Finite Group Databasebeta

← #30650270

Subgroups — #30650270

Order 16256.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2#22550545C211yesmaximal, characteristic, complemented
3#22550514C211yesmaximal, characteristic, complemented
4#22550262C211yesmaximal, characteristic, complemented
5#22550119C211yesmaximal, characteristic, complemented
6#22550112C211yesmaximal, characteristic, complemented
7#22550100C211yesmaximal, characteristic, complemented
8#22550069C211yesmaximal, characteristic, complemented
9#22550037C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22550021C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549977C211yesmaximal, characteristic, complemented
14#22549947C211yesmaximal, characteristic, complemented
15C2×#135×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18#16861236—24no—
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic, complemented
21C2×D1016C2211yescharacteristic
22C2×#14523233C2211yescharacteristic
23C2×#14523233C2211yescharacteristic
24C2×#14523231C2211yescharacteristic
25#16861205—21no—
26#16861205C2211yescharacteristic, complemented
27#16861198C2211yescharacteristic
28#16861192—21no—
29#16861185C2211yescharacteristic
30#16861183—21no—
31#16861183C2211yescharacteristic
32#16861183C2211yescharacteristic, complemented
33#16861181—21no—
34#16861180—21no—
35#16861180C2211yescharacteristic
36#16861177—21no—
37Q8×Dic127C2211yescharacteristic, complemented
38D4×Dic127C2211yescharacteristic
39C4×C22⋊D127—21no—
40C4×C22⋊D127C2211yescharacteristic
41C2×(C2×D127)⋊C4—21no—
42#16861143C2211yescharacteristic, complemented
43Dic127⋊C8C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861133C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic, complemented
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic, complemented
50C22⋊(C127⋊C8)C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic
52#16861119C2211yescharacteristic, complemented
53C2×#41×C127C2211yescharacteristic
54C2×#41×C127C2211yescharacteristic, complemented
55#135×C127—24no—
56C2×Q16×C127C2211yescharacteristic, complemented
57C2×SD16×C127C2211yescharacteristic
58C2×SD16×C127C2211yescharacteristic, complemented
59C2×D8×C127C2211yescharacteristic
60C22×C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64C2×D508—21no—
65C2×D508C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127D411yescharacteristic
74D4×D127—22no—
75D4×D127—41no—
76C22×Dic127—21no—
77C2×C4×D127—21no—
78C2×C4×D127C2311yescharacteristic
79#14523235—22no—
80#14523235—41no—
81D1016—22no—
82#14523233—22no—
83#14523233—41no—
84#14523231—22no—
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C4⋊Dic127—21no—
88C4⋊Dic127C2311yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91Dic127⋊C4D411yescharacteristic
92C22⋊Dic127—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127D411yescharacteristic
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4D411yescharacteristic
101C4×Dic127—21no—
102C4×Dic127C2311yescharacteristic
103#41×C127—22no—
104#41×C127—22no—
105#41×C127—41no—
106#41×C127—41no—
107C2×Q8×C127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127—21no—
110C2×D4×C127—21no—
111C2×D4×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116Q16×C127—22no—
117SD16×C127—22no—
118SD16×C127D414yescomplemented
119D8×C127—22no—
120C2×C8×C127—22no—
121C2×C8×C127C2311yescharacteristic
