Finite Group Databasebeta

← #30650440

Subgroups — #30650440

Order 16256.

504 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×#16861229C211yesmaximal, characteristic, complemented
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550533C211yesmaximal, characteristic, complemented
4#22550523C211yesmaximal, characteristic, complemented
5#22550259C211yesmaximal, characteristic, complemented
6#22550104C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550096C211yesmaximal, characteristic, complemented
9#22550090C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550033C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549961C211yesmaximal, characteristic, complemented
14(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861229—24no—
20C2×Dic508C2211yescharacteristic, complemented
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523237C2211yescharacteristic
23C2×#14523236C2211yescharacteristic
24C2×#14523236C2211yescharacteristic, complemented
25C2×#14523235C2211yescharacteristic
26SD16×D127—24no—
27C2×#14523233C2211yescharacteristic
28C22×C127⋊C8C2211yescharacteristic, complemented
29C2×C8×D127C2211yescharacteristic
30#16861196—21no—
31#16861196C2211yescharacteristic
32#16861196C2211yescharacteristic, complemented
33#16861191—21no—
34#16861189C2211yescharacteristic
35#16861188—21no—
36#16861188C2211yescharacteristic, complemented
37#16861180—21no—
38#16861180—21no—
39#16861180C2211yescharacteristic
40#16861177—21no—
41Q8×Dic127C2211yescharacteristic, complemented
42#16861171C2211yescharacteristic
43C4×C22⋊D127—21no—
44C4×C22⋊D127C2211yescharacteristic
45C22⋊C4×D127—21no—
46#16861140C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861132C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic
51#16861130C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53C8×Dic127C2211yescharacteristic, complemented
54C2×#41×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic
56C2×D8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60C22⋊C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523248—22no—
63#14523248—41no—
64C2×Dic254—21no—
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67Q8×D127—22no—
68Q8×D127—41no—
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127D411yescharacteristic
74D4×D127—22no—
75D4×D127—41no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79C2×C4×D127C2311yescharacteristic
80Dic508—41no—
81#14523238—22no—
82#14523237—22no—
83#14523236—22no—
84#14523236—22no—
85#14523235D414yescomplemented
86#14523233—22no—
87C2×C127⋊C8—22no—
88C2×C127⋊C8C2311yescharacteristic
89C2×C127⋊C8C2311yescharacteristic
90C8×D127—22no—
91C4⋊Dic127—21no—
92C4⋊Dic127—21no—
93C4⋊Dic127C2311yescharacteristic
94Dic127⋊C4—21no—
95Dic127⋊C4D411yescharacteristic
96C22⋊Dic127—21no—
97C22⋊Dic127—21no—
98C22⋊Dic127—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127D411yescharacteristic
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4D411yescharacteristic
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108#41×C127—22no—
109#41×C127—41no—
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127—21no—
116C22×C4×C127C2311yescharacteristic
117SD16×C127—22no—
118D8×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—42no—
126C22×D127—81no—
127C22×D127C2×D411yescharacteristic
128Dic254—21no—
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—42no—
133D508—21no—
134D508C2×D412yes—
135C22⋊D127—22no—
136C22⋊D127—42no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127C2×D411yescharacteristic
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C4×D127—21no—
146C4×D127—21no—
147C4×D127—42no—
148C4×D127—42no—
149C4×D127—81no—
150C127⋊C8—21no—
151C127⋊C8C2×D412yes—
152C23×C127—21no—
153C23×C127—21no—
154C23×C127C2×D411yescharacteristic
155Q8×C127—21no—
156Q8×C127C2×D412yes—
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—42no—
162D4×C127—81no—
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—21no—
174C22×C127—21no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—22no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—84no—
188C2×D127#13512yescomplemented
189C4×C127—21no—
190C4×C127—22no—
191C4×C127—41no—
192C4×C127C22×D411yescharacteristic
193C4×C127H(5,𝔽2)11yescharacteristic
194Dic127—21no—
195Dic127—21no—
196Dic127—21no—
197Dic127—41no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—41no—
201C2×C127—81no—
202C2×C127—81no—
203C2×C127D4211yescharacteristic
204C2×C127C2×#13511yescharacteristic
205C2×C127#57711yescharacteristic
