Finite Group Databasebeta

← #30650723

Subgroups — #30650723

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550535C211yesmaximal, characteristic, complemented
2#22550530C211yesmaximal, characteristic, complemented
3#22550236C211yesmaximal, characteristic, complemented
4#22550212C211yesmaximal, characteristic, complemented
5#22550190C211yesmaximal, characteristic, complemented
6#22550172C211yesmaximal, characteristic, complemented
7#22550122C211yesmaximal, characteristic, complemented
8#22550112C211yesmaximal, characteristic, complemented
9#22550092C211yesmaximal, characteristic, complemented
10#22550023C211yesmaximal, characteristic, complemented
11#22550017C211yesmaximal, characteristic, complemented
12#22549978C211yesmaximal, characteristic, complemented
13#22549878C211yesmaximal, characteristic, complemented
14C2×#16861134C211yesmaximal, characteristic, complemented
15#435×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523247—21no—
18C22×Dic254C2211yescharacteristic
19C2×#14523237C2211yescharacteristic
20C2×#14523237C2211yescharacteristic
21C2×D1016C2211yescharacteristic
22#16861209C2211yescharacteristic
23#16861197C2211yescharacteristic
24#16861195—21no—
25#16861193—21no—
26#16861193—21no—
27#16861189—21no—
28#16861188C2211yescharacteristic
29#16861187—21no—
30#16861183C2211yescharacteristic
31#16861183C2211yescharacteristic
32#16861183C2211yescharacteristic
33#16861181—21no—
34C4×Dic254C2211yescharacteristic
35C4×D508C2211yescharacteristic
36(C22⋊D127)⋊C4—21no—
37C2×C4⋊Dic127C2211yescharacteristic
38C2×C4⋊Dic127C2211yescharacteristic
39#16861143C2211yescharacteristic
40#16861143C2211yescharacteristic
41#16861143C2211yescharacteristic
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—41no—
61#14523247—44no—
62C2×Dic254—22no—
63C2×Dic254—21no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C22×Dic127—21no—
75C22×Dic127—21no—
76C22×Dic127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79#14523237—22no—
80#14523237—41no—
81D1016—41no—
82C4⋊Dic127—21no—
83C4⋊Dic127—21no—
84C4⋊Dic127—21no—
85C4⋊Dic127C2311yescharacteristic
86C4⋊Dic127C2311yescharacteristic
87C4⋊Dic127C2311yescharacteristic
88C4⋊Dic127C2311yescharacteristic
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98C4×Dic127—21no—
99C2×D4×C127D411yescharacteristic
100C22×C4×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102M16×C127—41no—
103C2×C8×C127—41no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C2×C8×C127C2311yescharacteristic
108C4⋊C4×C127D411yescharacteristic
109C22⋊C4×C127D411yescharacteristic
110C22⋊C4×C127D411yescharacteristic
111C42×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127—21no—
114Dic254—21no—
115Dic254—21no—
116Dic254—21no—
117Dic254—42no—
118Dic254—42no—
119Dic254#4112yescomplemented
120D508—21no—
121D508—21no—
122D508—21no—
123D508—21no—
124C22⋊D127—42no—
125C22⋊D127—42no—
126C22⋊D127—42no—
127C22⋊D127—42no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—42no—
135C2×Dic127—42no—
136C2×Dic127—42no—
137C4×D127—42no—
138C4×D127—42no—
139C23×C127C2×D411yescharacteristic
140C23×C127C2×D411yescharacteristic
141D4×C127—22no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2411yescharacteristic
151C8×C127—21no—
152C8×C127—21no—
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127AGL(1,ℤ/8)12yes—
161C22×C127C22×D411yescharacteristic
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C4×C127—21no—
169C4×C127—21no—
170C4×C127—21no—
171C4×C127—21no—
172C4×C127C2×#4111yescharacteristic
173C4×C127H(5,𝔽2)11yescharacteristic
174Dic127—22no—
175Dic127—41no—
176Dic127—41no—
177Dic127—41no—
178Dic127—41no—
179Dic127—41no—
180C2×C127—22no—
181C2×C127—41no—
182C2×C127#54611yescharacteristic
183C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C127#57811yescharacteristic
185D127—81no—
186D127—81no—
187#3308—1271nomaximal, Sylow
188C127#330811yescharacteristic, complemented, Sylow
189#548—1271no—
190#547—1271no—
191#489—1271no—
192#484—1271no—
193#480—1271no—
194#477—1271no—
195#468—1271no—
196#466—1271no—
197#463—1271no—