122C2×C8×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128C22×D127C2×D411yescharacteristic
129D508—21no—
130D508—21no—
131D508—21no—
132D508—21no—
133D508—42no—
134C22⋊D127—41no—
135C22⋊D127—42no—
136C22⋊D127—42no—
137C22⋊D127—81no—
138C22⋊D127—81no—
139C22⋊D127—81no—
140C22⋊D127—81no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—81no—
144C2×Dic127C2×D411yescharacteristic
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C4×D127—21no—
148C4×D127—21no—
149C4×D127—42no—
150C127⋊C8—21no—
151C127⋊C8—21no—
152C23×C127—21no—
153C23×C127—21no—
154C23×C127C2×D411yescharacteristic
155Q8×C127—21no—
156Q8×C127—21no—
157Q8×C127—21no—
158Q8×C127C2×D412yes—
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×C127—21no—
169C2×C4×C127—42no—
170C2×C4×C127—42no—
171C2×C4×C127C2×D411yescharacteristic
172C2×C4×C127C2411yescharacteristic
173C8×C127—21no—
174C8×C127C2×D412yes—
175C22×C127—21no—
176C22×C127—22no—
177C22×C127—21no—
178C22×C127—21no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127C22×D411yescharacteristic
185C2×D127—21no—
186C2×D127—21no—
187C2×D127—21no—
188C2×D127—41no—
189C2×D127—41no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—84no—
195C4×C127—21no—
196C4×C127—22no—
197C4×C127—41no—
198C4×C127C22×D411yescharacteristic
199C4×C127H(5,𝔽2)11yescharacteristic
200Dic127—21no—
201Dic127—21no—
202Dic127—21no—
203Dic127—41no—
204Dic127—41no—
205C2×C127—42no—
206C2×C127—41no—
207C2×C127—81no—
208C2×C127D4211yescharacteristic
209C2×C127#57511yescharacteristic
210C2×C127#57611yescharacteristic
211D127—41no—
212D127—81no—
213D127—81no—
214#3268—1271nomaximal, Sylow
215C127#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1271no—
217C2×#135D12711yescharacteristic, complemented
218#550—1271no—
219#546—1271no—
220#496—1271no—
221#468—1271no—
222#466—1271no—
223#464—1271no—
224#459—1271no—
225#452—1271no—
226#449—1271no—
227#449—1271no—
228#447—1271no—
229#440—1271no—
230#435—1271no—
231C2×#41—1271no—
232C2×#41C2×D12711yescharacteristic
233C2×#41C2×D12711yescharacteristic, complemented
234C22×D4—1271no—
235AGL(1,ℤ/8)—2544no—
236#135—24no—
237C2×Q16C2×D12711yescharacteristic, complemented
238C2×SD16—1271no—
239C2×SD16—1271no—
240C2×SD16C2×D12711yescharacteristic
241C2×SD16C2×D12711yescharacteristic, complemented
242C2×D8—1271no—
243C2×D8—1271no—
244C2×D8—1271no—
245C2×D8C2×D12711yescharacteristic
246C2×M16—1271no—
247C22×C8C2×D12711yescharacteristic, complemented
248#127—1271no—
249#127—2541no—
250#124—1271no—
251#123—2541no—
252C22⋊Q8—1271no—
253#121—1271no—
254#121—1271no—
255#121—1271no—
256#121—2541no—
257#121—2541no—
258#121—2541no—
259#120—2541no—
260C4×Q8—1271no—
261C4×D4—1271no—
262C4×D4—1271no—
263C4×D4—2541no—
264C2×C22⋊C4—2541no—
265#107—1271no—
266C4⋊C8—1271no—
267#103—1271no—
268#103—1271no—
269#102—1271no—
270#102—1271no—
271#102—1271no—
272#102—1271no—
273C22⋊C8—1271no—
274C22⋊C8—1271no—
275#97—1271no—
276C24—2541no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—5081no—
282#41—2542no—
283C2×Q8C22×D12711yescharacteristic
284C2×Q8C22×D12711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1271no—
288C2×D4—1271no—
289C2×D4—1271no—
290C2×D4—5081no—
291C2×D4—2541no—
292C2×D4—2541no—
293C2×D4—2541no—
294C2×D4—2541no—
295C2×D4—2541no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4—2542no—
299C2×D4C22×D12711yescharacteristic
