Finite Group Databasebeta

← #30648563

Subgroups — #30648563

Order 16256.

398 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 141 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#22550504C211yesmaximal, characteristic, complemented
2#22550351C211yesmaximal, characteristic
3#22550178C212yesmaximal, complemented
4#22550174C212yesmaximal, complemented
5#22550164C212yesmaximal, complemented
6#22550154C212yesmaximal, complemented
7#22549918C211yesmaximal, characteristic, complemented
8#22549912C211yesmaximal, characteristic, complemented
9#22549904C211yesmaximal, characteristic, complemented
10C2×#16861130C211yesmaximal, characteristic
11#418×C127C211yesmaximal, characteristic
12C22×C127⋊C8C2211yescharacteristic
13#16861209—21no—
14#16861208C2212yescomplemented
15#16861201—21no—
16#16861200C2212yescomplemented
17#16861186—22no—
18D4×Dic127—21no—
19D4×Dic127—21no—
20D4×Dic127C2212yes—
21D4×Dic127C2212yes—
22#16861168C2211yescharacteristic
23#16861168C2211yescharacteristic
24C2×C4⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127—22no—
26#16861143C2212yescomplemented
27#16861142C2211yescharacteristic, complemented
28#16861142C2211yescharacteristic, complemented
29#16861141C2212yescomplemented
30#16861140C2211yescharacteristic, complemented
31#16861140C2211yescharacteristic, complemented
32Dic127⋊C8C2212yescomplemented
33Dic127⋊C8C2212yescomplemented
34#16861130C2212yes—
35#16861130C2212yes—
36C22×D4×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38#117×C127C2211yescharacteristic
39#102×C127C2212yes—
40#102×C127C2212yes—
41C2×Dic254—22no—
42C22×Dic127—22no—
43C22×Dic127—22no—
44C22×Dic127—21no—
45C2×C127⋊C8—22no—
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2311yescharacteristic
48C4⋊Dic127—22no—
49C4⋊Dic127—21no—
50C4⋊Dic127D411yescharacteristic
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2312yes—
53C4⋊Dic127C2311yescharacteristic
54C4⋊Dic127C2311yescharacteristic
55Dic127⋊C4—22no—
56Dic127⋊C4—22no—
57C22⋊Dic127—22no—
58C22⋊Dic127—22no—
59C22⋊Dic127—21no—
60C22⋊Dic127D412yes—
61C4×Dic127—21no—
62C4×Dic127C2312yes—
63C24×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—22no—
66C2×D4×C127Q814yescomplemented
67C2×D4×C127C2312yes—
68C22×C4×C127C2311yescharacteristic
69M16×C127—22no—
70C2×C8×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—21no—
74C42×C127—21no—
75Dic254—82no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—22no—
79C2×Dic127—82no—
80C2×Dic127—82no—
81C2×Dic127—81no—
82C2×Dic127C2×D412yes—
83C2×Dic127C2×D412yes—
84C127⋊C8—21no—
85C127⋊C8—21no—
86C23×C127—22no—
87C23×C127—24no—
88C23×C127—81no—
89C23×C127—81no—
90C23×C127C2×D411yescharacteristic
91D4×C127—24no—
92D4×C127—22no—
93D4×C127C2×Q814yes—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×Q812yes—
98C2×C4×C127#4112yes—
99C2×C4×C127C2411yescharacteristic
100C8×C127—22no—
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—24no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127—84no—
107C22×C127C22⋊Q812yes—
108C22×C127C22×D411yescharacteristic
109C4×C127—41no—
110C4×C127—41no—
111C4×C127C22⋊Q812yes—
112C4×C127C22×Q811yescharacteristic
113C4×C127C2×#4111yescharacteristic
114Dic127—22no—
115Dic127—22no—
116Dic127—41no—
117Dic127—42no—
118Dic127—41no—
119C2×C127—22no—
120C2×C127—44no—
121C2×C127C2×C22⋊Q811yescharacteristic
