Finite Group Databasebeta

← #30639044

Subgroups — #30639044

Order 16256.

456 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16861180C211yesmaximal, characteristic
2C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic, complemented
4C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
5#22549639C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127—21no—
9C22×C4×D127—21no—
10C22×C4×D127C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861180—22no—
13#16861180C414yescomplemented
14C2×C4⋊Dic127C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4—21no—
17C4⋊C4×D127—22no—
18C4⋊C4×D127—22no—
19C2×C22⋊Dic127—21no—
20#16861130—22no—
21(C2×D127)⋊C8—22no—
22C22×D4×C127C2211yescharacteristic
23C22×C8×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25#102×C127—22no—
26C23×D127D411yescharacteristic
27C2×C22⋊D127—24no—
28C2×C22⋊D127—42no—
29C2×C22⋊D127—42no—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C2×C4×D127—24no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37C2×C4×D127—21no—
38C2×C4×D127—42no—
39C2×C4×D127—42no—
40C2×C4×D127C2×C412yes—
41C2×C4×D127D412yes—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C4⋊Dic127—21no—
46C4⋊Dic127—21no—
47C4⋊Dic127C2×C412yescomplemented
48C4⋊Dic127D412yescomplemented
49Dic127⋊C4—24no—
50C22⋊Dic127—24no—
51C24×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127C2×C412yescomplemented
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60C2×C8×C127—21no—
61C2×C8×C127—21no—
62C2×C8×C127—21no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127D412yescomplemented
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—21no—
68C22×D127—21no—
69C22×D127C22⋊C412yes—
70C22×D127D812yes—
71C22×D127SD1612yes—
72C22⋊D127—48no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C2×Dic127—41no—
81C2×Dic127C22⋊C412yes—
82C4×D127—22no—
83C4×D127—22no—
84C4×D127—22no—
85C4×D127—22no—
86C4×D127—48no—
87C127⋊C8—41no—
88C23×C127—22no—
89C23×C127—41no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127—41no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127C2×D411yescharacteristic
96D4×C127—22no—
97D4×C127—22no—
98D4×C127—41no—
99D4×C127—41no—
100C2×C4×C127—22no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127C22×C411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127#4111yescharacteristic
108C2×C4×C127#4111yescharacteristic
109C8×C127—41no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127—42no—
113C22×C127—81no—
114C22×C127—81no—
115C22×C127—81no—
116C22×C127—81no—
117C22×C127C2×C22⋊C411yescharacteristic
118C22×C127#12011yescharacteristic
119C22×C127#12311yescharacteristic
120C22×C127C2×D811yescharacteristic
121C22×C127C2×SD1611yescharacteristic
122C22×C127AGL(1,ℤ/8)11yescharacteristic
123C22×C127#13711yescharacteristic
124C2×D127—22no—
125C2×D127—22no—
126C2×D127—22no—
127C2×D127—22no—
128C2×D127—22no—
129C2×D127—22no—
130C2×D127#10214yescomplemented
131C4×C127—42no—
132C4×C127C4×D411yescharacteristic
133C4×C127C4×D411yescharacteristic
134C4×C127#12111yescharacteristic
135C4×C127#12111yescharacteristic
136Dic127—22no—
137Dic127—42no—
138Dic127—42no—
139C2×C127—82no—
140C2×C127#38611yescharacteristic
141C2×C127C2×#10211yescharacteristic
142C2×C127#41711yescharacteristic
143C2×C127#44711yescharacteristic
144C2×C127#44811yescharacteristic
145C2×C127#48011yescharacteristic
146C2×C127#48111yescharacteristic
147D127—24no—
148#1874—1271nomaximal, Sylow
149C127#187411yescharacteristic, complemented, Sylow
150C2×#121—1271no—
151C22×C4⋊C4—1271no—
152C2×#102—1271no—