198#449—1271no—
199#448—1271no—
200#440—1271no—
201#435D12711yescharacteristic, complemented
202#417—1271no—
203C2×#103—1271no—
204C2×#41—1271no—
205C2×#41—2541no—
206C22×Q8—1271no—
207C2×SD16—1271no—
208C2×SD16—1271no—
209C2×D8—1271no—
210C2×M16C2×D12711yescharacteristic
211C22×C8C2×D12711yescharacteristic
212C4⋊Q8—1271no—
213#124—1271no—
214#124—2541no—
215#123—2541no—
216#123—2541no—
217C22⋊Q8—1271no—
218C22⋊Q8—2541no—
219C22⋊Q8—2541no—
220#121—1271no—
221#121—1271no—
222#121—1271no—
223#121—2541no—
224C4×Q8—1271no—
225C4×D4—1271no—
226C4×D4—2541no—
227C4×D4C2×D12711yescharacteristic
228C2×C4⋊C4—1271no—
229C2×C4⋊C4—1271no—
230#107—1271no—
231#107—1271no—
232#107—1271no—
233C4⋊C8C2×D12711yescharacteristic
234#103—1271no—
235#103—1271no—
236#103—1271no—
237#103—1271no—
238#103—1271no—
239#103—1271no—
240#102—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244C22⋊C8C2×D12711yescharacteristic
245C22⋊C8C2×D12711yescharacteristic
246#97C2×D12711yescharacteristic
247#41—5081no—
248#41—5084no—
249#41—2542no—
250C2×Q8—1271no—
251C2×Q8—1271no—
252C2×Q8—5081no—
253C2×Q8—2542no—
254C2×Q8—2541no—
255C2×Q8—2541no—
256C2×D4—1271no—
257C2×D4—1271no—
258C2×D4—2541no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4D50811yescharacteristic
263C22×C4—2541no—
264C22×C4—2541no—
265C22×C4—2541no—
266C22×C4—2541no—
267C22×C4—2541no—
268C22×C4C22×D12711yescharacteristic
269C22×C4C22×D12711yescharacteristic
270SD16—5081no—
271SD16—2542no—
272D8—5081no—
273M16—41no—
274C2×C8—41no—
275C2×C8C22×D12711yescharacteristic
276C2×C8C22×D12711yescharacteristic
277C2×C8C22×D12711yescharacteristic
278C2×C8C22×D12711yescharacteristic
279C4⋊C4—1271no—
280C4⋊C4—1271no—
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—2541no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4D50811yescharacteristic
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4D50811yescharacteristic
297C22⋊C4D50811yescharacteristic
298C42—2541no—
299C42C22×D12711yescharacteristic
300C23—2541no—
301C23—2541no—
302C23C2×D50811yescharacteristic
303C23C2×D50811yescharacteristic
304Q8—1272no—
305Q8—5082no—
306Q8—5082no—
307Q8—2541no—
308Q8—2541no—
309Q8—2541no—
310D4—22no—
311D4—5082no—
312D4—5082no—
313D4—5082no—
314D4—5082no—
315D4—2541no—
316D4—2541no—
317D4—2541no—
318D4—2541no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—5082no—
324C2×C4—5082no—
325C2×C4—5082no—
326C2×C4—5082no—
327C2×C4—5082no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4C2×D50811yescharacteristic
335C2×C4C2×D50811yescharacteristic
336C2×C4C2×D50811yescharacteristic
337C2×C4C2×D50811yescharacteristic
338C2×C4C23×D12711yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—5081no—
349C22—5081no—
350C22—5081no—
351C22—5081no—
352C22—5081no—
353C22—5081no—
354C22#1686123712yes—
355C22C22×D50811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—5081no—
361C4—5081no—
362C4—5081no—
363C4—5081no—
364C4—5081no—
365C4—2542no—
366C4#41×D12711yescharacteristic
367C4#1686125411yescharacteristic
368C2—22no—
369C2—41no—
370C2—10161no—
371C2—10161no—
372C2#2255052111yescharacteristic
373C2C2×#1686123711yescharacteristic
374C2#2255069311yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#435×C127C2
socleC22×C127C22×D4
hypercenter#435D127
hypocenterC127#3308
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. #22550535C2semidirect
2. #22550530C2semidirect
3. #22550236C2semidirect
4. #22550212C2semidirect
5. #22550190C2semidirect
6. #22550172C2semidirect
7. #22550122C2semidirect
8. #22550112C2semidirect
9. #22550092C2semidirect
10. #22550023C2semidirect
11. #22550017C2semidirect
12. #22549978C2semidirect
13. #22549878C2semidirect
14. C2×#16861134C2semidirect
15. #435×C127C2semidirect
119. Dic254#41semidirect
188. C127#3308semidirect
201. #435D127semidirect

Maximal subgroups

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

subgroupindex
#225505352
#225505302
#225502362
#225502122
#225501902
#225501722
#225501222
#225501122
#225500922
#225500232
#225500172
#225499782
#225498782
C2×#168611342
#435×C1272
#3308127