Finite Group Databasebeta

← #30650260

Subgroups — #30650260

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×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861231C211yesmaximal, characteristic, complemented
3#22550531C211yesmaximal, characteristic, complemented
4#22550522C211yesmaximal, characteristic, complemented
5#22550270C211yesmaximal, characteristic, complemented
6#22550134C211yesmaximal, characteristic, complemented
7#22550105C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550082C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550040C211yesmaximal, characteristic, complemented
12#22550029C211yesmaximal, characteristic, complemented
13#22550023C211yesmaximal, characteristic, complemented
14#22549989C211yesmaximal, characteristic, complemented
15#435×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×#14523248C2211yescharacteristic, complemented
19C22×D508C2211yescharacteristic
20#16861237—24no—
21#16861231—24no—
22C2×Dic508C2211yescharacteristic, complemented
23C2×Dic508C2211yescharacteristic, complemented
24C2×#14523237C2211yescharacteristic
25C2×#14523237C2211yescharacteristic
26C2×#14523237C2211yescharacteristic
27C2×#14523237C2211yescharacteristic
28C2×#14523237C2211yescharacteristic, complemented
29C2×D1016C2211yescharacteristic
30C2×D1016C2211yescharacteristic
31#16861209C2211yescharacteristic, complemented
32#16861197C2211yescharacteristic
33#16861194—21no—
34#16861191—21no—
35#16861191—21no—
36#16861188C2211yescharacteristic
37#16861188C2211yescharacteristic
38#16861188C2211yescharacteristic, complemented
39#16861183C2211yescharacteristic
40#16861182—21no—
41#16861180—21no—
42#16861177—21no—
43C4×D508C2211yescharacteristic
44C4×D508C2211yescharacteristic, complemented
45C4⋊C4×D127—21no—
46C22⋊C4×D127—21no—
47#16861141C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861134C2211yescharacteristic, complemented
50#16861134C2211yescharacteristic, complemented
51#16861131C2211yescharacteristic
52#16861131C2211yescharacteristic
53#16861131C2211yescharacteristic, complemented
54C2×M16×C127C2211yescharacteristic
55C22×C8×C127C2211yescharacteristic, complemented
56C4×D4×C127C2211yescharacteristic
57C4⋊C8×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic
59C22⋊C8×C127C2211yescharacteristic, complemented
60#97×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523248—22no—
63#14523248—22no—
64#14523248—22no—
65#14523248—41no—
66#14523248—41no—
67#14523248—41no—
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70C2×Dic254C2311yescharacteristic
71C2×D508—22no—
72C2×D508—21no—
73C2×D508—41no—
74C2×D508C2311yescharacteristic
75C2×D508C2311yescharacteristic
76C2×D508C2311yescharacteristic
77C2×C22⋊D127—21no—
78C2×C22⋊D127—21no—
79C2×C22⋊D127—21no—
80C2×C22⋊D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85C2×C4×D127—21no—
86Dic508—22no—
87Dic508—41no—
88#14523237—22no—
89#14523237—22no—
90#14523237—22no—
91#14523237—22no—
92#14523237—41no—
93D1016—22no—
94D1016D414yescomplemented
95C4⋊Dic127—21no—
96C4⋊Dic127—21no—
97C4⋊Dic127C2311yescharacteristic
98C4⋊Dic127C2311yescharacteristic
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101Dic127⋊C4—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127—21no—
104C22⋊Dic127—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109(C2×D127)⋊C4—21no—
110C2×D4×C127D411yescharacteristic
111C22×C4×C127C2311yescharacteristic
112C22×C4×C127C2311yescharacteristic
113M16×C127—22no—
114C2×C8×C127—22no—
115C2×C8×C127C2311yescharacteristic
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127D411yescharacteristic
120C22⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—81no—
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—21no—
133Dic254—21no—
134Dic254—21no—
135D508—21no—
136D508—21no—
137D508—21no—
138D508—21no—
139D508—42no—
140D508C2×D412yes—
141D508C2×D412yes—
142C22⋊D127—42no—
143C22⋊D127—42no—
144C22⋊D127—42no—
145C22⋊D127—81no—
146C2×Dic127—21no—
147C2×Dic127—21no—
148C2×Dic127—21no—
149C2×Dic127—21no—
150C2×Dic127—21no—
151C4×D127—42no—
152C4×D127—42no—
153C4×D127—42no—
154C4×D127—42no—
155C4×D127—81no—
156C23×C127C2×D411yescharacteristic
157C23×C127C2×D411yescharacteristic
158D4×C127—41no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2×D411yescharacteristic
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2411yescharacteristic
168C8×C127—21no—
169C8×C127—21no—
170C8×C127—21no—
171C8×C127C2×D412yes—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—21no—
176C22×C127—21no—
177C22×C127—21no—
178C22×C127C22×D411yescharacteristic
179C2×D127—22no—
180C2×D127—22no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C2×D127—84no—
189C4×C127—21no—
190C4×C127—21no—
191C4×C127—21no—
192C4×C127—21no—
193C4×C127C22×D411yescharacteristic
194C4×C127H(5,𝔽2)11yescharacteristic
195Dic127—41no—
196Dic127—41no—
