Finite Group Databasebeta

← #30647570

Subgroups — #30647570

Order 16256.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861238C211yesmaximal, characteristic
2#22550355C211yesmaximal, characteristic
3#22550352C211yesmaximal, characteristic, complemented
4#22549984C211yesmaximal, characteristic, complemented
5#22549980C211yesmaximal, characteristic, complemented
6#22549973C211yesmaximal, characteristic, complemented
7#22549970C211yesmaximal, characteristic, complemented
8C4×#14523238C211yesmaximal, characteristic, complemented
9SD16×Dic127C211yesmaximal, characteristic, complemented
10C4×#14523236C211yesmaximal, characteristic, complemented
11#22549917C211yesmaximal, characteristic, complemented
12#22549869C211yesmaximal, characteristic
13#103×D127C211yesmaximal, characteristic, complemented
14#22549833C211yesmaximal, characteristic, complemented
15#440×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18#16861238C418yescomplemented
19C2×Dic508C2211yescharacteristic
20C2×#14523238C2211yescharacteristic
21C2×#14523236C2211yescharacteristic
22C2×#14523231C2211yescharacteristic
23C2×C8×D127C2211yescharacteristic
24Q8×Dic127C2211yescharacteristic
25C4×Dic254—21no—
26C4×Dic254C2211yescharacteristic
27C4×Dic254C2211yescharacteristic
28#16861174—21no—
29#16861174C2211yescharacteristic
30#16861174C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167—21no—
35#16861167C2211yescharacteristic
36#16861166—21no—
37#16861166C2211yescharacteristic
38#16861165—21no—
39C2×Dic127⋊C4—21no—
40C4⋊C4×D127C2211yescharacteristic
41#16861142C2211yescharacteristic
42#16861141C2211yescharacteristic
43#16861140C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861134C2211yescharacteristic
46#16861133C2211yescharacteristic
47#16861132C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861130C2211yescharacteristic
50#16861119C2211yescharacteristic
51C4×C127⋊C8C2211yescharacteristic
52C8×Dic127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C4×Q8×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56#107×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#97×C127C2211yescharacteristic
60#14523247—22no—
61#14523247C2×C414yes—
62C2×Dic254—21no—
63C2×Dic254C2311yescharacteristic
64C2×Dic254C2311yescharacteristic
65Q8×D127—22no—
66Q8×D127C2×C414yescomplemented
67C2×C22⋊D127—21no—
68C22×Dic127—21no—
69C22×Dic127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127C2311yescharacteristic
73Dic508C2×C414yescomplemented
74#14523238C2×C414yescomplemented
75#14523236C2×C414yescomplemented
76#14523231C2×C414yescomplemented
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C8×D127—22no—
80C4⋊Dic127—21no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4D411yescharacteristic
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4D411yescharacteristic
93(C2×D127)⋊C4D411yescharacteristic
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C4×Dic127—21no—
98C4×Dic127—21no—
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic
101C2×Q8×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104SD16×C127C2×C414yescomplemented
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C42×C127—21no—
112C42×C127C2311yescharacteristic
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116Dic254—24no—
117Dic254C22×C412yes—
118Dic254C22×C412yes—
119C22⋊D127—24no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—24no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—81no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—24no—
130C4×D127—81no—
131C4×D127C22×C412yes—
132C4×D127#4112yes—
133C127⋊C8—21no—
134C127⋊C8C22×C412yes—
135C23×C127—21no—
136Q8×C127—21no—
137Q8×C127C22×C412yes—
138D4×C127—21no—
139D4×C127C22×C412yes—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—81no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2411yescharacteristic
147C8×C127—21no—
148C8×C127C22×C412yes—
149C22×C127—22no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127C22×D411yescharacteristic
153C2×D127—21no—
154C2×D127—21no—
155C2×D127C4×D412yes—
156C4×C127—22no—
157C4×C127—21no—
158C4×C127—21no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127C23×C411yescharacteristic
162C4×C127C2×#4111yescharacteristic
163Dic127—22no—
164Dic127—22no—
165Dic127—21no—
166Dic127—21no—
167Dic127—41no—
168Dic127—41no—
169Dic127C4×D412yes—
170C2×C127—42no—
171C2×C127C2×C4×D411yescharacteristic
