Finite Group Databasebeta

← #30649082

Subgroups — #30649082

Order 16256.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#22550507C211yesmaximal, characteristic, complemented
2#22550443C211yesmaximal, characteristic
3#22550239C212yesmaximal, complemented
4#22550234C212yesmaximal, complemented
5#22550229C212yesmaximal, complemented
6#22550221C212yesmaximal, complemented
7#22549881C211yesmaximal, characteristic, complemented
8#22549875C211yesmaximal, characteristic, complemented
9#22549873C211yesmaximal, characteristic, complemented
10#22549831C211yesmaximal, characteristic, complemented
11C2×#102×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14#16861208—21no—
15#16861208C2212yescomplemented
16#16861205C2212yes—
17#16861200—21no—
18#16861200C2212yescomplemented
19#16861196C2212yes—
20#16861188—22no—
21#16861180—21no—
22#16861180—21no—
23#16861178—22no—
24#16861168C2211yescharacteristic
25#16861168C2211yescharacteristic
26#16861134C2211yescharacteristic, complemented
27#16861134C2211yescharacteristic, complemented
28#16861132C2212yescomplemented
29#16861131C2211yescharacteristic, complemented
30#16861131C2211yescharacteristic, complemented
31#16861130C2212yes—
32#16861130C2212yes—
33#16861129C2212yescomplemented
34#16861119C2212yescomplemented
35C8×Dic127C2212yescomplemented
36C22×D4×C127C2211yescharacteristic
37C22×C8×C127C2211yescharacteristic
38C2×C4⋊C4×C127C2211yescharacteristic
39#102×C127C2212yes—
40#102×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254—22no—
44C2×Dic254C2311yescharacteristic
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—21no—
49C2×C4×D127—21no—
50#14523232—22no—
51C2×C127⋊C8C2312yes—
52C4⋊Dic127—22no—
53C4⋊Dic127D411yescharacteristic
54C4⋊Dic127D411yescharacteristic
55C4⋊Dic127C2311yescharacteristic
56C4⋊Dic127C2311yescharacteristic
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59C22⋊Dic127—22no—
60C22⋊Dic127—22no—
61C22⋊Dic127—21no—
62C22⋊Dic127D412yes—
63C4×Dic127—21no—
64C4×Dic127C2312yes—
65C24×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127—22no—
68C2×D4×C127—22no—
69C2×D4×C127C2312yes—
70C22×C4×C127—21no—
71C22×C4×C127C2311yescharacteristic
72C2×C8×C127—22no—
73C2×C8×C127C2311yescharacteristic
74C2×C8×C127C2311yescharacteristic
75C4⋊C4×C127—21no—
76C4⋊C4×C127C2312yes—
77C22×D127—21no—
78Dic254—21no—
79Dic254—21no—
80Dic254—82no—
81D508—21no—
82D508—21no—
83C22⋊D127—42no—
84C22⋊D127—82no—
85C22⋊D127—82no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127C2×D412yes—
90C2×Dic127C2×D412yes—
91C4×D127—42no—
92C127⋊C8—22no—
93C23×C127—22no—
94C23×C127—42no—
95C23×C127—81no—
96C23×C127—81no—
97C23×C127C2×D411yescharacteristic
98D4×C127—22no—
99D4×C127—22no—
100D4×C127—42no—
101C2×C4×C127—22no—
102C2×C4×C127—81no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127#4112yes—
105C2×C4×C127#4112yes—
106C2×C4×C127C2411yescharacteristic
107C8×C127—21no—
108C8×C127—21no—
109C22×C127—21no—
110C22×C127—21no—
111C22×C127—42no—
112C22×C127—42no—
113C22×C127—42no—
114C22×C127—84no—
115C22×C127#12412yes—
116C22×C127C22×D411yescharacteristic
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—41no—
120C4×C127—42no—
121C4×C127#12412yes—
122C4×C127C2×#4111yescharacteristic
123C4×C127C2×#4111yescharacteristic
124Dic127—22no—
125Dic127—22no—
126Dic127—41no—
127Dic127—41no—
128Dic127—41no—
129C2×C127—22no—
130C2×C127—82no—
131C2×C127C2×#12411yescharacteristic
132C2×C127#57611yescharacteristic
133C2×C127#57711yescharacteristic
134D127—81no—