197Dic127—41no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—41no—
201C2×C127—41no—
202C2×C127D4211yescharacteristic
203C2×C127#57511yescharacteristic
204C2×C127#57711yescharacteristic
205D127—42no—
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#452—1271no—
221#450—1271no—
222#449—1271no—
223#442—1271no—
224#435D12711yescharacteristic, complemented
225C2×#41—1271no—
226C2×#41—1271no—
227C2×#41—1271no—
228C22×D4—1271no—
229AGL(1,ℤ/8)—2544no—
230#135—2544no—
231C2×Q16—1271no—
232C2×Q16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16—1271no—
237C2×SD16—1271no—
238C2×D8—1271no—
239C2×D8—1271no—
240C2×M16C2×D12711yescharacteristic
241C22×C8C2×D12711yescharacteristic, complemented
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×D4C2×D12711yescharacteristic
257C2×C4⋊C4—2541no—
258C2×C22⋊C4—2541no—
259#106—1271no—
260C4⋊C8C2×D12711yescharacteristic, complemented
261#103—1271no—
262#103—1271no—
263#103—1271no—
264#102—1271no—
265#102—1271no—
266#102—1271no—
267C22⋊C8C2×D12711yescharacteristic
268C22⋊C8C2×D12711yescharacteristic, complemented
269#97C2×D12711yescharacteristic, complemented
270C24—2541no—
271#41—5081no—
272#41—5081no—
273#41—5081no—
274#41—2542no—
275#41—2542no—
276#41—2542no—
277C2×Q8—1271no—
278C2×Q8—1271no—
279C2×Q8—1271no—
280C2×D4—1271no—
281C2×D4—1271no—
282C2×D4—1271no—
283C2×D4—5081no—
284C2×D4—2542no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4D50811yescharacteristic
291C22×C4—2541no—
292C22×C4—2541no—
293C22×C4—2541no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4C22×D12711yescharacteristic
297C22×C4C22×D12711yescharacteristic
298Q16—5081no—
299Q16—2542no—
300SD16—5081no—
301SD16—2542no—
302SD16—2542no—
303SD16—2542no—
304SD16—2542no—
305D8—1274no—
306D8—2542no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D12711yescharacteristic
310C2×C8C22×D12711yescharacteristic
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—1271no—
314C4⋊C4—1271no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C4⋊C4D50811yescharacteristic
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4D50811yescharacteristic
330C22⋊C4D50811yescharacteristic
331C42C22×D12711yescharacteristic
332C23—10161no—
333C23—5082no—
334C23—5082no—
335C23—2541no—
336C23—2541no—
337C23—2541no—
338C23C2×D50811yescharacteristic
339C23C2×D50811yescharacteristic
340Q8—2541no—
341Q8—2541no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345Q8—2541no—
346D4—41no—
347D4—1272no—
348D4—1272no—
349D4—10161no—
350D4—5082no—
351D4—5082no—
352D4—5082no—
353D4—5082no—
354D4—2541no—
355D4—2541no—
356D4—2541no—
357D4—2541no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—10161no—
363C2×C4—5082no—
364C2×C4—5082no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—2541no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4C2×D50811yescharacteristic
373C2×C4C2×D50811yescharacteristic
374C2×C4C2×D50811yescharacteristic
375C2×C4C2×D50811yescharacteristic
376C2×C4C23×D12711yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D12712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—10164no—
388C22—5081no—
389C22—5081no—
390C22—5081no—
391C22—5081no—
392C22—5081no—
393C22—5081no—
394C22—5081no—
395C22—2542no—
396C22—2542no—
397C22C22×D50811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—5081no—
403C4—5081no—
404C4—5081no—
405C4—5081no—
406C4—5081no—
407C4C2×D4×D12711yescharacteristic
408C4#1686125511yescharacteristic
409C2—41no—
410C2—41no—
411C2—10161no—
412C2—5082no—
413C2—5082no—
414C2D4×D50811yescharacteristic
415C2#2255067611yescharacteristic
416C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#435×C127C2
socleC22×C127C22×D4
hypercenter#435D127
hypocenterC127#3267
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C2×#16861231C2semidirect
3. #22550531C2semidirect
4. #22550522C2semidirect
5. #22550270C2semidirect
6. #22550134C2semidirect
7. #22550105C2semidirect
8. #22550092C2semidirect
9. #22550082C2semidirect
10. #22550040C2semidirect
11. #22550040C2semidirect
12. #22550029C2semidirect
13. #22550023C2semidirect
14. #22549989C2semidirect
15. #435×C127C2semidirect
18. C2×#14523248C22semidirect
22. C2×Dic508C22semidirect
23. C2×Dic508C22semidirect
28. C2×#14523237C22semidirect
31. #16861209C22semidirect
38. #16861188C22semidirect
44. C4×D508C22semidirect
47. #16861141C22semidirect
48. #16861134C22semidirect
49. #16861134C22semidirect
50. #16861134C22semidirect
53. #16861131C22semidirect
55. C22×C8×C127C22semidirect
57. C4⋊C8×C127C22semidirect
59. C22⋊C8×C127C22semidirect
60. #97×C127C22semidirect
94. D1016D4semidirect
209. C127#3267semidirect
224. #435D127semidirect
241. C22×C8C2×D127semidirect
260. C4⋊C8C2×D127semidirect
268. C22⋊C8C2×D127semidirect
269. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612312
#225505312
#225505222
#225502702
#225501342
#225501052
#225500922
#225500822
#225500402
#225500292
#225500232
#225499892
#435×C1272
#3267127