Finite Group Databasebeta

← #30651358

Subgroups — #30651358

Order 16256.

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

normal subgroups only

This group has 354 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#22550591C211yesmaximal, characteristic
2#22550586C211yesmaximal, characteristic, complemented
3#22550281C211yesmaximal, characteristic, complemented
4#22550216C212yesmaximal, complemented
5#22550188C212yesmaximal, complemented
6#22550171C211yesmaximal, characteristic, complemented
7#22550166C211yesmaximal, characteristic, complemented
8#22550119C212yesmaximal, complemented
9#22549955C211yesmaximal, characteristic, complemented
10#22549915C212yesmaximal, complemented
11C4×SD16×C127C211yesmaximal, characteristic, complemented
12C2×#14523231C2212yes—
13#16861209C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861201—21no—
16#16861195—22no—
17#16861187—22no—
18#16861185C2212yes—
19#16861183—21no—
20#16861180C2212yes—
21C4×D508—21no—
22C4×D508C2211yescharacteristic
23#16861168—21no—
24#16861167C2212yes—
25C4⋊C4×D127C2212yes—
26#16861143C2212yes—
27#16861143C2212yes—
28#16861142C2212yes—
29#16861140C2212yes—
30C4⋊(C127⋊C8)C2211yescharacteristic
31#16861133C2212yes—
32#16861130C2212yes—
33(C2×D127)⋊C8C2212yes—
34C2×SD16×C127C2211yescharacteristic
35C4×Q8×C127C2211yescharacteristic
36C4×D4×C127C2211yescharacteristic
37#106×C127C2211yescharacteristic
38#103×C127C2211yescharacteristic
39#102×C127C2211yescharacteristic
40C4×C8×C127C2211yescharacteristic
41C2×Dic254—21no—
42C2×Dic254—21no—
43C2×D508D411yescharacteristic
44C2×C22⋊D127—22no—
45C2×C4×D127—22no—
46C2×C4×D127C2312yes—
47#14523231—24no—
48C2×C127⋊C8C2312yes—
49C4⋊Dic127—21no—
50C4⋊Dic127—21no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2312yes—
53C4⋊Dic127C2312yes—
54Dic127⋊C4—22no—
55Dic127⋊C4—22no—
56C22⋊Dic127—22no—
57(C2×D127)⋊C4—22no—
58(C2×D127)⋊C4—22no—
59(C2×D127)⋊C4D412yes—
60C4×Dic127—22no—
61C2×Q8×C127C2311yescharacteristic
62C2×D4×C127C2311yescharacteristic
63C22×C4×C127—21no—
64SD16×C127—22no—
65C2×C8×C127C2311yescharacteristic
66C2×C8×C127C2311yescharacteristic
67C4⋊C4×C127—21no—
68C4⋊C4×C127C2311yescharacteristic
69C4⋊C4×C127C2311yescharacteristic
70C22⋊C4×C127—21no—
71C42×C127—21no—
72C42×C127C2311yescharacteristic
73C22×D127C2×D412yes—
74Dic254—81no—
75Dic254—81no—
76D508—22no—
77C22⋊D127—82no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127C2×D412yes—
81C4×D127—22no—
82C4×D127—22no—
83C4×D127—82no—
84C127⋊C8—22no—
85C23×C127—21no—
86Q8×C127—21no—
87Q8×C127—21no—
88D4×C127—21no—
89D4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—42no—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2411yescharacteristic
97C8×C127—21no—
98C8×C127#4112yes—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127C22×D411yescharacteristic
103C2×D127—22no—
104C2×D127—22no—
105C2×D127—22no—
106C4×C127—21no—
107C4×C127—41no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127#13712yes—
111C4×C127C2×#4111yescharacteristic
112C4×C1272−1+411yescharacteristic
113Dic127—22no—
114Dic127—42no—
115Dic127—42no—
116C2×C127—81no—
117C2×C127#54711yescharacteristic
118C2×C127C2×#13711yescharacteristic
119C2×C127#57611yescharacteristic
120D127—42no—
121#3357—1271nomaximal, Sylow
122C127#335711yescharacteristic, complemented, Sylow
123#555—1271no—
124#554—1271no—
125#500—1271no—
126#485—1272no—
127#480—1272no—
128#477—1271no—
129#476—1271no—
130#468—1272no—
131C4×SD16D12711yescharacteristic, complemented
132#436—1271no—
133#428—1272no—
134C2×SD16C2×D12711yescharacteristic
135C2×M16—1272no—
136C4⋊Q8—1271no—
137C4⋊Q8—1271no—
138#125—2541no—
139#124—2542no—
140C22⋊Q8—1272no—
141C22⋊Q8—2542no—
142#121—1272no—
143#121—2541no—
144C4×Q8C2×D12711yescharacteristic
145C4×D4—1271no—
146C4×D4—2541no—
147C4×D4C2×D12711yescharacteristic
148#117—1272no—
149#117—2541no—
150C2×C4⋊C4—1272no—
151#107—1272no—
152#107—1272no—
153#107—1272no—
154#106—1272no—
155#106C2×D12711yescharacteristic
156C4⋊C8—1271no—
157#103—1272no—
158#103C2×D12711yescharacteristic
159#102—1272no—
160#102C2×D12711yescharacteristic
161C22⋊C8—1272no—
162C4×C8C2×D12711yescharacteristic
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8C22×D12711yescharacteristic
166C2×D4—1271no—
167C2×D4—2542no—
168C2×D4C22×D12711yescharacteristic
169C22×C4—21no—
170C22×C4—1272no—
171C22×C4—2542no—
172SD16—22no—
173M16—2544no—
174C2×C8—1272no—
175C2×C8C22×D12711yescharacteristic
176C2×C8C22×D12711yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—1271no—
179C4⋊C4—1272no—
180C4⋊C4—1272no—
181C4⋊C4—2542no—
182C4⋊C4—2542no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C4⋊C4C22×D12711yescharacteristic
186C4⋊C4C22×D12711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—1272no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C22⋊C4—2542no—
192C42—21no—
193C42—2542no—
194C42C22×D12711yescharacteristic
195C23—21no—
196C23—1272no—
197Q8—21no—
198Q8—21no—
199Q8—10161no—
200Q8—10161no—
201D4—21no—
202D4—21no—
203D4—10162no—
204D4—2542no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—1272no—
210C2×C4—10162no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4D4×D12711yescharacteristic
216C2×C4D4×D12711yescharacteristic
217C2×C4C23×D12711yescharacteristic
218C8—21no—
219C8—2542no—
220C8#1452324812yes—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—2542no—
225C22—2542no—
226C22—2542no—
227C22C2×D4×D12711yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—5082no—
233C4—5082no—
234C4—2542no—
235C4#1686123812yes—
236C4C2×#1452324811yescharacteristic
237C4#1686125611yescharacteristic
238C2—81no—
239C2—5082no—
240C2#2255053111yescharacteristic
241C2C2×#1686123811yescharacteristic
242C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
prime radical#22550591C2
hypercenterC4×SD16D127
hypocenterC127#3357
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #22550586C2semidirect
3. #22550281C2semidirect
4. #22550216C2semidirect
5. #22550188C2semidirect
6. #22550171C2semidirect
7. #22550166C2semidirect
8. #22550119C2semidirect
9. #22549955C2semidirect
10. #22549915C2semidirect
11. C4×SD16×C127C2semidirect
122. C127#3357semidirect
131. C4×SD16D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225505862
#225502812
#225502162
#225501882
#225501712
#225501662
#225501192
#225499552
#225499152
C4×SD16×C1272
#3357127