Finite Group Databasebeta

← #30649284

Subgroups — #30649284

Order 16256.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550428C211yesmaximal, characteristic, complemented
2#22550421C211yesmaximal, characteristic
3#22550273C211yesmaximal, characteristic, complemented
4#22550247C211yesmaximal, characteristic, complemented
5#22550225C211yesmaximal, characteristic, complemented
6#22550221C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550103C211yesmaximal, characteristic, complemented
10#22549967C211yesmaximal, characteristic, complemented
11SD16×Dic127C211yesmaximal, characteristic, complemented
12#22549963C211yesmaximal, characteristic, complemented
13#22549961C211yesmaximal, characteristic, complemented
14C2×C8×Dic127C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×#14523237C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C22×C127⋊C8C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861208C2211yescharacteristic
23#16861207C2211yescharacteristic
24#16861205C2211yescharacteristic
25#16861199—21no—
26#16861195—21no—
27#16861188C2211yescharacteristic
28#16861186—21no—
29#16861183C2211yescharacteristic
30#16861181—21no—
31Q8×Dic127C2211yescharacteristic
32#16861174—21no—
33#16861174C2211yescharacteristic
34D4×Dic127C2211yescharacteristic
35#16861171C2211yescharacteristic
36(C22⋊D127)⋊C4—21no—
37#16861165—21no—
38C2×C4×Dic127C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861140C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861132C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861129C2211yescharacteristic
48C8×Dic127C2211yescharacteristic
49C8×Dic127C2211yescharacteristic
50C8×Dic127C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#106×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63C2×D508C2311yescharacteristic
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C22×Dic127—21no—
68C22×Dic127D411yescharacteristic
69C22×Dic127D411yescharacteristic
70C2×C4×D127—21no—
71#14523237—22no—
72#14523236—22no—
73#14523235—22no—
74C2×C127⋊C8—22no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C4⋊Dic127—21no—
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90C4×Dic127—21no—
91C4×Dic127—21no—
92C4×Dic127D411yescharacteristic
93C4×Dic127D411yescharacteristic
94C4×Dic127C2311yescharacteristic
95C4×Dic127C2311yescharacteristic
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127C2311yescharacteristic
100SD16×C127—22no—
101C2×C8×C127—22no—
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C4⋊C4×C127—21no—
105C4⋊C4×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22⋊C4×C127—21no—
109C22×D127—21no—
110Dic254—21no—
111Dic254—21no—
112Dic254—42no—
113Dic254—42no—
114D508—21no—
115D508—21no—
116C22⋊D127—42no—
117C22⋊D127—42no—
118C22⋊D127—42no—
119C2×Dic127—22no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—42no—
124C2×Dic127SD1614yescomplemented
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—42no—
130C127⋊C8—21no—
131C127⋊C8#4112yescomplemented
132C23×C127—21no—
133C23×C127C2×D411yescharacteristic
134Q8×C127—21no—
135Q8×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—42no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127#4112yescomplemented
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127C2×SD1612yes—
154C22×C127C22×D411yescharacteristic
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C4×C127—21no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127C2×#4111yescharacteristic
163C4×C127C2×#4111yescharacteristic
164Dic127—21no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168Dic127C2×SD1612yes—
169Dic127C2×SD1612yes—
170Dic127#13512yes—
171C2×C127—22no—
172C2×C127—81no—
173C2×C127#53211yescharacteristic
174C2×C127C22×SD1611yescharacteristic
175C2×C127C2×#13511yescharacteristic
176D127—81no—
177#3167—1271nomaximal, Sylow
178C127#316711yescharacteristic, complemented, Sylow
179#533—1271no—
180#532—1271no—
181#498—1271no—
182#492—1271no—
