Finite Group Databasebeta

← #30648898

Subgroups — #30648898

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#22550407C211yesmaximal, characteristic
2C2×#16861174C211yesmaximal, characteristic, complemented
3#22550211C211yesmaximal, characteristic, complemented
4#22550186C212yesmaximal, complemented
5#22550097C212yesmaximal, complemented
6#22550034C211yesmaximal, characteristic, complemented
7#22550014C211yesmaximal, characteristic
8#22549982C212yesmaximal, complemented
9#22549973C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#481×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13C2×Dic508C2212yes—
14C2×#14523236C2212yes—
15#16861202—21no—
16#16861200C2211yescharacteristic
17#16861196C2211yescharacteristic
18#16861190—21no—
19#16861188C2211yescharacteristic
20#16861174—21no—
21#16861174—21no—
22#16861174C2212yes—
23#16861174C2212yes—
24#16861174C2211yescharacteristic
25D4×Dic127—21no—
26D4×Dic127C2211yescharacteristic
27#16861165—21no—
28C2×Dic127⋊C4—21no—
29C2×C4×Dic127—21no—
30C2×C4×Dic127C2211yescharacteristic
31#16861142C2212yes—
32Dic127⋊C8C2212yes—
33#16861134C2212yes—
34#16861132C2212yes—
35#16861132C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yes—
38#121×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#106×C127C2212yes—
41#102×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43C2×Dic254—24no—
44C2×Dic254—21no—
45C2×Dic254C2311yescharacteristic
46C2×Dic254C2312yes—
47C22×Dic127—21no—
48C22×Dic127—21no—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51Dic508—24no—
52#14523236—24no—
53C2×C127⋊C8C2312yes—
54C4⋊Dic127—21no—
55C4⋊Dic127C2311yescharacteristic
56Dic127⋊C4—21no—
57Dic127⋊C4—21no—
58Dic127⋊C4—22no—
59Dic127⋊C4—22no—
60Dic127⋊C4—21no—
61C22⋊Dic127—21no—
62C22⋊Dic127—21no—
63C22⋊Dic127—21no—
64C22⋊Dic127—21no—
65C4×Dic127—22no—
66C4×Dic127—22no—
67C4×Dic127—21no—
68C4×Dic127D411yescharacteristic
69C4×Dic127D411yescharacteristic
70C4×Dic127C2311yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×D4×C127—21no—
73C2×D4×C127C2311yescharacteristic
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81Dic254—24no—
82Dic254—22no—
83Dic254—22no—
84Dic254#4114yes—
85C2×Dic127—24no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—21no—
89C2×Dic127—21no—
90C2×Dic127—41no—
91C2×Dic127—42no—
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C127⋊C8—22no—
97C23×C127—21no—
98C23×C127C2×D411yescharacteristic
99D4×C127—22no—
100D4×C127—42no—
101C2×C4×C127—22no—
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—41no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127#13712yes—
114C22×C127C22×D411yescharacteristic
115C4×C127—21no—
116C4×C127—41no—
117C4×C127—42no—
118C4×C127C2×#4112yes—
119Dic127—24no—
120Dic127—21no—
121Dic127—21no—
122Dic127—21no—
123Dic127—21no—
124Dic127—41no—
125Dic127—41no—
126C2×C127—22no—
127C2×C127—81no—
128C2×C127#52511yescharacteristic
129C2×C127C2×#13711yescharacteristic
130C2×C127#57511yescharacteristic
131#3117—1271nomaximal, Sylow
132C127#311711yescharacteristic, complemented, Sylow
133#529—1271no—
134C2×C4×Q8—1271no—
135#484—1271no—
136#481D12711yescharacteristic
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⋊Q8—1271no—
152#121C2×D12711yescharacteristic
153C4×Q8—1272no—
154C4×Q8—1272no—
155C4×Q8—1271no—
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#107—1272no—
166#106C2×D12712yes—
167C4⋊C8—1272no—
168#103—1272no—
169#103—1272no—
170#103—1272no—
171#102C2×D12712yes—
172C22⋊C8—1271no—
173C22⋊C8C2×D12711yescharacteristic
174#97—1272no—
175C2×Q8—1272no—
176C2×Q8—1271no—
177C2×Q8—2544no—
178C2×Q8—2541no—
179C2×D4—21no—
180C2×D4C22×D12711yescharacteristic
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—1271no—
193C4⋊C4—2542no—
194C4⋊C4—2542no—
195C4⋊C4—2541no—
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—21no—
214C23D4×D12711yescharacteristic
215Q8—1274no—
216Q8—2544no—
217Q8—2542no—
218Q8—2542no—
219D4—22no—
220D4—42no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—1271no—
226C2×C4—1271no—
227C2×C4—1271no—
228C2×C4—1271no—
229C2×C4—5082no—
230C2×C4—5081no—
231C2×C4—2541no—
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—41no—
243C22—41no—
244C22—41no—
245C22#1686123812yes—
246C22C2×D4×D12711yescharacteristic
247C4—21no—
248C4—41no—
249C4—42no—
250C4—5081no—
251C4—5081no—
252C4—2544no—
253C4—2541no—
254C4—2541no—
255C4—2541no—
256C4—2541no—
257C4#41×D12712yes—
258C2—22no—
259C2—81no—
260C2#2255038411yescharacteristic
261C2C2×#1686123811yescharacteristic
262C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#481×C127C2
socleC22×C127C22×D4
prime radical#121×C127C22
hypercenter#481D127
hypocenterC127#3117
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#16861174C2semidirect
3. #22550211C2semidirect
4. #22550186C2semidirect
5. #22550097C2semidirect
6. #22550034C2semidirect
8. #22549982C2semidirect
9. #22549973C2semidirect
10. #22549930C2semidirect
132. C127#3117semidirect

Maximal subgroups

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

subgroupindex
#225504072
C2×#168611742
#225502112
#225501862
#225500972
#225500342
#225500142
#225499822
#225499732
#225499302
#481×C1272
#3117127