122C2×C127#57611yescharacteristic
123C2×C127#57711yescharacteristic
124#3081—1271nomaximal, Sylow
125C127#308111yescharacteristic, complemented, Sylow
126#544—1271no—
127#518—1271no—
128#478—1272no—
129#478—1272no—
130#476—1272no—
131#474—1272no—
132#428—1271no—
133#427—1271no—
134#424—1271no—
135#418D12711yescharacteristic
136C2×#102—1271no—
137C22×D4C2×D12711yescharacteristic
138C2×M16C2×D12711yescharacteristic
139C22×C8—1271no—
140C4⋊Q8—1272no—
141C4⋊Q8—2541no—
142#125—1272no—
143#125—2541no—
144C22⋊Q8—2542no—
145C4×D4—1272no—
146C4×D4—1272no—
147C4×D4—2541no—
148C4×D4—2541no—
149#117—1271no—
150#117—1271no—
151#117C2×D12711yescharacteristic
152C2×C4⋊C4—1271no—
153C2×C22⋊C4—2542no—
154#107—1272no—
155#107—1271no—
156#107—1271no—
157#106—1272no—
158#106—1271no—
159#106—1271no—
160C4⋊C8—1272no—
161C4⋊C8—1272no—
162#102—1272no—
163#102—1272no—
164#102C2×D12712yes—
165#102C2×D12712yes—
166C24—21no—
167C2×Q8—2542no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic25414yescomplemented
171C2×D4C22×D12712yes—
172C22×C4—2542no—
173C22×C4—2542no—
174C22×C4—2541no—
175C22×C4C22×D12711yescharacteristic
176M16—22no—
177C2×C8—1271no—
178C2×C8—1271no—
179C2×C8—2542no—
180C2×C8C22×D12712yes—
181C4⋊C4—1271no—
182C4⋊C4—1271no—
183C4⋊C4—1272no—
184C4⋊C4—1271no—
185C4⋊C4—1271no—
186C4⋊C4—2542no—
187C4⋊C4—2542no—
188C4⋊C4—2542no—
189C4⋊C4—2541no—
190C4⋊C4C22×D12711yescharacteristic
191C4⋊C4C22×D12711yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—1272no—
194C22⋊C4—2542no—
195C22⋊C4—2542no—
196C22⋊C4—2541no—
197C42—21no—
198C42—1272no—
199C42—2541no—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23D4×D12711yescharacteristic
205Q8—10162no—
206D4—24no—
207D4—22no—
208D4C2×Dic25414yes—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—1272no—
212C2×C4—1272no—
213C2×C4—10161no—
214C2×C4—10162no—
215C2×C4—10162no—
216C2×C4—2541no—
217C2×C4—2541no—
218C2×C4—2542no—
219C2×C4D4×D12711yescharacteristic
220C2×C4C2×Dic25412yes—
221C2×C4#1452324712yes—
222C2×C4C23×D12711yescharacteristic
223C8—22no—
224C8—2541no—
225C8—2541no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#1686118612yes—
233C22C2×D4×D12711yescharacteristic
234C4—41no—
235C4—41no—
236C4—5081no—
237C4—5082no—
238C4—5081no—
239C4—2542no—
240C4—2542no—
241C4#1686118612yes—
242C4C22×Dic25411yescharacteristic
243C4C2×#1452324711yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#1686118611yescharacteristic
247C2#2255068011yescharacteristic
248C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
hypercenter#418D127
hypocenterC127#3081
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
1. #22550504C2semidirect
3. #22550178C2semidirect
4. #22550174C2semidirect
5. #22550164C2semidirect
6. #22550154C2semidirect
7. #22549918C2semidirect
8. #22549912C2semidirect
9. #22549904C2semidirect
14. #16861208C22semidirect
16. #16861200C22semidirect
26. #16861143C22semidirect
27. #16861142C22semidirect
28. #16861142C22semidirect
29. #16861141C22semidirect
30. #16861140C22semidirect
31. #16861140C22semidirect
32. Dic127⋊C8C22semidirect
33. Dic127⋊C8C22semidirect
66. C2×D4×C127Q8semidirect
125. C127#3081semidirect
170. C2×D4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225503512
#225501782
#225501742
#225501642
#225501542
#225499182
#225499122
#225499042
C2×#168611302
#418×C1272
#3081127