300C2×D4C22×D12711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—1271no—
304C22×C4—2541no—
305C22×C4—2541no—
306C22×C4C22×D12711yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—5081no—
310SD16—2542no—
311SD16C22⋊D12714yescomplemented
312D8—22no—
313D8—5081no—
314D8—2542no—
315D8—2542no—
316M16—2542no—
317C2×C8—22no—
318C2×C8—1271no—
319C2×C8—1271no—
320C2×C8C22×D12711yescharacteristic
321C2×C8C22×D12711yescharacteristic
322C4⋊C4—1271no—
323C4⋊C4—1271no—
324C4⋊C4—1271no—
325C4⋊C4—2541no—
326C4⋊C4—2541no—
327C22⋊C4—1271no—
328C22⋊C4—1271no—
329C22⋊C4—2541no—
330C22⋊C4—2541no—
331C22⋊C4—2541no—
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C42—1271no—
337C42—2541no—
338C23—21no—
339C23—21no—
340C23—1271no—
341C23—10161no—
342C23—5082no—
343C23—5082no—
344C23—2541no—
345C23—2541no—
346C23D4×D12711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D12712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—10161no—
357D4—10161no—
358D4—10161no—
359D4—10161no—
360D4—5082no—
361D4—5082no—
362D4—5081no—
363D4—5082no—
364D4—2541no—
365D4—2541no—
366D4—2541no—
367D4—2541no—
368D4C2×C22⋊D12712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—1271no—
376C2×C4—1271no—
377C2×C4—1271no—
378C2×C4—10161no—
379C2×C4—5082no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4D4×D12711yescharacteristic
385C2×C4C23×D12711yescharacteristic
386C8—21no—
387C8—2541no—
388C8—2541no—
389C8C2×C22⋊D12712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—10164no—
400C22—5081no—
401C22—5081no—
402C22—5081no—
403C22—5081no—
404C22—5081no—
405C22—5081no—
406C22—2541no—
407C22—2541no—
408C22—2541no—
409C22C2×D4×D12711yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—5081no—
414C4—5081no—
415C4—2541no—
416C4—2541no—
417C4—2541no—
418C4C22×C22⋊D12711yescharacteristic
419C4#1686125411yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—10161no—
424C2—10161no—
425C2—5081no—
426C2D4×C22⋊D12711yescharacteristic
427C2#2255067311yescharacteristic
428C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#135×C127C2
socleC22×C127C22×D4
hypercenterC2×#135D127
hypocenterC127#3268
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. #22550545C2semidirect
3. #22550514C2semidirect
4. #22550262C2semidirect
5. #22550119C2semidirect
6. #22550112C2semidirect
7. #22550100C2semidirect
8. #22550069C2semidirect
9. #22550037C2semidirect
10. #22550021C2semidirect
11. #22550021C2semidirect
12. #22550008C2semidirect
13. #22549977C2semidirect
14. #22549947C2semidirect
15. C2×#135×C127C2semidirect
20. C2×#14523235C22semidirect
26. #16861205C22semidirect
32. #16861183C22semidirect
37. Q8×Dic127C22semidirect
42. #16861143C22semidirect
43. Dic127⋊C8C22semidirect
44. #16861133C22semidirect
45. #16861133C22semidirect
47. #16861131C22semidirect
49. #16861130C22semidirect
50. C22⋊(C127⋊C8)C22semidirect
52. #16861119C22semidirect
54. C2×#41×C127C22semidirect
56. C2×Q16×C127C22semidirect
58. C2×SD16×C127C22semidirect
60. C22×C8×C127C22semidirect
118. SD16×C127D4semidirect
215. C127#3268semidirect
217. C2×#135D127semidirect
233. C2×#41C2×D127semidirect
237. C2×Q16C2×D127semidirect
241. C2×SD16C2×D127semidirect
247. C22×C8C2×D127semidirect
311. SD16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
#225505452
#225505142
#225502622
#225501192
#225501122
#225501002
#225500692
#225500372
#225500212
#225500082
#225499772
#225499472
C2×#135×C1272
#3268127