C2×C4×C271C2×D411yescharacteristic
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2×D411yescharacteristic
173C2×C4×C271C2411yescharacteristic
174C8×C271—21no—
175C8×C271—21no—
176C8×C271—21no—
177C8×C271C2×D412yescomplemented
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—21no—
181C22×C271—21no—
182C22×C271—21no—
183C22×C271AGL(1,ℤ/8)12yescomplemented
184C22×C271C22×D411yescharacteristic
185C2×D271—22no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—41no—
189C2×D271—41no—
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—21no—
201C4×C271—21no—
202C4×C271—21no—
203C4×C271C22×D411yescharacteristic
204C4×C271H(5,𝔽2)11yescharacteristic
205Dic271—22no—
206Dic271—41no—
207Dic271—41no—
208Dic271—41no—
209C2×C271—22no—
210C2×C271—41no—
211C2×C271D4211yescharacteristic
212C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C271#57711yescharacteristic
214D271—42no—
215D271—81no—
216D271—81no—
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#460—2711no—
229#450—2711no—
230#449—2711no—
231#448—2711no—
232#447—2711no—
233#440—2711no—
234#435D27111yescharacteristic, complemented
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—2711no—
246C2×SD16—2711no—
247C2×SD16—5424no—
248C2×D8—2711no—
249C2×D8—2711no—
250C2×D8—2711no—
251C2×M16C2×D27111yescharacteristic, complemented
252C22×C8C2×D27111yescharacteristic, complemented
253#127—2711no—
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×Q8—2711no—
266C4×D4—2711no—
267C4×D4—5421no—
268C4×D4C2×D27111yescharacteristic
269#107—2711no—
270C4⋊C8C2×D27111yescharacteristic, complemented
271#103—2711no—
272#103—2711no—
273#102—2711no—
274#102—2711no—
275#102—2711no—
276#102—2711no—
277C22⋊C8C2×D27111yescharacteristic, complemented
278C22⋊C8C2×D27111yescharacteristic, complemented
279#97C2×D27111yescharacteristic, complemented
280C24—5421no—
281C24—5421no—
282#41—5422no—
283#41—10841no—
284C2×Q8—2711no—
285C2×Q8—2711no—
286C2×Q8—5422no—
287C2×Q8—5421no—
288C2×Q8—10841no—
289C2×D4—2711no—
290C2×D4—2711no—
291C2×D4—2711no—
292C2×D4—2711no—
293C2×D4—5422no—
294C2×D4—5422no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4—5421no—
299C2×D4—5421no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—5421no—
303C2×D4—10841no—
304C2×D4—10841no—
305C2×D4—10844no—
306C2×D4D108411yescharacteristic
307C22×C4—5421no—
308C22×C4—5421no—
309C22×C4—5421no—
310C22×C4C22×D27111yescharacteristic
311C22×C4C22×D27111yescharacteristic
312SD16—2714no—
313SD16—5422no—
314SD16—5422no—
315SD16—5422no—
316SD16—5422no—
317SD16—10841no—
318D8—5422no—
319D8—5422no—
320D8—10841no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D27111yescharacteristic
324C2×C8C22×D27111yescharacteristic
325C2×C8C22×D27111yescharacteristic
326C2×C8C22×D27111yescharacteristic
327C4⋊C4—2711no—
328C4⋊C4—2711no—
329C4⋊C4—5421no—
330C4⋊C4—5421no—
331C4⋊C4D108411yescharacteristic
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C22⋊C4—5421no—
339C22⋊C4D108411yescharacteristic
340C22⋊C4D108411yescharacteristic
341C42—5421no—
342C42C22×D27111yescharacteristic
343C23—5421no—
344C23—5421no—
345C23—5421no—
346C23—5421no—
347C23—10842no—
348C23—10842no—
349C23—10842no—
350C23—10842no—
351C23—21681no—