135#3136—1271nomaximal, Sylow
136C127#313611yescharacteristic, complemented, Sylow
137#544—1271no—
138#535—1271no—
139#490—1272no—
140#489—1272no—
141#488—1272no—
142#486—1272no—
143#418—1271no—
144#417—1271no—
145#416—1271no—
146C2×#102D12711yescharacteristic, complemented
147#405—1271no—
148C2×#41—1271no—
149C22×D4C2×D12711yescharacteristic
150C2×M16—1271no—
151C22×C8C2×D12711yescharacteristic
152C4⋊Q8—1272no—
153C4⋊Q8—2541no—
154#127—1272no—
155#125—1272no—
156#125—2541no—
157#124—1272no—
158C22⋊Q8—2542no—
159#121—2541no—
160#121—2541no—
161#120—2542no—
162#117—1271no—
163#117—1271no—
164C2×C4⋊C4C2×D12711yescharacteristic
165#103—1271no—
166#103—1271no—
167#103—1272no—
168#102—1272no—
169#102—1272no—
170#102—1271no—
171#102—1271no—
172#102—1272no—
173#102C2×D12712yes—
174#102C2×D12712yes—
175#97—1272no—
176C4×C8—1272no—
177C24—21no—
178#41—2542no—
179#41—2542no—
180C2×Q8—1271no—
181C2×Q8—2542no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—1271no—
186C2×D4—2542no—
187C2×D4—2542no—
188C2×D4—2541no—
189C2×D4C22×D12712yes—
190C22×C4—21no—
191C22×C4—2541no—
192C22×C4C22×D12711yescharacteristic
193M16—2542no—
194C2×C8—22no—
195C2×C8—1272no—
196C2×C8C22×D12711yescharacteristic
197C2×C8C22×D12711yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—1271no—
200C4⋊C4—1271no—
201C4⋊C4—1271no—
202C4⋊C4—1271no—
203C4⋊C4—2542no—
204C4⋊C4—2542no—
205C4⋊C4—2542no—
206C4⋊C4C22×D12712yes—
207C22⋊C4—1272no—
208C22⋊C4—2542no—
209C22⋊C4—2542no—
210C22⋊C4—2541no—
211C42—1272no—
212C42—2541no—
213C23—22no—
214C23—42no—
215C23—81no—
216C23—81no—
217C23—2541no—
218C23D4×D12711yescharacteristic
219Q8—10162no—
220Q8—2541no—
221Q8—2541no—
222D4—22no—
223D4—22no—
224D4—42no—
225D4—10162no—
226D4—10162no—
227D4—5082no—
228D4—2541no—
229D4—2541no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—1272no—
233C2×C4—1272no—
234C2×C4—5082no—
235C2×C4—2541no—
236C2×C4—2541no—
237C2×C4—2541no—
238C2×C4D4×D12711yescharacteristic
239C2×C4#1452324712yes—
240C2×C4#1452324812yes—
241C2×C4C23×D12711yescharacteristic
242C8—21no—
243C8—21no—
244C8—2542no—
245C22—21no—
246C22—21no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—84no—
251C22—5081no—
252C22—5081no—
253C22—5081no—
254C22#1686119512yes—
255C22C2×D4×D12711yescharacteristic
256C4—42no—
257C4—5081no—
258C4—5081no—
259C4—5081no—
260C4—2542no—
261C4—2542no—
262C4#1686119512yes—
263C4C2×#1452324711yescharacteristic
264C4C2×#1452324811yescharacteristic
265C2—22no—
266C2—82no—
267C2—10161no—
268C2C2×#1686119511yescharacteristic
269C2#2255067711yescharacteristic
270C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC22×C127C22×D4
prime radical#22550443C2
hypercenterC2×#102D127
hypocenterC127#3136
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #22550507C2semidirect
3. #22550239C2semidirect
4. #22550234C2semidirect
5. #22550229C2semidirect
6. #22550221C2semidirect
7. #22549881C2semidirect
8. #22549875C2semidirect
9. #22549873C2semidirect
10. #22549831C2semidirect
11. C2×#102×C127C2semidirect
15. #16861208C22semidirect
18. #16861200C22semidirect
26. #16861134C22semidirect
27. #16861134C22semidirect
28. #16861132C22semidirect
29. #16861131C22semidirect
30. #16861131C22semidirect
33. #16861129C22semidirect
34. #16861119C22semidirect
35. C8×Dic127C22semidirect
136. C127#3136semidirect
146. C2×#102D127semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225504432
#225502392
#225502342
#225502292
#225502212
#225498812
#225498752
#225498732
#225498312
C2×#102×C1272
#3136127