Finite Group Databasebeta

← #30650237

Subgroups — #30650237

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
1C2×#16861241C211yesmaximal, characteristic, complemented
2C22×#14523237C211yesmaximal, characteristic, complemented
3#22550539C211yesmaximal, characteristic, complemented
4#22550517C211yesmaximal, characteristic
5#22550269C211yesmaximal, characteristic, complemented
6#22550123C211yesmaximal, characteristic, complemented
7#22550101C211yesmaximal, characteristic, complemented
8#22550080C211yesmaximal, characteristic, complemented
9#22550041C211yesmaximal, characteristic, complemented
10#22550031C211yesmaximal, characteristic, complemented
11#22550026C211yesmaximal, characteristic, complemented
12#22550016C211yesmaximal, characteristic, complemented
13#22549975C211yesmaximal, characteristic, complemented
14#22549947C211yesmaximal, characteristic, complemented
15#467×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic, complemented
19C22×D508C2211yescharacteristic
20#16861241—24no—
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523237—24no—
24C2×#14523237C2211yescharacteristic
25C2×#14523237C2211yescharacteristic
26C2×#14523237C2211yescharacteristic
27C2×#14523237C2211yescharacteristic
28C2×#14523235C2211yescharacteristic
29C2×#14523235C2211yescharacteristic
30C2×#14523231C2211yescharacteristic, complemented
31#16861208—21no—
32#16861207C2211yescharacteristic, complemented
33#16861198C2211yescharacteristic
34#16861194—21no—
35#16861189C2211yescharacteristic, complemented
36#16861187—21no—
37#16861186—21no—
38#16861184—21no—
39#16861184C2211yescharacteristic
40#16861179—21no—
41#16861174C2211yescharacteristic, complemented
42#16861171C2211yescharacteristic
43C4×C22⋊D127C2211yescharacteristic
44(C22⋊D127)⋊C4—21no—
45C22⋊C4×D127—21no—
46Dic127⋊C8C2211yescharacteristic, complemented
47#16861132C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic
50#16861129C2211yescharacteristic
51C22⋊(C127⋊C8)C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861119C2211yescharacteristic, complemented
54C2×Q16×C127C2211yescharacteristic, complemented
55C22×C8×C127C2211yescharacteristic
56C22⋊Q8×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic
58#107×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#103×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64C2×Dic254—21no—
65C2×Dic254—22no—
66C2×Dic254—21no—
67C2×Dic254—41no—
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70Q8×D127—22no—
71Q8×D127—41no—
72C2×D508—22no—
73C2×D508—21no—
74C2×D508—21no—
75C2×D508—21no—
76C2×D508—41no—
77C2×D508C2311yescharacteristic
78C2×D508C2311yescharacteristic
79C2×C22⋊D127D411yescharacteristic
80C22×Dic127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127—21no—
84C2×C4×D127C2311yescharacteristic
85#14523238—22no—
86#14523238—41no—
87#14523237—22no—
88#14523237—22no—
89#14523237—22no—
90#14523237D414yescomplemented
91#14523235—22no—
92#14523235—41no—
93#14523231—22no—
94C2×C127⋊C8C2311yescharacteristic
95C2×C127⋊C8C2311yescharacteristic
96C4⋊Dic127—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4D411yescharacteristic
101C22⋊Dic127D411yescharacteristic
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108C4×Dic127—21no—
109C4×Dic127C2311yescharacteristic
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C22×C4×C127C2311yescharacteristic