183#487—1271no—
184#486—1271no—
185#465—1271no—
186#465—1271no—
187#465—1271no—
188#465D12711yescharacteristic, complemented
189C4×SD16—1271no—
190C4×SD16—1271no—
191C4×SD16—1271no—
192C4×SD16—1271no—
193C2×C4×C8—1271no—
194C2×#41—2541no—
195C2×SD16—1271no—
196C2×SD16—1271no—
197C2×SD16—1271no—
198C2×SD16C2×D12711yescharacteristic
199C22×C8—1271no—
200C22×C8C2×D12711yescharacteristic
201C4⋊Q8—1271no—
202C4⋊Q8—1271no—
203C4⋊Q8—1271no—
204#127—1271no—
205#125—2541no—
206#124—2541no—
207C22⋊Q8—1271no—
208C22⋊Q8—2541no—
209C22⋊Q8C2×D12711yescharacteristic
210#121—1271no—
211#121—2541no—
212#121C2×D12711yescharacteristic
213C4×Q8—1271no—
214C4×Q8—1271no—
215C4×Q8—2541no—
216C4×D4—1271no—
217C4×D4—1271no—
218C4×D4—2541no—
219#117—2541no—
220C2×C42—1271no—
221#106—1271no—
222#106—1271no—
223#106—1271no—
224#106C2×D12711yescharacteristic
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#103C2×D12711yescharacteristic
229#102—1271no—
230#102—1271no—
231#102—1271no—
232#102C2×D12711yescharacteristic
233C4×C8—1271no—
234C4×C8—1271no—
235C4×C8—1271no—
236C4×C8—1271no—
237#41—5084no—
238C2×Q8—1271no—
239C2×Q8—2541no—
240C2×Q8—2541no—
241C2×Q8C22×D12711yescharacteristic
242C2×D4—21no—
243C2×D4—1271no—
244C2×D4—2541no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4C22×D12711yescharacteristic
248C22×C4—1271no—
249C22×C4—1271no—
250C22×C4—2541no—
251C22×C4—2541no—
252C22×C4C22×D12711yescharacteristic
253SD16—22no—
254SD16—2542no—
255SD16—2542no—
256SD16—2542no—
257C2×C8—22no—
258C2×C8—1271no—
259C2×C8—1271no—
260C2×C8—2542no—
261C2×C8C22×D12711yescharacteristic
262C2×C8C22×D12711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—2541no—
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—21no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C42—1271no—
282C42—1271no—
283C42—1271no—
284C42—1271no—
285C42—2541no—
286C42—2541no—
287C23—21no—
288C23—2541no—
289C23D4×D12711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5082no—
293Q8—5082no—
294Q8—2541no—
295Q8—2541no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—5082no—
300D4—5082no—
301D4—5082no—
302D4—2541no—
303D4—2541no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—1274no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—1271no—
314C2×C4—5082no—
315C2×C4—5082no—
316C2×C4—2542no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4D4×D12711yescharacteristic
321C2×C4C23×D12711yescharacteristic
322C8—21no—
323C8—1272no—
324C8—2541no—
325C8#1452324712yescomplemented
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—5081no—
332C22—5081no—
333C22—5081no—
334C22SD16×D12712yes—
335C22C2×D4×D12711yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—1272no—
341C4—1272no—
342C4—1272no—
343C4—5081no—
344C4—5081no—
345C4—5081no—
346C4—2541no—
347C4#41×D12711yescharacteristic
348C4C2×#1452324711yescharacteristic
349C2—22no—
350C2—81no—
351C2—10161no—
352C2#2255042111yescharacteristic
353C2C2×SD16×D12711yescharacteristic
354C2#135×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
prime radical#22550421C2
hypercenter#465D127
hypocenterC127#3167
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22550428C2semidirect
3. #22550273C2semidirect
4. #22550247C2semidirect
5. #22550225C2semidirect
6. #22550221C2semidirect
7. #22550107C2semidirect
8. #22550105C2semidirect
9. #22550103C2semidirect
10. #22549967C2semidirect
11. SD16×Dic127C2semidirect
12. #22549963C2semidirect
13. #22549961C2semidirect
14. C2×C8×Dic127C2semidirect
15. #465×C127C2semidirect
124. C2×Dic127SD16semidirect
131. C127⋊C8#41semidirect
147. C8×C127#41semidirect
178. C127#3167semidirect
188. #465D127semidirect
325. C8#14523247semidirect

Maximal subgroups

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

subgroupindex
#225504282
#225504212
#225502732
#225502472
#225502252
#225502212
#225501072
#225501052
#225501032
#225499672
SD16×Dic1272
#225499632
#225499612
C2×C8×Dic1272
#465×C1272
#3167127