Finite Group Databasebeta

← #30648395

Subgroups — #30648395

Order 16256.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861228C211yesmaximal, characteristic, complemented
2C2×#16861215C211yesmaximal, characteristic, complemented
3#22550455C212yesmaximal, complemented
4#22550121C212yesmaximal, complemented
5#22550120C212yesmaximal, complemented
6#22550118C211yesmaximal, characteristic, complemented
7#22550117C211yesmaximal, characteristic, complemented
8#22550109C211yesmaximal, characteristic, complemented
9#107×D127C211yesmaximal, characteristic, complemented
10#22549874C212yesmaximal, complemented
11#466×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×#14523247C2212yes—
14#16861228—24no—
15C2×Dic508C2211yescharacteristic
16C2×#14523236C2212yes—
17#16861215—24no—
18C2×#14523232C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20C2×#14523231C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861189C2212yes—
24#16861188C2212yes—
25#16861186—22no—
26#16861182—22no—
27#16861181—22no—
28#16861180—22no—
29#16861180C2212yes—
30C4⋊C4×D127C2212yes—
31#16861143C2211yescharacteristic
32#16861142C2212yes—
33#16861134C2212yes—
34#16861132C2212yes—
35#16861130C2212yes—
36C2×D8×C127C2211yescharacteristic
37C22×C8×C127C2211yescharacteristic
38#121×C127C2212yes—
39#107×C127C2211yescharacteristic
40#102×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43#14523247—24no—
44#14523247—24no—
45C2×Dic254D411yescharacteristic
46C2×Dic254C2312yes—
47C2×D508D411yescharacteristic
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50C2×C22⋊D127D411yescharacteristic
51C2×C22⋊D127D411yescharacteristic
52C22×Dic127—22no—
53C2×C4×D127—22no—
54C2×C4×D127C2311yescharacteristic
55C2×C4×D127C2311yescharacteristic
56Dic508—22no—
57#14523236—24no—
58#14523232—22no—
59#14523231—22no—
60#14523231—22no—
61C2×C127⋊C8C2311yescharacteristic
62C2×C127⋊C8C2311yescharacteristic
63C8×D127—22no—
64C8×D127D414yescomplemented
65C4⋊Dic127—22no—
66C4⋊Dic127C2312yes—
67Dic127⋊C4—22no—
68Dic127⋊C4—22no—
69C22⋊Dic127—22no—
70C22⋊Dic127—22no—
71(C2×D127)⋊C4—22no—
72C2×D4×C127—22no—
73C2×D4×C127C2312yes—
74C22×C4×C127C2311yescharacteristic
75D8×C127—22no—
76C2×C8×C127—22no—
77C2×C8×C127C2311yescharacteristic
78C2×C8×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22×D127C2×D411yescharacteristic
82C22×D127C2×D411yescharacteristic
83Dic254—22no—
84Dic254—22no—
85Dic254—22no—
86D508—22no—
87C22⋊D127—22no—
88C22⋊D127—22no—
89C22⋊D127—44no—
90C22⋊D127—82no—
91C2×Dic127—22no—
92C2×Dic127—22no—
93C2×Dic127—44no—
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—82no—
99C4×D127C2×D412yes—
100C4×D127#4112yes—
101C127⋊C8—21no—
102C127⋊C8C2×D412yes—
103C23×C127—22no—
104C23×C127C2×D411yescharacteristic
105D4×C127—22no—
106D4×C127—22no—
107D4×C127—82no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C8×C127—21no—
114C8×C127C2×D412yes—
115C22×C127—21no—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127C22×D411yescharacteristic
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—21no—
125C2×D127—21no—
126C2×D127—21no—
127C2×D127#12112yes—
128C4×C127—21no—
129C4×C127—42no—
130C4×C127C22×D411yescharacteristic
131C4×C127C2×#4111yescharacteristic
132Dic127—21no—
133Dic127—42no—
134Dic127—42no—
135Dic127#12112yes—
136C2×C127—41no—
137C2×C127—82no—
138C2×C127C2×#12111yescharacteristic
139C2×C127#57611yescharacteristic