206D127—22no—
207D127—42no—
208#3282—1271nomaximal, Sylow
209C127#328211yescharacteristic, complemented, Sylow
210C2×#135—1271no—
211C22×SD16—1271no—
212#548—1271no—
213#546—1271no—
214#495—1271no—
215#465—1271no—
216#465—1271no—
217#464—1271no—
218#463—1271no—
219#452—1271no—
220#451—1271no—
221#449D12711yescharacteristic, complemented
222#447—1271no—
223C4×SD16—1271no—
224C8×D4—1271no—
225C2×#41—1271no—
226C2×#41C2×D12711yescharacteristic, complemented
227C22×Q8—1271no—
228C22×D4—1271no—
229#135—2544no—
230C2×Q16—1271no—
231C2×Q16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16—2544no—
237C2×SD16C2×D12711yescharacteristic
238C2×D8—1271no—
239C2×D8C2×D12711yescharacteristic
240C22×C8—1271no—
241C22×C8—1271no—
242#124—1271no—
243#124—1271no—
244#124—2541no—
245#123—2541no—
246C22⋊Q8—1271no—
247C22⋊Q8—1271no—
248C22⋊Q8—2541no—
249#121—1271no—
250#121—2541no—
251#121—2541no—
252#121C2×D12711yescharacteristic
253#120—2541no—
254C4×Q8—1271no—
255C4×D4—1271no—
256C4×D4—1271no—
257C4×D4—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, complemented
267C22⋊C8—1271no—
268C22⋊C8C2×D12711yescharacteristic, complemented
269C4×C8—1271no—
270C24—2541no—
271#41—22no—
272#41—41no—
273#41—5081no—
274#41—2542no—
275C2×Q8—1271no—
276C2×Q8—1271no—
277C2×Q8—5081no—
278C2×Q8—2541no—
279C2×Q8—2542no—
280C2×Q8C22×D12711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—1271no—
284C2×D4—1271no—
285C2×D4—5081no—
286C2×D4—2541no—
287C2×D4—2542no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4C22×D12711yescharacteristic
291C2×D4C22×D12711yescharacteristic
292C22×C4—21no—
293C22×C4—1271no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4C22×D12711yescharacteristic
298Q16—5081no—
299Q16—2542no—
300SD16—22no—
301SD16—1274no—
302SD16—2542no—
303SD16—2542no—
304SD16—2542no—
305D8—41no—
306D8—2542no—
307C2×C8—1271no—
308C2×C8—1271no—
309C2×C8—2542no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—1271no—
314C4⋊C4—1271no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4C22×D12711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—1271no—
321C22⋊C4—1271no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C42—1271no—
331C42—2541no—
332C23—21no—
333C23—21no—
334C23—1271no—
335C23—10161no—
336C23—5082no—
337C23—5082no—
338C23—2541no—
339C23D4×D12711yescharacteristic
340Q8—21no—
341Q8—5082no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345Q8—2541no—
346Q8D4×D12712yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—81no—
353D4—1272no—
354D4—5082no—
355D4—5082no—
356D4—5082no—
357D4—2542no—
358D4—2541no—
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—2541no—
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—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—1272no—
391C22—10164no—
392C22—5081no—
393C22—5081no—
394C22—2542no—
395C22—2541no—
396C22—2541no—
397C22C2×D4×D12711yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—5081no—
402C4—5081no—
403C4—5081no—
404C4—2541no—
405C4—2541no—
406C4—2541no—
407C4C2×D4×D12711yescharacteristic
408C4#1686125411yescharacteristic
409C2—41no—
410C2—81no—
411C2—81no—
412C2—5082no—
413C2—2542no—
414C2#2255050811yescharacteristic
415C2#135×D12711yescharacteristic
416C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3282
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. C2×SD16×D127C2semidirect
3. #22550533C2semidirect
4. #22550523C2semidirect
5. #22550259C2semidirect
6. #22550104C2semidirect
7. #22550103C2semidirect
8. #22550096C2semidirect
9. #22550090C2semidirect
10. #22550040C2semidirect
11. #22550033C2semidirect
12. #22550008C2semidirect
13. #22549961C2semidirect
14. (C22⋊D127)⋊C8C2semidirect
15. #449×C127C2semidirect
20. C2×Dic508C22semidirect
21. C2×#14523238C22semidirect
24. C2×#14523236C22semidirect
28. C22×C127⋊C8C22semidirect
32. #16861196C22semidirect
36. #16861188C22semidirect
41. Q8×Dic127C22semidirect
46. #16861140C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861132C22semidirect
53. C8×Dic127C22semidirect
54. C2×#41×C127C22semidirect
58. #103×C127C22semidirect
59. #102×C127C22semidirect
60. C22⋊C8×C127C22semidirect
85. #14523235D4semidirect
188. C2×D127#135semidirect
209. C127#3282semidirect
221. #449D127semidirect
226. C2×#41C2×D127semidirect
263. #103C2×D127semidirect
266. #102C2×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×#168612292
C2×SD16×D1272
#225505332
#225505232
#225502592
#225501042
#225501032
#225500962
#225500902
#225500402
#225500332
#225500082
#225499612
(C22⋊D127)⋊C82
#449×C1272
#3282127