Finite Group Databasebeta

← #30650628

Subgroups — #30650628

Order 16256.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550574C211yesmaximal, characteristic, complemented
2#22550537C211yesmaximal, characteristic, complemented
3#22550232C211yesmaximal, characteristic, complemented
4#22550216C211yesmaximal, characteristic, complemented
5#22550194C211yesmaximal, characteristic, complemented
6#22550185C211yesmaximal, characteristic, complemented
7#22550112C211yesmaximal, characteristic, complemented
8#22550108C211yesmaximal, characteristic, complemented
9#22550094C211yesmaximal, characteristic, complemented
10#22550024C211yesmaximal, characteristic, complemented
11#22550015C211yesmaximal, characteristic, complemented
12#22549961C211yesmaximal, characteristic, complemented
13C8×C22⋊D127C211yesmaximal, characteristic, complemented
14#22549871C211yesmaximal, characteristic, complemented
15C2×#103×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×#14523235C2211yescharacteristic
19C2×D1016C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861204—21no—
22#16861200C2211yescharacteristic
23#16861198—21no—
24#16861198C2211yescharacteristic
25#16861194—21no—
26#16861185—21no—
27#16861185C2211yescharacteristic
28#16861184C2211yescharacteristic
29#16861183—21no—
30#16861183—21no—
31#16861183C2211yescharacteristic
32#16861183C2211yescharacteristic
33Q8×Dic127C2211yescharacteristic
34#16861171C2211yescharacteristic
35C4×C22⋊D127—21no—
36C4×C22⋊D127—21no—
37C4×C22⋊D127C2211yescharacteristic
38#16861167C2211yescharacteristic
39#16861143C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861140C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C22×Q8×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C2×C4⋊C4×C127C2211yescharacteristic
55#103×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—41no—
61C2×D508—21no—
62C2×D508—21no—
63C2×D508C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127D411yescharacteristic
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127C2311yescharacteristic
71#14523235—22no—
72#14523235—41no—
73D1016—41no—
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C8×D127—41no—
77C4⋊Dic127—21no—
78C4⋊Dic127—21no—
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4D411yescharacteristic
84C22⋊Dic127D411yescharacteristic
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4D411yescharacteristic
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97C2×Q8×C127—21no—
98C2×Q8×C127—22no—
99C2×Q8×C127—41no—
100C2×Q8×C127C2311yescharacteristic
101C2×Q8×C127C2311yescharacteristic
102C22×C4×C127—21no—
103C22×C4×C127—21no—
104C22×C4×C127C2311yescharacteristic
105C2×C8×C127—41no—
106C2×C8×C127C2311yescharacteristic
107C2×C8×C127C2311yescharacteristic
108C4⋊C4×C127—21no—
109C4⋊C4×C127C2311yescharacteristic
110C4⋊C4×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127—21no—
113C22×D127C2×D411yescharacteristic
114D508—21no—
115D508—21no—
116D508—21no—
117D508—21no—
118D508—42no—
119D508—81no—
120C22⋊D127—22no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C4×D127—21no—
129C4×D127—21no—
130C4×D127—42no—
131C4×D127—81no—
132C127⋊C8—21no—
133C127⋊C8—21no—
134C23×C127C2×D411yescharacteristic
135Q8×C127—21no—
136Q8×C127—21no—
137Q8×C127—21no—
138Q8×C127—42no—
139Q8×C127#4112yes—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—42no—
147C2×C4×C127—81no—
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127#13512yes—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—21no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—22no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127C2×#4111yescharacteristic
171C4×C127H(5,𝔽2)11yescharacteristic
172Dic127—21no—
173Dic127—21no—
174Dic127—21no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—22no—
178C2×C127#54611yescharacteristic
179C2×C127C2×#13511yescharacteristic
180C2×C127#57711yescharacteristic
181D127—41no—
182D127—81no—
183D127—81no—
