Finite Group Databasebeta

← #46847852

Subgroups — #46847852

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
2C22×#16864641C211yesmaximal, characteristic, complemented
3#31774558C211yesmaximal, characteristic, complemented
4#31774531C211yesmaximal, characteristic, complemented
5#31774283C211yesmaximal, characteristic, complemented
6#31774146C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774115C211yesmaximal, characteristic, complemented
9#31774095C211yesmaximal, characteristic, complemented
10#31774046C211yesmaximal, characteristic, complemented
11#31774045C211yesmaximal, characteristic, complemented
12#31774033C211yesmaximal, characteristic, complemented
13#31774000C211yesmaximal, characteristic, complemented
14#31773965C211yesmaximal, characteristic, complemented
15#448×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic, complemented
17C22×D1084C2211yescharacteristic
18C2×D4×D271—21no—
19C2×D4×D271C2211yescharacteristic
20#22576187—24no—
21C2×#16864641—24no—
22C2×#16864641C2211yescharacteristic
23C2×#16864641C2211yescharacteristic
24C2×#16864641C2211yescharacteristic, complemented
25C2×#16864641C2211yescharacteristic, complemented
26C2×D2168C2211yescharacteristic
27C2×D2168C2211yescharacteristic, complemented
28C2×#16864639C2211yescharacteristic, complemented
29C2×#16864637C2211yescharacteristic, complemented
30C22×C271⋊C8C2211yescharacteristic, complemented
31#22576156C2211yescharacteristic
32#22576149—21no—
33#22576149C2211yescharacteristic, complemented
34#22576136—21no—
35#22576136C2211yescharacteristic, complemented
36#22576135C2211yescharacteristic
37#22576134—21no—
38#22576134C2211yescharacteristic, complemented
39#22576133—21no—
40#22576130—21no—
41#22576128—21no—
42Q8×Dic271C2211yescharacteristic, complemented
43#22576122C2211yescharacteristic
44C4×C22⋊D271—21no—
45C4×C22⋊D271C2211yescharacteristic
46#22576093C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic
49#22576082C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51#22576080C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53#22576070C2211yescharacteristic, complemented
54C22×Q8×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic, complemented
56C2×SD16×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60C22⋊C8×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864655—22no—
64#16864655—41no—
65C2×D1084—22no—
66C2×D1084—21no—
67C2×D1084—21no—
68C2×D1084—21no—
69C2×D1084—41no—
70C2×D1084C2311yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271D411yescharacteristic
78D4×D271—22no—
79D4×D271—41no—
80D4×D271—44no—
81C2×C4×D271—21no—
82C2×C4×D271—21no—
83C2×C4×D271C2311yescharacteristic
84#16864641—22no—
85#16864641—22no—
86#16864641—22no—
87#16864641D414yescomplemented
88D2168—22no—
89D2168—41no—
90#16864639—22no—
91#16864637—22no—
92C2×C271⋊C8—22no—
93C2×C271⋊C8C2311yescharacteristic
94C2×C271⋊C8C2311yescharacteristic
95C4⋊Dic271—21no—
96C4⋊Dic271C2311yescharacteristic
97Dic271⋊C4—21no—
98Dic271⋊C4D411yescharacteristic
99C22⋊Dic271D411yescharacteristic
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4D411yescharacteristic
107C4×Dic271—21no—
108C4×Dic271C2311yescharacteristic
109C2×Q8×C271—21no—
110C2×Q8×C271—22no—
111C2×Q8×C271—41no—
112C2×Q8×C271C2311yescharacteristic
113C2×Q8×C271C2311yescharacteristic
114C2×D4×C271—21no—
115C2×D4×C271C2311yescharacteristic
116C22×C4×C271—21no—
117C22×C4×C271C2311yescharacteristic
118SD16×C271—22no—
119SD16×C271—41no—
120C2×C8×C271C2311yescharacteristic
121C2×C8×C271C2311yescharacteristic
122C4⋊C4×C271C2311yescharacteristic
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
134D1084—21no—
135D1084—21no—
136D1084—21no—
137D1084—21no—
138D1084—21no—
139D1084—42no—
140D1084—42no—
141D1084—42no—
142D1084C2×D412yes—
143C22⋊D271—22no—
144C22⋊D271—42no—
145C22⋊D271—42no—
146C22⋊D271—42no—
147C22⋊D271—81no—
148C2×Dic271—21no—
149C2×Dic271C2×D411yescharacteristic
150C2×Dic271C2×D411yescharacteristic
151C2×Dic271C2×D411yescharacteristic
152C4×D271—21no—
153C4×D271—21no—
154C4×D271—42no—
155C4×D271—42no—
156C271⋊C8—21no—
157C271⋊C8C2×D412yescomplemented
158C23×C271—21no—
159C23×C271C2×D411yescharacteristic
160Q8×C271—21no—
161Q8×C271—21no—
162Q8×C271—21no—
163Q8×C271—42no—
164Q8×C271C2×D412yescomplemented
165D4×C271—21no—
166D4×C271—21no—
167D4×C271—42no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—21no—
173C2×C4×C271—42no—
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2411yescharacteristic
176C8×C271—21no—
177C8×C271—21no—
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271AGL(1,ℤ/8)12yescomplemented
184C22×C271C22×D411yescharacteristic
185C2×D271—21no—
186C2×D271—22no—
187C2×D271—21no—
188C2×D271—21no—
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—22no—
201C4×C271—41no—
202C4×C271—41no—
203C4×C271C22×D411yescharacteristic
204C4×C271H(5,𝔽2)11yescharacteristic
205Dic271—21no—
206Dic271—21no—
207Dic271—21no—
208Dic271—41no—
209C2×C271—22no—
210C2×C271—81no—
211C2×C271D4211yescharacteristic
