Finite Group Databasebeta

← #30648394

Subgroups — #30648394

Order 16256.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16861232C211yesmaximal, characteristic, complemented
2C2×#16861217C211yesmaximal, characteristic, complemented
3#22550456C212yesmaximal, complemented
4#22550131C211yesmaximal, characteristic, complemented
5#22550130C211yesmaximal, characteristic, complemented
6#22550129C211yesmaximal, characteristic, complemented
7#22550108C212yesmaximal, complemented
8#22550107C212yesmaximal, complemented
9#106×D127C211yesmaximal, characteristic, complemented
10#22549876C212yesmaximal, complemented
11#470×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C2×#14523248C2211yescharacteristic
14#16861232—24no—
15C2×#14523235C2212yes—
16C2×D1016C2211yescharacteristic
17#16861217—24no—
18C2×#14523231C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20C22×C127⋊C8C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861187—22no—
24#16861185C2212yes—
25#16861184—22no—
26#16861184C2212yes—
27#16861183C2212yes—
28#16861182—22no—
29#16861182—22no—
30C4⋊C4×D127C2212yes—
31#16861141C2211yescharacteristic
32#16861140C2212yes—
33#16861133C2212yes—
34#16861131C2212yes—
35#16861129C2212yes—
36C2×Q16×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38C22⋊Q8×C127C2212yes—
39#106×C127C2211yescharacteristic
40#103×C127C2212yes—
41#14523249—24no—
42#14523249—24no—
43#14523248—22no—
44#14523248—22no—
45C2×Dic254D411yescharacteristic
46C2×D508—22no—
47C2×D508—22no—
48C2×D508D411yescharacteristic
49C2×D508C2312yes—
50C2×C22⋊D127—22no—
51C2×C22⋊D127D411yescharacteristic
52C2×C22⋊D127D411yescharacteristic
53C2×C4×D127—22no—
54C2×C4×D127—22no—
55C2×C4×D127C2311yescharacteristic
56C2×C4×D127C2311yescharacteristic
57#14523235—24no—
58D1016—22no—
59#14523231—22no—
60#14523231—22no—
61C2×C127⋊C8—22no—
62C2×C127⋊C8C2311yescharacteristic
63C2×C127⋊C8C2311yescharacteristic
64C8×D127—22no—
65C8×D127D414yescomplemented
66C4⋊Dic127C2312yes—
67Dic127⋊C4—22no—
68Dic127⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72C2×Q8×C127C2312yes—
73C22×C4×C127C2311yescharacteristic
74Q16×C127—22no—
75M16×C127—22no—
76C2×C8×C127C2311yescharacteristic
77C2×C8×C127C2311yescharacteristic
78C4⋊C4×C127—22no—
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22×D127—22no—
82C22×D127C2×D411yescharacteristic
83C22×D127C2×D411yescharacteristic
84Dic254—22no—
85D508—22no—
86D508—22no—
87D508—22no—
88D508—44no—
89D508—82no—
90C22⋊D127—22no—
91C22⋊D127—22no—
92C22⋊D127—82no—
93C2×Dic127—22no—
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—44no—
99C4×D127—82no—
100C4×D127C2×D412yes—
101C4×D127#4112yes—
102C127⋊C8—21no—
103C127⋊C8C2×D412yes—
104C23×C127C2×D411yescharacteristic
105Q8×C127—22no—
106Q8×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C8×C127—21no—
114C8×C127C2×D412yes—
115C22×C127—21no—
116C22×C127—22no—
117C22×C127C22×D411yescharacteristic
118C2×D127—22no—
119C2×D127—21no—
120C2×D127—22no—
121C2×D127—42no—
122C2×D127—42no—
123C2×D127—42no—
124C2×D127#12112yes—
125C4×C127—21no—
126C4×C127—42no—
127C4×C127—42no—
128C4×C127C22×D411yescharacteristic
129C4×C127C2×#4111yescharacteristic
130Dic127—21no—
131Dic127—42no—
132Dic127#12112yes—
133C2×C127—41no—
134C2×C127C2×#12111yescharacteristic
135C2×C127#57712yes—
136D127—22no—
