Finite Group Databasebeta

← #30648272

Subgroups — #30648272

Order 16256.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2C2×#16861232C211yesmaximal, characteristic, complemented
3#22550444C211yesmaximal, characteristic, complemented
4#22550358C211yesmaximal, characteristic, complemented
5#22550100C211yesmaximal, characteristic, complemented
6#22550094C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550076C211yesmaximal, characteristic, complemented
9#22550074C211yesmaximal, characteristic, complemented
10#22550072C211yesmaximal, characteristic, complemented
11#22550070C211yesmaximal, characteristic, complemented
12#22549901C211yesmaximal, characteristic, complemented
13#22549876C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861236—24no—
20#16861232—24no—
21C2×#14523235C2211yescharacteristic
22C2×#14523235C2211yescharacteristic, complemented
23C2×#14523235C2211yescharacteristic, complemented
24C2×D1016C2211yescharacteristic
25C2×D1016C2211yescharacteristic, complemented
26C2×#14523233C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861206C2211yescharacteristic
30#16861205C2211yescharacteristic
31#16861198C2211yescharacteristic
32#16861182—21no—
33#16861182—21no—
34#16861179—21no—
35#16861179—21no—
36Q8×Dic127C2211yescharacteristic, complemented
37C4×D508—21no—
38C4×D508C2211yescharacteristic, complemented
39#16861171—21no—
40#16861171C2211yescharacteristic, complemented
41#16861166—21no—
42#16861166C2211yescharacteristic
43C4⋊C4×D127—21no—
44C4⋊C4×D127C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46Dic127⋊C8C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861133C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic
52#16861129C2211yescharacteristic
53#16861129C2211yescharacteristic, complemented
54C2×Q16×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic
57C4×Q8×C127C2211yescharacteristic, complemented
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic
61C23×D127—21no—
62#14523249—22no—
63#14523249—22no—
64#14523249—22no—
65#14523249D414yescomplemented
66C2×D508—21no—
67C2×D508—21no—
68C2×D508—21no—
69C2×D508—21no—
70C2×D508C2311yescharacteristic
71C2×D508C2311yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76D4×D127—22no—
77D4×D127—22no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127C2311yescharacteristic
84#14523235—22no—
85#14523235—22no—
86#14523235—22no—
87D1016—22no—
88D1016—22no—
89#14523233—22no—
90#14523231—22no—
91C2×C127⋊C8C2311yescharacteristic
92C2×C127⋊C8C2311yescharacteristic
93C8×D127—22no—
94C4⋊Dic127—21no—
95C4⋊Dic127C2311yescharacteristic
96Dic127⋊C4—21no—
97Dic127⋊C4D411yescharacteristic
98Dic127⋊C4D411yescharacteristic
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4D411yescharacteristic
104(C2×D127)⋊C4D411yescharacteristic
105C4×Dic127—21no—
106C4×Dic127C2311yescharacteristic
107C2×Q8×C127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110Q16×C127—22no—
111SD16×C127—22no—
112C2×C8×C127C2311yescharacteristic
113C2×C8×C127C2311yescharacteristic
114C4⋊C4×C127—21no—
115C4⋊C4×C127C2311yescharacteristic
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C42×C127—21no—
119C42×C127C2311yescharacteristic
120C22×D127—21no—
121C22×D127—21no—
122C22×D127—21no—
123C22×D127—42no—
124C22×D127—81no—
