Finite Group Databasebeta

← #30639087

Subgroups — #30639087

Order 16256.

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

normal subgroups only

This group has 374 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
1C2×Q8×Dic127C211yesmaximal, characteristic, complemented
2C2×#16861133C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic
4#22549779C211yesmaximal, characteristic
5#22549639C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C22×SD16×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9Q8×Dic127—22no—
10Q8×Dic127—22no—
11C2×C4⋊Dic127—21no—
12C2×C4⋊Dic127C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127—21no—
15C2×C22⋊Dic127—21no—
16C2×C4×Dic127—21no—
17C2×C4×Dic127C2211yescharacteristic
18#16861133—22no—
19#16861130—22no—
20C22×Q8×C127C2211yescharacteristic
21C22×D4×C127C2211yescharacteristic
22C2×SD16×C127—22no—
23C2×SD16×C127—22no—
24C2×SD16×C127C414yescomplemented
25C22×C8×C127C2211yescharacteristic
26C22×Dic127—21no—
27C22×Dic127—21no—
28C22×Dic127D411yescharacteristic
29C22×Dic127D411yescharacteristic
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C4⋊Dic127—24no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D412yes—
38C22⋊Dic127—42no—
39C22⋊Dic127—42no—
40C4×Dic127—24no—
41C4×Dic127—21no—
42C4×Dic127—21no—
43C24×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127C2×C412yes—
48C2×Q8×C127D412yes—
49C2×Q8×C127D412yes—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127C2×C412yes—
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58SD16×C127—28no—
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C2×C8×C127C2×C412yes—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—22no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127SD1612yescomplemented
74C127⋊C8—41no—
75C23×C127—22no—
76C23×C127—41no—
77C23×C127—41no—
78C23×C127—41no—
79C23×C127—41no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—22no—
84Q8×C127—22no—
85Q8×C127—22no—
86Q8×C127C22⋊C414yescomplemented
87D4×C127—22no—
88D4×C127—22no—
89D4×C127—41no—
90D4×C127—41no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
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—22no—
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—24no—
116C4×C127C2×C22⋊C411yescharacteristic
117C4×C127C4×D411yescharacteristic
118C4×C127#12111yescharacteristic
119C4×C127#12311yescharacteristic
120Dic127—22no—
121Dic127—41no—
122Dic127—42no—
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×SD16D12711yescharacteristic
135C2×C4×Q8—1271no—
136C2×#103—1271no—
137C2×#102—1271no—
138#390—1271no—
139#340—1271no—
140#336—1271no—
141C22×Q8C2×D12711yescharacteristic
142C22×D4C2×D12711yescharacteristic
143C2×SD16—22no—
144C2×SD16—22no—
145C2×SD16Dic12714yescomplemented
146C22×C8—1271no—
147C22×C8C2×D12711yescharacteristic
148C4×Q8—2542no—
149C4×Q8—2542no—
150C2×C4⋊C4—1271no—
151C2×C4⋊C4—1271no—
152C2×C4⋊C4—2541no—
153C2×C22⋊C4—2541no—
154C2×C22⋊C4—2541no—
155C2×C42—1271no—
156C2×C42—2541no—
157#103—2542no—
158#102—2542no—
159C24—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8C2×Dic12712yes—
164C2×Q8C22⋊D12712yes—
165C2×Q8C22⋊D12712yes—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C2×Dic12712yes—
172C22×C4—21no—
173C22×C4—1271no—
174C22×C4—1271no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178SD16—28no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—2541no—
182C2×C8—2541no—
183C2×C8—2541no—
184C2×C8C2×Dic12712yes—
185C4⋊C4—1272no—
186C4⋊C4—2544no—
187C4⋊C4—2541no—
188C4⋊C4—2541no—
189C4⋊C4—2541no—
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—22no—
204Q8—22no—
205Q8—22no—
206Q8C22⋊Dic12714yescomplemented
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—1272no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—5081no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4—2542no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2541no—
226C2×C4C22×Dic12711yescharacteristic
227C2×C4D4×D12711yescharacteristic
228C2×C4C2×C22⋊D12711yescharacteristic
229C2×C4C2×C22⋊D12711yescharacteristic
230C2×C4#1452324711yescharacteristic
231C2×C4#1452324711yescharacteristic
232C8—22no—
233C8—5081no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22D4×Dic12711yescharacteristic
242C22#1686117711yescharacteristic
243C22#1686118111yescharacteristic
244C22SD16×D12711yescharacteristic
245C22#1686123011yescharacteristic
246C22#1686123611yescharacteristic
247C22#1686123811yescharacteristic
248C4—24no—
249C4—10161no—
250C4—5082no—
251C4—5081no—
252C4—2542no—
253C4C2×C22⋊Dic12711yescharacteristic
254C4D4×Dic12711yescharacteristic
255C4#1686118111yescharacteristic
256C4#1686119011yescharacteristic
257C2—82no—
258C2#2254975611yescharacteristic
259C2SD16×Dic12711yescharacteristic
260C2#2254997711yescharacteristic
261C2#2255001311yescharacteristic
262C2#2255002011yescharacteristic
263C2#2255007411yescharacteristic
264C2#2255009611yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC22×SD16D127
hypocenterC127#1880
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic127C2semidirect
2. C2×#16861133C2semidirect
5. #22549639C2semidirect
6. #22549630C2semidirect
24. C2×SD16×C127C4semidirect
73. C2×Dic127SD16semidirect
86. Q8×C127C22⋊C4semidirect
133. C127#1880semidirect
145. C2×SD16Dic127semidirect
206. Q8C22⋊Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1272
C2×#168611332
C2×#168611302
#225497792
#225496392
#225496302
C22×SD16×C1272
#1880127