113Q16×C127—22no—
114C2×C8×C127—22no—
115C2×C8×C127C2311yescharacteristic
116C2×C8×C127C2311yescharacteristic
117C4⋊C4×C127—21no—
118C4⋊C4×C127—21no—
119C4⋊C4×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128C22×D127C2×D411yescharacteristic
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—42no—
133Dic254—42no—
134Dic254C2×D412yes—
135D508—21no—
136D508—21no—
137D508—21no—
138D508—42no—
139D508—42no—
140D508—81no—
141D508C2×D412yes—
142C22⋊D127—22no—
143C2×Dic127—21no—
144C2×Dic127—21no—
145C2×Dic127—42no—
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C2×Dic127C2×D411yescharacteristic
149C4×D127—21no—
150C4×D127—21no—
151C4×D127—42no—
152C4×D127—42no—
153C4×D127—81no—
154C127⋊C8—21no—
155C127⋊C8—21no—
156C23×C127C2×D411yescharacteristic
157Q8×C127—21no—
158Q8×C127—21no—
159Q8×C127—21no—
160Q8×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2411yescharacteristic
169C8×C127—21no—
170C8×C127C2×D412yes—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127#13712yescomplemented
174C22×C127C22×D411yescharacteristic
175C2×D127—21no—
176C2×D127—22no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—84no—
185C4×C127—21no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127—41no—
189C4×C127—41no—
190C4×C127C22×D411yescharacteristic
191C4×C127H(5,𝔽2)11yescharacteristic
192Dic127—21no—
193Dic127—22no—
194Dic127—21no—
195Dic127—21no—
196Dic127—41no—
197C2×C127—22no—
198C2×C127D4211yescharacteristic
199C2×C127C2×#13711yescharacteristic
200C2×C127#57711yescharacteristic
201D127—41no—
202D127—42no—
203D127—81no—
204#3266—1271nomaximal, Sylow
205C127#326611yescharacteristic, complemented, Sylow
206C2×#137—1271no—
207C22×SD16—1271no—
208D4×Q8—1271no—
209#546—1271no—
210#497—1271no—
211#468—1271no—
212#467D12711yescharacteristic, complemented
213#464—1271no—
214#461—1271no—
215#452—1271no—
216#451—1271no—
217#450—1271no—
218#448—1271no—
219#440—1271no—
220#435—1271no—
221C2×#41—1271no—
222C22×Q8—1271no—
223C22×Q8—1271no—
224C22×D4—1271no—
225#137—2544no—
226C2×Q16—1271no—
227C2×Q16—1271no—
228C2×Q16C2×D12711yescharacteristic, complemented
229C2×SD16—1271no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—2544no—
236C2×M16—1271no—
237C22×C8C2×D12711yescharacteristic
238C4⋊Q8—1271no—
239C4⋊Q8—2541no—
240#124—1271no—
241#123—2541no—
242C22⋊Q8—1271no—
243C22⋊Q8—2541no—
244C22⋊Q8—2541no—
245C22⋊Q8C2×D12711yescharacteristic
246C22⋊Q8C2×D12711yescharacteristic
247#121—1271no—
248#121—2541no—
249#120—2541no—
250C4×Q8—1271no—
251C4×D4—1271no—
252C4×D4—1271no—
253C4×D4—2541no—
254C2×C22⋊C4—2541no—
255#107C2×D12711yescharacteristic, complemented
256C4⋊C8—1271no—
257#103—1271no—
258#103—1271no—
259#103C2×D12711yescharacteristic, complemented
260#103C2×D12711yescharacteristic, complemented
261#102—1271no—
262#102—1271no—
263C22⋊C8—1271no—
264C22⋊C8—1271no—
265#97—1271no—
266C24—2541no—
267#41—5081no—
268#41—2542no—
269C2×Q8—1271no—
270C2×Q8—1271no—
271C2×Q8—5081no—
272C2×Q8—5081no—
273C2×Q8—2541no—
274C2×Q8—2542no—
275C2×Q8—2542no—
276C2×Q8—2541no—
277C2×Q8C22×D12711yescharacteristic
278C2×Q8C22×D12711yescharacteristic
279C2×D4—1271no—
280C2×D4—1271no—
281C2×D4—1271no—
282C2×D4—5081no—
283C2×D4—2542no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4—2541no—
287C22×C4—1271no—
288C22×C4—2541no—
289C22×C4—2541no—
290C22×C4—2541no—
291C22×C4—2541no—
292C22×C4C22×D12711yescharacteristic
