Finite Group Databasebeta

← #30648935

Subgroups — #30648935

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#22550358C211yesmaximal, characteristic, complemented
3#22550196C211yesmaximal, characteristic, complemented
4#22550188C211yesmaximal, characteristic, complemented
5#22550185C211yesmaximal, characteristic, complemented
6#22550175C211yesmaximal, characteristic, complemented
7#22550074C211yesmaximal, characteristic, complemented
8#22550070C211yesmaximal, characteristic, complemented
9#22550024C211yesmaximal, characteristic, complemented
10#22550020C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549974C211yesmaximal, characteristic, complemented
13C4×D1016C211yesmaximal, characteristic, complemented
14#22549935C211yesmaximal, characteristic, complemented
15C4×SD16×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×D1016C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20#16861205C2211yescharacteristic
21#16861200C2211yescharacteristic
22#16861192—21no—
23#16861191—21no—
24#16861184C2211yescharacteristic
25#16861183—21no—
26#16861182—21no—
27#16861180C2211yescharacteristic
28Q8×Dic127C2211yescharacteristic
29C4×D508—21no—
30C4×D508C2211yescharacteristic
31C4×D508C2211yescharacteristic
32#16861171—21no—
33#16861171C2211yescharacteristic
34#16861166—21no—
35#16861166C2211yescharacteristic
36C4⋊C4×D127—21no—
37C4⋊C4×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C4×Q8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C4×C8×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—22no—
61C2×D508—21no—
62C2×D508—21no—
63C2×D508C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74#14523235—22no—
75D1016—22no—
76#14523233—22no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C4⋊Dic127—21no—
80C4⋊Dic127—21no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4D411yescharacteristic
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4D411yescharacteristic
93(C2×D127)⋊C4D411yescharacteristic
94C4×Dic127—21no—
95C4×Dic127C2311yescharacteristic
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127—21no—
99SD16×C127—22no—
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C42×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110C22×D127C2×D411yescharacteristic
111D508—24no—
112D508—21no—
113D508—21no—
114D508—21no—
115D508—81no—
116D508#4112yes—
117C22⋊D127—81no—
118C22⋊D127—81no—
119C22⋊D127—81no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C4×D127—24no—
126C4×D127—21no—
127C4×D127—21no—
128C4×D127—81no—
129C4×D127—81no—
130C4×D127—81no—
131C4×D127—81no—
132C127⋊C8—21no—
133C127⋊C8—21no—
134C23×C127—21no—
135Q8×C127—21no—
136Q8×C127#4112yes—
137D4×C127—21no—
138D4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—81no—
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127C22×D411yescharacteristic
152C2×D127—21no—
153C2×D127—22no—
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C4×C127—22no—
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127C2×#4111yescharacteristic
167C4×C127C2×#4111yescharacteristic
168Dic127—21no—
169Dic127—21no—
170Dic127—21no—
171Dic127—41no—
172Dic127—41no—
173C2×C127—81no—
174C2×C127#52511yescharacteristic
175C2×C127#57611yescharacteristic
176C2×C127#57711yescharacteristic
177D127—41no—
178D127—42no—
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#442—1271no—
193#440—1271no—
194C4×SD16D12711yescharacteristic, complemented
195C4×D8—1271no—
196#433—1271no—
197C2×#41—1271no—
198C2×SD16—1271no—
199C2×SD16C2×D12711yescharacteristic
200C2×D8—1271no—
201C2×D8—1271no—
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×Q8C2×D12711yescharacteristic
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215C4×D4—2541no—
216C4×D4—2541no—
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#103C2×D12711yescharacteristic
230#102—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102C2×D12711yescharacteristic
236C22⋊C8—1271no—
237C22⋊C8—1271no—
238#97—1271no—
239C4×C8C2×D12711yescharacteristic
240#41—2542no—
241#41—2542no—
242C2×Q8C22×D12711yescharacteristic
243C2×D4—1271no—
244C2×D4—1271no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4—2541no—
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—1271no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4—2541no—
258SD16—22no—
259SD16—2542no—
260D8—2542no—
261D8—2542no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8C22×D12711yescharacteristic
265C2×C8C22×D12711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—1271no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1271no—
279C22⋊C4—1271no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C42—21no—
286C42—1271no—
287C42—2541no—
288C42C22×D12711yescharacteristic
289C23—21no—
290C23—1271no—
291C23—2541no—
292C23—2541no—
293Q8—21no—
294Q8#1452324812yes—
295D4—21no—
296D4—21no—
297D4—1272no—
298D4—10161no—
299D4—10161no—
300D4—10161no—
301D4—10161no—
302D4—2544no—
303D4—2541no—
304D4—2541no—
305D4—2541no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—10161no—
314C2×C4—10161no—
315C2×C4—10161no—
316C2×C4—10161no—
317C2×C4—2544no—
318C2×C4—2541no—
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—41no—
330C22—41no—
331C22—41no—
332C22—5081no—
333C22—5081no—
334C22—5081no—
335C22—5081no—
336C22—5081no—
337C22—2542no—
338C22—2541no—
339C22—2541no—
340C22—2541no—
341C22C2×D4×D12711yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—5081no—
348C4—5081no—
349C4—2541no—
350C4—2541no—
351C4—2541no—
352C4#41×D12711yescharacteristic
353C4C2×#1452324811yescharacteristic
354C2—81no—
355C2—10161no—
356C2—5082no—
357C2—5081no—
358C2#2255038511yescharacteristic
359C2#2255067811yescharacteristic
360C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
hypercenterC4×SD16D127
hypocenterC127#3119
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #22550477C2semidirect
2. #22550358C2semidirect
3. #22550196C2semidirect
4. #22550188C2semidirect
5. #22550185C2semidirect
6. #22550175C2semidirect
7. #22550074C2semidirect
8. #22550070C2semidirect
9. #22550024C2semidirect
10. #22550020C2semidirect
11. #22549986C2semidirect
12. #22549974C2semidirect
13. C4×D1016C2semidirect
14. #22549935C2semidirect
15. C4×SD16×C127C2semidirect
181. C127#3119semidirect
194. C4×SD16D127semidirect

Maximal subgroups

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

subgroupindex
#225504772
#225503582
#225501962
#225501882
#225501852
#225501752
#225500742
#225500702
#225500242
#225500202
#225499862
#225499742
C4×D10162
#225499352
C4×SD16×C1272
#3119127