Finite Group Databasebeta

← #30639082

Subgroups — #30639082

Order 16256.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14523236C211yesmaximal, characteristic
2C2×#16861174C211yesmaximal, characteristic, complemented
3C2×#16861134C211yesmaximal, characteristic, complemented
4#22549779C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—22no—
10C2×#14523236—22no—
11C2×#14523236C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13#16861174—22no—
14#16861174—22no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4—21no—
17C2×C22⋊Dic127—21no—
18C2×C22⋊Dic127—21no—
19C2×C4×Dic127—21no—
20C2×C4×Dic127C2211yescharacteristic
21#16861134—22no—
22C22×D4×C127C2211yescharacteristic
23C22×C8×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25#102×C127—22no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C2×Dic254D412yes—
31C2×Dic254D412yes—
32C22×Dic127—21no—
33C22×Dic127—21no—
34C22×Dic127D411yescharacteristic
35C22×Dic127D411yescharacteristic
36#14523236—28no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8C2×C412yes—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42Dic127⋊C4—24no—
43C22⋊Dic127—42no—
44C22⋊Dic127—42no—
45C4×Dic127—24no—
46C4×Dic127—21no—
47C4×Dic127—21no—
48C24×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127C2×C412yes—
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59C2×C8×C127—21no—
60C4⋊C4×C127—21no—
61C4⋊C4×C127D412yes—
62Dic254—22no—
63Dic254—22no—
64Dic254—22no—
65Dic254C22⋊C414yescomplemented
66C2×Dic127—22no—
67C2×Dic127—21no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127SD1612yescomplemented
78C127⋊C8—22no—
79C23×C127—22no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127C2×D411yescharacteristic
87D4×C127—22no—
88D4×C127—22no—
89D4×C127—41no—
90D4×C127—41no—
91C2×C4×C127—22no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127C22×C411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C8×C127—41no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127—42no—
104C22×C127—81no—
105C22×C127—81no—
106C22×C127—81no—
107C22×C127—81no—
108C22×C127C4×D411yescharacteristic
109C22×C127#12011yescharacteristic
110C22×C127#12111yescharacteristic
111C22×C127C2×SD1611yescharacteristic
112C22×C127#13511yescharacteristic
113C22×C127AGL(1,ℤ/8)11yescharacteristic
114C22×C127#13711yescharacteristic
115C4×C127—42no—
116C4×C127C2×C22⋊C411yescharacteristic
117C4×C127C4×D411yescharacteristic
118C4×C127#12111yescharacteristic
119C4×C127#12311yescharacteristic
120Dic127—24no—
121Dic127—22no—
122Dic127—41no—
123Dic127—81no—
124C2×C127—82no—
125C2×C127#38611yescharacteristic
126C2×C127C4×SD1611yescharacteristic
127C2×C127#44011yescharacteristic
128C2×C127#44811yescharacteristic
129C2×C127#44911yescharacteristic
130C2×C127#46011yescharacteristic
131C2×C127#46411yescharacteristic
132#1880—1271nomaximal, Sylow
133C127#188011yescharacteristic, complemented, Sylow
134C22×SD16—1271no—
135C2×C4×Q8—1271no—
136C2×#103—1271no—
137C2×#102D12711yescharacteristic
138#390—1271no—
139#340—1271no—
140#336—1271no—
141C22×Q8—1271no—
142C22×D4C2×D12711yescharacteristic
143C2×SD16—1274no—
144C2×SD16—2542no—
145C2×SD16—2542no—
146C22×C8—1271no—
147C22×C8C2×D12711yescharacteristic
148C4×Q8—2542no—
149C4×Q8—2542no—
150C2×C4⋊C4—1271no—
151C2×C4⋊C4—2541no—
152C2×C4⋊C4C2×D12711yescharacteristic
153C2×C22⋊C4—2541no—
154C2×C22⋊C4—2541no—
155C2×C42—1271no—
156C2×C42—2541no—
157#103—2542no—
158#102—22no—
159C24—21no—
160C2×Q8—1272no—
161C2×Q8—1272no—
162C2×Q8—1272no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C4×D12712yes—
172C22×C4—21no—
173C22×C4—1271no—
174C22×C4—1271no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178SD16—2548no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—1272no—
183C2×C8—2541no—
184C2×C8—2541no—
185C4⋊C4—21no—
186C4⋊C4—2544no—
187C4⋊C4—2541no—
188C4⋊C4—2541no—
189C4⋊C4C22⋊D12712yes—
190C22⋊C4—5082no—
191C22⋊C4—5082no—
192C42—2544no—
193C42—2541no—
194C42—2541no—
195C23—22no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23D4×D12711yescharacteristic
203Q8—1274no—
204Q8—2542no—
205Q8—2542no—
206Q8—2542no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—1272no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—2541no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4—2542no—
223C2×C4—2541no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4C2×C4×D12711yescharacteristic
227C2×C4D4×D12711yescharacteristic
228C2×C4D4×D12711yescharacteristic
229C2×C4C2×C22⋊D12711yescharacteristic
230C2×C4#1452324711yescharacteristic
231C2×C4#1452324811yescharacteristic
232C8—41no—
233C8—2542no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22(C22⋊D127)⋊C411yescharacteristic
242C22#1686117711yescharacteristic
243C22#1686118211yescharacteristic
244C22SD16×D12711yescharacteristic
245C22#1686122811yescharacteristic
246C22#1686123311yescharacteristic
247C22#1686123811yescharacteristic
248C4—42no—
249C4—10161no—
250C4—5081no—
251C4—2544no—
252C4—2542no—
253C4C22⋊C4×D12711yescharacteristic
254C4C4×C22⋊D12711yescharacteristic
255C4#1686118011yescharacteristic
256C4#1686119111yescharacteristic
257C2—82no—
258C2#2254975211yescharacteristic
259C2#2254996711yescharacteristic
260C2#2254997311yescharacteristic
261C2#2255001311yescharacteristic
262C2#2255001811yescharacteristic
263C2#2255007511yescharacteristic
264C2#2255009711yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1880
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16861174C2semidirect
3. C2×#16861134C2semidirect
5. #22549638C2semidirect
6. #22549630C2semidirect
11. C2×#14523236C4semidirect
65. Dic254C22⋊C4semidirect
77. C2×Dic127SD16semidirect
133. C127#1880semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168611742
C2×#168611342
#225497792
#225496382
#225496302
C2×#102×C1272
#1880127