Finite Group Databasebeta

← #46846415

Subgroups — #46846415

Order 34688.

454 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#31774434C211yesmaximal, characteristic, complemented
2C4×D4×D271C211yesmaximal, characteristic, complemented
3#31774231C211yesmaximal, characteristic, complemented
4#31774217C211yesmaximal, characteristic, complemented
5#31774196C211yesmaximal, characteristic, complemented
6#31774195C211yesmaximal, characteristic, complemented
7#31774115C211yesmaximal, characteristic, complemented
8#31774114C211yesmaximal, characteristic, complemented
9#31774046C211yesmaximal, characteristic, complemented
10#31774032C211yesmaximal, characteristic
11C4×#16864641C211yesmaximal, characteristic, complemented
12C4×D2168C211yesmaximal, characteristic, complemented
13C4×#16864639C211yesmaximal, characteristic, complemented
14#31773948C211yesmaximal, characteristic, complemented
15C4×SD16×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C22×C4×D271—21no—
18C2×#16864641C2211yescharacteristic
19C2×D2168C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21#22576155—21no—
22#22576151C2211yescharacteristic, complemented
23#22576149C2211yescharacteristic, complemented
24#22576145—21no—
25#22576136C2211yescharacteristic
26#22576135C2211yescharacteristic
27C4×D1084—21no—
28C4×D1084C2211yescharacteristic
29C4×D1084C2211yescharacteristic, complemented
30D4×Dic271C2211yescharacteristic
31#22576122C2211yescharacteristic
32C4×C22⋊D271—21no—
33(C22⋊D271)⋊C4—21no—
34#22576117—21no—
35C4⋊C4×D271C2211yescharacteristic
36C22⋊C4×D271—21no—
37C42×D271C2211yescharacteristic
38#22576094C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42Dic271⋊C8C2211yescharacteristic, complemented
43#22576084C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic, complemented
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic, complemented
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C4×C271⋊C8C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic
53C4×Q8×C271C2211yescharacteristic, complemented
54C4×D4×C271C2211yescharacteristic
55#106×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58C4×C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C2×D1084—21no—
61C2×D1084C2311yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64D4×D271—22no—
65D4×D271—22no—
66C22×Dic271—21no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—44no—
72C2×C4×D271D411yescharacteristic
73C2×C4×D271D411yescharacteristic
74C2×C4×D271C2311yescharacteristic
75#16864641—22no—
76D2168—22no—
77#16864639—22no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C4⋊Dic271—21no—
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94C4×Dic271D411yescharacteristic
95C4×Dic271D411yescharacteristic
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271C2311yescharacteristic
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271—21no—
100SD16×C271—22no—
101C2×C8×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C4⋊C4×C271—21no—
104C4⋊C4×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C42×C271C2311yescharacteristic
109C22×D271—24no—
110C22×D271—21no—
111C22×D271—21no—
112C22×D271—42no—
113C22×D271—42no—
114C22×D271C2×D411yescharacteristic
115D1084—21no—
116D1084—21no—
117D1084—21no—
118D1084—42no—
119D1084#4112yes—
120C22⋊D271—24no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—42no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—22no—
128C4×D271—22no—
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—42no—
132C4×D271—42no—
133C4×D271—42no—
134C4×D271—42no—
135C271⋊C8—21no—
136C271⋊C8—21no—
137C23×C271—21no—
138Q8×C271—21no—
139Q8×C271—21no—
140D4×C271—21no—
141D4×C271#4112yes—
142C2×C4×C271—21no—
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×C271C22×D411yescharacteristic
155C2×D271—21no—
156C2×D271—21no—
157C2×D271—22no—
158C2×D271—48no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271#13512yes—
165C4×C271—21no—
166C4×C271—41no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271AGL(1,ℤ/8)12yes—
170C4×C271C2×#4111yescharacteristic
171C4×C271C2×#4111yescharacteristic
172Dic271—21no—
173Dic271—21no—
174Dic271—21no—
175Dic271—41no—
