Finite Group Databasebeta

← #30649286

Subgroups — #30649286

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#22550430C211yesmaximal, characteristic, complemented
2#22550423C211yesmaximal, characteristic
3#22550273C211yesmaximal, characteristic, complemented
4#22550247C211yesmaximal, characteristic, complemented
5#22550225C211yesmaximal, characteristic, complemented
6#22550221C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550106C211yesmaximal, characteristic, complemented
9#22550104C211yesmaximal, characteristic, complemented
10#22550102C211yesmaximal, characteristic, complemented
11C4×#14523237C211yesmaximal, characteristic, complemented
12C4×#14523235C211yesmaximal, characteristic, complemented
13C4×#14523236C211yesmaximal, characteristic, complemented
14C4×C8×D127C211yesmaximal, characteristic, complemented
15C4×SD16×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C2×#14523237C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22#16861208C2211yescharacteristic
23#16861208C2211yescharacteristic
24#16861207C2211yescharacteristic
25#16861205C2211yescharacteristic
26#16861199—21no—
27#16861195—21no—
28#16861189C2211yescharacteristic
29#16861187—21no—
30#16861185C2211yescharacteristic
31#16861184C2211yescharacteristic
32#16861182—21no—
33#16861180C2211yescharacteristic
34C4×Dic254—21no—
35C4×Dic254C2211yescharacteristic
36C4×D508C2211yescharacteristic
37C4×C22⋊D127—21no—
38#16861166—21no—
39C42×D127C2211yescharacteristic
40#16861141C2211yescharacteristic
41#16861140C2211yescharacteristic
42#16861140C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49C4×C127⋊C8C2211yescharacteristic
50C8×Dic127C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C4×Q8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C4×C8×C127C2211yescharacteristic
59#14523248—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127D411yescharacteristic
70C2×C4×D127D411yescharacteristic
71C2×C4×D127C2311yescharacteristic
72#14523237—22no—
73#14523236—22no—
74#14523235—22no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C8×D127—22no—
78C8×D127—22no—
79C4⋊Dic127—21no—
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
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—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—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—21no—
112Dic254—21no—
113Dic254—42no—
114Dic254—42no—
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—21no—
128C4×D127—21no—
129C4×D127—22no—
130C4×D127—42no—
131C4×D127SD1614yescomplemented
132C127⋊C8—21no—
133C127⋊C8#4112yescomplemented
134C23×C127—21no—
135Q8×C127—21no—
136Q8×C127—21no—
137D4×C127—21no—
138D4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—42no—
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127#4112yescomplemented
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127C22×D411yescharacteristic
152C2×D127—21no—
153C2×D127—21no—
154C2×D127—41no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127C2×SD1612yes—
158C4×C127—21no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127C2×SD1612yes—
163C4×C127C2×#4111yescharacteristic
164C4×C127C2×#4111yescharacteristic
165Dic127—21no—
166Dic127—41no—
167Dic127—41no—
168Dic127—41no—
169Dic127C2×SD1612yes—
170Dic127#13512yes—
171C2×C127—81no—
172C2×C127#53211yescharacteristic
173C2×C127C22×SD1611yescharacteristic
174C2×C127C2×#13511yescharacteristic
175D127—22no—
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#465—1271no—
189C4×SD16—1271no—
190C4×SD16—1271no—
191C4×SD16—1271no—
192C4×SD16D12711yescharacteristic, complemented
193C2×C4×C8—1271no—
194C2×#41—2541no—
195C2×SD16—1271no—
196C2×SD16—1271no—
197C2×SD16—1271no—
198C2×SD16C2×D12711yescharacteristic
199C22×C8—1271no—
200C22×C8—1271no—
201C4⋊Q8—1271no—
202C4⋊Q8—1271no—
203C4⋊Q8—1271no—
204#127—1271no—
205#125—2541no—
206#124—2541no—
207C22⋊Q8—1271no—
208C22⋊Q8—1271no—
209C22⋊Q8—2541no—
210#121—1271no—
211#121—1271no—
212#121—2541no—
213C4×Q8—1271no—
214C4×Q8—2541no—
215C4×Q8C2×D12711yescharacteristic
216C4×D4—1271no—
217C4×D4—2541no—
218C4×D4C2×D12711yescharacteristic
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×C8C2×D12711yescharacteristic
237#41—5084no—
238C2×Q8—1271no—
239C2×Q8—2541no—
240C2×Q8—2541no—
241C2×Q8C22×D12711yescharacteristic
242C2×D4—1271no—
243C2×D4—2541no—
244C2×D4—2541no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4C22×D12711yescharacteristic
248C22×C4—21no—
249C22×C4—1271no—
250C22×C4—1271no—
251C22×C4—1271no—
252C22×C4—2541no—
253SD16—22no—
254SD16—2542no—
255SD16—2542no—
256SD16—2542no—
257C2×C8—1271no—
258C2×C8—1271no—
259C2×C8—2542no—
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—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C42—21no—
282C42—1271no—
283C42—1271no—
284C42—1271no—
285C42—2541no—
286C42C22×D12711yescharacteristic
287C23—21no—
288C23—1271no—
289C23—2541no—
290Q8—21no—
291Q8—21no—
292Q8—5082no—
293Q8—5082no—
294Q8—2541no—
295Q8—2541no—
296D4—21no—
297D4—21no—
298D4—5082no—
299D4—5082no—
300D4—5082no—
301D4—5082no—
302D4—2541no—
303D4—2541no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—1271no—
309C2×C4—1271no—
310C2×C4—1274no—
311C2×C4—1271no—
312C2×C4—5082no—
313C2×C4—2542no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4D4×D12711yescharacteristic
320C2×C4D4×D12711yescharacteristic
321C2×C4C23×D12711yescharacteristic
322C8—21no—
323C8—1272no—
324C8—2541no—
325C8#1452324812yescomplemented
326C22—41no—
327C22—41no—
328C22—41no—
329C22—1272no—
330C22—5081no—
331C22—5081no—
332C22—5081no—
333C22—2541no—
334C22—2541no—
335C22C2×D4×D12711yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—1272no—
341C4—1272no—
342C4—5081no—
343C4—5081no—
344C4—5081no—
345C4—2541no—
346C4SD16×D12712yes—
347C4#41×D12711yescharacteristic
348C4C2×#1452324811yescharacteristic
349C2—81no—
350C2—10161no—
351C2—2542no—
352C2#2255042311yescharacteristic
353C2C2×SD16×D12711yescharacteristic
354C2#135×D12711yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
prime radical#22550423C2
hypercenterC4×SD16D127
hypocenterC127#3167
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #22550430C2semidirect
3. #22550273C2semidirect
4. #22550247C2semidirect
5. #22550225C2semidirect
6. #22550221C2semidirect
7. #22550108C2semidirect
8. #22550106C2semidirect
9. #22550104C2semidirect
10. #22550102C2semidirect
11. C4×#14523237C2semidirect
12. C4×#14523235C2semidirect
13. C4×#14523236C2semidirect
14. C4×C8×D127C2semidirect
15. C4×SD16×C127C2semidirect
131. C4×D127SD16semidirect
133. C127⋊C8#41semidirect
147. C8×C127#41semidirect
178. C127#3167semidirect
192. C4×SD16D127semidirect
325. C8#14523248semidirect

Maximal subgroups

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

subgroupindex
#225504302
#225504232
#225502732
#225502472
#225502252
#225502212
#225501082
#225501062
#225501042
#225501022
C4×#145232372
C4×#145232352
C4×#145232362
C4×C8×D1272
C4×SD16×C1272
#3167127