Finite Group Databasebeta

← #30650539

Subgroups — #30650539

Order 16256.

494 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
1#22550527C211yesmaximal, characteristic, complemented
2D4×C22⋊D127C211yesmaximal, characteristic, complemented
3#22550221C211yesmaximal, characteristic, complemented
4#22550189C211yesmaximal, characteristic, complemented
5#22550188C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550112C211yesmaximal, characteristic, complemented
8#22550111C211yesmaximal, characteristic, complemented
9#22550070C211yesmaximal, characteristic, complemented
10#22550011C211yesmaximal, characteristic, complemented
11#22550010C211yesmaximal, characteristic, complemented
12#22549956C211yesmaximal, characteristic, complemented
13C8×C22⋊D127C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15C2×#102×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C22×C22⋊D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×D1016C2211yescharacteristic, complemented
20C2×#14523233C2211yescharacteristic
21C2×#14523233C2211yescharacteristic, complemented
22C2×C8×D127C2211yescharacteristic, complemented
23#16861208C2211yescharacteristic, complemented
24#16861205C2211yescharacteristic
25#16861194—21no—
26#16861192—21no—
27#16861187—21no—
28#16861184C2211yescharacteristic, complemented
29#16861183C2211yescharacteristic
30#16861183C2211yescharacteristic, complemented
31#16861181—21no—
32#16861180C2211yescharacteristic
33#16861178—21no—
34#16861177—21no—
35D4×Dic127C2211yescharacteristic
36#16861171C2211yescharacteristic, complemented
37C4×C22⋊D127C2211yescharacteristic
38C4⋊C4×D127C2211yescharacteristic, complemented
39#16861143C2211yescharacteristic, complemented
40#16861142C2211yescharacteristic, complemented
41#16861142C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic, complemented
47#16861129C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic, complemented
49C22⋊(C127⋊C8)C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic, complemented
54C2×C4⋊C4×C127C2211yescharacteristic, complemented
55#102×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—44no—
61C2×Dic254—21no—
62C2×D508—21no—
63C2×D508C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—44no—
72C2×C22⋊D127D411yescharacteristic
73D4×D127—22no—
74D4×D127—41no—
75C22×Dic127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127C2311yescharacteristic
79D1016—41no—
80#14523233—22no—
81#14523233—41no—
82C2×C127⋊C8C2311yescharacteristic
83C2×C127⋊C8C2311yescharacteristic
84C8×D127—41no—
85C4⋊Dic127C2311yescharacteristic
86C4⋊Dic127C2311yescharacteristic
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127D411yescharacteristic
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127C2311yescharacteristic
99C24×C127—21no—
100C2×D4×C127—22no—
101C2×D4×C127—21no—
102C2×D4×C127—41no—
103C2×D4×C127C2311yescharacteristic
104C2×D4×C127C2311yescharacteristic
105C22×C4×C127—21no—
106C22×C4×C127C2311yescharacteristic
107C2×C8×C127—41no—
108C2×C8×C127C2311yescharacteristic
109C2×C8×C127C2311yescharacteristic
110C4⋊C4×C127—21no—
111C4⋊C4×C127C2311yescharacteristic
112C4⋊C4×C127C2311yescharacteristic
113C22×D127—21no—
114C22×D127—21no—
115C22×D127—42no—
116C22×D127—42no—
117C22×D127—81no—
118C22×D127C2×D411yescharacteristic
119Dic254—42no—
120D508—21no—
121D508—21no—
122D508—21no—
123D508—21no—
124D508—42no—
125C22⋊D127—42no—
126C22⋊D127—42no—
127C22⋊D127—42no—
128C22⋊D127—42no—
129C22⋊D127—42no—