212C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C271#57711yescharacteristic
214D271—41no—
215D271—42no—
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#448D27111yescharacteristic, complemented
232#447—2711no—
233#440—2711no—
234#435—2711no—
235C2×#41—2711no—
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—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#127—2711no—
254#124—2711no—
255#124—5421no—
256C22⋊Q8—2711no—
257C22⋊Q8—5421no—
258#121—2711no—
259#121—2711no—
260#121—5421no—
261#121—5421no—
262#121C2×D27111yescharacteristic
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#103C2×D27111yescharacteristic, complemented
272#103C2×D27111yescharacteristic, complemented
273#102—2711no—
274#102—2711no—
275#102—2711no—
276#102—2711no—
277C22⋊C8—2711no—
278C22⋊C8C2×D27111yescharacteristic, complemented
279#97—2711no—
280C24—5421no—
281C24—5421no—
282#41—5422no—
283#41—10841no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D27111yescharacteristic
288C2×Q8C22×D27111yescharacteristic
289C2×D4—21no—
290C2×D4—2711no—
291C2×D4—2711no—
292C2×D4—2711no—
293C2×D4—2711no—
294C2×D4—5422no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4—5421no—
299C2×D4—5421no—
300C2×D4—5422no—
301C2×D4—5421no—
302C2×D4—5421no—
303C2×D4—10841no—
304C2×D4—10841no—
305C2×D4—10844no—
306C2×D4C22×D27111yescharacteristic
307C22×C4—21no—
308C22×C4—2711no—
309C22×C4—5421no—
310C22×C4—5421no—
311C22×C4C22×D27111yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—2714no—
315SD16—5422no—
316SD16—5422no—
317SD16—5422no—
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—2711no—
328C4⋊C4—2711no—
329C4⋊C4—5421no—
330C4⋊C4—5421no—
331C4⋊C4C22×D27111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—2711no—
334C22⋊C4—2711no—
335C22⋊C4—5421no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C22⋊C4—5421no—
339C22⋊C4—5421no—
340C22⋊C4—5421no—
341C42—2711no—
342C42—5421no—
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—21no—
358Q8—42no—
359Q8D4×D27112yescomplemented
360D4—21no—
361D4—21no—
362D4—42no—
363D4—2712no—
364D4—5422no—
365D4—5421no—
366D4—5421no—
367D4—5421no—
368D4—5421no—
369D4—5421no—
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—21no—
382C2×C4—42no—
383C2×C4—2711no—
384C2×C4—2711no—
385C2×C4—2711no—
386C2×C4—5421no—
387C2×C4—5421no—
388C2×C4—5421no—
389C2×C4—10842no—
390C2×C4—10842no—
391C2×C4D4×D27111yescharacteristic
392C2×C4C23×D27111yescharacteristic
393C8—21no—
394C8—21no—
395C8—2712no—
396C8—5421no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—5421no—
403C22—5422no—
404C22—5421no—
405C22—5421no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22—10841no—
410C22—10841no—
411C22—10841no—
412C22—10841no—
413C22—10841no—
414C22—21684no—
415C22—21684no—
416C22#2257618712yescomplemented
417C22C2×D4×D27111yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—5421no—
423C4—5421no—
424C4—5421no—
425C4—10841no—
426C4C2×D4×D27111yescharacteristic
427C4#2257620511yescharacteristic
428C2—22no—
429C2—81no—
430C2—10841no—
431C2—10842no—
432C2—21681no—
433C2—21681no—
434C2#3177452911yescharacteristic
435C2C2×#2257618711yescharacteristic
436C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#448×C271C2
socleC22×C271C22×D4
hypercenter#448D271
hypocenterC271#3265
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C22×#16864641C2semidirect
3. #31774558C2semidirect
4. #31774531C2semidirect
5. #31774283C2semidirect
6. #31774146C2semidirect
7. #31774131C2semidirect
8. #31774115C2semidirect
9. #31774095C2semidirect
10. #31774046C2semidirect
11. #31774045C2semidirect
12. #31774033C2semidirect
13. #31774000C2semidirect
14. #31773965C2semidirect
15. #448×C271C2semidirect
16. C2×#16864655C22semidirect
24. C2×#16864641C22semidirect
25. C2×#16864641C22semidirect
27. C2×D2168C22semidirect
28. C2×#16864639C22semidirect
29. C2×#16864637C22semidirect
30. C22×C271⋊C8C22semidirect
33. #22576149C22semidirect
35. #22576136C22semidirect
38. #22576134C22semidirect
42. Q8×Dic271C22semidirect
46. #22576093C22semidirect
47. Dic271⋊C8C22semidirect
49. #22576082C22semidirect
50. #22576081C22semidirect
51. #22576080C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. #22576070C22semidirect
54. C22×Q8×C271C22semidirect
55. C2×SD16×C271C22semidirect
56. C2×SD16×C271C22semidirect
58. #103×C271C22semidirect
59. #103×C271C22semidirect
60. C22⋊C8×C271C22semidirect
87. #16864641D4semidirect
157. C271⋊C8C2×D4semidirect
164. Q8×C271C2×D4semidirect
183. C22×C271AGL(1,ℤ/8)semidirect
218. C271#3265semidirect
231. #448D271semidirect
236. C22×Q8C2×D271semidirect
246. C2×SD16C2×D271semidirect
247. C2×SD16C2×D271semidirect
271. #103C2×D271semidirect
272. #103C2×D271semidirect
278. C22⋊C8C2×D271semidirect
359. Q8D4×D271semidirect
416. C22#22576187semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C22×#168646412
#317745582
#317745312
#317742832
#317741462
#317741312
#317741152
#317740952
#317740462
#317740452
#317740332
#317740002
#317739652
#448×C2712
#3265271