Finite Group Databasebeta

← #30650708

Subgroups — #30650708

Order 16256.

394 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#22550572C211yesmaximal, characteristic, complemented
2#22550531C211yesmaximal, characteristic, complemented
3#22550245C211yesmaximal, characteristic, complemented
4#22550218C211yesmaximal, characteristic, complemented
5#22550207C211yesmaximal, characteristic, complemented
6#22550177C211yesmaximal, characteristic, complemented
7#22550122C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550082C211yesmaximal, characteristic, complemented
10#22550023C211yesmaximal, characteristic, complemented
11#22550023C211yesmaximal, characteristic, complemented
12#22549989C211yesmaximal, characteristic, complemented
13#22549878C211yesmaximal, characteristic, complemented
14#22549873C211yesmaximal, characteristic, complemented
15#435×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523237C2211yescharacteristic
20C2×D1016C2211yescharacteristic
21#16861209C2211yescharacteristic
22#16861204—21no—
23#16861201C2211yescharacteristic
24#16861191—21no—
25#16861191—21no—
26#16861191—21no—
27#16861188C2211yescharacteristic
28#16861188C2211yescharacteristic
29#16861183C2211yescharacteristic
30#16861183C2211yescharacteristic
31#16861181—21no—
32#16861180—21no—
33C4×D508C2211yescharacteristic
34C4×D508C2211yescharacteristic
35(C22⋊D127)⋊C4—21no—
36#16861168C2211yescharacteristic
37C2×C4⋊Dic127C2211yescharacteristic
38C4⋊C4×D127—21no—
39#16861143C2211yescharacteristic
40#16861141C2211yescharacteristic
41#16861141C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861134C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861134C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861131C2211yescharacteristic
52C2×M16×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55C4⋊C8×C127C2211yescharacteristic
56C22⋊C8×C127C2211yescharacteristic
57C22⋊C8×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—22no—
61#14523248—41no—
62#14523248—41no—
63C2×Dic254C2311yescharacteristic
64C2×Dic254C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76#14523237—41no—
77#14523237—41no—
78D1016—22no—
79C4⋊Dic127—21no—
80C4⋊Dic127—21no—
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84C4⋊Dic127C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97C4×Dic127—21no—
98C2×D4×C127D411yescharacteristic
99C22×C4×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101M16×C127—41no—
102C2×C8×C127—41no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127D411yescharacteristic
108C22⋊C4×C127D411yescharacteristic
109C22⋊C4×C127D411yescharacteristic
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—21no—
116Dic254—21no—
117D508—21no—
118D508—21no—
119D508—21no—
120D508#4112yes—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C22⋊D127—42no—
124C22⋊D127—81no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127—81no—
132C4×D127—42no—
133C4×D127—42no—
134C4×D127—42no—
135C4×D127—81no—
136C23×C127C2×D411yescharacteristic
137C23×C127C2×D411yescharacteristic
138D4×C127—41no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C8×C127—21no—
149C8×C127—21no—
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127C22×D411yescharacteristic
159C2×D127—22no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127C2×#4111yescharacteristic
170C4×C127H(5,𝔽2)11yescharacteristic
171Dic127—41no—
172Dic127—41no—
173Dic127—41no—
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—41no—
178C2×C127—41no—
179C2×C127#54611yescharacteristic
180C2×C127#57511yescharacteristic
181C2×C127#57711yescharacteristic
182D127—42no—
183D127—81no—
184#3307—1271nomaximal, Sylow
185C127#330711yescharacteristic, complemented, Sylow
186#553—1271no—
187#547—1271no—
188#491—1271no—
189#485—1271no—