130C22⋊D127—81no—
131C22⋊D127D814yescomplemented
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—42no—
135C2×Dic127C2×D411yescharacteristic
136C2×Dic127C2×D411yescharacteristic
137C2×Dic127C2×D411yescharacteristic
138C4×D127—21no—
139C4×D127—21no—
140C4×D127—42no—
141C4×D127—42no—
142C127⋊C8—21no—
143C127⋊C8—21no—
144C23×C127—21no—
145C23×C127—21no—
146C23×C127—42no—
147C23×C127—42no—
148C23×C127—81no—
149C23×C127C2×D411yescharacteristic
150D4×C127—21no—
151D4×C127—21no—
152D4×C127—21no—
153D4×C127—42no—
154D4×C127#4112yescomplemented
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—42no—
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2411yescharacteristic
162C8×C127—21no—
163C8×C127—21no—
164C22×C127—21no—
165C22×C127—21no—
166C22×C127—22no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—84no—
173C22×C127C2×D812yes—
174C22×C127C22×D411yescharacteristic
175C2×D127—21no—
176C2×D127—21no—
177C2×D127—41no—
178C2×D127—41no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—84no—
184C2×D127C2×D812yes—
185C4×C127—21no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127C2×#4111yescharacteristic
189C4×C127H(5,𝔽2)11yescharacteristic
190Dic127—21no—
191Dic127—21no—
192Dic127—41no—
193Dic127—41no—
194Dic127C2×D812yes—
195C2×C127—22no—
196C2×C127—42no—
197C2×C127—81no—
198C2×C127#54611yescharacteristic
199C2×C127C22×D811yescharacteristic
200C2×C127#57711yescharacteristic
201D127—22no—
202D127—81no—
203D127—81no—
204#3294—1271nomaximal, Sylow
205C127#329411yescharacteristic, complemented, Sylow
206#547—1271no—
207D42—1271no—
208#486—1271no—
209#480—1271no—
210#480—1271no—
211#474—1271no—
212#466—1271no—
213#466—1271no—
214#459—1271no—
215#447—1271no—
216#447—1271no—
217C4×D8—1271no—
218C8×D4—1271no—
219C2×#102—1271no—
220C2×#102D12711yescharacteristic, complemented
221C2×#41—2541no—
222C22×D4—1271no—
223C22×D4—2541no—
224C22×D4C2×D12711yescharacteristic
225C2×D8—1271no—
226C2×D8—1271no—
227C2×D8—1271no—
228C22×C8—1271no—
229C22×C8C2×D12711yescharacteristic, complemented
230C4⋊Q8—1271no—
231#127—1271no—
232#123—2541no—
233#123—2541no—
234C22⋊Q8—2541no—
235#121—1271no—
236#121—1271no—
237#121—1271no—
238#121—1271no—
239#121—2541no—
240#120—2541no—
241#120—2541no—
242C4×D4—1271no—
243C4×D4—1271no—
244C4×D4—1271no—
245C2×C4⋊C4—1271no—
246C2×C4⋊C4C2×D12711yescharacteristic, complemented
247#107—1271no—
248#107—1271no—
249#107—1271no—
250C4⋊C8—1271no—
251#102—1271no—
252#102—1271no—
253#102—1271no—
254#102—1271no—
255#102—1271no—
256#102—1271no—
257#102C2×D12711yescharacteristic
258#102C2×D12711yescharacteristic
259#102C2×D12711yescharacteristic, complemented
260#102C2×D12711yescharacteristic, complemented
261C22⋊C8—1271no—
262C22⋊C8—1271no—
263C4×C8—1271no—
264C24—21no—
265C24—2541no—
266#41—5084no—
267C2×Q8—2541no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—1271no—
272C2×D4—1271no—
273C2×D4—1271no—
274C2×D4—5084no—
275C2×D4—5081no—
276C2×D4—2541no—
277C2×D4—2541no—
278C2×D4—2541no—
279C2×D4—2541no—
280C2×D4—2541no—
281C2×D4—2541no—
282C2×D4—2541no—
283C2×D4—2542no—
284C2×D4C22×D12711yescharacteristic
285C2×D4C22×D12711yescharacteristic
286C22×C4—21no—
287C22×C4—1271no—
288C22×C4—2541no—
289C22×C4—2541no—
290C22×C4—2541no—
291C22×C4C22×D12711yescharacteristic
292D8—5081no—
293D8—5081no—
294D8—2542no—
295C2×C8—41no—
296C2×C8—1271no—
297C2×C8—1271no—
298C2×C8—5081no—
299C2×C8C22×D12711yescharacteristic
300C2×C8C22×D12711yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—1271no—