184#3301—1271nomaximal, Sylow
185C127#330111yescharacteristic, complemented, Sylow
186#553—1271no—
187#548—1271no—
188#488—1271no—
189#485—1271no—
190#481—1271no—
191#479—1271no—
192#466—1271no—
193#465—1271no—
194#463—1271no—
195#449—1271no—
196#448—1271no—
197C4×SD16—1271no—
198C8×D4—1271no—
199#416—1271no—
200C2×#103D12711yescharacteristic, complemented
201C2×#41—1271no—
202C22×Q8C2×D12711yescharacteristic
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C22×C8—1271no—
207C22×C8C2×D12711yescharacteristic
208#126—2541no—
209#125—1271no—
210#124—1271no—
211#124—2541no—
212#123—2541no—
213C22⋊Q8—1271no—
214C22⋊Q8—2541no—
215#121—1271no—
216#121—1271no—
217#121—1271no—
218#121—2541no—
219#121—2541no—
220C4×Q8—1271no—
221C4×D4—1271no—
222C4×D4—1271no—
223C4×D4—2541no—
224C4×D4—2541no—
225#117—1271no—
226C2×C4⋊C4C2×D12711yescharacteristic
227#107—1271no—
228#106—1271no—
229#106—1271no—
230C4⋊C8—1271no—
231#103—1271no—
232#103—1271no—
233#103C2×D12711yescharacteristic
234#103C2×D12711yescharacteristic
235#103C2×D12711yescharacteristic
236#103C2×D12711yescharacteristic
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241C22⋊C8—1271no—
242C22⋊C8—1271no—
243C4×C8—1271no—
244#41—5081no—
245#41—2542no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D12711yescharacteristic
250C2×Q8C22×D12711yescharacteristic
251C2×D4—1271no—
252C2×D4—1271no—
253C2×D4—1271no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4—2541no—
257C2×D4—2541no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1271no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4C22×D12711yescharacteristic
264SD16—5081no—
265SD16—2542no—
266D8—5081no—
267C2×C8—41no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—5081no—
271C2×C8C22×D12711yescharacteristic
272C2×C8C22×D12711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4C22×D12711yescharacteristic
282C4⋊C4C22×D12711yescharacteristic
283C22⋊C4—1271no—
284C22⋊C4—1271no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C42—1271no—
294C42—2541no—
295C42—2541no—
296C23—1271no—
297C23—2541no—
298C23—2541no—
299C23D4×D12711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1452324812yes—
305D4—10161no—
306D4—5082no—
307D4—5082no—
308D4—5082no—
309D4—2542no—
310D4—2541no—
311D4—2541no—
312D4—2541no—
313D4—2541no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—1271no—
323C2×C4—1271no—
324C2×C4—1271no—
325C2×C4—10161no—
326C2×C4—5082no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4D4×D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2541no—
336C8—2541no—
337C22—21no—
338C22—21no—
339C22—5081no—
340C22—5081no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—2541no—
346C22—2541no—
347C22—2541no—
348C22#1686123212yes—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—5081no—
356C4—5081no—
357C4—2541no—
358C4—2541no—
359C4—2541no—
360C4C2×#1452324811yescharacteristic
361C4#1686125411yescharacteristic
362C2—22no—
363C2—10161no—
364C2—10161no—
365C2—5081no—
366C2#2255051611yescharacteristic
367C2C2×#1686123211yescharacteristic
368C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC22×C127C22×D4
hypercenterC2×#103D127
hypocenterC127#3301
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #22550574C2semidirect
2. #22550537C2semidirect
3. #22550232C2semidirect
4. #22550216C2semidirect
5. #22550194C2semidirect
6. #22550185C2semidirect
7. #22550112C2semidirect
8. #22550108C2semidirect
9. #22550094C2semidirect
10. #22550024C2semidirect
11. #22550015C2semidirect
12. #22549961C2semidirect
13. C8×C22⋊D127C2semidirect
14. #22549871C2semidirect
15. C2×#103×C127C2semidirect
185. C127#3301semidirect
200. C2×#103D127semidirect

Maximal subgroups

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

subgroupindex
#225505742
#225505372
#225502322
#225502162
#225501942
#225501852
#225501122
#225501082
#225500942
#225500242
#225500152
#225499612
C8×C22⋊D1272
#225498712
C2×#103×C1272
#3301127