Finite Group Databasebeta

← #46846501

Subgroups — #46846501

Order 34688.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774426C211yesmaximal, characteristic, complemented
2C4×D4×D271C211yesmaximal, characteristic, complemented
3#31774226C211yesmaximal, characteristic, complemented
4#31774209C211yesmaximal, characteristic, complemented
5#31774198C211yesmaximal, characteristic, complemented
6#31774197C211yesmaximal, characteristic, complemented
7#31774109C211yesmaximal, characteristic, complemented
8#31774046C211yesmaximal, characteristic, complemented
9#31774029C211yesmaximal, characteristic
10#31774009C211yesmaximal, characteristic, complemented
11#31774006C211yesmaximal, characteristic, complemented
12#31774000C211yesmaximal, characteristic, complemented
13#31773998C211yesmaximal, characteristic, complemented
14#31773953C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C22×C4×D271—21no—
18C2×#16864641C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20#22576153—21no—
21#22576152C2211yescharacteristic, complemented
22#22576142—21no—
23#22576136C2211yescharacteristic
24#22576131C2211yescharacteristic
25Q8×Dic271C2211yescharacteristic, complemented
26C4×D1084C2211yescharacteristic
27D4×Dic271C2211yescharacteristic
28D4×Dic271C2211yescharacteristic, complemented
29#22576122C2211yescharacteristic
30C4×C22⋊D271—21no—
31(C22⋊D271)⋊C4—21no—
32(C22⋊D271)⋊C4—21no—
33#22576116—21no—
34C4⋊C4×D271C2211yescharacteristic
35C22⋊C4×D271—21no—
36C42×D271C2211yescharacteristic
37#22576094C2211yescharacteristic, complemented
38#22576093C2211yescharacteristic, complemented
39#22576092C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41C4⋊(C271⋊C8)C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic, complemented
48(C2×D271)⋊C8C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51#22576070C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic
53C2×D8×C271C2211yescharacteristic
54#124×C271C2211yescharacteristic, complemented
55C4×D4×C271C2211yescharacteristic
56C4⋊C8×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C2×D1084C2311yescharacteristic
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63D4×D271—22no—
64D4×D271—22no—
65C22×Dic271—21no—
66C22×Dic271—21no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—44no—
71C2×C4×D271D411yescharacteristic
72C2×C4×D271D411yescharacteristic
73C2×C4×D271C2311yescharacteristic
74#16864641—22no—
75#16864639—22no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92C4×Dic271—21no—
93C4×Dic271D411yescharacteristic
94C4×Dic271D411yescharacteristic
95C4×Dic271C2311yescharacteristic
96C2×Q8×C271C2311yescharacteristic
97C2×D4×C271C2311yescharacteristic
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271—21no—
100SD16×C271—22no—
101D8×C271—22no—
102C2×C8×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C22⋊C4×C271—21no—
108C42×C271C2311yescharacteristic
109C22×D271—24no—
110C22×D271—21no—
111C22×D271—42no—
112C22×D271—42no—
113C22×D271C2×D411yescharacteristic
114D1084—21no—
115D1084#4112yes—
116C22⋊D271—24no—
117C22⋊D271—42no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—42no—
122C2×Dic271—42no—
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C4×D271—22no—
127C4×D271—21no—
128C4×D271—21no—
129C4×D271—41no—
130C4×D271—42no—
131C4×D271—42no—
132C4×D271—42no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271—21no—
136C23×C271—21no—
137Q8×C271—21no—
138Q8×C271—21no—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—21no—
142D4×C271#4112yes—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—42no—
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2411yescharacteristic
149C8×C271—21no—
150C8×C271—21no—
151C22×C271—22no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271C22×D411yescharacteristic
158C2×D271—21no—
159C2×D271—21no—
160C2×D271—22no—
161C2×D271—48no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271AGL(1,ℤ/8)12yes—
165C4×C271—21no—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271C2×#4111yescharacteristic
171Dic271—21no—
172Dic271—21no—
173Dic271—21no—
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177C2×C271—42no—
178C2×C271—81no—
179C2×C271#52511yescharacteristic
180C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C271#57511yescharacteristic
182D271—22no—
183D271—42no—
184C271#311411yescharacteristic, complemented, Sylow
185#3114—2711nomaximal, Sylow
186#529—2711no—
187C2×C4×D4—2711no—
