Finite Group Databasebeta

← #30650727

Subgroups — #30650727

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#22550532C211yesmaximal, characteristic, complemented
2#22550526C211yesmaximal, characteristic, complemented
3#22550234C211yesmaximal, characteristic, complemented
4#22550213C211yesmaximal, characteristic, complemented
5#22550187C211yesmaximal, characteristic, complemented
6#22550170C211yesmaximal, characteristic, complemented
7#22550121C211yesmaximal, characteristic, complemented
8#22550089C211yesmaximal, characteristic, complemented
9#22550018C211yesmaximal, characteristic, complemented
10#22550014C211yesmaximal, characteristic, complemented
11#22549973C211yesmaximal, characteristic, complemented
12#22549947C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14C2×#16861134C211yesmaximal, characteristic, complemented
15#466×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×#14523247C2211yescharacteristic
18C22×Dic254C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523236C2211yescharacteristic
21C2×#14523231C2211yescharacteristic
22#16861208C2211yescharacteristic
23#16861196C2211yescharacteristic
24#16861195—21no—
25#16861191—21no—
26#16861190—21no—
27#16861189C2211yescharacteristic
28#16861186—21no—
29#16861186—21no—
30#16861181—21no—
31#16861180C2211yescharacteristic
32#16861174C2211yescharacteristic
33D4×Dic127—21no—
34D4×Dic127C2211yescharacteristic
35C4×C22⋊D127C2211yescharacteristic
36C2×C4⋊Dic127C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861142C2211yescharacteristic
40Dic127⋊C8C2211yescharacteristic
41#16861134C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861119C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54#121×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#107×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—41no—
61#14523247—44no—
62C2×Dic254—21no—
63C2×Dic254—22no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127D411yescharacteristic
71C22×Dic127—21no—
72C22×Dic127—21no—
73C22×Dic127—21no—
74C22×Dic127—21no—
75C2×C4×D127—21no—
76C2×C4×D127C2311yescharacteristic
77#14523236—22no—
78#14523236—41no—
79#14523231—41no—
80C2×C127⋊C8C2311yescharacteristic
81C2×C127⋊C8C2311yescharacteristic
82C4⋊Dic127—21no—
83C4⋊Dic127—21no—
84C4⋊Dic127C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127D411yescharacteristic
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4D411yescharacteristic
97C4×Dic127—21no—
98C4×Dic127C2311yescharacteristic
99C2×D4×C127—21no—
100C2×D4×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127C2311yescharacteristic
104D8×C127—41no—
105C2×C8×C127—41no—
106C2×C8×C127C2311yescharacteristic
107C2×C8×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C22⋊C4×C127—21no—
112C22×D127C2×D411yescharacteristic
113Dic254—21no—
114Dic254—21no—
115Dic254—21no—
116Dic254—42no—
117Dic254—42no—
118Dic254#4112yescomplemented
119C22⋊D127—22no—
120C22⋊D127—42no—
121C22⋊D127—42no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—42no—
127C2×Dic127—42no—
128C2×Dic127—42no—
129C2×Dic127—42no—
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—42no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127C2×D411yescharacteristic
141D4×C127—21no—
142D4×C127—21no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127—42no—
146D4×C127—42no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127AGL(1,ℤ/8)12yes—
164C22×C127C22×D411yescharacteristic
165C2×D127—21no—
166C2×D127—21no—
167C2×D127—21no—
168C4×C127—21no—
169C4×C127—41no—
170C4×C127—41no—
171C4×C127C2×#4111yescharacteristic
172C4×C127H(5,𝔽2)11yescharacteristic
173Dic127—21no—
174Dic127—22no—
175Dic127—21no—
176Dic127—21no—
177Dic127—41no—
178Dic127—41no—
179Dic127—41no—
180C2×C127—22no—
181C2×C127—81no—
182C2×C127—81no—
183C2×C127#54611yescharacteristic
184C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C127#57811yescharacteristic
186D127—41no—
