Finite Group Databasebeta

← #30648353

Subgroups — #30648353

Order 16256.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861238C211yesmaximal, characteristic, complemented
2C22×#14523231C211yesmaximal, characteristic, complemented
3#22550386C211yesmaximal, characteristic
4C22⋊Q8×D127C211yesmaximal, characteristic, complemented
5#22550135C211yesmaximal, characteristic, complemented
6#22550133C211yesmaximal, characteristic, complemented
7#22550121C211yesmaximal, characteristic, complemented
8#22550119C211yesmaximal, characteristic, complemented
9#22550117C211yesmaximal, characteristic, complemented
10#22550115C211yesmaximal, characteristic, complemented
11#22550103C211yesmaximal, characteristic, complemented
12#22549915C211yesmaximal, characteristic, complemented
13#22549883C211yesmaximal, characteristic, complemented
14#22549874C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19#16861238—24no—
20C2×Dic508C2211yescharacteristic, complemented
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523236C2211yescharacteristic, complemented
23C2×#14523231—24no—
24C2×#14523231C2211yescharacteristic
25C2×#14523231C2211yescharacteristic
26C2×#14523231C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C22×C127⋊C8C2211yescharacteristic
29#16861191—21no—
30#16861189C2211yescharacteristic
31#16861189C2211yescharacteristic
32#16861188C2211yescharacteristic
33#16861188C2211yescharacteristic
34#16861187—21no—
35#16861186—21no—
36#16861185C2211yescharacteristic
37#16861180C2211yescharacteristic
38#16861177—21no—
39D4×Dic127—21no—
40(C22⋊D127)⋊C4—21no—
41#16861167C2211yescharacteristic
42C4⋊C4×D127—21no—
43C4⋊C4×D127C2211yescharacteristic
44C22⋊C4×D127—21no—
45#16861143C2211yescharacteristic, complemented
46#16861142C2211yescharacteristic, complemented
47#16861140C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861134C2211yescharacteristic, complemented
50#16861133C2211yescharacteristic, complemented
51#16861132C2211yescharacteristic, complemented
52#16861132C2211yescharacteristic, complemented
53#16861130C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C22×C8×C127C2211yescharacteristic
56C22⋊Q8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#106×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127D411yescharacteristic
62#14523247—22no—
63#14523247—22no—
64C2×Dic254—21no—
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67Q8×D127—22no—
68Q8×D127—22no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C22×Dic127—21no—
72C22×Dic127D411yescharacteristic
73C2×C4×D127—22no—
74C2×C4×D127—22no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127D411yescharacteristic
78C2×C4×D127D411yescharacteristic
79C2×C4×D127C2311yescharacteristic
80C2×C4×D127C2311yescharacteristic
81Dic508—22no—
82#14523238—22no—
83#14523236—22no—
84#14523231—22no—
85#14523231—22no—
86#14523231—22no—
87#14523231D414yescomplemented
88C2×C127⋊C8—22no—
89C2×C127⋊C8C2311yescharacteristic
90C2×C127⋊C8C2311yescharacteristic
91C4⋊Dic127—21no—
92C4⋊Dic127—21no—
93C4⋊Dic127C2311yescharacteristic
94C4⋊Dic127C2311yescharacteristic
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106C4×Dic127—21no—
107C2×Q8×C127C2311yescharacteristic
108C2×D4×C127—21no—
109C2×D4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111SD16×C127—22no—
112C2×C8×C127—22no—
113C2×C8×C127C2311yescharacteristic