176Dic271—41no—
177C2×C271—42no—
178C2×C271#52511yescharacteristic
179C2×C271C2×#13511yescharacteristic
180C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
181D271—22no—
182D271—42no—
183D271—81no—
184C271#310511yescharacteristic, complemented, Sylow
185#3105—2711nomaximal, Sylow
186#529—2711no—
187C2×C4×D4—2711no—
188#484—2711no—
189#481—2711no—
190#478—2711no—
191#478—2711no—
192#463—2711no—
193#463—2711no—
194#450—2711no—
195#447—2711no—
196C4×SD16—2711no—
197C4×SD16D27111yescharacteristic, complemented
198C4×D8—2711no—
199C4×D8—2711no—
200#431—2711no—
201C22×D4—2711no—
202C23×C4—5421no—
203C2×SD16—2711no—
204C2×SD16C2×D27111yescharacteristic
205C2×D8—2711no—
206C2×D8—2711no—
207#126—5421no—
208#125—2711no—
209#124—2711no—
210#123—5421no—
211C22⋊Q8—2711no—
212#121—2711no—
213C4×Q8C2×D27111yescharacteristic, complemented
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#106C2×D27111yescharacteristic, complemented
230C4⋊C8—2711no—
231C4⋊C8—2711no—
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
240C22⋊C8—2711no—
241C22⋊C8—2711no—
242C4×C8—2711no—
243C4×C8C2×D27111yescharacteristic, complemented
244C24—5421no—
245C2×Q8C22×D27111yescharacteristic
246C2×D4—2711no—
247C2×D4—2711no—
248C2×D4—5421no—
249C2×D4—5421no—
250C2×D4—5422no—
251C2×D4—5422no—
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—5422no—
266D8—5422no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8C22×D27111yescharacteristic
270C2×C8C22×D27111yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4C22×D27111yescharacteristic
279C4⋊C4C22×D27111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C42—21no—
290C42—2711no—
291C42—2711no—
292C42—2711no—
293C42C22×D27111yescharacteristic
294C23—21no—
295C23—2711no—
296C23—5424no—
297C23—5421no—
298C23—5421no—
299C23—10842no—
300C23—10842no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—2712no—
305D4—5424no—
306D4—5421no—
307D4—5421no—
308D4—5421no—
309D4—10842no—
310D4#1686465412yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—2711no—
316C2×C4—2711no—
317C2×C4—2711no—
318C2×C4—5422no—
319C2×C4—5422no—
320C2×C4—5421no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
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—2712no—
340C22—5421no—
341C22—5421no—
342C22—5422no—
343C22—10848no—
344C22—10841no—
345C22—10841no—
346C22—10841no—
347C22—10841no—
348C22—10841no—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—5421no—
355C4—5421no—
356C4—5421no—
357C4—10841no—
358C4—10841no—
359C4#2257618712yes—
360C4#41×D27111yescharacteristic
361C4C2×#1686465411yescharacteristic
362C2—42no—
363C2—5422no—
364C2—10842no—
365C2—21681no—
366C2#3177440611yescharacteristic
367C2#135×D27111yescharacteristic
368C2C2×#2257618711yescharacteristic
subgroupG/H
centerC2×C4D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC4×SD16×C271C2
socleC22×C271C22×D4
prime radical#31774032C2
hypercenterC4×SD16D271
hypocenterC271#3105
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #31774434C2semidirect
2. C4×D4×D271C2semidirect
3. #31774231C2semidirect
4. #31774217C2semidirect
5. #31774196C2semidirect
6. #31774195C2semidirect
7. #31774115C2semidirect
8. #31774114C2semidirect
9. #31774046C2semidirect
11. C4×#16864641C2semidirect
12. C4×D2168C2semidirect
13. C4×#16864639C2semidirect
14. #31773948C2semidirect
15. C4×SD16×C271C2semidirect
22. #22576151C22semidirect
23. #22576149C22semidirect
29. C4×D1084C22semidirect
38. #22576094C22semidirect
39. #22576093C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. Dic271⋊C8C22semidirect
43. #22576084C22semidirect
45. #22576082C22semidirect
48. #22576080C22semidirect
51. C4×C271⋊C8C22semidirect
53. C4×Q8×C271C22semidirect
55. #106×C271C22semidirect
56. #103×C271C22semidirect
58. C4×C8×C271C22semidirect
184. C271#3105semidirect
197. C4×SD16D271semidirect
213. C4×Q8C2×D271semidirect
229. #106C2×D271semidirect
233. #103C2×D271semidirect
243. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744342
C4×D4×D2712
#317742312
#317742172
#317741962
#317741952
#317741152
#317741142
#317740462
#317740322
C4×#168646412
C4×D21682
C4×#168646392
#317739482
C4×SD16×C2712
#3105271