188#483—2711no—
189#480—2711no—
190#478—2711no—
191#478—2711no—
192#463—2711no—
193#463D27111yescharacteristic, complemented
194#450—2711no—
195#447—2711no—
196#442—2711no—
197#442—2711no—
198#440—2711no—
199#440—2711no—
200#432—2711no—
201C22×D4—2711no—
202C23×C4—5421no—
203C2×SD16—2711no—
204C2×SD16C2×D27111yescharacteristic
205C2×D8—2711no—
206C2×D8C2×D27111yescharacteristic
207#126—5421no—
208#125—2711no—
209#124C2×D27111yescharacteristic, complemented
210#123—5421no—
211C22⋊Q8—2711no—
212#121—2711no—
213C4×Q8—2711no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—2711no—
217C4×D4—2711no—
218C4×D4—5421no—
219C4×D4—5421no—
220C4×D4—5421no—
221C4×D4C2×D27111yescharacteristic
222#117—5421no—
223C2×C4⋊C4—2711no—
224C2×C22⋊C4—5421no—
225C2×C42—2711no—
226#107—2711no—
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8—2711no—
231C4⋊C8C2×D27111yescharacteristic, complemented
232#103—2711no—
233#103C2×D27111yescharacteristic, complemented
234#102—2711no—
235#102—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102C2×D27111yescharacteristic, complemented
240C22⋊C8—2711no—
241C22⋊C8—2711no—
242#97—2711no—
243#97—2711no—
244C24—5421no—
245C2×Q8C22×D27111yescharacteristic
246C2×D4—2711no—
247C2×D4—5421no—
248C2×D4—5422no—
249C2×D4—5422no—
250C2×D4—5421no—
251C2×D4C22×D27111yescharacteristic
252C2×D4C22×D27111yescharacteristic
253C22×C4—21no—
254C22×C4—2711no—
255C22×C4—2711no—
256C22×C4—2711no—
257C22×C4—5421no—
258C22×C4—5421no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—10844no—
263SD16—22no—
264SD16—5422no—
265D8—22no—
266D8—5422no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8C22×D27111yescharacteristic
270C2×C8C22×D27111yescharacteristic
271C4⋊C4—2711no—
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4C22×D27111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C42—2711no—
290C42—2711no—
291C42—2711no—
292C42—5421no—
293C42C22×D27111yescharacteristic
294C23—21no—
295C23—21no—
296C23—2711no—
297C23—5424no—
298C23—5421no—
299C23—10842no—
300C23—10842no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—2712no—
307D4—5424no—
308D4—5421no—
309D4—10842no—
310D4#1686465312yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—2711no—
315C2×C4—2711no—
316C2×C4—2711no—
317C2×C4—5422no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—10841no—
324C2×C4—10842no—
325C2×C4—10842no—
326C2×C4—10842no—
327C2×C4—10842no—
328C2×C4—10842no—
329C2×C4D4×D27111yescharacteristic
330C2×C4D4×D27111yescharacteristic
331C2×C4C23×D27111yescharacteristic
332C8—21no—
333C8—21no—
334C8—5421no—
335C8—5421no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—2712no—
343C22—5421no—
344C22—5421no—
345C22—5422no—
346C22—10848no—
347C22—10841no—
348C22—10841no—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—5421no—
355C4—5421no—
356C4—5421no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4#41×D27111yescharacteristic
361C4C2×#1686465311yescharacteristic
362C2—42no—
363C2—81no—
364C2—5422no—
365C2—10842no—
366C2#3177440711yescharacteristic
367C2AGL(1,ℤ/8)×D27111yescharacteristic
368C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
prime radical#31774029C2
hypercenter#463D271
hypocenterC271#3114
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #31774426C2semidirect
2. C4×D4×D271C2semidirect
3. #31774226C2semidirect
4. #31774209C2semidirect
5. #31774198C2semidirect
6. #31774197C2semidirect
7. #31774109C2semidirect
8. #31774046C2semidirect
10. #31774009C2semidirect
11. #31774006C2semidirect
12. #31774000C2semidirect
13. #31773998C2semidirect
14. #31773953C2semidirect
15. #463×C271C2semidirect
21. #22576152C22semidirect
25. Q8×Dic271C22semidirect
28. D4×Dic271C22semidirect
37. #22576094C22semidirect
38. #22576093C22semidirect
39. #22576092C22semidirect
40. #22576091C22semidirect
41. C4⋊(C271⋊C8)C22semidirect
42. #22576084C22semidirect
47. #22576081C22semidirect
50. #22576070C22semidirect
51. #22576070C22semidirect
54. #124×C271C22semidirect
56. C4⋊C8×C271C22semidirect
57. #103×C271C22semidirect
58. #102×C271C22semidirect
184. C271#3114semidirect
193. #463D271semidirect
209. #124C2×D271semidirect
231. C4⋊C8C2×D271semidirect
233. #103C2×D271semidirect
239. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744262
C4×D4×D2712
#317742262
#317742092
#317741982
#317741972
#317741092
#317740462
#317740292
#317740092
#317740062
#317740002
#317739982
#317739532
#463×C2712
#3114271