Finite Group Databasebeta

← #30648192

Subgroups — #30648192

Order 16256.

548 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2C2×#16861233C211yesmaximal, characteristic, complemented
3#22550425C211yesmaximal, characteristic, complemented
4C4×D4×D127C211yesmaximal, characteristic, complemented
5#22550092C211yesmaximal, characteristic, complemented
6#22550091C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550080C211yesmaximal, characteristic, complemented
9#22550079C211yesmaximal, characteristic, complemented
10#22550070C211yesmaximal, characteristic, complemented
11#22550069C211yesmaximal, characteristic, complemented
12#22549898C211yesmaximal, characteristic, complemented
13#22549876C211yesmaximal, characteristic, complemented
14#22549874C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19C22×C4×D127—21no—
20#16861236—24no—
21#16861233—24no—
22C2×#14523237C2211yescharacteristic
23C2×#14523236C2211yescharacteristic
24C2×#14523235C2211yescharacteristic
25C2×D1016C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29C2×#14523231C2211yescharacteristic
30#16861207C2211yescharacteristic, complemented
31#16861205C2211yescharacteristic, complemented
32#16861197C2211yescharacteristic, complemented
33#16861182—21no—
34#16861182—21no—
35C4×D508C2211yescharacteristic
36D4×Dic127C2211yescharacteristic
37#16861171C2211yescharacteristic
38C4×C22⋊D127—21no—
39(C22⋊D127)⋊C4—21no—
40(C22⋊D127)⋊C4—21no—
41(C22⋊D127)⋊C4—21no—
42C4⋊C4×D127C2211yescharacteristic
43C22⋊C4×D127—21no—
44C42×D127C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46Dic127⋊C8C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861133C2211yescharacteristic, complemented
50#16861132C2211yescharacteristic, complemented
51#16861131C2211yescharacteristic, complemented
52#16861130C2211yescharacteristic, complemented
53#16861129C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—22no—
64#14523247—22no—
65#14523247—22no—
66C2×Dic254C2311yescharacteristic
67C2×D508—21no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74D4×D127—22no—
75D4×D127D414yescomplemented
76C22×Dic127—21no—
77C22×Dic127—21no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—44no—
83C2×C4×D127D411yescharacteristic
84C2×C4×D127D411yescharacteristic
85C2×C4×D127C2311yescharacteristic
86#14523237—22no—
87#14523236—22no—
88#14523235—22no—
89D1016—22no—
90#14523233—22no—
91#14523233—22no—
92#14523231—22no—
93#14523231—22no—
94C2×C127⋊C8C2311yescharacteristic
95C2×C127⋊C8C2311yescharacteristic
96C4⋊Dic127C2311yescharacteristic
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100C22⋊Dic127—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105C4×Dic127D411yescharacteristic
106C4×Dic127D411yescharacteristic
107C4×Dic127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110C2×D4×C127C2311yescharacteristic
111C22×C4×C127—21no—
112SD16×C127—22no—
113D8×C127—22no—
114C2×C8×C127C2311yescharacteristic
115C2×C8×C127C2311yescharacteristic
116C4⋊C4×C127C2311yescharacteristic
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C42×C127C2311yescharacteristic
121C22×D127—21no—
122C22×D127—24no—
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—42no—
126C22×D127C2×D411yescharacteristic
127Dic254—21no—
128Dic254—21no—
129D508—21no—
130D508—21no—
131D508—21no—
132D508—42no—
133D508C2×D412yes—
134C22⋊D127—24no—
135C22⋊D127—42no—
136C22⋊D127—42no—
137C22⋊D127—42no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127—42no—
141C2×Dic127—42no—
142C2×Dic127C2×D411yescharacteristic
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C4×D127—22no—
146C4×D127—41no—
147C4×D127—42no—
148C4×D127—42no—
149C4×D127—42no—
150C4×D127—42no—
151C4×D127C2×D412yes—
152C4×D127#4112yes—
153C127⋊C8—21no—
154C127⋊C8—21no—
155C23×C127—21no—
156C23×C127—21no—
157Q8×C127—21no—
158Q8×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127C2×D412yes—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—42no—
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2411yescharacteristic
169C8×C127—21no—
170C8×C127—21no—
171C22×C127—22no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127C22×D411yescharacteristic
178C2×D127—21no—
179C2×D127—21no—
180C2×D127—22no—
181C2×D127—48no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127#12112yes—
188C4×C127—21no—
189C4×C127—21no—
190C4×C127—41no—
191C4×C127—41no—
192C4×C127C22×D411yescharacteristic
193C4×C127C2×#4111yescharacteristic
194Dic127—21no—
195Dic127—41no—
196Dic127—41no—
197Dic127#12112yes—
198Dic127AGL(1,ℤ/8)12yescomplemented
199C2×C127—42no—
200C2×C127—81no—
201C2×C127C2×#12111yescharacteristic
202C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C127#57511yescharacteristic
204D127—22no—
205D127—42no—
206D127—81no—
207#3036—1271nomaximal, Sylow
208C127#303611yescharacteristic, complemented, Sylow
209C2×AGL(1,ℤ/8)—1271no—
210C2×AGL(1,ℤ/8)—1271no—