293Q16—22no—
294Q16—5081no—
295Q16—2542no—
296SD16—1274no—
297SD16—5081no—
298SD16—2542no—
299SD16—2542no—
300SD16—2542no—
301SD16—2542no—
302M16—2542no—
303C2×C8—22no—
304C2×C8—1271no—
305C2×C8—1271no—
306C2×C8C22×D12711yescharacteristic
307C2×C8C22×D12711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—1271no—
311C4⋊C4—2541no—
312C4⋊C4—2541no—
313C4⋊C4—2541no—
314C4⋊C4—2541no—
315C4⋊C4C22×D12711yescharacteristic
316C4⋊C4C22×D12711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—1271no—
320C22⋊C4—1271no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C42—1271no—
327C42—2541no—
328C23—1271no—
329C23—10161no—
330C23—5082no—
331C23—5082no—
332C23—2541no—
333C23—2541no—
334C23D4×D12711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—1272no—
340Q8—5082no—
341Q8—5082no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345D4—1272no—
346D4—10161no—
347D4—5082no—
348D4—5082no—
349D4—2542no—
350D4—2541no—
351D4—2541no—
352D4—2541no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—1271no—
360C2×C4—1271no—
361C2×C4—1271no—
362C2×C4—10161no—
363C2×C4—5082no—
364C2×C4—5082no—
365C2×C4—5082no—
366C2×C4—2541no—
367C2×C4—2541no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4D4×D12711yescharacteristic
371C2×C4C23×D12711yescharacteristic
372C8—21no—
373C8—2541no—
374C8—2541no—
375C8D4×D12712yes—
376C22—21no—
377C22—21no—
378C22—10164no—
379C22—5081no—
380C22—5081no—
381C22—5081no—
382C22—5081no—
383C22—5081no—
384C22—2541no—
385C22—2541no—
386C22—2541no—
387C22—2542no—
388C22#1686124112yescomplemented
389C22C2×D4×D12711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—5081no—
396C4—2541no—
397C4—2541no—
398C4—2541no—
399C4—2542no—
400C4C2×D4×D12711yescharacteristic
401C4#1686125511yescharacteristic
402C2—22no—
403C2—10161no—
404C2—5082no—
405C2—5081no—
406C2#2255051011yescharacteristic
407C2C2×#1686124111yescharacteristic
408C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#467×C127C2
socleC22×C127C22×D4
prime radical#22550517C2
hypercenter#467D127
hypocenterC127#3266
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
2. C22×#14523237C2semidirect
3. #22550539C2semidirect
5. #22550269C2semidirect
6. #22550123C2semidirect
7. #22550101C2semidirect
8. #22550080C2semidirect
9. #22550041C2semidirect
10. #22550031C2semidirect
11. #22550026C2semidirect
12. #22550016C2semidirect
13. #22549975C2semidirect
14. #22549947C2semidirect
15. #467×C127C2semidirect
18. C2×Q8×D127C22semidirect
21. C2×#14523238C22semidirect
22. C2×#14523238C22semidirect
30. C2×#14523231C22semidirect
32. #16861207C22semidirect
35. #16861189C22semidirect
41. #16861174C22semidirect
46. Dic127⋊C8C22semidirect
47. #16861132C22semidirect
48. #16861132C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861119C22semidirect
54. C2×Q16×C127C22semidirect
58. #107×C127C22semidirect
59. #103×C127C22semidirect
60. #103×C127C22semidirect
90. #14523237D4semidirect
173. C22×C127#137semidirect
205. C127#3266semidirect
212. #467D127semidirect
228. C2×Q16C2×D127semidirect
255. #107C2×D127semidirect
259. #103C2×D127semidirect
260. #103C2×D127semidirect
388. C22#16861241semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C22×#145232372
#225505392
#225505172
#225502692
#225501232
#225501012
#225500802
#225500412
#225500312
#225500262
#225500162
#225499752
#225499472
#467×C1272
#3266127