125C22×D127—81no—
126C22×D127C2×D411yescharacteristic
127D508—24no—
128D508—21no—
129D508—21no—
130D508—21no—
131D508—21no—
132D508—21no—
133D508—42no—
134D508—81no—
135D508—81no—
136D508C2×D412yes—
137C22⋊D127—42no—
138C22⋊D127—81no—
139C22⋊D127—81no—
140C2×Dic127—21no—
141C2×Dic127C2×D411yescharacteristic
142C2×Dic127C2×D411yescharacteristic
143C2×Dic127C2×D411yescharacteristic
144C4×D127—24no—
145C4×D127—42no—
146C4×D127—81no—
147C4×D127—81no—
148C4×D127—81no—
149C4×D127C2×D412yes—
150C4×D127#4112yes—
151C127⋊C8—21no—
152C127⋊C8—21no—
153C23×C127—21no—
154Q8×C127—21no—
155Q8×C127—21no—
156Q8×C127—21no—
157Q8×C127C2×D412yes—
158D4×C127—21no—
159D4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—21no—
167C8×C127—21no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127C22×D411yescharacteristic
172C2×D127—21no—
173C2×D127—21no—
174C2×D127—22no—
175C2×D127—41no—
176C2×D127—41no—
177C2×D127—41no—
178C2×D127—41no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—84no—
184C2×D127#12112yes—
185C4×C127—22no—
186C4×C127—21no—
187C4×C127—21no—
188C4×C127—41no—
189C4×C127—41no—
190C4×C127C22×D411yescharacteristic
191C4×C127C2×#4111yescharacteristic
192Dic127—21no—
193Dic127—21no—
194Dic127—41no—
195Dic127#12112yes—
196C2×C127—81no—
197C2×C127C2×#12111yescharacteristic
198C2×C127#57611yescharacteristic
199C2×C127#57711yescharacteristic
200D127—22no—
201D127—42no—
202D127—81no—
203D127—81no—
204#3045—1271nomaximal, Sylow
205C127#304511yescharacteristic, complemented, Sylow
206C2×AGL(1,ℤ/8)—1271no—
207C2×#135—1271no—
208#535—1271no—
209#520—1271no—
210#464—1271no—
211#464D12711yescharacteristic, complemented
212#463—1271no—
213#463—1271no—
214#460—1271no—
215#460—1271no—
216#459—1271no—
217#459—1271no—
218#424—1271no—
219#417—1271no—
220C2×#102—1271no—
221C2×#41—1271no—
222C2×#41—1271no—
223C22×D4—1271no—
224AGL(1,ℤ/8)—2544no—
225#135—2544no—
226C2×Q16C2×D12711yescharacteristic, complemented
227C2×SD16—1271no—
228C2×SD16—1271no—
229C2×SD16—1271no—
230C2×SD16C2×D12711yescharacteristic
231C2×D8—1271no—
232C2×D8—1271no—
233C2×D8—1271no—
234C2×M16—1271no—
235C22×C8—1271no—
236#127—1271no—
237#127—1271no—
238#124—1271no—
239#124C2×D12711yescharacteristic
240#121—2541no—
241#121—2541no—
242#120—2541no—
243#120—2541no—
244C4×Q8—1271no—
245C4×Q8C2×D12711yescharacteristic, complemented
246C4×D4—1271no—
247C4×D4—1271no—
248C4×D4—2541no—
249C4×D4—2541no—
250#117—1271no—
251#117—2541no—
252C2×C4⋊C4—1271no—
253C2×C4⋊C4—2541no—
254C4⋊C8—1271no—
255C4⋊C8—1271no—
256C4⋊C8—1271no—
257C4⋊C8C2×D12711yescharacteristic, complemented
258#103—1271no—
259#103C2×D12711yescharacteristic, complemented
260#102—1271no—
261#102—1271no—
262#102—1271no—
263#102—1271no—
264#102—1271no—
265#102C2×D12711yescharacteristic
266C24—2541no—
267#41—1274no—
268#41—2542no—
269#41—2542no—
270#41—2542no—
271C2×Q8C22×D12711yescharacteristic
272C2×Q8C22×D12711yescharacteristic
273C2×D4—1271no—
274C2×D4—1271no—
275C2×D4—1271no—
276C2×D4—2541no—
277C2×D4—2541no—
278C2×D4—2542no—
279C2×D4—2542no—
280C2×D4—2541no—
281C2×D4—2541no—
282C2×D4—2541no—
283C2×D4—2541no—
284C2×D4—2541no—
285C2×D4C22×D12711yescharacteristic
286C22×C4—1271no—
287C22×C4—2541no—
288C22×C4—2541no—
289C22×C4—2541no—
290C22×C4—2541no—
291C22×C4—2541no—
292Q16—22no—
