Finite Group Databasebeta

← #30639261

Subgroups — #30639261

Order 16256.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#22550445C211yesmaximal, characteristic
2#22550350C211yesmaximal, characteristic, complemented
3#22549894C211yesmaximal, characteristic, complemented
4#22549892C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861206C412yescomplemented
12#16861197C412yescomplemented
13#16861184—22no—
14#16861179—22no—
15C4×D508—22no—
16#16861171—22no—
17#16861168C2211yescharacteristic
18C22⋊C4×D127—22no—
19#16861137—22no—
20#16861135—22no—
21#16861125—22no—
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25C23×D127D412yes—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30C2×Dic254C2×C411yescharacteristic, complemented
31C2×D508—24no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508C2×C412yes—
37C2×D508C2×C411yescharacteristic, complemented
38C2×D508D412yes—
39C2×C22⋊D127—21no—
40C2×C4×D127—24no—
41C2×C4×D127—21no—
42#14523232—21no—
43#14523232—21no—
44C2×C127⋊C8—21no—
45C4⋊Dic127—21no—
46C22⋊Dic127—21no—
47(C2×D127)⋊C4—24no—
48(C2×D127)⋊C4—24no—
49C4×Dic127D412yescomplemented
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53C2×C8×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—21no—
58C42×C127C2×C412yescomplemented
59C42×C127D412yescomplemented
60C22×D127—22no—
61C22×D127—22no—
62C22×D127—21no—
63C22×D127—82no—
64C22×D127C22⋊C414yes—
65Dic254—21no—
66Dic254—21no—
67D508—22no—
68D508—22no—
69D508—21no—
70D508—21no—
71D508—22no—
72D508—22no—
73D508—84no—
74C22⋊D127—41no—
75C22⋊D127—41no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C4×D127—41no—
79C4×D127—41no—
80C4×D127—84no—
81C127⋊C8—41no—
82C23×C127C2×D411yescharacteristic
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127C22×C411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C2×C4×C127#4111yescharacteristic
91C8×C127—41no—
92C22×C127—41no—
93C22×C127C2×C22⋊C411yescharacteristic
94C22×C127#12011yescharacteristic
95C22×C127#12311yescharacteristic
96C2×D127—24no—
97C2×D127—44no—
98C2×D127—44no—
99C2×D127—42no—
100C2×D127—41no—
101C2×D127—82no—
102C4×C127—42no—
103C4×C127—42no—
104C4×C127C4×D411yescharacteristic
105C4×C127C4×D411yescharacteristic
106C4×C127#12111yescharacteristic
107C4×C127#12111yescharacteristic
108Dic127—42no—
109Dic127—41no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—21no—
113C2×C127#38611yescharacteristic
114D127—44no—
115D127—81no—
116#1911—1271nomaximal, Sylow
117C127#191111yescharacteristic, complemented, Sylow
118#535—1271no—
119#518—1271no—
120#421—1271no—
121#421—1271no—
122C2×#100—1271no—
123H(3,ℤ/4)—1271no—
124H(3,ℤ/4)D12711yescharacteristic, complemented
125C2×#41—1271no—
126C22×D4—1271no—
127C2×M16—1271no—
128C2×M16C2×D12711yescharacteristic
129#127—1272no—
130#124—1272no—
131#121—2542no—
132#120—2542no—
133C4×D4—2542no—
134C4×D4—2542no—
135#117—1271no—
136#117C2×D12711yescharacteristic
137#117C2×D12711yescharacteristic
138C2×C22⋊C4—2542no—
139#104—2542no—
140#104—2542no—
141#100—2542no—
142C24—1272no—
143#41—2541no—
144#41—2541no—
145#41—2541no—
146#41—2541no—
147C2×Q8—1271no—
148C2×D4—1272no—
149C2×D4—1271no—
150C2×D4—1272no—
151C2×D4—2544no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4—2541no—
157C22×C4—2544no—
158C22×C4—2541no—
159C22×C4C22×D12711yescharacteristic
160M16—21no—
161M16—21no—
162M16—2541no—
163M16—2541no—
164C2×C8—21no—
165C2×C8—2541no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—2541no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—2544no—
172C22⋊C4—2544no—
173C22⋊C4—2541no—
174C42—1272no—
175C42C4×D12712yescomplemented
176C42D50812yescomplemented
177C23—1274no—
178C23—10162no—
179C23—2542no—
180C23—2541no—
181C23—2542no—
182C23D4×D12711yescharacteristic
183Q8—2541no—
184Q8—2541no—
185D4—10164no—
186D4—5081no—
187D4—5081no—
188D4—2542no—
189D4—2541no—
190D4—2542no—
191D4—2541no—
192D4—2542no—
193D4—2542no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—10164no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—2541no—
200C2×C4—2542no—
201C2×C4C2×C4×D12711yescharacteristic
202C2×C4D4×D12711yescharacteristic
203C2×C4D4×D12711yescharacteristic
204C2×C4C2×D50811yescharacteristic
205C2×C4#1452324811yescharacteristic
206C2×C4#1452324911yescharacteristic
207C8—41no—
208C8—5081no—
209C22—41no—
210C22—10162no—
211C22—5082no—
212C22—5084no—
213C22—5084no—
214C22—5081no—
215C22—2544no—
216C22C22⋊C4×D12711yescharacteristic
217C22#1686117911yescharacteristic
218C22#1686119411yescharacteristic
219C4—42no—
220C4—42no—
221C4—5082no—
222C4—5081no—
223C4#1686117111yescharacteristic
224C4C4×D50811yescharacteristic
225C4#1686118211yescharacteristic
226C4#1686118411yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—10161no—
231C2—5084no—
232C2#2254975411yescharacteristic
subgroupG/H
centerC2#22549754
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingH(3,ℤ/4)×C127C2
socleC2×C127#386
prime radical#22550445C2
cosocleC22×C4×C127C23
hypercenterH(3,ℤ/4)D127
hypocenterC127#1911
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #22550350C2semidirect
3. #22549894C2semidirect
4. #22549892C2semidirect
5. C2×#16861125C2semidirect
6. #22549632C2semidirect
7. H(3,ℤ/4)×C127C2semidirect
11. #16861206C4semidirect
12. #16861197C4semidirect
30. C2×Dic254C2×C4semidirect
37. C2×D508C2×C4semidirect
49. C4×Dic127D4semidirect
58. C42×C127C2×C4semidirect
59. C42×C127D4semidirect
117. C127#1911semidirect
124. H(3,ℤ/4)D127semidirect
175. C42C4×D127semidirect
176. C42D508semidirect

Maximal subgroups

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

subgroupindex
#225504452
#225503502
#225498942
#225498922
C2×#168611252
#225496322
H(3,ℤ/4)×C1272
#1911127