Finite Group Databasebeta

← #30650249

Subgroups — #30650249

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×#16861228C211yesmaximal, characteristic, complemented
2#22550525C211yesmaximal, characteristic, complemented
3#22550520C211yesmaximal, characteristic, complemented
4#22550267C211yesmaximal, characteristic, complemented
5#22550134C211yesmaximal, characteristic, complemented
6#22550102C211yesmaximal, characteristic, complemented
7#22550089C211yesmaximal, characteristic, complemented
8#22550079C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550035C211yesmaximal, characteristic, complemented
11#22550027C211yesmaximal, characteristic, complemented
12#22550018C211yesmaximal, characteristic, complemented
13#22549988C211yesmaximal, characteristic, complemented
14#22549945C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523247C2211yescharacteristic, complemented
18#16861228—24no—
19C2×Dic508C2211yescharacteristic, complemented
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523236C2211yescharacteristic
22C2×#14523236C2211yescharacteristic
23C2×#14523236C2211yescharacteristic, complemented
24C2×C8×D127C2211yescharacteristic, complemented
25#16861207C2211yescharacteristic, complemented
26#16861196C2211yescharacteristic
27#16861192—21no—
28#16861190—21no—
29#16861190—21no—
30#16861188C2211yescharacteristic
31#16861188C2211yescharacteristic
32#16861185C2211yescharacteristic, complemented
33#16861181—21no—
34#16861181—21no—
35#16861180C2211yescharacteristic
36#16861178—21no—
37D4×Dic127C2211yescharacteristic
38D4×Dic127C2211yescharacteristic, complemented
39C4×C22⋊D127C2211yescharacteristic
40C2×Dic127⋊C4—21no—
41C2×C22⋊Dic127—21no—
42#16861141C2211yescharacteristic, complemented
43Dic127⋊C8C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861134C2211yescharacteristic, complemented
46#16861133C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic, complemented
50C22⋊(C127⋊C8)C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52#16861119C2211yescharacteristic, complemented
53C2×#41×C127C2211yescharacteristic
54C22×D4×C127C2211yescharacteristic
55AGL(1,ℤ/8)×C127—24no—
56C2×SD16×C127C2211yescharacteristic
57C2×SD16×C127C2211yescharacteristic
58C2×D8×C127C2211yescharacteristic
59C2×D8×C127C2211yescharacteristic
60C2×M16×C127C2211yescharacteristic
61#14523247—22no—
62#14523247—22no—
63#14523247—41no—
64#14523247—41no—
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127D411yescharacteristic
71C22×Dic127—21no—
72C22×Dic127—21no—
73C22×Dic127—21no—
74C22×Dic127—21no—
75C2×C4×D127C2311yescharacteristic
76Dic508—22no—
77#14523238—41no—
78#14523236—22no—
79#14523236—22no—
80#14523236—41no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C8×D127—22no—
84C4⋊Dic127C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91Dic127⋊C4D411yescharacteristic
92C22⋊Dic127—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95C22⋊Dic127—21no—
96C22⋊Dic127—21no—
97C22⋊Dic127—21no—
98C22⋊Dic127—21no—
99C22⋊Dic127D411yescharacteristic
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4D411yescharacteristic
102C4×Dic127C2311yescharacteristic
103C24×C127—21no—
104#41×C127—22no—
105#41×C127—41no—
106C2×Q8×C127C2311yescharacteristic
107C2×D4×C127—22no—
108C2×D4×C127—21no—
109C2×D4×C127—21no—
110C2×D4×C127—41no—
111C2×D4×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116SD16×C127—22no—
117SD16×C127—22no—
118D8×C127—22no—
119D8×C127D414yescomplemented
120M16×C127—22no—
121C2×C8×C127C2311yescharacteristic
122C2×C8×C127C2311yescharacteristic
123C22×D127C2×D411yescharacteristic
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128C22⋊D127—41no—
129C22⋊D127—42no—
130C22⋊D127—42no—
131C22⋊D127—81no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—42no—
137C2×Dic127—42no—
138C2×Dic127—42no—
139C2×Dic127—81no—
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C2×Dic127C2×D411yescharacteristic
143C4×D127—21no—
144C4×D127—21no—
145C127⋊C8—21no—
146C127⋊C8—21no—
147C23×C127—21no—
148C23×C127—21no—
149C23×C127—21no—
150C23×C127—42no—
151C23×C127—42no—
152C23×C127—81no—
153C23×C127C2×D411yescharacteristic
154Q8×C127—21no—
155Q8×C127—21no—
156D4×C127—21no—
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—42no—
161D4×C127—42no—
162D4×C127C2×D412yes—
163D4×C127C2×D412yes—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—42no—
