Finite Group Databasebeta

← #30638921

Subgroups — #30638921

Order 16256.

360 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4⋊C4×D127C211yesmaximal, characteristic
2C2×#16861140C211yesmaximal, characteristic, complemented
3C2×(C2×D127)⋊C8C211yesmaximal, characteristic
4#22549733C211yesmaximal, characteristic
5#22549639C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#106×C127C211yesmaximal, characteristic, complemented
8C22×C4×D127—21no—
9C22×C4×D127C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4—21no—
14C4⋊C4×D127—22no—
15C4⋊C4×D127—22no—
16C2×(C2×D127)⋊C4—21no—
17#16861140—22no—
18(C2×D127)⋊C8C414yescomplemented
19C127⋊#95—21no—
20C22×C8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#106×C127—22no—
24C23×D127D411yescharacteristic
25C22×Dic127—21no—
26C22×Dic127—21no—
27C22×Dic127D411yescharacteristic
28C2×C4×D127—24no—
29C2×C4×D127—21no—
30C2×C4×D127—21no—
31C2×C4×D127—21no—
32C2×C4×D127—21no—
33C2×C4×D127—42no—
34C2×C4×D127—42no—
35C2×C4×D127C2×C412yes—
36C2×C4×D127Q812yes—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8C2×C412yescomplemented
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127D412yes—
44Dic127⋊C4—24no—
45(C2×D127)⋊C4—42no—
46C22×C4×C127—21no—
47C22×C4×C127—21no—
48C22×C4×C127C2311yescharacteristic
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C2×C8×C127C2×C412yescomplemented
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127D412yes—
56C22×D127—21no—
57C22×D127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127C4⋊C412yes—
61C22×D127SD1612yes—
62C22×D127SD1612yes—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—22no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127C4⋊C412yes—
72C4×D127—22no—
73C4×D127—22no—
74C4×D127—22no—
75C4×D127—22no—
76C4×D127—48no—
77C127⋊C8—22no—
78C23×C127C2×D411yescharacteristic
79C2×C4×C127—22no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127C22×C411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×Q811yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C2×C4×C127#4111yescharacteristic
91C8×C127—22no—
92C22×C127C2×C4⋊C411yescharacteristic
93C22×C127#12011yescharacteristic
94C22×C127#12311yescharacteristic
95C22×C127C2×SD1611yescharacteristic
96C22×C127C2×SD1611yescharacteristic
97C22×C127AGL(1,ℤ/8)11yescharacteristic
98C22×C127#13711yescharacteristic
99C2×D127—22no—
100C2×D127—22no—
101C2×D127—22no—
102C2×D127—22no—
103C2×D127—22no—
104C2×D127—22no—
105C2×D127#10614yescomplemented
106C4×C127—42no—
107C4×C127—81no—
108C4×C127C4×D411yescharacteristic
109C4×C127C4×D411yescharacteristic
110C4×C127C22⋊Q811yescharacteristic
111C4×C127C22⋊Q811yescharacteristic
112Dic127—22no—
113Dic127—42no—
114Dic127—81no—
115C2×C127#38511yescharacteristic
116C2×C127C2×#10611yescharacteristic
117C2×C127#42811yescharacteristic
118C2×C127#44811yescharacteristic
119C2×C127#45011yescharacteristic
120C2×C127#48111yescharacteristic
121C2×C127#48311yescharacteristic
122D127—24no—
123#1853—1271nomaximal, Sylow
124C127#185311yescharacteristic, complemented, Sylow
125C22×C4⋊C4—1271no—
126C2×#106—1271no—
127C2×#106D12711yescharacteristic, complemented
128C2×C22⋊C8—1271no—
129#380—1271no—
130#340—1271no—
131#340—1271no—
132C23×C4—1271no—
133C23×C4—2541no—
134C22×C8—1271no—
135C22×C8C2×D12711yescharacteristic
136C2×C4⋊C4—1271no—
137C2×C4⋊C4—1271no—
138C2×C4⋊C4—2542no—
139C2×C4⋊C4—2541no—
140C2×C4⋊C4—2542no—
141C2×C4⋊C4C2×D12711yescharacteristic
142C2×C4⋊C4C2×D12711yescharacteristic
143C2×C22⋊C4—2541no—
144#106—22no—
145#106—2542no—
146C22⋊C8—1274no—
147#95—2541no—
148C24—1271no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1271no—
152C22×C4—1272no—
153C22×C4—1272no—
154C22×C4—5082no—
155C22×C4—5082no—
156C22×C4—2544no—
157C22×C4—2541no—
158C22×C4—2541no—
159C22×C4—2541no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4—2541no—
163C22×C4C22×D12711yescharacteristic
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—1272no—
167C2×C8—2541no—
168C2×C8—2541no—
169C2×C8C4×D12712yescomplemented
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—1272no—
174C4⋊C4—2544no—
175C4⋊C4—2541no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C4⋊C4C22⋊D12712yes—
179C22⋊C4—5082no—
180C23—1272no—
181C23—1272no—
182C23—1272no—
183C23—2541no—
184C23—2541no—
185C23—2541no—
186C23—2541no—
187C23D4×D12711yescharacteristic
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—1272no—
195C2×C4—5088no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—5081no—
200C2×C4—5081no—
201C2×C4—2542no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4—2542no—
207C2×C4—2542no—
208C2×C4C2×C4×D12711yescharacteristic
209C2×C4D4×D12711yescharacteristic
210C2×C4C2×C22⋊D12711yescharacteristic
211C2×C4Q8×D12711yescharacteristic
212C2×C4#1452324811yescharacteristic
213C2×C4#1452324911yescharacteristic
214C8—22no—
215C8—2542no—
216C22—1274no—
217C22—2542no—
218C22—2542no—
219C22—2542no—
220C22—2542no—
221C22—2542no—
222C22—2542no—
223C22C4⋊C4×D12711yescharacteristic
224C22#1686117711yescharacteristic
225C22#1686119411yescharacteristic
226C22SD16×D12711yescharacteristic
227C22SD16×D12711yescharacteristic
228C22#1686123611yescharacteristic
229C22#1686123811yescharacteristic
230C4—42no—
231C4—81no—
232C4—10161no—
233C4—5082no—
234C4—2542no—
235C4C4×C22⋊D12711yescharacteristic
236C4#1686117111yescharacteristic
237C4#1686118511yescharacteristic
238C4#1686118711yescharacteristic
239C2—2544no—
240C2#2254974711yescharacteristic
241C2#106×D12711yescharacteristic
242C2#2254991511yescharacteristic
243C2#2255001311yescharacteristic
244C2#2255002511yescharacteristic
245C2#2255019611yescharacteristic
246C2#2255020811yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#106×C127C2
socleC23×C127C2×D4
prime radicalC23×D127D4
cosocleC22×C4×C127C23
hypercenterC2×#106D127
hypocenterC127#1853
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
2. C2×#16861140C2semidirect
5. #22549639C2semidirect
6. #22549638C2semidirect
7. C2×#106×C127C2semidirect
18. (C2×D127)⋊C8C4semidirect
39. C2×C127⋊C8C2×C4semidirect
51. C2×C8×C127C2×C4semidirect
105. C2×D127#106semidirect
124. C127#1853semidirect
127. C2×#106D127semidirect
169. C2×C8C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1272
C2×#168611402
C2×(C2×D127)⋊C82
#225497332
#225496392
#225496382
C2×#106×C1272
#1853127