137D127—41no—
138D127—82no—
139#3060—1271nomaximal, Sylow
140C127#306011yescharacteristic, complemented, Sylow
141C2×#135—1271no—
142C2×#131—1271no—
143#537—1272no—
144#470—1271no—
145#470D12711yescharacteristic, complemented
146#469—1271no—
147#469—1271no—
148#465—1272no—
149#465—1272no—
150C2×#106—1271no—
151#417—1272no—
152C2×#41—1271no—
153C2×#41—1272no—
154#135—2544no—
155C2×Q16C2×D12711yescharacteristic
156C2×SD16—1272no—
157C2×D8—1271no—
158#131—2544no—
159C2×M16—1271no—
160C2×M16—1271no—
161C2×M16C2×D12711yescharacteristic
162C22×C8—1271no—
163C22×C8—1271no—
164C22×C8—1271no—
165C22⋊Q8—1272no—
166C22⋊Q8—2542no—
167C22⋊Q8C2×D12712yes—
168#121—1272no—
169#121—1272no—
170#121—2542no—
171#121—2542no—
172#121—2542no—
173C2×C4⋊C4—1272no—
174#106—1271no—
175#106—1272no—
176#106C2×D12711yescharacteristic
177#103—1272no—
178#103C2×D12712yes—
179#102—1272no—
180#102—1272no—
181#41—2542no—
182#41—2542no—
183#41—2544no—
184#41—2544no—
185C2×Q8—1271no—
186C2×Q8C22×D12712yes—
187C2×D4—1271no—
188C2×D4—1271no—
189C2×D4—1271no—
190C2×D4—1272no—
191C2×D4—2542no—
192C2×D4—2542no—
193C2×D4—2542no—
194C22×C4—1271no—
195C22×C4—1271no—
196C22×C4—2542no—
197C22×C4—2542no—
198C22×C4C22×D12711yescharacteristic
199Q16—22no—
200SD16—2544no—
201D8—2542no—
202M16—22no—
203M16—2542no—
204M16—2542no—
205C2×C8—1271no—
206C2×C8—1274no—
207C2×C8—1271no—
208C2×C8—2542no—
209C2×C8—2542no—
210C2×C8C22×D12711yescharacteristic
211C2×C8C22×D12711yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—1272no—
214C4⋊C4—2542no—
215C4⋊C4—2542no—
216C4⋊C4C22×D12712yes—
217C22⋊C4—22no—
218C22⋊C4—2542no—
219C22⋊C4—2542no—
220C22⋊C4—2542no—
221C23—1271no—
222C23—1271no—
223C23—2542no—
224C23D4×D12711yescharacteristic
225Q8—22no—
226Q8—22no—
227Q8—2542no—
228D4—10162no—
229D4—10162no—
230D4—5084no—
231D4—2542no—
232D4—2542no—
233D4—2542no—
234D4—2542no—
235D4—2542no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—1271no—
241C2×C4—1272no—
242C2×C4—1272no—
243C2×C4—1271no—
244C2×C4—10162no—
245C2×C4—5084no—
246C2×C4—2542no—
247C2×C4—2541no—
248C2×C4—2541no—
249C2×C4D4×D12711yescharacteristic
250C2×C4C23×D12711yescharacteristic
251C8—21no—
252C8—1272no—
253C8—2541no—
254C8D4×D12712yes—
255C22—21no—
256C22—22no—
257C22—1272no—
258C22—5082no—
259C22—5082no—
260C22—5082no—
261C22—2541no—
262C22—2542no—
263C22—2542no—
264C22C2×D4×D12711yescharacteristic
265C4—21no—
266C4—42no—
267C4—42no—
268C4—1272no—
269C4—5082no—
270C4—2541no—
271C4C2×D4×D12711yescharacteristic
272C4#41×D12711yescharacteristic
273C2—41no—
274C2—10162no—
275C2—5081no—
276C2—2542no—
277C2#121×D12711yescharacteristic
278C2#2255068512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#470×C127C2
socleC22×C127C22×D4
hypercenter#470D127
hypocenterC127#3060
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. C2×#16861217C2semidirect
3. #22550456C2semidirect
4. #22550131C2semidirect
5. #22550130C2semidirect
6. #22550129C2semidirect
7. #22550108C2semidirect
8. #22550107C2semidirect
9. #106×D127C2semidirect
10. #22549876C2semidirect
11. #470×C127C2semidirect
65. C8×D127D4semidirect
140. C127#3060semidirect
145. #470D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612322
C2×#168612172
#225504562
#225501312
#225501302
#225501292
#225501082
#225501072
#106×D1272
#225498762
#470×C1272
#3060127