293SD16—22no—
294SD16—2542no—
295SD16—2542no—
296SD16—2542no—
297D8—2542no—
298D8—2542no—
299D8—2542no—
300M16—2542no—
301C2×C8—1271no—
302C2×C8—1271no—
303C2×C8—2542no—
304C2×C8C22×D12711yescharacteristic
305C2×C8C22×D12711yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—1271no—
308C4⋊C4—1271no—
309C4⋊C4—1271no—
310C4⋊C4—2541no—
311C4⋊C4—2541no—
312C4⋊C4C22×D12711yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—1271no—
316C22⋊C4—1271no—
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C42—21no—
322C42—1271no—
323C42—2541no—
324C42C22×D12711yescharacteristic
325C23—21no—
326C23—1271no—
327C23—10161no—
328C23—10161no—
329C23—5082no—
330C23—2541no—
331C23—2541no—
332C23—2541no—
333Q8—21no—
334Q8—21no—
335Q8—21no—
336Q8D4×D12712yes—
337D4—21no—
338D4—21no—
339D4—1272no—
340D4—10161no—
341D4—10161no—
342D4—10161no—
343D4—10161no—
344D4—5082no—
345D4—5082no—
346D4—2544no—
347D4—2541no—
348D4—2541no—
349D4—2541no—
350D4—2541no—
351D4—2541no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—1271no—
356C2×C4—1272no—
357C2×C4—1272no—
358C2×C4—1271no—
359C2×C4—1271no—
360C2×C4—10161no—
361C2×C4—10161no—
362C2×C4—10161no—
363C2×C4—5082no—
364C2×C4—2544no—
365C2×C4—2541no—
366C2×C4D4×D12711yescharacteristic
367C2×C4D4×D12711yescharacteristic
368C2×C4C23×D12711yescharacteristic
369C8—21no—
370C8—21no—
371C8—2541no—
372C8—2541no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—1272no—
377C22—10164no—
378C22—5081no—
379C22—5081no—
380C22—5081no—
381C22—5081no—
382C22—5081no—
383C22—5081no—
384C22—5081no—
385C22—5081no—
386C22—2542no—
387C22—2541no—
388C22—2541no—
389C22C2×D4×D12711yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—1272no—
396C4—5081no—
397C4—2541no—
398C4—2541no—
399C4C2×D4×D12711yescharacteristic
400C4#41×D12711yescharacteristic
401C2—81no—
402C2—10161no—
403C2—10161no—
404C2—5082no—
405C2—2542no—
406C2#121×D12711yescharacteristic
407C2#2255067811yescharacteristic
408C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
hypercenter#464D127
hypocenterC127#3045
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861232C2semidirect
3. #22550444C2semidirect
4. #22550358C2semidirect
5. #22550100C2semidirect
6. #22550094C2semidirect
7. #22550088C2semidirect
8. #22550076C2semidirect
9. #22550074C2semidirect
10. #22550072C2semidirect
11. #22550070C2semidirect
12. #22549901C2semidirect
13. #22549876C2semidirect
14. #102×D127C2semidirect
15. #464×C127C2semidirect
22. C2×#14523235C22semidirect
23. C2×#14523235C22semidirect
25. C2×D1016C22semidirect
36. Q8×Dic127C22semidirect
38. C4×D508C22semidirect
40. #16861171C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. Dic127⋊C8C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861133C22semidirect
50. #16861131C22semidirect
53. #16861129C22semidirect
54. C2×Q16×C127C22semidirect
57. C4×Q8×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
65. #14523249D4semidirect
205. C127#3045semidirect
211. #464D127semidirect
226. C2×Q16C2×D127semidirect
245. C4×Q8C2×D127semidirect
257. C4⋊C8C2×D127semidirect
259. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×#168612322
#225504442
#225503582
#225501002
#225500942
#225500882
#225500762
#225500742
#225500722
#225500702
#225499012
#225498762
#102×D1272
#464×C1272
#3045127