Finite Group Databasebeta

← #30648904

Subgroups — #30648904

Order 16256.

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

normal subgroups only

This group has 386 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
1#22550412C211yesmaximal, characteristic
2C2×#16861174C211yesmaximal, characteristic, complemented
3#22550194C211yesmaximal, characteristic
4#22550182C212yesmaximal, complemented
5#22550101C212yesmaximal, complemented
6#22550031C211yesmaximal, characteristic, complemented
7#22550017C211yesmaximal, characteristic
8#22549980C212yesmaximal, complemented
9#22549975C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#484×C127C211yesmaximal, characteristic, complemented
12C22×Dic254C2211yescharacteristic
13C2×#14523238C2212yes—
14C2×#14523237C2212yes—
15#16861204—21no—
16#16861200C2211yescharacteristic
17#16861198C2211yescharacteristic
18#16861193—21no—
19#16861183C2211yescharacteristic
20Q8×Dic127C2211yescharacteristic
21#16861174—21no—
22#16861174—21no—
23#16861174C2212yes—
24#16861174C2212yes—
25#16861171C2211yescharacteristic
26(C22⋊D127)⋊C4—21no—
27#16861167—21no—
28C2×Dic127⋊C4—21no—
29C2×C4×Dic127—21no—
30C2×C4×Dic127C2211yescharacteristic
31#16861140C2212yes—
32Dic127⋊C8C2212yes—
33#16861134C2212yes—
34#16861132C2212yes—
35#16861129C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yes—
38C22⋊Q8×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#107×C127C2212yes—
41#103×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43C2×Dic254—24no—
44C2×Dic254—21no—
45C2×Dic254C2312yes—
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C22×Dic127—21no—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51C2×C4×D127—21no—
52#14523238—24no—
53#14523237—24no—
54C2×C127⋊C8C2312yes—
55C4⋊Dic127—21no—
56C4⋊Dic127—21no—
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—21no—
59Dic127⋊C4—22no—
60Dic127⋊C4—22no—
61(C2×D127)⋊C4—21no—
62(C2×D127)⋊C4—21no—
63(C2×D127)⋊C4—21no—
64(C2×D127)⋊C4—21no—
65C4×Dic127—22no—
66C4×Dic127—22no—
67C4×Dic127—21no—
68C4×Dic127D411yescharacteristic
69C4×Dic127D411yescharacteristic
70C4×Dic127C2311yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127C2311yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127C2311yescharacteristic
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C22×D127—21no—
82Dic254—24no—
83Dic254—22no—
84Dic254#4114yes—
85D508—22no—
86C22⋊D127—42no—
87C2×Dic127—24no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C2×Dic127—41no—
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—42no—
97C127⋊C8—22no—
98C23×C127C2×D411yescharacteristic
99Q8×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—22no—
104C2×C4×C127—42no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127#13712yes—
111C22×C127C22×D411yescharacteristic
112C2×D127—41no—
113C2×D127—41no—
114C2×D127—41no—
115C4×C127—21no—
116C4×C127—41no—
117C4×C127—42no—
118C4×C127—41no—
119C4×C127C2×#4112yes—
120Dic127—24no—
121Dic127—21no—
122Dic127—21no—
123Dic127—21no—
124Dic127—21no—
125Dic127—41no—
126C2×C127—22no—
127C2×C127#52511yescharacteristic
128C2×C127C2×#13711yescharacteristic
129C2×C127#57511yescharacteristic
130D127—81no—
131#3117—1271nomaximal, Sylow
132C127#311711yescharacteristic, complemented, Sylow
133#529—1271no—
134C2×C4×Q8—1271no—
135#484D12711yescharacteristic, complemented
136#481—1271no—
137#479—1272no—
138#464—1272no—
139#451—1271no—
140#448—1271no—
141#441—1272no—
142#440—1272no—
143#432—1271no—
144C22×Q8—1271no—
145C2×Q16—1272no—
146C2×SD16—1272no—
147#126—2541no—
148#125—1271no—
149#124—1271no—
150#123—2541no—
151C22⋊Q8C2×D12711yescharacteristic
152#121—1271no—
153C4×Q8—1271no—
154C4×Q8—1272no—
155C4×Q8—1272no—
156C4×Q8—2541no—
157C4×Q8—2541no—
158C4×D4—1271no—
159C4×D4—2541no—
160#117—2541no—
161C2×C4⋊C4—2541no—
162C2×C4⋊C4C2×D12711yescharacteristic
163C2×C42—1271no—
164C2×C42—2541no—
165#107C2×D12712yes—
166#106—1272no—
167C4⋊C8—1272no—
168#103—1272no—
169#103—1272no—
170#103C2×D12712yes—
171#102—1272no—
172C22⋊C8—1271no—
173C22⋊C8C2×D12711yescharacteristic
174#97—1272no—
175C2×Q8—1272no—
176C2×Q8—2544no—
177C2×Q8—2541no—
178C2×Q8C22×D12711yescharacteristic
179C2×D4—1271no—
180C2×D4—2541no—
181C22×C4—21no—
182C22×C4—1271no—
183C22×C4—1271no—
184C22×C4—2541no—
185C22×C4—2541no—
186C22×C4C22×D12711yescharacteristic
187Q16—2544no—
188SD16—2544no—
189C2×C8—1272no—
190C2×C8C22×D12712yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—1271no—
194C4⋊C4—2542no—
195C4⋊C4—2542no—
196C4⋊C4—2541no—
197C4⋊C4—2541no—
198C4⋊C4—2541no—
199C4⋊C4C22×D12712yes—
200C4⋊C4C22×D12711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2541no—
203C22⋊C4—2541no—
204C22⋊C4—2541no—
205C22⋊C4—2541no—
206C42—1271no—
207C42—1271no—
208C42—1271no—
209C42—1271no—
210C42—2542no—
211C42—2542no—
212C42—2541no—
213C23—2541no—
214C23D4×D12711yescharacteristic
215Q8—22no—
216Q8—1274no—
217Q8—2544no—
218Q8—2542no—
219D4—5082no—
220D4—2542no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—1271no—
227C2×C4—1271no—
228C2×C4—1271no—
229C2×C4—1271no—
230C2×C4—5082no—
231C2×C4—5081no—
232C2×C4—2541no—
233C2×C4—2544no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4D4×D12711yescharacteristic
237C2×C4C23×D12711yescharacteristic
238C8—22no—
239C8—2542no—
240C22—21no—
241C22—21no—
242C22—5081no—
243C22—5081no—
244C22—5081no—
245C22#1686124112yes—
246C22C2×D4×D12711yescharacteristic
247C4—21no—
248C4—41no—
249C4—42no—
250C4—41no—
251C4—5081no—
252C4—2544no—
253C4—2541no—
254C4—2541no—
255C4—2541no—
256C4—2541no—
257C4#41×D12712yes—
258C2—22no—
259C2—10161no—
260C2#2255038411yescharacteristic
261C2C2×#1686124111yescharacteristic
262C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#484×C127C2
socleC22×C127C22×D4
prime radical#16861183C22
hypercenter#484D127
hypocenterC127#3117
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#16861174C2semidirect
4. #22550182C2semidirect
5. #22550101C2semidirect
6. #22550031C2semidirect
8. #22549980C2semidirect
9. #22549975C2semidirect
10. #22549930C2semidirect
11. #484×C127C2semidirect
132. C127#3117semidirect
135. #484D127semidirect

Maximal subgroups

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

subgroupindex
#225504122
C2×#168611742
#225501942
#225501822
#225501012
#225500312
#225500172
#225499802
#225499752
#225499302
#484×C1272
#3117127