190#483—1271no—
191#478—1271no—
192#468—1271no—
193#465—1271no—
194#461—1271no—
195#449—1271no—
196#449—1271no—
197#442—1271no—
198#435D12711yescharacteristic, complemented
199#417—1271no—
200#416—1271no—
201C2×#41—1271no—
202C2×#41—1271no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C2×M16C2×D12711yescharacteristic
207C22×C8C2×D12711yescharacteristic
208C4⋊Q8—1271no—
209#126—2541no—
210#125—1271no—
211#123—2541no—
212#123—2541no—
213#123—2541no—
214C22⋊Q8—1271no—
215C22⋊Q8—1271no—
216#121—1271no—
217#121—1271no—
218#121—2541no—
219#121—2541no—
220C4×D4—1271no—
221C4×D4—1271no—
222C4×D4—2541no—
223C4×D4C2×D12711yescharacteristic
224#117—1271no—
225C2×C4⋊C4—1271no—
226C2×C4⋊C4—2541no—
227#107—1271no—
228#106—1271no—
229#106—1271no—
230C4⋊C8C2×D12711yescharacteristic
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#103—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241C22⋊C8C2×D12711yescharacteristic
242C22⋊C8C2×D12711yescharacteristic
243#97C2×D12711yescharacteristic
244#41—5081no—
245#41—5081no—
246#41—2542no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8—1271no—
250C2×D4—1271no—
251C2×D4—1271no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4D50811yescharacteristic
257C22×C4—2541no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4C22×D12711yescharacteristic
263C22×C4C22×D12711yescharacteristic
264SD16—5081no—
265SD16—5081no—
266D8—2542no—
267M16—41no—
268C2×C8—41no—
269C2×C8C22×D12711yescharacteristic
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C2×C8C22×D12711yescharacteristic
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4—2541no—
283C4⋊C4—2541no—
284C4⋊C4D50811yescharacteristic
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4D50811yescharacteristic
293C22⋊C4D50811yescharacteristic
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—2541no—
297C23—2541no—
298C23C2×D50811yescharacteristic
299C23C2×D50811yescharacteristic
300Q8—2541no—
301Q8—2541no—
302Q8—2541no—
303Q8—2541no—
304D4—41no—
305D4—1272no—
306D4—10161no—
307D4—5082no—
308D4—5082no—
309D4—5082no—
310D4—2541no—
311D4—2541no—
312D4—2541no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—10161no—
318C2×C4—10161no—
319C2×C4—5082no—
320C2×C4—5082no—
321C2×C4—5082no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4C2×D50811yescharacteristic
329C2×C4C2×D50811yescharacteristic
330C2×C4C2×D50811yescharacteristic
331C2×C4C2×D50811yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—5081no—
344C22—5081no—
345C22—5081no—
346C22—5081no—
347C22—5081no—
348C22—2542no—
349C22C22×D50811yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—5081no—
355C4—5081no—
356C4—5081no—
357C4—5081no—
358C4—5081no—
359C4—5081no—
360C4#41×D12711yescharacteristic
361C4#1686125411yescharacteristic
362C2—41no—
363C2—41no—
364C2—10161no—
365C2—5082no—
366C2#2255052111yescharacteristic
367C2#2255067611yescharacteristic
368C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#435×C127C2
socleC22×C127C22×D4
hypercenter#435D127
hypocenterC127#3307
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. #22550572C2semidirect
2. #22550531C2semidirect
3. #22550245C2semidirect
4. #22550218C2semidirect
5. #22550207C2semidirect
6. #22550177C2semidirect
7. #22550122C2semidirect
8. #22550105C2semidirect
9. #22550082C2semidirect
10. #22550023C2semidirect
11. #22550023C2semidirect
12. #22549989C2semidirect
13. #22549878C2semidirect
14. #22549873C2semidirect
15. #435×C127C2semidirect
185. C127#3307semidirect
198. #435D127semidirect

Maximal subgroups

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

subgroupindex
#225505722
#225505312
#225502452
#225502182
#225502072
#225501772
#225501222
#225501052
#225500822
#225500232
#225499892
#225498782
#225498732
#435×C1272
#3307127