303C4⋊C4—1271no—
304C4⋊C4—1271no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4C22×D12711yescharacteristic
308C4⋊C4C22×D12711yescharacteristic
309C22⋊C4—1271no—
310C22⋊C4—1271no—
311C22⋊C4—2541no—
312C22⋊C4—2541no—
313C22⋊C4—2541no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4—2541no—
317C42—1271no—
318C23—21no—
319C23—21no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—1271no—
324C23—10161no—
325C23—5082no—
326C23—5082no—
327C23—2541no—
328C23—2541no—
329C23D4×D12711yescharacteristic
330Q8—5082no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—1274no—
336D4—10161no—
337D4—5082no—
338D4—5082no—
339D4—5082no—
340D4—5082no—
341D4—5082no—
342D4—5082no—
343D4—2541no—
344D4—2541no—
345D4—2541no—
346D4—2541no—
347D4#1452324812yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—1271no—
354C2×C4—1271no—
355C2×C4—1271no—
356C2×C4—5082no—
357C2×C4—5082no—
358C2×C4—5082no—
359C2×C4—2541no—
360C2×C4—2541no—
361C2×C4—2541no—
362C2×C4—2541no—
363C2×C4D4×D12711yescharacteristic
364C2×C4C23×D12711yescharacteristic
365C8—21no—
366C8—21no—
367C8—2541no—
368C8—2541no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—1272no—
379C22—10164no—
380C22—5081no—
381C22—5081no—
382C22—5081no—
383C22—5081no—
384C22—5081no—
385C22—5081no—
386C22—2541no—
387C22—2541no—
388C22D8×D12712yes—
389C22C2×D4×D12711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—1272no—
394C4—5081no—
395C4—5081no—
396C4—2541no—
397C4—2541no—
398C4C2×#1452324811yescharacteristic
399C4#1686125411yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—10161no—
404C2—10161no—
405C2—2542no—
406C2#2255051611yescharacteristic
407C2C2×D8×D12711yescharacteristic
408C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC22×C127C22×D4
hypercenterC2×#102D127
hypocenterC127#3294
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #22550527C2semidirect
2. D4×C22⋊D127C2semidirect
3. #22550221C2semidirect
4. #22550189C2semidirect
5. #22550188C2semidirect
6. #22550154C2semidirect
7. #22550112C2semidirect
8. #22550111C2semidirect
9. #22550070C2semidirect
10. #22550011C2semidirect
11. #22550010C2semidirect
12. #22549956C2semidirect
13. C8×C22⋊D127C2semidirect
14. #102×D127C2semidirect
15. C2×#102×C127C2semidirect
19. C2×D1016C22semidirect
21. C2×#14523233C22semidirect
22. C2×C8×D127C22semidirect
23. #16861208C22semidirect
28. #16861184C22semidirect
30. #16861183C22semidirect
36. #16861171C22semidirect
38. C4⋊C4×D127C22semidirect
39. #16861143C22semidirect
40. #16861142C22semidirect
41. #16861142C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861131C22semidirect
44. #16861131C22semidirect
46. #16861130C22semidirect
47. #16861129C22semidirect
48. #16861129C22semidirect
49. C22⋊(C127⋊C8)C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. C8×Dic127C22semidirect
53. C22×C8×C127C22semidirect
54. C2×C4⋊C4×C127C22semidirect
57. #102×C127C22semidirect
58. #102×C127C22semidirect
131. C22⋊D127D8semidirect
154. D4×C127#41semidirect
205. C127#3294semidirect
220. C2×#102D127semidirect
229. C22×C8C2×D127semidirect
246. C2×C4⋊C4C2×D127semidirect
259. #102C2×D127semidirect
260. #102C2×D127semidirect
347. D4#14523248semidirect

Maximal subgroups

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

subgroupindex
#225505272
D4×C22⋊D1272
#225502212
#225501892
#225501882
#225501542
#225501122
#225501112
#225500702
#225500112
#225500102
#225499562
C8×C22⋊D1272
#102×D1272
C2×#102×C1272
#3294127