Finite Group Databasebeta

← #30648850

Subgroups — #30648850

Order 16256.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550423C211yesmaximal, characteristic
2C4×Q8×D127C211yesmaximal, characteristic, complemented
3#22550191C211yesmaximal, characteristic, complemented
4#22550188C211yesmaximal, characteristic, complemented
5#22550170C211yesmaximal, characteristic, complemented
6#22550169C211yesmaximal, characteristic, complemented
7#22550075C211yesmaximal, characteristic, complemented
8#22550074C211yesmaximal, characteristic, complemented
9#22550016C211yesmaximal, characteristic, complemented
10#22550013C211yesmaximal, characteristic, complemented
11#22549978C211yesmaximal, characteristic, complemented
12#22549976C211yesmaximal, characteristic, complemented
13#22549974C211yesmaximal, characteristic, complemented
14#22549931C211yesmaximal, characteristic, complemented
15#440×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C2×Q8×D127C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861205C2211yescharacteristic
23#16861195—21no—
24#16861184C2211yescharacteristic
25#16861182—21no—
26#16861180C2211yescharacteristic
27Q8×Dic127C2211yescharacteristic
28C4×Dic254—21no—
29C4×Dic254C2211yescharacteristic
30#16861174—21no—
31#16861174C2211yescharacteristic
32C4×D508C2211yescharacteristic
33C4×C22⋊D127—21no—
34C4⋊C4×D127—21no—
35C4⋊C4×D127C2211yescharacteristic
36C42×D127—21no—
37C42×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861142C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C4×Q8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#107×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic
59#14523248—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63Q8×D127—22no—
64Q8×D127—22no—
65C2×D508—21no—
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127D411yescharacteristic
74C2×C4×D127D411yescharacteristic
75C2×C4×D127C2311yescharacteristic
76#14523237—22no—
77#14523236—22no—
78#14523235—22no—
79C2×C127⋊C8C2311yescharacteristic
80C2×C127⋊C8C2311yescharacteristic
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88C22⋊Dic127—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91C4×Dic127—21no—
92C4×Dic127—21no—
93C4×Dic127—21no—
94C4×Dic127D411yescharacteristic
95C4×Dic127D411yescharacteristic
96C4×Dic127C2311yescharacteristic
97C2×Q8×C127C2311yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100SD16×C127—22no—
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127—21no—
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127—21no—
108C42×C127C2311yescharacteristic
109C22×D127—21no—
110C22×D127C2×D411yescharacteristic
111Dic254—24no—
112Dic254—21no—
113Dic254—42no—
114Dic254#4112yescomplemented
115D508—21no—
116D508—21no—
117D508—42no—
118C22⋊D127—42no—
119C22⋊D127—42no—
120C22⋊D127—42no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C4×D127—24no—
128C4×D127—22no—
129C4×D127—21no—
130C4×D127—21no—
131C4×D127—22no—
132C4×D127—42no—
133C4×D127—42no—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127—21no—
137Q8×C127—21no—
138Q8×C127#4112yescomplemented
139D4×C127—21no—
140D4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—42no—
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C8×C127—21no—
149C8×C127—21no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127C22×D411yescharacteristic
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C2×D127AGL(1,ℤ/8)12yes—
160C4×C127—22no—
161C4×C127—21no—
162C4×C127—21no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127C2×#4111yescharacteristic
166C4×C127C2×#4111yescharacteristic
167Dic127—22no—
168Dic127—21no—
169Dic127—21no—
170Dic127—41no—
171Dic127—41no—
172Dic127#13712yes—
173C2×C127—81no—
174C2×C127#52511yescharacteristic
175C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C127C2×#13711yescharacteristic
177D127—22no—
178D127—81no—
179#3112—1271nomaximal, Sylow
180C127#311211yescharacteristic, complemented, Sylow
181#532—1271no—