114C2×C8×C127C2311yescharacteristic
115C4⋊C4×C127—21no—
116C4⋊C4×C127C2311yescharacteristic
117C4⋊C4×C127C2311yescharacteristic
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22×D127—22no—
121C22×D127—22no—
122C22×D127—41no—
123C22×D127C2×D411yescharacteristic
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—42no—
130C22⋊D127—42no—
131C22⋊D127—42no—
132C2×Dic127—22no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—42no—
138C2×Dic127C2×D411yescharacteristic
139C2×Dic127C2×D411yescharacteristic
140C4×D127—22no—
141C4×D127—21no—
142C4×D127—21no—
143C4×D127—22no—
144C4×D127—42no—
145C4×D127—42no—
146C4×D127C2×D412yes—
147C4×D127#4112yes—
148C127⋊C8—21no—
149C127⋊C8C2×D412yes—
150C23×C127—21no—
151C23×C127C2×D411yescharacteristic
152Q8×C127—21no—
153Q8×C127—21no—
154D4×C127—21no—
155D4×C127—21no—
156D4×C127—42no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127C2×D411yescharacteristic
163C2×C4×C127C2411yescharacteristic
164C8×C127—21no—
165C8×C127C2×D412yes—
166C22×C127—21no—
167C22×C127—21no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127#13712yescomplemented
172C22×C127C22×D411yescharacteristic
173C2×D127—21no—
174C2×D127—21no—
175C2×D127—21no—
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—44no—
179C2×D127#12112yes—
180C4×C127—21no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127—41no—
184C4×C127C22×D411yescharacteristic
185C4×C127C2×#4111yescharacteristic
186Dic127—21no—
187Dic127—41no—
188Dic127—41no—
189Dic127—41no—
190Dic127—41no—
191Dic127#12112yes—
192C2×C127—22no—
193C2×C127—81no—
194C2×C127C2×#12111yescharacteristic
195C2×C127C2×#13711yescharacteristic
196C2×C127#57511yescharacteristic
197D127—22no—
198D127—41no—
199#3056—1271nomaximal, Sylow
200C127#305611yescharacteristic, complemented, Sylow
201C2×#137—1271no—
202C22×M16—1271no—
203#525—1271no—
204C2×C22⋊Q8—1271no—
205#470—1271no—
206#470—1271no—
207#468—1271no—
208#468—1271no—
209#467—1271no—
210#467—1271no—
211#465—1271no—
212#465D12711yescharacteristic, complemented
213#428—1271no—
214#419—1271no—
215#417—1271no—
216C2×#41—1271no—
217C22×Q8—1271no—
218C23×C4—1271no—
219#137—2544no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×SD16—1271no—
223C2×SD16C2×D12711yescharacteristic, complemented
224C2×M16—1271no—
225C2×M16—1271no—
226C2×M16—1271no—
227C2×M16—1271no—
228C2×M16—2544no—
229C22×C8—1271no—
230C22×C8C2×D12711yescharacteristic
231#123—2541no—
232C22⋊Q8—1271no—
233C22⋊Q8—1271no—
234C22⋊Q8—1271no—
235C22⋊Q8—1271no—
236C22⋊Q8—1271no—
237C22⋊Q8—2541no—
238C22⋊Q8—2541no—
239C22⋊Q8C2×D12711yescharacteristic
240#121—1271no—
241#121C2×D12711yescharacteristic
242#120—2541no—
243C4×D4—2541no—
244C4×D4—2541no—
245#117—1271no—
246C2×C4⋊C4—1271no—
247C2×C4⋊C4—2541no—
248C2×C22⋊C4—2541no—
249#107—1271no—
250#107—1271no—
251#106—1271no—
252#106C2×D12711yescharacteristic, complemented
253#103—1271no—
254#103—1271no—
255#103—1271no—
256#103—1271no—
257#103—1271no—
258#103C2×D12711yescharacteristic, complemented
259#102—1271no—
260#102C2×D12711yescharacteristic, complemented
261C24—1271no—
262#41—2542no—
263#41—2542no—
264C2×Q8—1271no—
265C2×Q8—1271no—
266C2×Q8—2541no—
267C2×Q8—2542no—
268C2×Q8—2542no—
269C2×Q8C22×D12711yescharacteristic
270C2×D4—21no—
271C2×D4—2541no—
272C2×D4—2541no—
273C2×D4C22×D12711yescharacteristic
274C22×C4—1271no—
275C22×C4—1271no—
276C22×C4—1271no—
277C22×C4—1271no—
