Finite Group Databasebeta

← #30650424

Subgroups — #30650424

Order 16256.

546 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
1C22×#14523236C211yesmaximal, characteristic, complemented
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550539C211yesmaximal, characteristic, complemented
4D4×C22⋊D127C211yesmaximal, characteristic
5#22550247C211yesmaximal, characteristic, complemented
6#22550107C211yesmaximal, characteristic, complemented
7#22550104C211yesmaximal, characteristic, complemented
8#22550079C211yesmaximal, characteristic, complemented
9#22550075C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22550017C211yesmaximal, characteristic, complemented
12#22550013C211yesmaximal, characteristic, complemented
13#22549967C211yesmaximal, characteristic, complemented
14(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
15#450×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic, complemented
17C2×Q8×D127C2211yescharacteristic, complemented
18C22×C22⋊D127—21no—
19C2×D4×D127C2211yescharacteristic
20C2×#14523237C2211yescharacteristic, complemented
21C2×#14523237C2211yescharacteristic, complemented
22C2×#14523236—24no—
23C2×#14523236C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic, complemented
26C2×#14523236C2211yescharacteristic, complemented
27C2×#14523235C2211yescharacteristic, complemented
28SD16×D127—24no—
29C22×C127⋊C8C2211yescharacteristic, complemented
30C2×C8×D127C2211yescharacteristic, complemented
31#16861208—21no—
32#16861207C2211yescharacteristic, complemented
33#16861205C2211yescharacteristic
34#16861189C2211yescharacteristic, complemented
35#16861187—21no—
36#16861186—21no—
37#16861183C2211yescharacteristic
38#16861181—21no—
39#16861180C2211yescharacteristic
40#16861178—21no—
41#16861177—21no—
42#16861174C2211yescharacteristic, complemented
43D4×Dic127C2211yescharacteristic
44C4×C22⋊D127C2211yescharacteristic
45(C22⋊D127)⋊C4—21no—
46#16861140C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861134C2211yescharacteristic, complemented
50#16861133C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C8×Dic127C2211yescharacteristic, complemented
54C22×D4×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56C2×SD16×C127C2211yescharacteristic, complemented
57C22⋊Q8×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic, complemented
60C22⋊C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C2×Dic254—21no—
63C2×Dic254—22no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—41no—
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—44no—
77C2×C22⋊D127D411yescharacteristic
78D4×D127—22no—
79D4×D127—41no—
80C22×Dic127—21no—
81C22×Dic127—21no—
82C2×C4×D127—21no—
83C2×C4×D127C2311yescharacteristic
84#14523237—22no—
85#14523237—41no—
86#14523236—22no—
87#14523236—22no—
88#14523236—22no—
89#14523236D414yescomplemented
90#14523235—22no—
91C2×C127⋊C8—22no—
92C2×C127⋊C8C2311yescharacteristic
93C2×C127⋊C8C2311yescharacteristic
94C8×D127—22no—
95C4⋊Dic127—21no—
96C4⋊Dic127C2311yescharacteristic
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4D411yescharacteristic
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4D411yescharacteristic
107C4×Dic127—21no—
108C4×Dic127C2311yescharacteristic
109C24×C127—21no—
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—22no—
112C2×D4×C127—21no—
113C2×D4×C127—41no—
114C2×D4×C127C2311yescharacteristic
115C2×D4×C127C2311yescharacteristic
116C22×C4×C127C2311yescharacteristic
117SD16×C127—22no—
118SD16×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127—21no—
122C4⋊C4×C127C2311yescharacteristic
123C22⋊C4×C127—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×D412yescomplemented
