Finite Group Databasebeta

← #30648686

Subgroups — #30648686

Order 16256.

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

normal subgroups only

This group has 418 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#22550443C211yesmaximal, characteristic
2#22550356C211yesmaximal, characteristic, complemented
3#22550199C212yesmaximal, complemented
4#22550198C212yesmaximal, complemented
5#22550193C212yesmaximal, complemented
6#22550188C212yesmaximal, complemented
7#22549918C211yesmaximal, characteristic, complemented
8#22549912C211yesmaximal, characteristic, complemented
9C2×#16861130C211yesmaximal, characteristic, complemented
10#22549844C211yesmaximal, characteristic, complemented
11#418×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861205—21no—
15#16861196—21no—
16#16861180C2212yes—
17#16861180C2212yes—
18#16861178—22no—
19C4×Dic254—21no—
20C4×D508—21no—
21(C22⋊D127)⋊C4—22no—
22#16861168C2211yescharacteristic
23#16861167C2212yescomplemented
24C2×C4⋊Dic127C2211yescharacteristic
25C4⋊C4×D127C2212yescomplemented
26#16861143C2212yescomplemented
27#16861142C2211yescharacteristic, complemented
28#16861142C2211yescharacteristic, complemented
29#16861141C2212yescomplemented
30#16861140C2211yescharacteristic, complemented
31#16861140C2211yescharacteristic, complemented
32#16861130C2212yes—
33#16861130C2212yes—
34(C2×D127)⋊C8C2212yescomplemented
35(C2×D127)⋊C8C2212yescomplemented
36C22×D4×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38#117×C127C2211yescharacteristic
39#102×C127C2212yes—
40#102×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254D411yescharacteristic
44C2×D508D411yescharacteristic
45C2×C22⋊D127—22no—
46C2×C22⋊D127—22no—
47C2×C22⋊D127D412yes—
48C22×Dic127—21no—
49C2×C4×D127—22no—
50C2×C4×D127C2312yes—
51C2×C127⋊C8—22no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C4⋊Dic127—21no—
55C4⋊Dic127C2312yes—
56C4⋊Dic127C2311yescharacteristic
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—22no—
59C22⋊Dic127—22no—
60C22⋊Dic127—22no—
61C22⋊Dic127—21no—
62(C2×D127)⋊C4—22no—
63C4×Dic127—22no—
64C4×Dic127—21no—
65C24×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127—22no—
68C2×D4×C127—22no—
69C2×D4×C127C2312yes—
70C22×C4×C127C2311yescharacteristic
71M16×C127—22no—
72C2×C8×C127C2312yes—
73C4⋊C4×C127C2311yescharacteristic
74C4⋊C4×C127C2311yescharacteristic
75C22⋊C4×C127—21no—
76C42×C127—21no—
77C22×D127C2×D412yes—
78Dic254—22no—
79D508—22no—
80C22⋊D127—24no—
81C22⋊D127—82no—
82C22⋊D127—82no—
83C2×Dic127—22no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—81no—
87C2×Dic127C2×D412yes—
88C4×D127—22no—
89C4×D127—22no—
90C4×D127—82no—
91C127⋊C8—21no—
92C127⋊C8—21no—
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—21no—
102C2×C4×C127—21no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127#4112yes—
105C2×C4×C127#4112yes—
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127—42no—
113C22×C127—84no—
114C22×C127#12312yes—
115C22×C127C22×D411yescharacteristic
116C2×D127—22no—
117C2×D127—22no—
118C2×D127—22no—
119C4×C127—41no—
120C4×C127—41no—
121C4×C127#12312yes—
122C4×C127C2×#4111yescharacteristic
123C4×C127C2×#4111yescharacteristic
124Dic127—22no—
125Dic127—42no—
126Dic127—41no—
127Dic127—41no—
128C2×C127—22no—
129C2×C127—82no—
130C2×C127C2×#12311yescharacteristic
131C2×C127#57611yescharacteristic
132C2×C127#57711yescharacteristic
133D127—42no—
134#3096—1271nomaximal, Sylow
135C127#309611yescharacteristic, complemented, Sylow
