Finite Group Databasebeta

← #30650251

Subgroups — #30650251

Order 16256.

500 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×#16861235C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550531C211yesmaximal, characteristic, complemented
4#22550271C211yesmaximal, characteristic, complemented
5#22550132C211yesmaximal, characteristic, complemented
6#22550103C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550081C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550036C211yesmaximal, characteristic, complemented
11#22550027C211yesmaximal, characteristic, complemented
12#22550019C211yesmaximal, characteristic, complemented
13#22549986C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15#546×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic, complemented
18#16861235—24no—
19#16861229—24no—
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523236C2211yescharacteristic
23C2×#14523236C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523235C2211yescharacteristic, complemented
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523232C2211yescharacteristic
30C22×C127⋊C8C2211yescharacteristic, complemented
31#16861209C2211yescharacteristic, complemented
32#16861191—21no—
33#16861191—21no—
34#16861188C2211yescharacteristic
35#16861188C2211yescharacteristic, complemented
36#16861180—21no—
37C4×D508C2211yescharacteristic, complemented
38C4⋊C4×D127—21no—
39#16861140C2211yescharacteristic, complemented
40C4⋊(C127⋊C8)C2211yescharacteristic, complemented
41#16861133C2211yescharacteristic, complemented
42#16861132C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861130C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861129C2211yescharacteristic, complemented
47C22⋊(C127⋊C8)C2211yescharacteristic
48C22⋊(C127⋊C8)C2211yescharacteristic, complemented
49#16861120C2211yescharacteristic, complemented
50C2×#41×C127C2211yescharacteristic
51C22×D4×C127C2211yescharacteristic
52#124×C127C2211yescharacteristic
53#123×C127—21no—
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127—21no—
56#121×C127C2211yescharacteristic
57#120×C127—21no—
58C4×D4×C127C2211yescharacteristic
59C4×D4×C127C2211yescharacteristic
60C2×C22⋊C4×C127—21no—
61#14523248—22no—
62#14523248—22no—
63#14523248—41no—
64#14523248—41no—
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73#14523238—22no—
74#14523238—41no—
75#14523236—22no—
76#14523236—22no—
77#14523236—22no—
78#14523236—41no—
79#14523235—22no—
80#14523233—22no—
81#14523233D414yescomplemented
82#14523232—22no—
83C2×C127⋊C8—22no—
84C2×C127⋊C8C2311yescharacteristic
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C2×C127⋊C8C2311yescharacteristic
88C4⋊Dic127—21no—
89C4⋊Dic127—21no—
90C4⋊Dic127C2311yescharacteristic
91Dic127⋊C4—21no—
92Dic127⋊C4—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95(C2×D127)⋊C4—21no—
96C24×C127—21no—
97#41×C127—22no—
98#41×C127—41no—
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—22no—
101C2×D4×C127—21no—
102C2×D4×C127—21no—
103C2×D4×C127—21no—
104C2×D4×C127—41no—
105C2×D4×C127D411yescharacteristic
106C2×D4×C127C2311yescharacteristic
107C2×D4×C127C2311yescharacteristic
108C22×C4×C127—21no—
109C22×C4×C127—21no—
110C22×C4×C127C2311yescharacteristic
111C22×C4×C127C2311yescharacteristic
112C4⋊C4×C127—21no—
113C4⋊C4×C127D411yescharacteristic
114C4⋊C4×C127C2311yescharacteristic
115C22⋊C4×C127—21no—
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128D508—21no—
129D508C2×D412yes—
130C22⋊D127—42no—
131C22⋊D127—42no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C4×D127—42no—
136C4×D127—42no—
137C4×D127—42no—
138C127⋊C8—21no—
139C127⋊C8—21no—
140C127⋊C8—21no—
141C127⋊C8C2×D412yes—
142C23×C127—21no—
143C23×C127—21no—
144C23×C127—42no—
145C23×C127—42no—
146C23×C127—81no—
147C23×C127C2×D411yescharacteristic
148C23×C127C2×D411yescharacteristic
149Q8×C127—21no—
150Q8×C127—21no—
151D4×C127—21no—
152D4×C127—21no—
153D4×C127—21no—
154D4×C127—41no—
155D4×C127—42no—
156D4×C127—42no—
157D4×C127—81no—
158D4×C127C2×D412yes—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—42no—
166C2×C4×C127—81no—
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2×D411yescharacteristic
170C2×C4×C127C2×D411yescharacteristic
171C2×C4×C127C2411yescharacteristic
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—22no—
175C22×C127—21no—
176C22×C127—21no—
177C22×C127—21no—
178C22×C127—21no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—84no—
185C22×C127C22×D411yescharacteristic
186C2×D127—22no—
187C2×D127—41no—
188C2×D127—41no—
189C4×C127—21no—
190C4×C127—21no—
191C4×C127—21no—
192C4×C127—21no—
193C4×C127—41no—
194C4×C127—41no—
195C4×C127C22×D411yescharacteristic
196C4×C127H(5,𝔽2)11yescharacteristic
197Dic127—41no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—41no—
201C2×C127—41no—