135D508—21no—
136D508—21no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C22⋊D127—81no—
142C22⋊D127SD1614yescomplemented
143C2×Dic127—21no—
144C2×Dic127—21no—
145C2×Dic127—21no—
146C2×Dic127—42no—
147C2×Dic127—42no—
148C2×Dic127C2×D411yescharacteristic
149C2×Dic127C2×D411yescharacteristic
150C2×Dic127C2×D411yescharacteristic
151C4×D127—21no—
152C4×D127—21no—
153C4×D127—42no—
154C127⋊C8—21no—
155C127⋊C8C2×D412yescomplemented
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—42no—
159C23×C127—42no—
160C23×C127—81no—
161C23×C127C2×D411yescharacteristic
162Q8×C127—21no—
163Q8×C127—21no—
164D4×C127—21no—
165D4×C127—21no—
166D4×C127—21no—
167D4×C127—42no—
168D4×C127C2×D412yes—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127C2×D411yescharacteristic
174C2×C4×C127C2411yescharacteristic
175C8×C127—21no—
176C8×C127—21no—
177C22×C127—21no—
178C22×C127—21no—
179C22×C127—22no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—84no—
186C22×C127C2×SD1612yescomplemented
187C22×C127C22×D411yescharacteristic
188C2×D127—21no—
189C2×D127—21no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—84no—
194C2×D127C2×SD1612yescomplemented
195C4×C127—21no—
196C4×C127—41no—
197C4×C127—41no—
198C4×C127C22×D411yescharacteristic
199C4×C127H(5,𝔽2)11yescharacteristic
200Dic127—21no—
201Dic127—22no—
202Dic127—21no—
203Dic127—41no—
204Dic127—41no—
205Dic127C2×SD1612yescomplemented
206C2×C127—22no—
207C2×C127—42no—
208C2×C127—81no—
209C2×C127D4211yescharacteristic
210C2×C127C22×SD1611yescharacteristic
211C2×C127#57711yescharacteristic
212D127—22no—
213D127—81no—
214D4×SD16—1271nomaximal, Sylow
215C127D4×SD1611yescharacteristic, complemented, Sylow
216C22×SD16—1271no—
217C22×SD16—1271no—
218D4×Q8—1271no—
219D42—1271no—
220#492—1271no—
221#465—1271no—
222#465—1271no—
223#461—1271no—
224#460—1271no—
225#450—1271no—
226#450D12711yescharacteristic, complemented
227#448—1271no—
228#448—1271no—
229C4×SD16—1271no—
230C8×D4—1271no—
231C22×Q8—1271no—
232C22×Q8—1271no—
233C22×D4—1271no—
234C22×D4—2541no—
235C22×D4C2×D12711yescharacteristic
236C2×SD16—1271no—
237C2×SD16—1271no—
238C2×SD16—1271no—
239C2×SD16—1271no—
240C2×SD16—1271no—
241C2×SD16—1271no—
242C2×SD16—1271no—
243C2×SD16—2544no—
244C2×SD16—2544no—
245C2×SD16C2×D12711yescharacteristic
246C2×SD16C2×D12711yescharacteristic, complemented
247C22×C8—1271no—
248C22×C8—1271no—
249C4⋊Q8—1271no—
250C4⋊Q8—2541no—
251#127—1271no—
252C22⋊Q8—1271no—
253C22⋊Q8—2541no—
254C22⋊Q8—2541no—
255C22⋊Q8C2×D12711yescharacteristic, complemented
256#121—1271no—
257#121—1271no—
258#121—2541no—
259#120—2541no—
260#120—2541no—
261C4×Q8—1271no—
262C4×D4—1271no—
263C4×D4—1271no—
264C4×D4—2541no—
265#106—1271no—
266C4⋊C8—1271no—
267#103—1271no—
268#103—1271no—
269#103—1271no—
270#102—1271no—
271#102C2×D12711yescharacteristic
272#102C2×D12711yescharacteristic, complemented
273C22⋊C8—1271no—
274C22⋊C8C2×D12711yescharacteristic, complemented
275C4×C8—1271no—
276C24—21no—
277C24—2541no—
278C2×Q8—1271no—
279C2×Q8—1271no—
280C2×Q8—5081no—
281C2×Q8—5081no—
282C2×Q8—2541no—
283C2×Q8—2542no—
284C2×Q8—2542no—
285C2×Q8—2541no—
286C2×Q8C22×D12711yescharacteristic
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—1271no—
291C2×D4—1271no—
292C2×D4—5084no—
293C2×D4—5081no—
294C2×D4—2541no—
295C2×D4—2541no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4—2541no—
299C2×D4—2542no—
300C2×D4C22×D12711yescharacteristic
301C2×D4C22×D12711yescharacteristic
302C22×C4—1271no—
303C22×C4—2541no—
304C22×C4—2541no—
305C22×C4—2541no—
306C22×C4C22×D12711yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—1274no—
310SD16—5081no—
311SD16—2542no—
312SD16—2542no—
313SD16—2542no—
