Finite Group Databasebeta

← #30638728

Subgroups — #30638728

Order 16256.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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×D4×Dic127C211yesmaximal, characteristic
2C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic
4#22549745C211yesmaximal, characteristic, complemented
5#22549639C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C23×Dic127—21no—
9C22×C127⋊C8C2211yescharacteristic
10D4×Dic127—22no—
11D4×Dic127C414yescomplemented
12C2×C4⋊Dic127C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—21no—
15C2×Dic127⋊C4—21no—
16C2×C22⋊Dic127—21no—
17C2×C4×Dic127C2211yescharacteristic
18Dic127⋊C8—22no—
19#16861130—22no—
20C22×D4×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#102×C127—22no—
24C22×Dic127—24no—
25C22×Dic127—21no—
26C22×Dic127—21no—
27C22×Dic127—42no—
28C22×Dic127—42no—
29C22×Dic127D411yescharacteristic
30C22×Dic127D411yescharacteristic
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127C2×C412yes—
38Dic127⋊C4—42no—
39Dic127⋊C4—42no—
40C22⋊Dic127—24no—
41C4×Dic127C2×C412yes—
42C4×Dic127D412yes—
43C24×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127C2×C412yes—
48C2×D4×C127D412yes—
49C2×D4×C127Q812yes—
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C2×C8×C127—21no—
53C2×C8×C127—21no—
54C2×C8×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127—21no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—22no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C2×Dic127—48no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C2×Dic127C22⋊C412yes—
67C2×Dic127C22⋊C412yes—
68C2×Dic127D812yes—
69C2×Dic127SD1612yes—
70C127⋊C8—41no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127—41no—
75C23×C127—41no—
76C23×C127—41no—
77C23×C127—41no—
78C23×C127C2×D411yescharacteristic
79D4×C127—22no—
80D4×C127—22no—
81D4×C127—22no—
82D4×C127C4⋊C414yescomplemented
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127C22×C411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×Q811yescharacteristic
90C2×C4×C127#4111yescharacteristic
91C2×C4×C127#4111yescharacteristic
92C8×C127—41no—
93C22×C127—24no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127—42no—
99C22×C127—42no—
100C22×C127C2×C22⋊C411yescharacteristic
101C22×C127#12111yescharacteristic
102C22×C127C22⋊Q811yescharacteristic
103C22×C127C2×D811yescharacteristic
104C22×C127C2×SD1611yescharacteristic
105C22×C127AGL(1,ℤ/8)11yescharacteristic
106C22×C127#13711yescharacteristic
107C4×C127—81no—
108C4×C127C2×C4⋊C411yescharacteristic
109C4×C127#11711yescharacteristic
110C4×C127#12111yescharacteristic
111C4×C127C22⋊Q811yescharacteristic
112Dic127—22no—
113Dic127—42no—
114Dic127—81no—
115Dic127#10214yescomplemented
116C2×C127—44no—
117C2×C127#38011yescharacteristic
118C2×C127C2×#10211yescharacteristic
119C2×C127#41711yescharacteristic
120C2×C127#45911yescharacteristic
121C2×C127#46111yescharacteristic
122C2×C127#47411yescharacteristic
123C2×C127#47611yescharacteristic
124#1802—1271nomaximal, Sylow
125C127#180211yescharacteristic, complemented, Sylow
126C2×C4×D4—1271no—
127C2×C4⋊C8—1271no—
128C2×#102—1271no—
129C2×#102D12711yescharacteristic
130#384—1271no—
131#340—1271no—
132#340—1271no—
133C22×D4C2×D12711yescharacteristic
134C23×C4—2541no—
135C22×C8—1271no—
136C22×C8C2×D12711yescharacteristic
137C4×D4—1274no—
138C4×D4—2542no—
139C2×C4⋊C4—1271no—
140C2×C4⋊C4—1271no—
141C2×C4⋊C4—2541no—
142C2×C4⋊C4—2541no—
143C2×C4⋊C4C2×D12711yescharacteristic
144C2×C22⋊C4—2541no—
145C2×C42—1271no—
146C4⋊C8—2542no—
147#102—22no—
148#102—2542no—
149C24—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C4×D12712yes—
154C2×D4C22⋊D12712yes—
155C2×D4Dic25412yes—
156C22×C4—21no—
157C22×C4—1271no—
158C22×C4—1271no—
159C22×C4—5082no—
160C22×C4—5082no—
161C22×C4—2544no—
162C22×C4—2541no—
163C22×C4—2541no—
164C22×C4C22×D12711yescharacteristic
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—2541no—
169C2×C8—2541no—
170C2×C8—2541no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—1272no—
174C4⋊C4—5082no—
175C4⋊C4—5082no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C4⋊C4—2541no—
179C22⋊C4—2544no—
180C42—1272no—
181C42—1272no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23D4×D12711yescharacteristic
190D4—22no—
191D4—22no—
192D4—22no—
193D4Dic127⋊C414yescomplemented
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—1272no—
198C2×C4—1272no—
199C2×C4—1272no—
200C2×C4—1272no—
201C2×C4—5088no—
202C2×C4—5081no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—5081no—
206C2×C4—5081no—
207C2×C4—2542no—
208C2×C4—2541no—
209C2×C4—2541no—
210C2×C4C2×C4×D12711yescharacteristic
211C2×C4D4×D12711yescharacteristic
212C2×C4C2×C22⋊D12711yescharacteristic
213C2×C4C2×Dic25411yescharacteristic
214C2×C4#1452324711yescharacteristic
215C2×C4#1452324711yescharacteristic
216C8—41no—
217C8—5081no—
218C22—24no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22C22⋊C4×D12711yescharacteristic
226C22#1686118111yescharacteristic
227C22#1686118611yescharacteristic
228C22D8×D12711yescharacteristic
229C22SD16×D12711yescharacteristic
230C22#1686123311yescharacteristic
231C22#1686123811yescharacteristic
232C4—81no—
233C4—1274no—
234C4—10161no—
235C4—5082no—
236C4—2542no—
237C4C2×Dic127⋊C411yescharacteristic
238C4#1686116511yescharacteristic
239C4#1686118111yescharacteristic
240C4#1686118611yescharacteristic
241C2—44no—
242C2#2254973411yescharacteristic
243C2#102×D12711yescharacteristic
244C2#2254987411yescharacteristic
245C2#2255006911yescharacteristic
246C2#2255007911yescharacteristic
247C2#2255015411yescharacteristic
248C2#2255016411yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1802
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×Dic127⋊C8C2semidirect
4. #22549745C2semidirect
5. #22549639C2semidirect
6. #22549638C2semidirect
11. D4×Dic127C4semidirect
82. D4×C127C4⋊C4semidirect
115. Dic127#102semidirect
125. C127#1802semidirect
193. D4Dic127⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
C2×Dic127⋊C82
C2×#168611302
#225497452
#225496392
#225496382
C2×#102×C1272
#1802127