153C2×#102D12711yescharacteristic, complemented
154C2×C22⋊C8—1271no—
155#340—1271no—
156#340—1271no—
157C22×D4—2541no—
158C22×D4C2×D12711yescharacteristic
159C23×C4—1271no—
160C23×C4—2541no—
161C22×C8—1271no—
162C22×C8C2×D12711yescharacteristic
163#121—1274no—
164#121—2542no—
165C2×C4⋊C4—1271no—
166C2×C4⋊C4—1271no—
167C2×C4⋊C4—2542no—
168C2×C4⋊C4—2541no—
169C2×C4⋊C4—2542no—
170C2×C4⋊C4C2×D12711yescharacteristic
171C2×C22⋊C4—2541no—
172#102—22no—
173#102—2542no—
174C22⋊C8—2542no—
175C24—21no—
176C24—1271no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—5082no—
183C2×D4—5082no—
184C2×D4—2544no—
185C2×D4C4×D12712yescomplemented
186C22×C4—21no—
187C22×C4—1271no—
188C22×C4—1272no—
189C22×C4—1272no—
190C22×C4—5082no—
191C22×C4—5082no—
192C22×C4—2544no—
193C22×C4—2541no—
194C22×C4—2541no—
195C22×C4—2541no—
196C22×C4—2541no—
197C22×C4—2541no—
198C22×C4—2541no—
199C22×C4C22×D12711yescharacteristic
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—2541no—
204C2×C8—2541no—
205C2×C8—2541no—
206C4⋊C4—21no—
207C4⋊C4—1272no—
208C4⋊C4—1272no—
209C4⋊C4—2544no—
210C4⋊C4—2541no—
211C4⋊C4—2541no—
212C4⋊C4C22⋊D12712yescomplemented
213C22⋊C4—2544no—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—1272no—
222C23—1272no—
223C23—1272no—
224C23—2541no—
225C23—2541no—
226C23—2541no—
227C23—2541no—
228C23D4×D12711yescharacteristic
229D4—22no—
230D4—22no—
231D4—41no—
232D4—41no—
233D4—5088no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—1272no—
238C2×C4—5088no—
239C2×C4—5081no—
240C2×C4—5081no—
241C2×C4—5081no—
242C2×C4—5081no—
243C2×C4—2542no—
244C2×C4—2542no—
245C2×C4—2541no—
246C2×C4—2541no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4—2542no—
251C2×C4C2×C4×D12711yescharacteristic
252C2×C4D4×D12711yescharacteristic
253C2×C4D4×D12711yescharacteristic
254C2×C4C2×C22⋊D12711yescharacteristic
255C2×C4#1452324711yescharacteristic
256C2×C4#1452324811yescharacteristic
257C8—41no—
258C8—5081no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22—1274no—
267C22—2542no—
268C22—2542no—
269C22—2542no—
270C22—2542no—
271C22—2542no—
272C22—2542no—
273C22C22⋊C4×D12711yescharacteristic
274C22#1686117711yescharacteristic
275C22#1686119111yescharacteristic
276C22D8×D12711yescharacteristic
277C22SD16×D12711yescharacteristic
278C22#1686123311yescharacteristic
279C22#1686123811yescharacteristic
280C4—42no—
281C4—5082no—
282C4—5082no—
283C4—2542no—
284C4(C22⋊D127)⋊C411yescharacteristic
285C4C4×C22⋊D12711yescharacteristic
286C4#1686118011yescharacteristic
287C4#1686118211yescharacteristic
288C2—82no—
289C2—2544no—
290C2#2254975211yescharacteristic
291C2#102×D12711yescharacteristic
292C2#2254987411yescharacteristic
293C2#2255000811yescharacteristic
294C2#2255001311yescharacteristic
295C2#2255018811yescharacteristic
296C2#2255019311yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861180C2
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1874
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×C4⋊C4×D127C2semidirect
3. C2×#16861130C2semidirect
4. C2×(C2×D127)⋊C8C2semidirect
5. #22549639C2semidirect
6. #22549638C2semidirect
7. C2×#102×C127C2semidirect
13. #16861180C4semidirect
47. C4⋊Dic127C2×C4semidirect
48. C4⋊Dic127D4semidirect
57. C2×D4×C127C2×C4semidirect
64. C4⋊C4×C127D4semidirect
130. C2×D127#102semidirect
149. C127#1874semidirect
153. C2×#102D127semidirect
185. C2×D4C4×D127semidirect
212. C4⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611802
C2×C4⋊C4×D1272
C2×#168611302
C2×(C2×D127)⋊C82
#225496392
#225496382
C2×#102×C1272
#1874127