172C2×C127#57711yescharacteristic
173C2×C127#57811yescharacteristic
174D127—22no—
175#2947—1271nomaximal, Sylow
176C127#294711yescharacteristic, complemented, Sylow
177C2×#137—1271no—
178#520—1271no—
179#519—1271no—
180#441—1271no—
181#441—1271no—
182#440—1271no—
183#440D12711yescharacteristic, complemented
184C4×Q16—1271no—
185C4×Q16—1271no—
186C4×SD16—1271no—
187C4×SD16—1271no—
188#428—1271no—
189#416—1271no—
190C2×#103—1271no—
191#405—1271no—
192C2×#41—1271no—
193C22×Q8—1271no—
194#137—1278no—
195C2×Q16—1271no—
196C2×Q16—1271no—
197C2×SD16—1271no—
198C2×SD16C2×D12711yescharacteristic
199C2×M16—1271no—
200C22×C8—1271no—
201C4×Q8—1271no—
202C4×Q8—1271no—
203C4×Q8—1271no—
204C4×Q8—1271no—
205C4×Q8—1271no—
206C4×Q8—2541no—
207C4×Q8—2541no—
208C4×Q8C2×D12711yescharacteristic
209C4×D4—1271no—
210C4×D4—2541no—
211C4×D4—2541no—
212C4×D4C2×D12711yescharacteristic
213#117—1271no—
214#117—1271no—
215#117—2541no—
216#117—2541no—
217#117—2541no—
218C2×C4⋊C4—1271no—
219C2×C4⋊C4—2541no—
220#107—1271no—
221#107C2×D12711yescharacteristic
222#106—1271no—
223#106—1271no—
224#103—1271no—
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#103—1271no—
229#103C2×D12711yescharacteristic
230#102—1271no—
231#102C2×D12711yescharacteristic
232#97—1271no—
233#97C2×D12711yescharacteristic
234C4×C8—1271no—
235C4×C8—1271no—
236#41—1274no—
237#41—2542no—
238C2×Q8—1274no—
239C2×Q8—1271no—
240C2×Q8—1271no—
241C2×Q8—2541no—
242C2×Q8—2542no—
243C2×Q8C22×D12711yescharacteristic
244C2×D4—2541no—
245C2×D4C22×D12711yescharacteristic
246C22×C4—21no—
247C22×C4—1271no—
248C22×C4—2541no—
249C22×C4—2541no—
250C22×C4—2541no—
251C22×C4—2541no—
252Q16—1274no—
253Q16—1274no—
254SD16—1274no—
255SD16C4×D12714yescomplemented
256M16—1274no—
257C2×C8—1271no—
258C2×C8—1271no—
259C2×C8—2542no—
260C2×C8C22×D12711yescharacteristic
261C2×C8C22×D12711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—1271no—
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4C22×D12711yescharacteristic
274C4⋊C4C22×D12711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—1271no—
277C22⋊C4—1271no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C42—21no—
281C42—1271no—
282C42—2541no—
283C42—2541no—
284C42—2541no—
285C42—2541no—
286C42—2541no—
287C42—2541no—
288C42C22×D12711yescharacteristic
289C23—21no—
290C23—1271no—
291Q8—21no—
292Q8—1272no—
293Q8—1272no—
294Q8—2544no—
295Q8—2541no—
296Q8—2541no—
297Q8C2×C4×D12712yes—
298D4—21no—
299D4—2544no—
300D4C2×C4×D12712yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—1271no—
306C2×C4—1272no—
307C2×C4—1272no—
308C2×C4—1271no—
309C2×C4—1271no—
310C2×C4—10161no—
311C2×C4—10161no—
312C2×C4—2544no—
313C2×C4—2544no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4D4×D12711yescharacteristic
319C2×C4D4×D12711yescharacteristic
320C2×C4C23×D12711yescharacteristic
321C8—21no—
322C8—1272no—
323C8—2541no—
324C8C2×C4×D12712yes—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—1272no—
329C22—2541no—
330C22—2541no—
331C22C2×D4×D12711yescharacteristic
332C4—22no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—1272no—
338C4—5081no—
339C4—5081no—
340C4—2542no—
341C4—2542no—
342C4—2541no—
343C4—2541no—
344C4C22×C4×D12711yescharacteristic
345C4#41×D12711yescharacteristic
346C2—42no—
347C2—2542no—
348C2C4×D4×D12711yescharacteristic
349C2#2255068611yescharacteristic
350C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
Fitting#440×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523247C22
cosocleC2×C4×C127C24
hypercenter#440D127
hypocenterC127#2947
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
3. #22550352C2semidirect
4. #22549984C2semidirect
5. #22549980C2semidirect
6. #22549973C2semidirect
7. #22549970C2semidirect
8. C4×#14523238C2semidirect
9. SD16×Dic127C2semidirect
10. C4×#14523236C2semidirect
11. #22549917C2semidirect
13. #103×D127C2semidirect
14. #22549833C2semidirect
15. #440×C127C2semidirect
18. #16861238C4semidirect
66. Q8×D127C2×C4semidirect
73. Dic508C2×C4semidirect
74. #14523238C2×C4semidirect
75. #14523236C2×C4semidirect
76. #14523231C2×C4semidirect
104. SD16×C127C2×C4semidirect
176. C127#2947semidirect
183. #440D127semidirect
255. SD16C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
#225503552
#225503522
#225499842
#225499802
#225499732
#225499702
C4×#145232382
SD16×Dic1272
C4×#145232362
#225499172
#225498692
#103×D1272
#225498332
#440×C1272
#2947127