Finite Group Databasebeta

← #30650259

Subgroups — #30650259

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×#16861236C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550528C211yesmaximal, characteristic, complemented
4#22550514C211yesmaximal, characteristic, complemented
5#22550269C211yesmaximal, characteristic, complemented
6#22550133C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550094C211yesmaximal, characteristic, complemented
9#22550079C211yesmaximal, characteristic, complemented
10#22550037C211yesmaximal, characteristic, complemented
11#22550028C211yesmaximal, characteristic, complemented
12#22550023C211yesmaximal, characteristic, complemented
13#22549985C211yesmaximal, characteristic, complemented
14#22549944C211yesmaximal, characteristic, complemented
15#452×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic, complemented
18C2×D4×D127C2211yescharacteristic
19#16861236—24no—
20#16861229—24no—
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523237C2211yescharacteristic
23C2×#14523236C2211yescharacteristic
24C2×#14523235C2211yescharacteristic
25C2×#14523235C2211yescharacteristic, complemented
26C2×D1016C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29C22×C127⋊C8C2211yescharacteristic, complemented
30#16861207C2211yescharacteristic, complemented
31#16861198C2211yescharacteristic
32#16861194—21no—
33#16861192—21no—
34#16861191—21no—
35#16861189C2211yescharacteristic
36#16861185C2211yescharacteristic
37#16861183C2211yescharacteristic
38#16861182—21no—
39#16861181—21no—
40#16861177—21no—
41D4×Dic127C2211yescharacteristic
42#16861171C2211yescharacteristic, complemented
43C4×C22⋊D127C2211yescharacteristic
44C4⋊C4×D127—21no—
45C2×(C2×D127)⋊C4—21no—
46#16861140C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861133C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53#16861119C2211yescharacteristic, complemented
54C2×#41×C127C2211yescharacteristic
55C2×Q16×C127C2211yescharacteristic, complemented
56C2×SD16×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic, complemented
58#103×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64#14523248—22no—
65#14523248—41no—
66C2×Dic254C2311yescharacteristic
67C2×D508—21no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
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×D127—21no—
79C2×C4×D127—21no—
80C2×C4×D127C2311yescharacteristic
81#14523238—22no—
82#14523237—41no—
83#14523236—22no—
84#14523235—22no—
85#14523235—22no—
86D1016—22no—
87#14523233D414yescomplemented
88#14523231—22no—
89C2×C127⋊C8—22no—
90C2×C127⋊C8C2311yescharacteristic
91C2×C127⋊C8C2311yescharacteristic
92C4⋊Dic127—21no—
93C4⋊Dic127C2311yescharacteristic
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4D411yescharacteristic
98C22⋊Dic127—21no—
99C22⋊Dic127D411yescharacteristic
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4D411yescharacteristic
107C4×Dic127C2311yescharacteristic
108#41×C127—22no—
109#41×C127—41no—
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116Q16×C127—41no—
117SD16×C127—22no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127—21no—
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128C22×D127C2×D411yescharacteristic
129Dic254—21no—
130Dic254—21no—
131D508—21no—
132D508—21no—
133D508—21no—
134D508—42no—
135D508C2×D412yes—
136C22⋊D127—41no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C22⋊D127—81no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127—81no—
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C2×Dic127C2×D411yescharacteristic
146C4×D127—21no—
147C4×D127—21no—
148C4×D127—42no—
149C4×D127—42no—
150C4×D127—42no—
151C127⋊C8—21no—
152C127⋊C8C2×D412yes—
153C23×C127—21no—
154C23×C127C2×D411yescharacteristic
155Q8×C127—21no—
156Q8×C127—21no—
157Q8×C127—21no—
158Q8×C127—21no—
159D4×C127—21no—
160D4×C127—42no—
161D4×C127C2×D412yes—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
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×C127—21no—
172C22×C127—21no—
173C22×C127—22no—
174C22×C127—21no—
175C22×C127—21no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127C22×D411yescharacteristic
179C2×D127—21no—
180C2×D127—22no—
181C2×D127—21no—
182C2×D127—21no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C2×D127—84no—
189C4×C127—21no—
190C4×C127—41no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127C22×D411yescharacteristic
194C4×C127H(5,𝔽2)11yescharacteristic
195Dic127—21no—
196Dic127—21no—