278C22×C4—1271no—
279C22×C4—2542no—
280C22×C4—2542no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284C22×C4C22×D12711yescharacteristic
285Q16—2542no—
286Q16—2542no—
287SD16—22no—
288SD16—2542no—
289M16—1274no—
290M16—2542no—
291M16—2542no—
292M16—2542no—
293C2×C8—22no—
294C2×C8—1271no—
295C2×C8—1271no—
296C2×C8—2542no—
297C2×C8C22×D12711yescharacteristic
298C2×C8C22×D12711yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—1271no—
301C4⋊C4—1271no—
302C4⋊C4—2541no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4—2541no—
308C4⋊C4C22×D12711yescharacteristic
309C4⋊C4C22×D12711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2541no—
313C22⋊C4—2541no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4—2541no—
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C42—2541no—
320C23—21no—
321C23—1271no—
322C23—1271no—
323C23—5081no—
324C23—2542no—
325C23—2542no—
326C23D4×D12711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—5082no—
330Q8—2541no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—5082no—
338D4—5082no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—1271no—
345C2×C4—1272no—
346C2×C4—1272no—
347C2×C4—1271no—
348C2×C4—5082no—
349C2×C4—5082no—
350C2×C4—5082no—
351C2×C4—2542no—
352C2×C4—2542no—
353C2×C4—2542no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4D4×D12711yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—21no—
363C8—1272no—
364C8—2541no—
365C8D4×D12712yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—1272no—
372C22—5084no—
373C22—2541no—
374C22—2541no—
375C22—2541no—
376C22—2541no—
377C22—2541no—
378C22#1686123812yescomplemented
379C22C2×D4×D12711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—1272no—
385C4—5081no—
386C4—5081no—
387C4—5081no—
388C4—5081no—
389C4—2541no—
390C4C2×D4×D12711yescharacteristic
391C4#41×D12711yescharacteristic
392C2—22no—
393C2—81no—
394C2—5081no—
395C2—2542no—
396C2#121×D12711yescharacteristic
397C2C2×#1686123811yescharacteristic
398C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
prime radical#22550386C2
hypercenter#465D127
hypocenterC127#3056
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. C22×#14523231C2semidirect
4. C22⋊Q8×D127C2semidirect
5. #22550135C2semidirect
6. #22550133C2semidirect
7. #22550121C2semidirect
8. #22550119C2semidirect
9. #22550117C2semidirect
10. #22550115C2semidirect
11. #22550103C2semidirect
12. #22549915C2semidirect
13. #22549883C2semidirect
14. #22549874C2semidirect
15. #465×C127C2semidirect
20. C2×Dic508C22semidirect
21. C2×#14523238C22semidirect
22. C2×#14523236C22semidirect
45. #16861143C22semidirect
46. #16861142C22semidirect
47. #16861140C22semidirect
48. #16861134C22semidirect
49. #16861134C22semidirect
50. #16861133C22semidirect
51. #16861132C22semidirect
52. #16861132C22semidirect
53. #16861130C22semidirect
54. C2×SD16×C127C22semidirect
58. #106×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
87. #14523231D4semidirect
171. C22×C127#137semidirect
200. C127#3056semidirect
212. #465D127semidirect
223. C2×SD16C2×D127semidirect
252. #106C2×D127semidirect
258. #103C2×D127semidirect
260. #102C2×D127semidirect
378. C22#16861238semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C22×#145232312
#225503862
C22⋊Q8×D1272
#225501352
#225501332
#225501212
#225501192
#225501172
#225501152
#225501032
#225499152
#225498832
#225498742
#465×C1272
#3056127