Finite Group Databasebeta

← #30650441

Subgroups — #30650441

Order 16256.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861228C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3#22550532C211yesmaximal, characteristic, complemented
4#22550520C211yesmaximal, characteristic, complemented
5#22550259C211yesmaximal, characteristic, complemented
6#22550104C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550097C211yesmaximal, characteristic, complemented
9#22550089C211yesmaximal, characteristic, complemented
10#22550039C211yesmaximal, characteristic, complemented
11#22550034C211yesmaximal, characteristic, complemented
12#22550018C211yesmaximal, characteristic, complemented
13#22549967C211yesmaximal, characteristic, complemented
14(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
15#447×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523247C2211yescharacteristic, complemented
18C22×Dic254C2211yescharacteristic
19#16861228—24no—
20C2×Dic508C2211yescharacteristic, complemented
21C2×Dic508C2211yescharacteristic, complemented
22C2×#14523236—24no—
23C2×#14523236C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523236C2211yescharacteristic
27C2×#14523236C2211yescharacteristic, complemented
28C22×C127⋊C8C2211yescharacteristic
29C2×C8×D127C2211yescharacteristic, complemented
30#16861196C2211yescharacteristic
31#16861196C2211yescharacteristic, complemented
32#16861195—21no—
33#16861190—21no—
34#16861189C2211yescharacteristic, complemented
35#16861188C2211yescharacteristic
36#16861186—21no—
37#16861181—21no—
38#16861181—21no—
39#16861180C2211yescharacteristic
40#16861178—21no—
41#16861174C2211yescharacteristic, complemented
42D4×Dic127—21no—
43D4×Dic127C2211yescharacteristic
44C4×C22⋊D127C2211yescharacteristic
45C2×C22⋊Dic127—21no—
46#16861140C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861134C2211yescharacteristic, complemented
50#16861132C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C8×Dic127C2211yescharacteristic, complemented
54C22×D4×C127C2211yescharacteristic
55C2×D8×C127C2211yescharacteristic
56C2×D8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61#14523247—22no—
62#14523247—22no—
63#14523247—41no—
64#14523247—41no—
65C2×Dic254—22no—
66C2×Dic254—21no—
67C2×Dic254—41no—
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70C2×Dic254C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127D411yescharacteristic
75C22×Dic127—21no—
76C22×Dic127—21no—
77C22×Dic127—21no—
78C22×Dic127—21no—
79C2×C4×D127C2311yescharacteristic
80Dic508—22no—
81Dic508—41no—
82#14523236—22no—
83#14523236—22no—
84#14523236—22no—
85#14523236—22no—
86#14523236D414yescomplemented
87C2×C127⋊C8—22no—
88C2×C127⋊C8C2311yescharacteristic
89C2×C127⋊C8C2311yescharacteristic
90C8×D127—22no—
91C4⋊Dic127—21no—
92C4⋊Dic127C2311yescharacteristic
93Dic127⋊C4—21no—
94Dic127⋊C4—21no—
95Dic127⋊C4D411yescharacteristic
96C22⋊Dic127—21no—
97C22⋊Dic127—21no—
98C22⋊Dic127—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4D411yescharacteristic
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108C24×C127—21no—
109C2×D4×C127—22no—
110C2×D4×C127—21no—
111C2×D4×C127—21no—
112C2×D4×C127—41no—
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C2×D4×C127C2311yescharacteristic
116C22×C4×C127C2311yescharacteristic
117D8×C127—22no—
118D8×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127C2×D411yescharacteristic
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—42no—
130Dic254C2×D412yes—
131C22⋊D127—22no—
132C22⋊D127—42no—
133C22⋊D127—42no—
134C22⋊D127—81no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—42no—
140C2×Dic127—42no—
141C2×Dic127—42no—
142C2×Dic127—81no—
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C2×Dic127C2×D411yescharacteristic
146C4×D127—21no—
147C4×D127—21no—
148C127⋊C8—21no—
149C127⋊C8C2×D412yes—
150C23×C127—21no—
151C23×C127—21no—
152C23×C127—21no—
153C23×C127—42no—
154C23×C127—42no—
155C23×C127—81no—
156C23×C127C2×D411yescharacteristic
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—42no—
163D4×C127—42no—
164D4×C127C2×D412yes—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127—21no—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—22no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—84no—
184C22×C127#13512yescomplemented
185C22×C127C22×D411yescharacteristic
186C2×D127—21no—
187C2×D127—21no—
188C2×D127—21no—
189C4×C127—21no—
190C4×C127—41no—
191C4×C127C22×D411yescharacteristic
192C4×C127H(5,𝔽2)11yescharacteristic
193Dic127—21no—
194Dic127—22no—
195Dic127—21no—
196Dic127—21no—
197Dic127—41no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—22no—
201C2×C127—42no—
202C2×C127—81no—
203C2×C127—81no—
204C2×C127D4211yescharacteristic
205C2×C127C2×#13511yescharacteristic
206C2×C127#57711yescharacteristic
207D127—41no—
208#3282—1271nomaximal, Sylow