182C2×C4×Q8—1271no—
183#480—1271no—
184#480—1271no—
185#477—1271no—
186#477—1271no—
187#460—1271no—
188#460—1271no—
189#448—1271no—
190#448—1271no—
191#440—1271no—
192#440—1271no—
193#440—1271no—
194#440D12711yescharacteristic, complemented
195#432—1271no—
196C2×#41—2541no—
197C22×Q8—1271no—
198C2×SD16—1271no—
199C2×SD16—1271no—
200C2×SD16—1271no—
201C2×SD16C2×D12711yescharacteristic
202C4⋊Q8—1271no—
203#127—1271no—
204#124—2541no—
205#121—1271no—
206#121—1271no—
207#121—2541no—
208C4×Q8—1271no—
209C4×Q8—1271no—
210C4×Q8—1271no—
211C4×Q8—2541no—
212C4×Q8—2541no—
213C4×Q8C2×D12711yescharacteristic
214C4×D4—1271no—
215C4×D4—2541no—
216C4×D4C2×D12711yescharacteristic
217C2×C4⋊C4—1271no—
218C2×C4⋊C4—2541no—
219C2×C42—1271no—
220C2×C42—2541no—
221#107—1271no—
222#107—1271no—
223#107—1271no—
224#107C2×D12711yescharacteristic
225C4⋊C8—1271no—
226C4⋊C8—1271no—
227#103—1271no—
228#103—1271no—
229#103—1271no—
230#103C2×D12711yescharacteristic
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102C2×D12711yescharacteristic
235C22⋊C8—1271no—
236C22⋊C8—1271no—
237#97—1271no—
238#97C2×D12711yescharacteristic
239#41—5084no—
240C2×Q8—1271no—
241C2×Q8—2541no—
242C2×Q8—2541no—
243C2×Q8—2542no—
244C2×Q8—2542no—
245C2×Q8C22×D12711yescharacteristic
246C2×D4—1271no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4—2541no—
250C2×D4—2541no—
251C2×D4C22×D12711yescharacteristic
252C22×C4—21no—
253C22×C4—1271no—
254C22×C4—1271no—
255C22×C4—1271no—
256C22×C4—2541no—
257C22×C4—2541no—
258C22×C4—2541no—
259SD16—22no—
260SD16—2542no—
261SD16—2542no—
262SD16—2542no—
263C2×C8—1271no—
264C2×C8—1271no—
265C2×C8C22×D12711yescharacteristic
266C2×C8C22×D12711yescharacteristic
267C4⋊C4—21no—
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—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C42—21no—
282C42—1271no—
283C42—1271no—
284C42—1271no—
285C42—2541no—
286C42—2541no—
287C42—2541no—
288C42C22×D12711yescharacteristic
289C23—21no—
290C23—1271no—
291C23—2541no—
292Q8—21no—
293Q8—1272no—
294Q8—5082no—
295Q8—2544no—
296Q8—2541no—
297Q8#1452324812yescomplemented
298D4—21no—
299D4—21no—
300D4—5082no—
301D4—5082no—
302D4—5082no—
303D4—5082no—
304D4—2541no—
305D4—2541no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—5082no—
314C2×C4—5082no—
315C2×C4—2542no—
316C2×C4—2542no—
317C2×C4—2544no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4D4×D12711yescharacteristic
324C2×C4D4×D12711yescharacteristic
325C2×C4C23×D12711yescharacteristic
326C8—21no—
327C8—21no—
328C8—2541no—
329C8—2541no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—1272no—
334C22—5081no—
335C22—5081no—
336C22—5081no—
337C22—2541no—
338C22—2541no—
339C22C2×D4×D12711yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—1272no—
346C4—5081no—
347C4—5081no—
348C4—2541no—
349C4—2542no—
350C4—2541no—
351C4#41×D12711yescharacteristic
352C4C2×#1452324811yescharacteristic
353C2—81no—
354C2—10161no—
355C2—2542no—
356C2#2255038511yescharacteristic
357C2AGL(1,ℤ/8)×D12711yescharacteristic
358C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#440×C127C2
socleC22×C127C22×D4
prime radical#22550423C2
hypercenter#440D127
hypocenterC127#3112
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D127C2semidirect
3. #22550191C2semidirect
4. #22550188C2semidirect
5. #22550170C2semidirect
6. #22550169C2semidirect
7. #22550075C2semidirect
8. #22550074C2semidirect
9. #22550016C2semidirect
10. #22550013C2semidirect
11. #22549978C2semidirect
12. #22549976C2semidirect
13. #22549974C2semidirect
14. #22549931C2semidirect
15. #440×C127C2semidirect
114. Dic254#41semidirect
138. Q8×C127#41semidirect
180. C127#3112semidirect
194. #440D127semidirect
297. Q8#14523248semidirect

Maximal subgroups

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

subgroupindex
#225504232
C4×Q8×D1272
#225501912
#225501882
#225501702
#225501692
#225500752
#225500742
#225500162
#225500132
#225499782
#225499762
#225499742
#225499312
#440×C1272
#3112127