352C23—21681no—
353C23C2×D108411yescharacteristic
354C23C2×D108411yescharacteristic
355Q8—2712no—
356Q8—5421no—
357Q8—5421no—
358Q8—5421no—
359Q8—10842no—
360D4—22no—
361D4—2712no—
362D4—5421no—
363D4—5421no—
364D4—5421no—
365D4—5421no—
366D4—5421no—
367D4—5421no—
368D4—5421no—
369D4—10842no—
370D4—10842no—
371D4—10842no—
372D4—10842no—
373D4—10842no—
374D4—10842no—
375D4—10842no—
376D4—21681no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—5421no—
382C2×C4—5421no—
383C2×C4—5421no—
384C2×C4—5421no—
385C2×C4—10842no—
386C2×C4—10842no—
387C2×C4—10842no—
388C2×C4C2×D108411yescharacteristic
389C2×C4C2×D108411yescharacteristic
390C2×C4C2×D108411yescharacteristic
391C2×C4C2×D108411yescharacteristic
392C2×C4C23×D27111yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D27112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—5422no—
403C22—10841no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22—10841no—
410C22—10841no—
411C22—10841no—
412C22—10841no—
413C22—10841no—
414C22—21684no—
415C22—21684no—
416C22#2257618812yescomplemented
417C22C22×D108411yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—5422no—
423C4—10841no—
424C4—10841no—
425C4—10841no—
426C4C2×D4×D27111yescharacteristic
427C4#2257620611yescharacteristic
428C2—22no—
429C2—41no—
430C2—10842no—
431C2—21681no—
432C2—21681no—
433C2—21681no—
434C2D4×D108411yescharacteristic
435C2C2×#2257618811yescharacteristic
436C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
hypercenter#435D271
hypocenterC271#3265
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C22×#16864643C2semidirect
3. #31774556C2semidirect
4. D4×D1084C2semidirect
5. #31774286C2semidirect
6. #31774143C2semidirect
7. #31774133C2semidirect
8. #31774113C2semidirect
9. #31774098C2semidirect
10. #31774050C2semidirect
11. #31774044C2semidirect
12. #31774038C2semidirect
13. #31774033C2semidirect
14. #31773999C2semidirect
15. #435×C271C2semidirect
16. C2×#16864654C22semidirect
17. C22×Dic542C22semidirect
25. C2×#16864643C22semidirect
26. C2×#16864643C22semidirect
27. C2×#16864643C22semidirect
28. C2×#16864643C22semidirect
30. C2×D2168C22semidirect
31. C2×D2168C22semidirect
33. #22576148C22semidirect
35. #22576139C22semidirect
40. #22576134C22semidirect
44. C4×Dic542C22semidirect
47. #22576094C22semidirect
48. #22576085C22semidirect
49. #22576085C22semidirect
51. #22576082C22semidirect
52. #22576082C22semidirect
53. #22576082C22semidirect
54. C2×M16×C271C22semidirect
55. C22×C8×C271C22semidirect
57. C4⋊C8×C271C22semidirect
58. C22⋊C8×C271C22semidirect
59. C22⋊C8×C271C22semidirect
60. #97×C271C22semidirect
95. #16864643D4semidirect
138. Dic542C2×D4semidirect
177. C8×C271C2×D4semidirect
183. C22×C271AGL(1,ℤ/8)semidirect
218. C271#3265semidirect
234. #435D271semidirect
251. C2×M16C2×D271semidirect
252. C22×C8C2×D271semidirect
270. C4⋊C8C2×D271semidirect
277. C22⋊C8C2×D271semidirect
278. C22⋊C8C2×D271semidirect
279. #97C2×D271semidirect
396. C8D4×D271semidirect
416. C22#22576188semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
C22×#168646432
#317745562
D4×D10842
#317742862
#317741432
#317741332
#317741132
#317740982
#317740502
#317740442
#317740382
#317740332
#317739992
#435×C2712
#3265271