136#535—1271no—
137#520—1271no—
138#482—1272no—
139#482—1272no—
140#481—1272no—
141#480—1272no—
142#428—1271no—
143#427—1271no—
144#418D12711yescharacteristic, complemented
145C2×#102—1271no—
146#408—1271no—
147C2×#41—1271no—
148C22×D4C2×D12711yescharacteristic
149C2×M16C2×D12711yescharacteristic
150C22×C8—1271no—
151#127—2541no—
152#124—2541no—
153#121—1272no—
154#121—1272no—
155#120—2542no—
156C4×Q8—2541no—
157C4×D4—2541no—
158C4×D4—2542no—
159#117—1271no—
160#117—1272no—
161#117C2×D12711yescharacteristic
162C2×C4⋊C4—1271no—
163C2×C4⋊C4—1272no—
164#107—1272no—
165#107—1271no—
166#107—1271no—
167#106—1272no—
168#106—1271no—
169#106—1271no—
170#102—1272no—
171#102—1272no—
172#102C2×D12712yes—
173#102C2×D12712yes—
174C22⋊C8—1272no—
175C22⋊C8—1272no—
176C24—21no—
177#41—2542no—
178#41—2542no—
179C2×Q8—1271no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—1272no—
184C2×D4—1271no—
185C2×D4—2542no—
186C2×D4—2542no—
187C2×D4C22×D12712yes—
188C22×C4—1272no—
189C22×C4—2542no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192M16—22no—
193C2×C8—1271no—
194C2×C8—1271no—
195C2×C8—2542no—
196C2×C8C22×D12712yes—
197C4⋊C4—1272no—
198C4⋊C4—1271no—
199C4⋊C4—1271no—
200C4⋊C4—2542no—
201C4⋊C4—2541no—
202C4⋊C4C22×D12711yescharacteristic
203C4⋊C4C22×D12711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—2542no—
206C22⋊C4—2542no—
207C22⋊C4—2542no—
208C22⋊C4—2541no—
209C42—21no—
210C42—2542no—
211C42—2541no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1272no—
217C23D4×D12711yescharacteristic
218Q8—2542no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—10162no—
223D4—10162no—
224D4—2544no—
225D4—2542no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—1272no—
229C2×C4—10162no—
230C2×C4—10161no—
231C2×C4—2541no—
232C2×C4—2541no—
233C2×C4—2542no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4D4×D12711yescharacteristic
237C2×C4#1452324712yes—
238C2×C4#1452324812yes—
239C2×C4C23×D12711yescharacteristic
240C8—22no—
241C8—2541no—
242C8—2541no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—2542no—
250C22—2542no—
251C22—2542no—
252C22#1686119112yes—
253C22C2×D4×D12711yescharacteristic
254C4—41no—
255C4—41no—
256C4—5082no—
257C4—5081no—
258C4—5081no—
259C4—2542no—
260C4#1686119112yes—
261C4C2×#1452324711yescharacteristic
262C4C2×#1452324811yescharacteristic
263C2—22no—
264C2—82no—
265C2—5082no—
266C2C2×#1686119111yescharacteristic
267C2#2255068011yescharacteristic
268C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
prime radical#22550443C2
hypercenter#418D127
hypocenterC127#3096
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #22550356C2semidirect
3. #22550199C2semidirect
4. #22550198C2semidirect
5. #22550193C2semidirect
6. #22550188C2semidirect
7. #22549918C2semidirect
8. #22549912C2semidirect
9. C2×#16861130C2semidirect
10. #22549844C2semidirect
11. #418×C127C2semidirect
23. #16861167C22semidirect
25. C4⋊C4×D127C22semidirect
26. #16861143C22semidirect
27. #16861142C22semidirect
28. #16861142C22semidirect
29. #16861141C22semidirect
30. #16861140C22semidirect
31. #16861140C22semidirect
34. (C2×D127)⋊C8C22semidirect
35. (C2×D127)⋊C8C22semidirect
135. C127#3096semidirect
144. #418D127semidirect

Maximal subgroups

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

subgroupindex
#225504432
#225503562
#225501992
#225501982
#225501932
#225501882
#225499182
#225499122
C2×#168611302
#225498442
#418×C1272
#3096127