202C2×C127—42no—
203C2×C127—81no—
204C2×C127D4211yescharacteristic
205C2×C127#57511yescharacteristic
206C2×C127#57711yescharacteristic
207D127—42no—
208#3267—1271nomaximal, Sylow
209C127#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1271no—
211C2×#135—1271no—
212#547—1271no—
213#546D12711yescharacteristic, complemented
214#497—1271no—
215#470—1271no—
216#465—1271no—
217#463—1271no—
218#461—1271no—
219#452—1271no—
220#452—1271no—
221#450—1271no—
222#449—1271no—
223#442—1271no—
224#435—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C2×#41C2×D12711yescharacteristic
228C22×D4C2×D12711yescharacteristic
229AGL(1,ℤ/8)—2544no—
230#135—2544no—
231C2×Q16—1271no—
232C2×Q16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16—1271no—
237C2×SD16—1271no—
238C2×D8—1271no—
239C2×D8—1271no—
240C2×M16—1271no—
241C22×C8—1271no—
242C4⋊Q8—1271no—
243#124C2×D12711yescharacteristic
244#123—21no—
245#123—2541no—
246#123—2541no—
247C22⋊Q8—1271no—
248C22⋊Q8—1271no—
249C22⋊Q8C2×D12711yescharacteristic
250#121—21no—
251#121—2541no—
252#121C2×D12711yescharacteristic
253#120—21no—
254C4×D4—1271no—
255C4×D4C2×D12711yescharacteristic
256C4×D4C2×D12711yescharacteristic
257C2×C4⋊C4—2541no—
258C2×C22⋊C4—21no—
259#106—1271no—
260C4⋊C8—1271no—
261#103—1271no—
262#103—1271no—
263#103—1271no—
264#102—1271no—
265#102—1271no—
266#102—1271no—
267C22⋊C8—1271no—
268C22⋊C8—1271no—
269#97—1271no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—5081no—
274#41—5081no—
275#41—2542no—
276#41—2542no—
277C2×Q8—1271no—
278C2×Q8—1271no—
279C2×Q8C22×D12711yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—1271no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4C22⋊D12711yescharacteristic
289C2×D4C22×D12711yescharacteristic
290C2×D4C22×D12711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—2541no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4C22×D12711yescharacteristic
297C22×C4C22×D12711yescharacteristic
298Q16—5081no—
299Q16—2542no—
300SD16—5081no—
301SD16—2542no—
302SD16—2542no—
303SD16—2542no—
304SD16—2542no—
305D8—1274no—
306D8—2542no—
307M16—2542no—
308C2×C8—1271no—
309C2×C8—1271no—
310C2×C8—1271no—
311C2×C8—1271no—
312C2×C8—2542no—
313C4⋊C4—21no—
314C4⋊C4—1271no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4C22⋊D12711yescharacteristic
320C4⋊C4C22×D12711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4C22⋊D12711yescharacteristic
330C22⋊C4C22⋊D12711yescharacteristic
331C42C22×D12711yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—2541no—
338C23C2×C22⋊D12711yescharacteristic
339C23C2×C22⋊D12711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345Q8—2541no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—1272no—
354D4—5082no—
355D4—5082no—
356D4—2541no—
357D4D4×D12712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—5082no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4C2×C22⋊D12711yescharacteristic
373C2×C4C2×C22⋊D12711yescharacteristic
374C2×C4C2×C22⋊D12711yescharacteristic
375C2×C4C2×C22⋊D12711yescharacteristic
376C2×C4C23×D12711yescharacteristic
377C8—1272no—
378C8—2541no—
379C8—2541no—
380C8—2541no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—5081no—
395C22—5081no—
396C22—2542no—
397C22C22×C22⋊D12711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—5081no—
405C4—5081no—
406C4—5081no—
407C4C2×D4×D12711yescharacteristic
408C4#1686125411yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—5082no—
414C2D4×C22⋊D12711yescharacteristic
415C2#2255067411yescharacteristic
416C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#546×C127C2
socleC22×C127C22×D4
hypercenter#546D127
hypocenterC127#3267
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. C2×#16861229C2semidirect
3. #22550531C2semidirect
4. #22550271C2semidirect
5. #22550132C2semidirect
6. #22550103C2semidirect
7. #22550088C2semidirect
8. #22550081C2semidirect
9. #22550038C2semidirect
10. #22550036C2semidirect
11. #22550027C2semidirect
12. #22550019C2semidirect
13. #22549986C2semidirect
14. #22549948C2semidirect
15. #546×C127C2semidirect
17. C2×#14523248C22semidirect
20. C2×#14523238C22semidirect
21. C2×#14523238C22semidirect
26. C2×#14523235C22semidirect
30. C22×C127⋊C8C22semidirect
31. #16861209C22semidirect
35. #16861188C22semidirect
37. C4×D508C22semidirect
39. #16861140C22semidirect
40. C4⋊(C127⋊C8)C22semidirect
41. #16861133C22semidirect
42. #16861132C22semidirect
43. #16861132C22semidirect
46. #16861129C22semidirect
48. C22⋊(C127⋊C8)C22semidirect
49. #16861120C22semidirect
81. #14523233D4semidirect
209. C127#3267semidirect
213. #546D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C2×#168612292
#225505312
#225502712
#225501322
#225501032
#225500882
#225500812
#225500382
#225500362
#225500272
#225500192
#225499862
#225499482
#546×C1272
#3267127