Finite Group Databasebeta

← #46847860

Subgroups — #46847860

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×#22576185C211yesmaximal, characteristic, complemented
2C22×#16864641C211yesmaximal, characteristic, complemented
3D4×D1084C211yesmaximal, characteristic, complemented
4#31774287C211yesmaximal, characteristic, complemented
5#31774145C211yesmaximal, characteristic, complemented
6#31774131C211yesmaximal, characteristic, complemented
7#31774109C211yesmaximal, characteristic, complemented
8#31774097C211yesmaximal, characteristic, complemented
9#31774047C211yesmaximal, characteristic, complemented
10#31774042C211yesmaximal, characteristic, complemented
11#31774036C211yesmaximal, characteristic, complemented
12#31774030C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14#31773969C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C2×D4×D271—21no—
19#22576185—24no—
20C2×#16864641—24no—
21C2×#16864641C2211yescharacteristic
22C2×#16864641C2211yescharacteristic
23C2×#16864641C2211yescharacteristic, complemented
24C2×#16864641C2211yescharacteristic, complemented
25C2×#16864641C2211yescharacteristic, complemented
26C2×#16864641C2211yescharacteristic, complemented
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic, complemented
29C2×#16864639C2211yescharacteristic, complemented
30C2×#16864638C2211yescharacteristic, complemented
31C22×C271⋊C8C2211yescharacteristic, complemented
32#22576157C2211yescharacteristic
33#22576135—21no—
34#22576134C2211yescharacteristic
35#22576134C2211yescharacteristic
36#22576130—21no—
37#22576130—21no—
38C4×D1084C2211yescharacteristic
39#22576093C2211yescharacteristic, complemented
40C4⋊(C271⋊C8)C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576081C2211yescharacteristic, complemented
44#22576080C2211yescharacteristic
45#22576080C2211yescharacteristic, complemented
46#22576080C2211yescharacteristic, complemented
47C22⋊(C271⋊C8)C2211yescharacteristic, complemented
48C22⋊(C271⋊C8)C2211yescharacteristic, complemented
49#22576071C2211yescharacteristic, complemented
50C2×#41×C271C2211yescharacteristic, complemented
51C22×Q8×C271C2211yescharacteristic, complemented
52#124×C271—21no—
53#124×C271C2211yescharacteristic, complemented
54C22⋊Q8×C271—21no—
55C22⋊Q8×C271C2211yescharacteristic, complemented
56#121×C271—21no—
57#121×C271C2211yescharacteristic, complemented
58C4×Q8×C271C2211yescharacteristic, complemented
59C4×D4×C271—21no—
60C4×D4×C271C2211yescharacteristic
61C23×D271—21no—
62C23×D271—21no—
63C2×D1084—22no—
64C2×D1084—22no—
65C2×D1084—21no—
66C2×D1084—21no—
67C2×D1084—21no—
68C2×D1084—21no—
69C2×D1084—41no—
70C2×D1084—41no—
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×D1084C2311yescharacteristic
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76D4×D271—44no—
77C2×C4×D271—21no—
78#16864641—22no—
79#16864641—22no—
80#16864641—22no—
81#16864641—22no—
82#16864641—41no—
83#16864641D414yescomplemented
84#16864639—22no—
85#16864639—22no—
86#16864639—41no—
87#16864638—22no—
88C2×C271⋊C8—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C2×C271⋊C8C2311yescharacteristic
92C2×C271⋊C8C2311yescharacteristic
93C4⋊Dic271C2311yescharacteristic
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271—21no—
100C2×Q8×C271—22no—
101C2×Q8×C271—41no—
102C2×Q8×C271C2311yescharacteristic
103C2×Q8×C271C2311yescharacteristic
104C2×D4×C271—21no—
105C2×D4×C271—21no—
106C2×D4×C271D411yescharacteristic
107C2×D4×C271C2311yescharacteristic
108C22×C4×C271—21no—
109C22×C4×C271—21no—
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C4⋊C4×C271—21no—
113C4⋊C4×C271—21no—
114C4⋊C4×C271D411yescharacteristic
115C4⋊C4×C271C2311yescharacteristic
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
122C42×C271—21no—
123C42×C271C2311yescharacteristic
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—
133D1084—21no—
134D1084—21no—
135D1084—21no—
136D1084—21no—
137D1084—21no—
138D1084—42no—
139D1084—42no—
140D1084—42no—
141D1084—81no—
142D1084C2×D412yes—
143C22⋊D271—42no—
144C22⋊D271—42no—
145C2×Dic271—21no—
146C4×D271—42no—
147C271⋊C8—21no—
148C271⋊C8—21no—
149C271⋊C8—21no—
150C271⋊C8C2×D412yescomplemented
151C23×C271—21no—
152C23×C271C2×D411yescharacteristic
153C23×C271C2×D411yescharacteristic
154Q8×C271—21no—
155Q8×C271—21no—
156Q8×C271—21no—
157Q8×C271—42no—
158Q8×C271C2×D412yescomplemented
159D4×C271—22no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—42no—
163D4×C271—42no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—42no—
172C2×C4×C271—42no—
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2×D411yescharacteristic
176C2×C4×C271C2×D411yescharacteristic
177C2×C4×C271C2411yescharacteristic
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—21no—
181C22×C271—21no—
182C22×C271—21no—
183C22×C271—41no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271AGL(1,ℤ/8)12yescomplemented
187C22×C271C22×D411yescharacteristic
188C2×D271—22no—
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—22no—
202C4×C271—21no—
203C4×C271—21no—
204C4×C271—41no—
205C4×C271—41no—
206C4×C271C22×D411yescharacteristic
207C4×C271H(5,𝔽2)11yescharacteristic
208Dic271—41no—
209C2×C271—22no—
210C2×C271—41no—
211C2×C271—81no—
212C2×C271D4211yescharacteristic
213C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C271#57711yescharacteristic
