Finite Group Databasebeta

← #30651359

Subgroups — #30651359

Order 16256.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550587C211yesmaximal, characteristic, complemented
3#22550279C211yesmaximal, characteristic, complemented
4#22550220C212yesmaximal, complemented
5#22550188C212yesmaximal, complemented
6#22550172C211yesmaximal, characteristic, complemented
7#22550121C212yesmaximal, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9#22549953C211yesmaximal, characteristic, complemented
10#22549915C212yesmaximal, complemented
11#476×C127C211yesmaximal, characteristic, complemented
12C2×#14523236C2211yescharacteristic
13C2×#14523231C2212yes—
14#16861209C2211yescharacteristic
15#16861200—21no—
16#16861195—22no—
17#16861189C2212yes—
18#16861185—22no—
19#16861183—21no—
20#16861180C2212yes—
21C4×Dic254C2211yescharacteristic
22C4×D508C2211yescharacteristic
23#16861171—21no—
24#16861168—21no—
25#16861167C2212yes—
26C4⋊C4×D127C2212yes—
27#16861143C2212yes—
28#16861142C2212yes—
29#16861142C2212yes—
30#16861140C2211yescharacteristic
31#16861134C2212yes—
32#16861132C2211yescharacteristic
33#16861130C2211yescharacteristic
34(C2×D127)⋊C8C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C4⋊Q8×C127C2211yescharacteristic
37C4×D4×C127C2211yescharacteristic
38#106×C127C2212yes—
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41C2×Dic254—21no—
42C2×Dic254C2311yescharacteristic
43C2×D508D411yescharacteristic
44C2×C22⋊D127—22no—
45C2×C4×D127—22no—
46C2×C4×D127C2312yes—
47#14523236—22no—
48#14523231—24no—
49C2×C127⋊C8C2311yescharacteristic
50C2×C127⋊C8C2311yescharacteristic
51C4⋊Dic127—22no—
52C4⋊Dic127—21no—
53C4⋊Dic127D411yescharacteristic
54C4⋊Dic127C2312yes—
55C4⋊Dic127C2311yescharacteristic
56Dic127⋊C4—22no—
57Dic127⋊C4—21no—
58C22⋊Dic127—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4D412yes—
62C4×Dic127—22no—
63C4×Dic127—21no—
64C2×Q8×C127—21no—
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—21no—
67C2×C8×C127C2312yes—
68C4⋊C4×C127—21no—
69C4⋊C4×C127C2312yes—
70C4⋊C4×C127C2311yescharacteristic
71C22⋊C4×C127—21no—
72C42×C127C2311yescharacteristic
73C22×D127C2×D412yes—
74Dic254—21no—
75Dic254—21no—
76Dic254—81no—
77D508—22no—
78C22⋊D127—82no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127—21no—
82C2×Dic127C2×D412yes—
83C4×D127—22no—
84C4×D127—22no—
85C4×D127—82no—
86C127⋊C8—21no—
87C127⋊C8#4112yes—
88C23×C127—21no—
89Q8×C127—81no—
90D4×C127—21no—
91D4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—42no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2411yescharacteristic
98C8×C127—22no—
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—42no—
109C4×C127#13712yes—
110C4×C127C2×#4111yescharacteristic
111C4×C1272−1+411yescharacteristic
112Dic127—22no—
113Dic127—41no—
114Dic127—41no—
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#476D12711yescharacteristic, complemented
130#468—1272no—
131C4×SD16—1271no—
132#436—1271no—
133#428—1272no—
134C2×SD16—1271no—
135C2×M16—1272no—
136C4⋊Q8—1271no—
137C4⋊Q8C2×D12711yescharacteristic
138#125—2541no—
139#124—2542no—
140C22⋊Q8—1272no—
141C22⋊Q8—2542no—
142#121—1272no—
143#121—2541no—
144C4×Q8—1271no—
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—1271no—
155#106C2×D12712yes—
156C4⋊C8C2×D12711yescharacteristic
157#103—1272no—
158#103—1271no—
159#102—1271no—
160#102C2×D12712yes—
161C22⋊C8—1272no—
162C4×C8—1271no—
163C2×Q8—21no—
164C2×Q8—1271no—
165C2×Q8—2541no—
166C2×D4—1271no—
167C2×D4—2542no—
168C2×D4C22×D12711yescharacteristic
169C22×C4—21no—
170C22×C4—1272no—
171C22×C4—2542no—
172SD16—2542no—
173M16—2544no—
174C2×C8—1271no—
175C2×C8—1271no—
176C2×C8C22×D12712yes—
177C4⋊C4—21no—
178C4⋊C4—1271no—
179C4⋊C4—1271no—
180C4⋊C4—1272no—
181C4⋊C4—2542no—
182C4⋊C4—2542no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C4⋊C4C22×D12712yes—
186C4⋊C4C22×D12711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—1272no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C22⋊C4—2542no—
192C42—2542no—
193C42—2541no—
194C42C22×D12711yescharacteristic
195C23—21no—
196C23—1272no—
197Q8—81no—
198Q8—10161no—
199Q8—2541no—
200Q8—2541no—
201D4—21no—
202D4—21no—
203D4—10162no—
204D4—2542no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—1272no—
209C2×C4—10162no—
210C2×C4—2542no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4D4×D12711yescharacteristic
216C2×C4D4×D12711yescharacteristic
217C2×C4C23×D12711yescharacteristic
218C8—22no—
219C8—1272no—
220C8—2541no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—2542no—
225C22—2542no—
226C22—2542no—
227C22C2×D4×D12711yescharacteristic
228C4—21no—
229C4—41no—
230C4—42no—
231C4—5081no—
232C4—5081no—
233C4—5082no—
234C4—2542no—
235C4#1686123812yes—
236C4#41×D12711yescharacteristic
237C4#1686125711yescharacteristic
238C2—81no—
239C2—5082no—
240C2#2255052811yescharacteristic
241C2C2×#1686123811yescharacteristic
242C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#476×C127C2
socleC22×C127C22×D4
prime radical#22550591C2
hypercenter#476D127
hypocenterC127#3357
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #22550587C2semidirect
3. #22550279C2semidirect
4. #22550220C2semidirect
5. #22550188C2semidirect
6. #22550172C2semidirect
7. #22550121C2semidirect
8. C4×#14523236C2semidirect
9. #22549953C2semidirect
10. #22549915C2semidirect
11. #476×C127C2semidirect
122. C127#3357semidirect
129. #476D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225505872
#225502792
#225502202
#225501882
#225501722
#225501212
C4×#145232362
#225499532
#225499152
#476×C1272
#3357127