197Dic127—21no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—42no—
201C2×C127—41no—
202C2×C127D4211yescharacteristic
203C2×C127#57511yescharacteristic
204C2×C127#57711yescharacteristic
205D127—41no—
206D127—42no—
207D127—81no—
208#3267—1271nomaximal, Sylow
209C127#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1271no—
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#452D12711yescharacteristic, complemented
221#450—1271no—
222#449—1271no—
223#442—1271no—
224#435—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C2×#41C2×D12711yescharacteristic
228C22×D4—1271no—
229AGL(1,ℤ/8)—2544no—
230#135—2544no—
231C2×Q16—1271no—
232C2×Q16C2×D12711yescharacteristic, complemented
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16—1271no—
237C2×SD16C2×D12711yescharacteristic
238C2×D8—1271no—
239C2×D8—1271no—
240C2×M16—1271no—
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⋊Q8C2×D12711yescharacteristic, complemented
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#103C2×D12711yescharacteristic, complemented
264#102—1271no—
265#102—1271no—
266#102C2×D12711yescharacteristic
267C22⋊C8—1271no—
268C22⋊C8C2×D12711yescharacteristic, complemented
269#97—1271no—
270C24—2541no—
271#41—22no—
272#41—41no—
273#41—5081no—
274#41—5081no—
275#41—2542no—
276#41—2542no—
277C2×Q8—1271no—
278C2×Q8C22×D12711yescharacteristic
279C2×Q8C22×D12711yescharacteristic
280C2×D4—21no—
281C2×D4—1271no—
282C2×D4—1271no—
283C2×D4—1271no—
284C2×D4—5081no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2542no—
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—41no—
299Q16—2542no—
300SD16—22no—
301SD16—5081no—
302SD16—2542no—
303SD16—2542no—
304SD16—2542no—
305D8—1274no—
306D8—2542no—
307M16—2542no—
308C2×C8—1271no—
309C2×C8—1271no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—1271no—
315C4⋊C4—1271no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C4⋊C4C22×D12711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—1271no—
323C22⋊C4—1271no—
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—1271no—
334C23—10161no—
335C23—5082no—
336C23—5082no—
337C23—2541no—
338C23—2541no—
339C23D4×D12711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2541no—
345Q8—2541no—
346D4—21no—
347D4—42no—
348D4—1272no—
349D4—10161no—
350D4—5082no—
351D4—5082no—
352D4—5082no—
353D4—5081no—
354D4—2541no—
355D4—2541no—
356D4—2541no—
357D4D4×D12712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—1271no—
365C2×C4—1271no—
366C2×C4—1271no—
367C2×C4—10161no—
368C2×C4—5082no—
369C2×C4—5082no—
370C2×C4—5082no—
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—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—10164no—
388C22—5081no—
389C22—5081no—
390C22—5081no—
391C22—5081no—
392C22—5081no—
393C22—2541no—
394C22—2541no—
395C22—2542no—
396C22—2541no—
397C22C2×D4×D12711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—5081no—
403C4—5081no—
404C4—2541no—
405C4—2541no—
406C4—2541no—
407C4C2×D4×D12711yescharacteristic
408C4#1686125411yescharacteristic
409C2—41no—
410C2—42no—
411C2—10161no—
412C2—5081no—
413C2—5082no—
414C2#2255050811yescharacteristic
415C2#2255067311yescharacteristic
416C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#452×C127C2
socleC22×C127C22×D4
hypercenter#452D127
hypocenterC127#3267
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861229C2semidirect
3. #22550528C2semidirect
4. #22550514C2semidirect
5. #22550269C2semidirect
6. #22550133C2semidirect
7. #22550107C2semidirect
8. #22550094C2semidirect
9. #22550079C2semidirect
10. #22550037C2semidirect
11. #22550028C2semidirect
12. #22550023C2semidirect
13. #22549985C2semidirect
14. #22549944C2semidirect
15. #452×C127C2semidirect
17. C2×#14523248C22semidirect
21. C2×#14523238C22semidirect
25. C2×#14523235C22semidirect
29. C22×C127⋊C8C22semidirect
30. #16861207C22semidirect
42. #16861171C22semidirect
46. #16861140C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861133C22semidirect
50. #16861131C22semidirect
53. #16861119C22semidirect
55. C2×Q16×C127C22semidirect
57. C22⋊Q8×C127C22semidirect
58. #103×C127C22semidirect
60. C22⋊C8×C127C22semidirect
87. #14523233D4semidirect
209. C127#3267semidirect
220. #452D127semidirect
232. C2×Q16C2×D127semidirect
249. C22⋊Q8C2×D127semidirect
263. #103C2×D127semidirect
268. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×#168612292
#225505282
#225505142
#225502692
#225501332
#225501072
#225500942
#225500792
#225500372
#225500282
#225500232
#225499852
#225499442
#452×C1272
#3267127