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127C2×D412yes—
172C22×C127—21no—
173C22×C127—22no—
174C22×C127—22no—
175C22×C127—21no—
176C22×C127—21no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—84no—
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—21no—
195Dic127—21no—
196Dic127—41no—
197Dic127—41no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—41no—
201C2×C127—42no—
202C2×C127—42no—
203C2×C127—81no—
204C2×C127D4211yescharacteristic
205C2×C127#57511yescharacteristic
206C2×C127#57711yescharacteristic
207D127—41no—
208#3267—1271nomaximal, Sylow
209C127#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
211C2×#135—1271no—
212#547—1271no—
213#546—1271no—
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)—24no—
230#135—2544no—
231C2×Q16—1271no—
232C2×Q16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16C2×D12711yescharacteristic
237C2×SD16C2×D12711yescharacteristic
238C2×D8C2×D12711yescharacteristic
239C2×D8C2×D12711yescharacteristic
240C2×M16C2×D12711yescharacteristic
241C22×C8—1271no—
242C4⋊Q8—1271no—
243#124—1271no—
244#123—2541no—
245#123—2541no—
246#123—2541no—
247C22⋊Q8—1271no—
248C22⋊Q8—1271no—
249C22⋊Q8—1271no—
250#121—1271no—
251#121—2541no—
252#121—2541no—
253#120—2541no—
254C4×D4—1271no—
255C4×D4—1271no—
256C4×D4—1271no—
257C2×C4⋊C4—2541no—
258C2×C22⋊C4—2541no—
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—41no—
284C2×D4—1271no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4C22×D12711yescharacteristic
289C2×D4C22×D12711yescharacteristic
290C2×D4C22×D12711yescharacteristic
291C22×C4—21no—
292C22×C4—1271no—
293C22×C4—2541no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4C22×D12711yescharacteristic
298Q16—5081no—
299Q16—2542no—
300SD16—22no—
301SD16—22no—
302SD16—5081no—
303SD16—2542no—
304SD16—2542no—
305D8—22no—
306D8C22⋊D12714yescomplemented
307M16—22no—
308C2×C8—1271no—
309C2×C8—1271no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—1271no—
314C4⋊C4—1271no—
315C4⋊C4—1271no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C4⋊C4—2541no—
321C22⋊C4—1271no—
322C22⋊C4—1271no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C22⋊C4—2541no—
331C42—1271no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1271no—
339C23D4×D12711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345Q8—2541no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—10161no—
353D4—5082no—
354D4—5082no—
355D4—5081no—
356D4C2×C22⋊D12712yes—
357D4C2×C22⋊D12712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
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—2541no—
379C8—2541no—
380C8C2×C22⋊D12712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—2541no—
395C22—2541no—
396C22—2541no—
397C22C2×D4×D12711yescharacteristic
398C4—21no—
399C4—41no—
400C4—5081no—
401C4—5081no—
402C4—5081no—
403C4—5081no—
404C4—2541no—
405C4—2541no—
406C4—2541no—
407C4C22×C22⋊D12711yescharacteristic
408C4#1686125411yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—5081no—
414C2D4×C22⋊D12711yescharacteristic
415C2#2255067211yescharacteristic
416C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3267
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861228C2semidirect
2. #22550525C2semidirect
3. #22550520C2semidirect
4. #22550267C2semidirect
5. #22550134C2semidirect
6. #22550102C2semidirect
7. #22550089C2semidirect
8. #22550079C2semidirect
9. #22550038C2semidirect
10. #22550035C2semidirect
11. #22550027C2semidirect
12. #22550018C2semidirect
13. #22549988C2semidirect
14. #22549945C2semidirect
15. C2×AGL(1,ℤ/8)×C127C2semidirect
17. C2×#14523247C22semidirect
19. C2×Dic508C22semidirect
20. C2×#14523238C22semidirect
23. C2×#14523236C22semidirect
24. C2×C8×D127C22semidirect
25. #16861207C22semidirect
32. #16861185C22semidirect
38. D4×Dic127C22semidirect
42. #16861141C22semidirect
43. Dic127⋊C8C22semidirect
44. #16861134C22semidirect
45. #16861134C22semidirect
46. #16861133C22semidirect
49. #16861130C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. #16861119C22semidirect
119. D8×C127D4semidirect
209. C127#3267semidirect
210. C2×AGL(1,ℤ/8)D127semidirect
306. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612282
#225505252
#225505202
#225502672
#225501342
#225501022
#225500892
#225500792
#225500382
#225500352
#225500272
#225500182
#225499882
#225499452
C2×AGL(1,ℤ/8)×C1272
#3267127