Finite Group Databasebeta

← #30648933

Subgroups — #30648933

Order 16256.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550477C211yesmaximal, characteristic, complemented
2#22550355C211yesmaximal, characteristic, complemented
3#22550196C211yesmaximal, characteristic, complemented
4#22550188C211yesmaximal, characteristic, complemented
5#22550186C211yesmaximal, characteristic, complemented
6#22550174C211yesmaximal, characteristic, complemented
7#22550075C211yesmaximal, characteristic, complemented
8#22550069C211yesmaximal, characteristic, complemented
9#22550022C211yesmaximal, characteristic, complemented
10#22550018C211yesmaximal, characteristic, complemented
11#22549978C211yesmaximal, characteristic, complemented
12C4×#14523236C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549938C211yesmaximal, characteristic, complemented
15#442×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523237C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20#16861205C2211yescharacteristic
21#16861200C2211yescharacteristic
22#16861192—21no—
23#16861191—21no—
24#16861184C2211yescharacteristic
25#16861183—21no—
26#16861182—21no—
27#16861180C2211yescharacteristic
28C4×Dic254C2211yescharacteristic
29#16861174C2211yescharacteristic
30C4×D508C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861166C2211yescharacteristic
35#16861165—21no—
36C2×Dic127⋊C4—21no—
37C4⋊C4×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861132C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C4×C127⋊C8C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—22no—
61C2×Dic254C2311yescharacteristic
62C2×D508—21no—
63C2×D508C2311yescharacteristic
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C22×Dic127—21no—
69C22×Dic127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127C2311yescharacteristic
73#14523237—22no—
74#14523236—22no—
75#14523233—22no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C4⋊Dic127—21no—
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4D411yescharacteristic
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4D411yescharacteristic
92(C2×D127)⋊C4D411yescharacteristic
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127C2311yescharacteristic
96C2×D4×C127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100D8×C127—22no—
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127C2×D411yescharacteristic
110Dic254—21no—
111Dic254#4112yes—
112D508—21no—
113D508—21no—
114D508—81no—
115C22⋊D127—24no—
116C22⋊D127—81no—
117C22⋊D127—81no—
118C22⋊D127—81no—
119C2×Dic127—24no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—81no—
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C4×D127—21no—
128C4×D127—21no—
129C4×D127—81no—
130C4×D127—81no—
131C127⋊C8—21no—
132C127⋊C8—21no—
133C23×C127—21no—
134C23×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127#4112yes—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—81no—
142C2×C4×C127—81no—
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—22no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127C22×D411yescharacteristic
155C2×D127—21no—
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C4×C127—21no—
162C4×C127—21no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127C2×#4111yescharacteristic
166C4×C127C2×#4111yescharacteristic
167Dic127—21no—
168Dic127—22no—
169Dic127—21no—
170Dic127—21no—
171Dic127—41no—
172Dic127—41no—
173C2×C127—42no—
174C2×C127—81no—
175C2×C127#52511yescharacteristic
176C2×C127#57611yescharacteristic
177C2×C127#57711yescharacteristic
178D127—41no—
179D127—81no—
180#3119—1271nomaximal, Sylow
181C127#311911yescharacteristic, complemented, Sylow
182#540—1271no—
183#520—1271no—
184#481—1271no—
185#480—1271no—
186#479—1271no—
187#478—1271no—
188#460—1271no—
189#459—1271no—
190#449—1271no—
191#449—1271no—
192#442D12711yescharacteristic, complemented
193#440—1271no—
194C4×SD16—1271no—
195C4×D8—1271no—
196#433—1271no—
197C2×#41—1271no—
198C2×SD16—1271no—
199C2×SD16—1271no—
200C2×D8—1271no—
201C2×D8C2×D12711yescharacteristic
202#127—1271no—
203#125—1271no—
204#123—2541no—
205#123—2541no—
206#121—1271no—
207#121—1271no—
208#121—2541no—
209#121—2541no—
210C4×Q8—1271no—
211C4×Q8—1271no—
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—2541no—
215C4×D4—2541no—
216C4×D4C2×D12711yescharacteristic
217C4×D4C2×D12711yescharacteristic
218#117—1271no—
219#117—2541no—
220C2×C4⋊C4—1271no—
221C2×C4⋊C4—2541no—
222#107—1271no—
223#107—1271no—
224#106—1271no—
225#106C2×D12711yescharacteristic
226C4⋊C8—1271no—
227C4⋊C8—1271no—
228#103—1271no—
229#103—1271no—
230#102—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102C2×D12711yescharacteristic
235#102C2×D12711yescharacteristic
236C22⋊C8—1271no—
237C22⋊C8—1271no—
238#97C2×D12711yescharacteristic
239C4×C8—1271no—
240#41—2542no—
241#41—2542no—
242C2×Q8—1271no—
243C2×D4—1271no—
244C2×D4—2541no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4C22×D12711yescharacteristic
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—1271no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4—2541no—
258SD16—2542no—
259SD16—2542no—
260D8—22no—
261D8—2542no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8C22×D12711yescharacteristic
265C2×C8C22×D12711yescharacteristic
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1271no—
280C22⋊C4—1271no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C42—1271no—
286C42—2541no—
287C42—2541no—
288C42C22×D12711yescharacteristic
289C23—21no—
290C23—21no—
291C23—1271no—
292C23—2541no—
293Q8—1272no—
294Q8—2541no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—10161no—
299D4—10161no—
300D4—10161no—
301D4—10161no—
302D4—2544no—
303D4—2541no—
304D4—2541no—
305D4#1452324812yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—10161no—
314C2×C4—10161no—
315C2×C4—10161no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2544no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4D4×D12711yescharacteristic
323C2×C4D4×D12711yescharacteristic
324C2×C4C23×D12711yescharacteristic
325C8—21no—
326C8—21no—
327C8—2541no—
328C8—2541no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—5081no—
336C22—5081no—
337C22—5081no—
338C22—2541no—
339C22—2541no—
340C22—2541no—
341C22C2×D4×D12711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—5081no—
347C4—5081no—
348C4—2541no—
349C4—2542no—
350C4—2541no—
351C4—2541no—
352C4#41×D12711yescharacteristic
353C4C2×#1452324811yescharacteristic
354C2—42no—
355C2—81no—
356C2—10161no—
357C2—5081no—
358C2#2255038511yescharacteristic
359C2#2255068011yescharacteristic
360C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#442×C127C2
socleC22×C127C22×D4
hypercenter#442D127
hypocenterC127#3119
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #22550477C2semidirect
2. #22550355C2semidirect
3. #22550196C2semidirect
4. #22550188C2semidirect
5. #22550186C2semidirect
6. #22550174C2semidirect
7. #22550075C2semidirect
8. #22550069C2semidirect
9. #22550022C2semidirect
10. #22550018C2semidirect
11. #22549978C2semidirect
12. C4×#14523236C2semidirect
13. C4×#14523233C2semidirect
14. #22549938C2semidirect
15. #442×C127C2semidirect
181. C127#3119semidirect
192. #442D127semidirect

Maximal subgroups

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

subgroupindex
#225504772
#225503552
#225501962
#225501882
#225501862
#225501742
#225500752
#225500692
#225500222
#225500182
#225499782
C4×#145232362
C4×#145232332
#225499382
#442×C1272
#3119127