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#466D12711yescharacteristic, complemented
197#463—1271no—
198#449—1271no—
199#448—1271no—
200#440—1271no—
201#435—1271no—
202#417—1271no—
203C2×#103—1271no—
204C2×#41—1271no—
205C2×#41—2541no—
206C22×Q8—1271no—
207C2×SD16—1271no—
208C2×SD16—1271no—
209C2×D8C2×D12711yescharacteristic
210C2×M16—1271no—
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—2541no—
222#121C2×D12711yescharacteristic
223#121C2×D12711yescharacteristic
224C4×Q8—1271no—
225C4×D4—1271no—
226C4×D4—1271no—
227C4×D4—2541no—
228C2×C4⋊C4—1271no—
229C2×C4⋊C4—1271no—
230#107—1271no—
231#107—1271no—
232#107C2×D12711yescharacteristic
233C4⋊C8—1271no—
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#102C2×D12711yescharacteristic
243#102C2×D12711yescharacteristic
244C22⋊C8—1271no—
245C22⋊C8—1271no—
246#97—1271no—
247#41—5084no—
248#41—5081no—
249#41—2542no—
250C2×Q8—1271no—
251C2×Q8—1271no—
252C2×Q8—5081no—
253C2×Q8—2541no—
254C2×Q8—2542no—
255C2×Q8—2541no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—1271no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4C22×D12711yescharacteristic
262C2×D4C22×D12711yescharacteristic
263C22×C4—1271no—
264C22×C4—2541no—
265C22×C4—2541no—
266C22×C4—2541no—
267C22×C4—2541no—
268C22×C4—2541no—
269C22×C4C22×D12711yescharacteristic
270SD16—5081no—
271SD16—2542no—
272D8—41no—
273M16—5081no—
274C2×C8—41no—
275C2×C8—1271no—
276C2×C8—1271no—
277C2×C8C22×D12711yescharacteristic
278C2×C8C22×D12711yescharacteristic
279C4⋊C4—1271no—
280C4⋊C4—1271no—
281C4⋊C4—1271no—
282C4⋊C4—2541no—
283C4⋊C4—2541no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4C22×D12711yescharacteristic
288C4⋊C4C22×D12711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—1271no—
292C22⋊C4—1271no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C42—1271no—
299C42—2541no—
300C23—21no—
301C23—21no—
302C23—1271no—
303C23D4×D12711yescharacteristic
304Q8—1272no—
305Q8—5082no—
306Q8—5082no—
307Q8—2541no—
308Q8—2541no—
309Q8—2541no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—42no—
316D4—5082no—
317D4—5082no—
318D4—2542no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—1271no—
324C2×C4—1271no—
325C2×C4—1271no—
326C2×C4—5082no—
327C2×C4—5082no—
328C2×C4—5082no—
329C2×C4—5082no—
330C2×C4—5082no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4—2541no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4D4×D12711yescharacteristic
338C2×C4C23×D12711yescharacteristic
339C8—21no—
340C8—21no—
341C8—2541no—
342C8—2541no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—2541no—
352C22—2541no—
353C22—2541no—
354C22#1686123312yes—
355C22C2×D4×D12711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—5081no—
360C4—5081no—
361C4—5081no—
362C4—2541no—
363C4—2541no—
364C4—2541no—
365C4—2542no—
366C4#41×D12711yescharacteristic
367C4#1686125411yescharacteristic
368C2—22no—
369C2—81no—
370C2—81no—
371C2—5081no—
372C2#2255051511yescharacteristic
373C2C2×#1686123311yescharacteristic
374C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#466×C127C2
socleC22×C127C22×D4
hypercenter#466D127
hypocenterC127#3308
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #22550532C2semidirect
2. #22550526C2semidirect
3. #22550234C2semidirect
4. #22550213C2semidirect
5. #22550187C2semidirect
6. #22550170C2semidirect
7. #22550121C2semidirect
8. #22550089C2semidirect
9. #22550018C2semidirect
10. #22550014C2semidirect
11. #22549973C2semidirect
12. #22549947C2semidirect
13. #22549874C2semidirect
14. C2×#16861134C2semidirect
15. #466×C127C2semidirect
118. Dic254#41semidirect
188. C127#3308semidirect
196. #466D127semidirect

Maximal subgroups

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

subgroupindex
#225505322
#225505262
#225502342
#225502132
#225501872
#225501702
#225501212
#225500892
#225500182
#225500142
#225499732
#225499472
#225498742
C2×#168611342
#466×C1272
#3308127