215D271—42no—
216D271—81no—
217D271—81no—
218C271#326511yescharacteristic, complemented, Sylow
219#3265—2711nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2711no—
221C22×SD16—2711no—
222#548D27111yescharacteristic, complemented
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#435—2711no—
235C2×#41C2×D27111yescharacteristic, complemented
236C22×Q8C2×D27111yescharacteristic, complemented
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×M16—2711no—
252C22×C8—2711no—
253#127—2711no—
254#124—21no—
255#124C2×D27111yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D27111yescharacteristic, complemented
258#121—21no—
259#121—2711no—
260#121—2711no—
261#121—5421no—
262#121C2×D27111yescharacteristic, complemented
263#120—5421no—
264#120—5421no—
265C4×Q8C2×D27111yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—2711no—
268C4×D4C2×D27111yescharacteristic
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#97—2711no—
280C24—5421no—
281C24—5421no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D27111yescharacteristic
288C2×Q8C22×D27111yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2711no—
292C2×D4—2711no—
293C2×D4—2711no—
294C2×D4—5422no—
295C2×D4—5422no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4—5421no—
299C2×D4—5421no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—10841no—
303C2×D4—10841no—
304C2×D4—10844no—
305C2×D4C22⋊D27111yescharacteristic
306C2×D4C22×D27111yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
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—5422no—
322C2×C8—2711no—
323C2×C8—2711no—
324C2×C8—2711no—
325C2×C8—2711no—
326C2×C8—5422no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—2711no—
330C4⋊C4C22⋊D27111yescharacteristic
331C4⋊C4C22×D27111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C22⋊C4—5421no—
339C22⋊C4C22⋊D27111yescharacteristic
340C22⋊C4C22⋊D27111yescharacteristic
341C42—21no—
342C42C22×D27111yescharacteristic
343C23—21no—
344C23—5421no—
345C23—5421no—
346C23—5421no—
347C23—10842no—
348C23—10842no—
349C23—10842no—
350C23—10842no—
351C23—21681no—
352C23—21681no—
353C23C2×C22⋊D27111yescharacteristic
354C23C2×C22⋊D27111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D27112yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—2712no—
366D4—5421no—
367D4—5421no—
368D4—5421no—
369D4—5421no—
370D4—5421no—
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—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—5421no—
387C2×C4—10842no—
388C2×C4C2×C22⋊D27111yescharacteristic
389C2×C4C2×C22⋊D27111yescharacteristic
390C2×C4C2×C22⋊D27111yescharacteristic
391C2×C4C2×C22⋊D27111yescharacteristic
392C2×C4C23×D27111yescharacteristic
393C8—2712no—
394C8—5421no—
395C8—5421no—
396C8—5421no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—5422no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22—10841no—
410C22—10841no—
411C22—10841no—
412C22—10841no—
413C22—10841no—
414C22—21684no—
415C22—21684no—
416C22#2257618512yescomplemented
417C22C22×C22⋊D27111yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—10841no—
426C4C2×D4×D27111yescharacteristic
427C4#2257620511yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—10842no—
432C2—21681no—
433C2—21681no—
434C2D4×C22⋊D27111yescharacteristic
435C2C2×#2257618511yescharacteristic
436C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#548×C271C2
socleC22×C271C22×D4
hypercenter#548D271
hypocenterC271#3265
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22576185C2semidirect
2. C22×#16864641C2semidirect
3. D4×D1084C2semidirect
4. #31774287C2semidirect
5. #31774145C2semidirect
6. #31774131C2semidirect
7. #31774109C2semidirect
8. #31774097C2semidirect
9. #31774047C2semidirect
10. #31774042C2semidirect
11. #31774036C2semidirect
12. #31774030C2semidirect
13. #31773995C2semidirect
14. #31773969C2semidirect
15. #548×C271C2semidirect
23. C2×#16864641C22semidirect
24. C2×#16864641C22semidirect
25. C2×#16864641C22semidirect
26. C2×#16864641C22semidirect
28. C2×#16864639C22semidirect
29. C2×#16864639C22semidirect
30. C2×#16864638C22semidirect
31. C22×C271⋊C8C22semidirect
39. #22576093C22semidirect
40. C4⋊(C271⋊C8)C22semidirect
41. #22576084C22semidirect
42. #22576084C22semidirect
43. #22576081C22semidirect
45. #22576080C22semidirect
46. #22576080C22semidirect
47. C22⋊(C271⋊C8)C22semidirect
48. C22⋊(C271⋊C8)C22semidirect
49. #22576071C22semidirect
50. C2×#41×C271C22semidirect
51. C22×Q8×C271C22semidirect
53. #124×C271C22semidirect
55. C22⋊Q8×C271C22semidirect
57. #121×C271C22semidirect
58. C4×Q8×C271C22semidirect
83. #16864641D4semidirect
150. C271⋊C8C2×D4semidirect
158. Q8×C271C2×D4semidirect
186. C22×C271AGL(1,ℤ/8)semidirect
218. C271#3265semidirect
222. #548D271semidirect
235. C2×#41C2×D271semidirect
236. C22×Q8C2×D271semidirect
255. #124C2×D271semidirect
257. C22⋊Q8C2×D271semidirect
262. #121C2×D271semidirect
265. C4×Q8C2×D271semidirect
359. Q8D4×D271semidirect
416. C22#22576185semidirect

Maximal subgroups

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

subgroupindex
C2×#225761852
C22×#168646412
D4×D10842
#317742872
#317741452
#317741312
#317741092
#317740972
#317740472
#317740422
#317740362
#317740302
#317739952
#317739692
#548×C2712
#3265271