209C127#328211yescharacteristic, complemented, Sylow
210C2×#135—1271no—
211C22×SD16—1271no—
212#548—1271no—
213#546—1271no—
214#495—1271no—
215#465—1271no—
216#465—1271no—
217#464—1271no—
218#463—1271no—
219#452—1271no—
220#451—1271no—
221#449—1271no—
222#447D12711yescharacteristic, complemented
223C4×SD16—1271no—
224C8×D4—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C22×Q8—1271no—
228C22×D4C2×D12711yescharacteristic
229#135—2544no—
230C2×Q16—1271no—
231C2×Q16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16—1271no—
237C2×SD16—2544no—
238C2×D8C2×D12711yescharacteristic
239C2×D8C2×D12711yescharacteristic
240C22×C8—1271no—
241C22×C8—1271no—
242#124—1271no—
243#124—1271no—
244#124—2541no—
245#123—2541no—
246C22⋊Q8—1271no—
247C22⋊Q8—1271no—
248C22⋊Q8—2541no—
249#121—1271no—
250#121—2541no—
251#121—2541no—
252#121C2×D12711yescharacteristic
253#120—2541no—
254C4×Q8—1271no—
255C4×D4—1271no—
256C4×D4—1271no—
257C4×D4—2541no—
258C2×C22⋊C4—2541no—
259#106—1271no—
260C4⋊C8—1271no—
261#103—1271no—
262#103—1271no—
263#103—1271no—
264#102—1271no—
265#102C2×D12711yescharacteristic
266#102C2×D12711yescharacteristic
267C22⋊C8—1271no—
268C22⋊C8C2×D12711yescharacteristic
269C4×C8—1271no—
270C24—21no—
271#41—5081no—
272#41—5081no—
273#41—2542no—
274#41—2542no—
275C2×Q8—1271no—
276C2×Q8—1271no—
277C2×Q8—1271no—
278C2×Q8—5081no—
279C2×Q8—2542no—
280C2×Q8—2541no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—1271no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4C22×D12711yescharacteristic
290C2×D4C22×D12711yescharacteristic
291C2×D4C22×D12711yescharacteristic
292C22×C4—1271no—
293C22×C4—2541no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4C22×D12711yescharacteristic
298Q16—5081no—
299Q16—2542no—
300SD16—1274no—
301SD16—2542no—
302SD16—2542no—
303SD16—2542no—
304SD16—2542no—
305D8—22no—
306D8—41no—
307C2×C8—1271no—
308C2×C8—1271no—
309C2×C8—2542no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—1271no—
314C4⋊C4—1271no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4C22×D12711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—1271no—
321C22⋊C4—1271no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C42—1271no—
331C42—2541no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1271no—
339C23D4×D12711yescharacteristic
340Q8—1272no—
341Q8—5082no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345Q8—2541no—
346Q8—2541no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—10161no—
355D4—5082no—
356D4—5082no—
357D4—2542no—
358D4D4×D12712yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—1271no—
363C2×C4—1271no—
364C2×C4—1271no—
365C2×C4—10161no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—5082no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4—2541no—
375C2×C4D4×D12711yescharacteristic
376C2×C4C23×D12711yescharacteristic
377C8—21no—
378C8—21no—
379C8—1272no—
380C8—2541no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—84no—
393C22—2541no—
394C22—2541no—
395C22—2541no—
396C22#1686122812yescomplemented
397C22C2×D4×D12711yescharacteristic
398C4—21no—
399C4—41no—
400C4—5081no—
401C4—5081no—
402C4—5081no—
403C4—2541no—
404C4—2541no—
405C4—2541no—
406C4—2542no—
407C4C2×D4×D12711yescharacteristic
408C4#1686125411yescharacteristic
409C2—22no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—5081no—
414C2#2255050811yescharacteristic
415C2C2×#1686122811yescharacteristic
416C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#447×C127C2
socleC22×C127C22×D4
hypercenter#447D127
hypocenterC127#3282
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#16861228C2semidirect
2. C22×#14523236C2semidirect
3. #22550532C2semidirect
4. #22550520C2semidirect
5. #22550259C2semidirect
6. #22550104C2semidirect
7. #22550103C2semidirect
8. #22550097C2semidirect
9. #22550089C2semidirect
10. #22550039C2semidirect
11. #22550034C2semidirect
12. #22550018C2semidirect
13. #22549967C2semidirect
14. (C22⋊D127)⋊C8C2semidirect
15. #447×C127C2semidirect
17. C2×#14523247C22semidirect
20. C2×Dic508C22semidirect
21. C2×Dic508C22semidirect
27. C2×#14523236C22semidirect
29. C2×C8×D127C22semidirect
31. #16861196C22semidirect
34. #16861189C22semidirect
41. #16861174C22semidirect
46. #16861140C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861134C22semidirect
50. #16861132C22semidirect
51. #16861130C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C8×Dic127C22semidirect
86. #14523236D4semidirect
184. C22×C127#135semidirect
209. C127#3282semidirect
222. #447D127semidirect
396. C22#16861228semidirect

Maximal subgroups

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

subgroupindex
C2×#168612282
C22×#145232362
#225505322
#225505202
#225502592
#225501042
#225501032
#225500972
#225500892
#225500392
#225500342
#225500182
#225499672
(C22⋊D127)⋊C82
#447×C1272
#3282127