Finite Group Databasebeta

← #30639034

Subgroups — #30639034

Order 16256.

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

normal subgroups only

This group has 404 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×#16861185C211yesmaximal, characteristic
2C2×C4⋊C4×D127C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×(C2×D127)⋊C8C211yesmaximal, characteristic
5#22549639C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#103×C127C211yesmaximal, characteristic, complemented
8C22×C4×D127—21no—
9C22×C4×D127C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11#16861185—22no—
12#16861185C414yescomplemented
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4—21no—
16C2×Dic127⋊C4—21no—
17C4⋊C4×D127—22no—
18C4⋊C4×D127—22no—
19C2×(C2×D127)⋊C4—21no—
20#16861133—22no—
21(C2×D127)⋊C8—22no—
22C22×Q8×C127C2211yescharacteristic
23C22×C8×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25#103×C127—22no—
26C23×D127D411yescharacteristic
27C22×Dic127—21no—
28C22×Dic127—21no—
29C22×Dic127D411yescharacteristic
30C2×C4×D127—24no—
31C2×C4×D127—21no—
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34C2×C4×D127—21no—
35C2×C4×D127—42no—
36C2×C4×D127—42no—
37C2×C4×D127C2×C412yes—
38C2×C4×D127D412yes—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127C2×C412yescomplemented
45C4⋊Dic127D412yes—
46Dic127⋊C4—24no—
47Dic127⋊C4—24no—
48(C2×D127)⋊C4—24no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127C2×C412yescomplemented
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58C2×C8×C127—21no—
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C4⋊C4×C127—21no—
62C4⋊C4×C127D412yes—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127C22⋊C412yes—
68C22×D127SD1612yes—
69C22×D127Q1612yes—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C2×Dic127C22⋊C412yes—
79C4×D127—22no—
80C4×D127—22no—
81C4×D127—22no—
82C4×D127—22no—
83C4×D127—48no—
84C127⋊C8—41no—
85C23×C127C2×D411yescharacteristic
86Q8×C127—22no—
87Q8×C127—22no—
88Q8×C127—41no—
89Q8×C127—41no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127C22×C411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C8×C127—41no—
103C22×C127C2×C22⋊C411yescharacteristic
104C22×C127#12011yescharacteristic
105C22×C127#12311yescharacteristic
106C22×C127C2×SD1611yescharacteristic
107C22×C127C2×Q1611yescharacteristic
108C22×C127AGL(1,ℤ/8)11yescharacteristic
109C22×C127#13711yescharacteristic
110C2×D127—22no—
111C2×D127—22no—
112C2×D127—22no—
113C2×D127—22no—
114C2×D127—22no—
115C2×D127—22no—
116C2×D127#10314yescomplemented
117C4×C127—42no—
118C4×C127—42no—
119C4×C127C4×D411yescharacteristic
120C4×C127C4×D411yescharacteristic
121C4×C127#12111yescharacteristic
122C4×C127#12111yescharacteristic
123Dic127—22no—
124Dic127—42no—
125Dic127—42no—
126C2×C127#38611yescharacteristic
127C2×C127C2×#10311yescharacteristic
128C2×C127#41711yescharacteristic
129C2×C127#45011yescharacteristic
130C2×C127#45111yescharacteristic
131C2×C127#48311yescharacteristic
132C2×C127#48411yescharacteristic
133D127—24no—
134#1872—1271nomaximal, Sylow
135C127#187211yescharacteristic, complemented, Sylow
136C2×C22⋊Q8—1271no—
137C22×C4⋊C4—1271no—
138C2×#103—1271no—
139C2×#103D12711yescharacteristic, complemented
140C2×C22⋊C8—1271no—
141#340—1271no—
142#340—1271no—
143C22×Q8C2×D12711yescharacteristic
144C23×C4—1271no—
145C23×C4—2541no—
146C22×C8—1271no—
147C22×C8C2×D12711yescharacteristic
148C22⋊Q8—1274no—
149C22⋊Q8—2542no—
150C2×C4⋊C4—1271no—
151C2×C4⋊C4—1271no—
152C2×C4⋊C4—2542no—
153C2×C4⋊C4—2541no—
154C2×C4⋊C4—2541no—
155C2×C4⋊C4—2542no—
156C2×C4⋊C4C2×D12711yescharacteristic
157C2×C22⋊C4—2541no—
158#103—22no—
159#103—2542no—
160C22⋊C8—2542no—
161C24—1271no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8C4×D12712yescomplemented
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—1271no—
171C22×C4—1272no—
172C22×C4—1272no—
173C22×C4—5082no—
174C22×C4—5082no—
175C22×C4—2544no—
176C22×C4—2541no—
177C22×C4—2541no—
178C22×C4—2541no—
179C22×C4—2541no—
180C22×C4—2541no—
181C22×C4—2541no—
182C22×C4C22×D12711yescharacteristic
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—2541no—
187C2×C8—2541no—
188C2×C8—2541no—
189C4⋊C4—21no—
190C4⋊C4—1272no—
191C4⋊C4—1272no—
192C4⋊C4—2544no—
193C4⋊C4—2544no—
194C4⋊C4—2541no—
195C4⋊C4—2541no—
196C4⋊C4C22⋊D12712yes—
197C22⋊C4—2544no—
198C23—1272no—
199C23—1272no—
200C23—1272no—
201C23—2541no—
202C23—2541no—
203C23—2541no—
204C23—2541no—
205C23D4×D12711yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—41no—
209Q8—41no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—1272no—
217C2×C4—5088no—
218C2×C4—5081no—
219C2×C4—5081no—
220C2×C4—5081no—
221C2×C4—5081no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4—2541no—
225C2×C4—2541no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4—2542no—
229C2×C4—2542no—
230C2×C4C2×C4×D12711yescharacteristic
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4#1452324711yescharacteristic
235C2×C4#1452324811yescharacteristic
236C8—41no—
237C8—5081no—
238C22—1274no—
239C22—2542no—
240C22—2542no—
241C22—2542no—
242C22—2542no—
243C22—2542no—
244C22—2542no—
245C22C22⋊C4×D12711yescharacteristic
246C22#1686117711yescharacteristic
247C22#1686119111yescharacteristic
248C22SD16×D12711yescharacteristic
249C22Q16×D12711yescharacteristic
250C22#1686123611yescharacteristic
251C22#1686124111yescharacteristic
252C4—42no—
253C4—42no—
254C4—5082no—
255C4—5082no—
256C4—2542no—
257C4(C22⋊D127)⋊C411yescharacteristic
258C4C4×C22⋊D12711yescharacteristic
259C4#1686118011yescharacteristic
260C4#1686118211yescharacteristic
261C2—2544no—
262C2#2254975211yescharacteristic
263C2#103×D12711yescharacteristic
264C2#2254987611yescharacteristic
265C2#2255002511yescharacteristic
266C2#2255003011yescharacteristic
267C2#2255020511yescharacteristic
268C2#2255021011yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
prime radicalC23×D127D4
cosocleC22×C4×C127C23
hypercenterC2×#103D127
hypocenterC127#1872
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#16861133C2semidirect
5. #22549639C2semidirect
6. #22549638C2semidirect
7. C2×#103×C127C2semidirect
12. #16861185C4semidirect
44. C4⋊Dic127C2×C4semidirect
54. C2×Q8×C127C2×C4semidirect
116. C2×D127#103semidirect
135. C127#1872semidirect
139. C2×#103D127semidirect
167. C2×Q8C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611852
C2×C4⋊C4×D1272
C2×#168611332
C2×(C2×D127)⋊C82
#225496392
#225496382
C2×#103×C1272
#1872127