140C2×C127#57811yescharacteristic
141D127—22no—
142D127—41no—
143#3061—1271nomaximal, Sylow
144C127#306111yescharacteristic, complemented, Sylow
145C2×#135—1271no—
146C2×#131—1271no—
147#537—1272no—
148#468—1272no—
149#468—1272no—
150#467—1271no—
151#467—1271no—
152#466—1271no—
153#466D12711yescharacteristic, complemented
154C2×#107—1271no—
155#417—1272no—
156C2×#41—1271no—
157C2×#41—1272no—
158#135—2544no—
159C2×Q16—1271no—
160C2×SD16—1272no—
161C2×D8C2×D12711yescharacteristic
162#131—2544no—
163C2×M16—1271no—
164C2×M16—1271no—
165C2×M16—1271no—
166C22×C8—1271no—
167C22×C8—1271no—
168C22×C8C2×D12711yescharacteristic
169C22⋊Q8—1272no—
170C22⋊Q8—1272no—
171C22⋊Q8—2542no—
172#121—1272no—
173#121—2542no—
174#121—2542no—
175#121—2542no—
176#121C2×D12712yes—
177C2×C4⋊C4—1272no—
178#107—1271no—
179#107—1272no—
180#107C2×D12711yescharacteristic
181#103—1272no—
182#103—1272no—
183#102—1272no—
184#102C2×D12712yes—
185#41—2542no—
186#41—2542no—
187#41—2544no—
188#41—2544no—
189C2×Q8—1271no—
190C2×Q8—1272no—
191C2×D4—22no—
192C2×D4—1271no—
193C2×D4—1271no—
194C2×D4—1271no—
195C2×D4—2542no—
196C2×D4—2542no—
197C2×D4C22×D12712yes—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—2542no—
201C22×C4—2542no—
202C22×C4C22×D12711yescharacteristic
203Q16—2542no—
204SD16—2544no—
205D8—22no—
206M16—2542no—
207M16—2542no—
208M16—2542no—
209C2×C8—22no—
210C2×C8—1271no—
211C2×C8—1274no—
212C2×C8—1271no—
213C2×C8—2542no—
214C2×C8C22×D12711yescharacteristic
215C2×C8C22×D12711yescharacteristic
216C4⋊C4—1272no—
217C4⋊C4—2542no—
218C4⋊C4—2542no—
219C4⋊C4—2542no—
220C4⋊C4C22×D12712yes—
221C22⋊C4—22no—
222C22⋊C4—2542no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C23—22no—
226C23—1271no—
227C23—1271no—
228C23D4×D12711yescharacteristic
229Q8—2542no—
230Q8—2542no—
231Q8—2542no—
232D4—22no—
233D4—22no—
234D4—82no—
235D4—10162no—
236D4—5084no—
237D4—2542no—
238D4—2542no—
239D4—2542no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—1271no—
244C2×C4—1272no—
245C2×C4—1272no—
246C2×C4—1271no—
247C2×C4—10162no—
248C2×C4—5084no—
249C2×C4—2542no—
250C2×C4—2542no—
251C2×C4—2541no—
252C2×C4—2541no—
253C2×C4D4×D12711yescharacteristic
254C2×C4C23×D12711yescharacteristic
255C8—21no—
256C8—1272no—
257C8—2541no—
258C8D4×D12712yes—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—1272no—
266C22—2541no—
267C22—2541no—
268C22—2541no—
269C22—2541no—
270C22—2541no—
271C22C2×D4×D12711yescharacteristic
272C4—21no—
273C4—42no—
274C4—1272no—
275C4—5082no—
276C4—5082no—
277C4—2541no—
278C4C2×D4×D12711yescharacteristic
279C4#41×D12711yescharacteristic
280C2—41no—
281C2—82no—
282C2—5081no—
283C2—2542no—
284C2#121×D12711yescharacteristic
285C2#2255067711yescharacteristic
286C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#466×C127C2
socleC22×C127C22×D4
hypercenter#466D127
hypocenterC127#3061
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16861228C2semidirect
2. C2×#16861215C2semidirect
3. #22550455C2semidirect
4. #22550121C2semidirect
5. #22550120C2semidirect
6. #22550118C2semidirect
7. #22550117C2semidirect
8. #22550109C2semidirect
9. #107×D127C2semidirect
10. #22549874C2semidirect
11. #466×C127C2semidirect
64. C8×D127D4semidirect
144. C127#3061semidirect
153. #466D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612282
C2×#168612152
#225504552
#225501212
#225501202
#225501182
#225501172
#225501092
#107×D1272
#225498742
#466×C1272
#3061127