314SD16—2542no—
315SD16—2542no—
316C2×C8—1271no—
317C2×C8—1271no—
318C2×C8—2542no—
319C2×C8—2542no—
320C2×C8C22×D12711yescharacteristic
321C2×C8C22×D12711yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—1271no—
324C4⋊C4—1271no—
325C4⋊C4—2541no—
326C4⋊C4—2541no—
327C4⋊C4—2541no—
328C4⋊C4—2541no—
329C4⋊C4C22×D12711yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—1271no—
332C22⋊C4—1271no—
333C22⋊C4—2541no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C22⋊C4—2541no—
337C42—1271no—
338C42—2541no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—1271no—
345C23—10161no—
346C23—5082no—
347C23—5082no—
348C23—2541no—
349C23D4×D12711yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—1272no—
353Q8—5082no—
354Q8—5082no—
355Q8—2541no—
356Q8—2541no—
357Q8—2541no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—1274no—
363D4—10161no—
364D4—5082no—
365D4—5082no—
366D4—5082no—
367D4—5082no—
368D4—2541no—
369D4—2541no—
370D4D4×D12712yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—1271no—
376C2×C4—1271no—
377C2×C4—1271no—
378C2×C4—5082no—
379C2×C4—5082no—
380C2×C4—5082no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4—2541no—
385C2×C4—2541no—
386C2×C4D4×D12711yescharacteristic
387C2×C4C23×D12711yescharacteristic
388C8—21no—
389C8—21no—
390C8—1272no—
391C8—2541no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—1272no—
402C22—10164no—
403C22—5081no—
404C22—5081no—
405C22—5081no—
406C22—2541no—
407C22—2541no—
408C22SD16×D12712yescomplemented
409C22C2×D4×D12711yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—1272no—
414C4—5081no—
415C4—5081no—
416C4—2541no—
417C4—2541no—
418C4—2542no—
419C4C2×D4×D12711yescharacteristic
420C4#1686125411yescharacteristic
421C2—22no—
422C2—42no—
423C2—81no—
424C2—10161no—
425C2—2542no—
426C2#2255050811yescharacteristic
427C2C2×SD16×D12711yescharacteristic
428C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radicalD4×C22⋊D127C2
hypercenter#450D127
hypocenterC127D4×SD16
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. C2×SD16×D127C2semidirect
3. #22550539C2semidirect
5. #22550247C2semidirect
6. #22550107C2semidirect
7. #22550104C2semidirect
8. #22550079C2semidirect
9. #22550075C2semidirect
10. #22550028C2semidirect
11. #22550017C2semidirect
12. #22550013C2semidirect
13. #22549967C2semidirect
14. (C22⋊D127)⋊C8C2semidirect
15. #450×C127C2semidirect
16. C22×Dic254C22semidirect
17. C2×Q8×D127C22semidirect
20. C2×#14523237C22semidirect
21. C2×#14523237C22semidirect
25. C2×#14523236C22semidirect
26. C2×#14523236C22semidirect
27. C2×#14523235C22semidirect
29. C22×C127⋊C8C22semidirect
30. C2×C8×D127C22semidirect
32. #16861207C22semidirect
34. #16861189C22semidirect
42. #16861174C22semidirect
46. #16861140C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861134C22semidirect
50. #16861133C22semidirect
51. #16861129C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C8×Dic127C22semidirect
56. C2×SD16×C127C22semidirect
57. C22⋊Q8×C127C22semidirect
59. #102×C127C22semidirect
60. C22⋊C8×C127C22semidirect
89. #14523236D4semidirect
134. Dic254C2×D4semidirect
142. C22⋊D127SD16semidirect
155. C127⋊C8C2×D4semidirect
186. C22×C127C2×SD16semidirect
194. C2×D127C2×SD16semidirect
205. Dic127C2×SD16semidirect
215. C127D4×SD16semidirect
226. #450D127semidirect
246. C2×SD16C2×D127semidirect
255. C22⋊Q8C2×D127semidirect
272. #102C2×D127semidirect
274. C22⋊C8C2×D127semidirect
408. C22SD16×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×SD16×D1272
#225505392
D4×C22⋊D1272
#225502472
#225501072
#225501042
#225500792
#225500752
#225500282
#225500172
#225500132
#225499672
(C22⋊D127)⋊C82
#450×C1272
D4×SD16127