211#532—1271no—
212C2×C4×D4—1271no—
213#463—1271no—
214#463—1271no—
215#463—1271no—
216#463D12711yescharacteristic, complemented
217#461—1271no—
218#461—1271no—
219#459—1271no—
220#459—1271no—
221#423—1271no—
222#417—1271no—
223#417—1271no—
224C2×#41—1271no—
225C2×#41—1271no—
226C22×D4—1271no—
227C23×C4—2541no—
228AGL(1,ℤ/8)—2544no—
229AGL(1,ℤ/8)—2544no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×SD16—1271no—
233C2×SD16C2×D12711yescharacteristic
234C2×D8—1271no—
235C2×D8—1271no—
236C2×D8—1271no—
237C2×D8C2×D12711yescharacteristic
238C2×M16—1271no—
239C2×M16—1271no—
240C4⋊Q8—1271no—
241#127—1271no—
242#124—1271no—
243#124C2×D12711yescharacteristic, complemented
244#121—2541no—
245#121—2541no—
246C4×D4—1271no—
247C4×D4—1271no—
248C4×D4—1271no—
249C4×D4—2541no—
250C4×D4—2541no—
251C4×D4—2541no—
252C4×D4—2541no—
253C4×D4C2×D12711yescharacteristic
254C2×C4⋊C4—1271no—
255C2×C22⋊C4—2541no—
256C2×C42—1271no—
257C4⋊C8—1271no—
258C4⋊C8—1271no—
259C4⋊C8—1271no—
260C4⋊C8C2×D12711yescharacteristic, complemented
261#103—1271no—
262#103—1271no—
263#103—1271no—
264#103C2×D12711yescharacteristic, complemented
265#102—1271no—
266#102—1271no—
267#102—1271no—
268#102C2×D12711yescharacteristic, complemented
269C24—2541no—
270#41—2542no—
271#41—2542no—
272#41—2542no—
273#41—2542no—
274C2×Q8—1271no—
275C2×Q8C22×D12711yescharacteristic
276C2×D4—1274no—
277C2×D4—1271no—
278C2×D4—1271no—
279C2×D4—2541no—
280C2×D4—2542no—
281C2×D4—2541no—
282C2×D4—2541no—
283C2×D4—2541no—
284C2×D4—2541no—
285C2×D4C22×D12711yescharacteristic
286C2×D4C22×D12711yescharacteristic
287C22×C4—21no—
288C22×C4—1271no—
289C22×C4—1271no—
290C22×C4—1271no—
291C22×C4—5084no—
292C22×C4—2541no—
293C22×C4—2541no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298SD16—22no—
299SD16—2542no—
300SD16—2542no—
301SD16—2542no—
302D8—22no—
303D8—2542no—
304D8—2542no—
305D8—2542no—
306M16—2542no—
307M16—2542no—
308C2×C8—1271no—
309C2×C8—1271no—
310C2×C8C22×D12711yescharacteristic
311C2×C8C22×D12711yescharacteristic
312C4⋊C4—1271no—
313C4⋊C4—2541no—
314C4⋊C4—2541no—
315C4⋊C4—2541no—
316C4⋊C4C22×D12711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C42—1271no—
326C42—1271no—
327C42—1271no—
328C42C22×D12711yescharacteristic
329C23—21no—
330C23—21no—
331C23—1271no—
332C23—5082no—
333C23—5082no—
334C23—2544no—
335C23—2541no—
336C23—2541no—
337Q8—21no—
338Q8—21no—
339Q8—2541no—
340Q8—2541no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—1272no—
345D4—5082no—
346D4—5082no—
347D4—5082no—
348D4—5082no—
349D4—2544no—
350D4—2541no—
351D4—2541no—
352D4—2541no—
353D4D4×D12712yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—1271no—
358C2×C4—1271no—
359C2×C4—1272no—
360C2×C4—1272no—
361C2×C4—1271no—
362C2×C4—5082no—
363C2×C4—5081no—
364C2×C4—5082no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—5082no—
369C2×C4—2542no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4D4×D12711yescharacteristic
373C2×C4D4×D12711yescharacteristic
374C2×C4C23×D12711yescharacteristic
375C8—21no—
376C8—21no—
377C8—2541no—
378C8—2541no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—1272no—
386C22—5088no—
387C22—5081no—
388C22—5081no—
389C22—5081no—
390C22—5081no—
391C22—5081no—
392C22—2542no—
393C22—2541no—
394C22—2541no—
395C22C2×D4×D12711yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—1272no—
401C4—1272no—
402C4—5081no—
403C4—5081no—
404C4—2541no—
405C4C2×D4×D12711yescharacteristic
406C4#41×D12711yescharacteristic
407C2—42no—
408C2—81no—
409C2—10161no—
410C2—5082no—
411C2—2542no—
412C2#121×D12711yescharacteristic
413C2AGL(1,ℤ/8)×D12711yescharacteristic
414C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3036
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861233C2semidirect
3. #22550425C2semidirect
4. C4×D4×D127C2semidirect
5. #22550092C2semidirect
6. #22550091C2semidirect
7. #22550088C2semidirect
8. #22550080C2semidirect
9. #22550079C2semidirect
10. #22550070C2semidirect
11. #22550069C2semidirect
12. #22549898C2semidirect
13. #22549876C2semidirect
14. #22549874C2semidirect
15. #463×C127C2semidirect
30. #16861207C22semidirect
31. #16861205C22semidirect
32. #16861197C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. Dic127⋊C8C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861133C22semidirect
50. #16861132C22semidirect
51. #16861131C22semidirect
52. #16861130C22semidirect
53. #16861129C22semidirect
56. #124×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
75. D4×D127D4semidirect
198. Dic127AGL(1,ℤ/8)semidirect
208. C127#3036semidirect
216. #463D127semidirect
243. #124C2×D127semidirect
260. C4⋊C8C2×D127semidirect
264. #103C2×D127semidirect
268. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×#168612332
#225504252
C4×D4×D1272
#225500922
#225500912
#225500882
#225500802
#225500792
#225500702
#225500692
#225498982
#225498762
#225498742
#463×C1272
#3036127