Finite Group Databasebeta

← #30639672

Subgroups — #30639672

Order 16256.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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#22550429C211yesmaximal, characteristic
2M16×Dic127C211yesmaximal, characteristic
3C4×Dic127⋊C4C211yesmaximal, characteristic
4#22549637C211yesmaximal, characteristic
5#22549636C211yesmaximal, characteristic
6#22549635C211yesmaximal, characteristic
7#339×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9#16861208C411yescharacteristic, complemented
10#16861208C411yescharacteristic, complemented
11#16861200C411yescharacteristic, complemented
12#16861200C411yescharacteristic, complemented
13#16861186—21no—
14#16861186—21no—
15C4×Dic254—21no—
16C4×Dic254—21no—
17#16861168C2211yescharacteristic
18C2×Dic127⋊C4—21no—
19C2×C4×Dic127C2211yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21#16861119—21no—
22C8×Dic127—21no—
23C127⋊#95—21no—
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—21no—
29C22×Dic127—21no—
30C22×Dic127D411yescharacteristic
31C22×Dic127D411yescharacteristic
32#14523232—21no—
33#14523232Q812yescomplemented
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127C2×C411yescharacteristic
38C4⋊Dic127C2×C411yescharacteristic
39Dic127⋊C4—24no—
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44C22⋊Dic127—21no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C4×Dic127—21no—
48C4×Dic127C2×C411yescharacteristic
49C4×Dic127C2×C411yescharacteristic
50C4×Dic127D411yescharacteristic
51C4×Dic127D411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127Q812yescomplemented
56C2×C8×C127—21no—
57C4⋊C4×C127C2×C411yescharacteristic
58C4⋊C4×C127C2×C411yescharacteristic
59C22⋊C4×C127—21no—
60C42×C127—21no—
61C42×C127—21no—
62C42×C127—21no—
63Dic254—42no—
64Dic254—42no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127C22⋊C411yescharacteristic
73C2×Dic127C22⋊C411yescharacteristic
74C2×Dic127C22⋊C411yescharacteristic
75C2×Dic127C22⋊C411yescharacteristic
76C127⋊C8—22no—
77C23×C127C2×D411yescharacteristic
78C2×C4×C127—22no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127C22×C411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×Q811yescharacteristic
86C2×C4×C127C2×Q811yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C8×C127—22no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127C2×C22⋊C411yescharacteristic
93C22×C127#12411yescharacteristic
94C22×C127C4⋊Q811yescharacteristic
95C4×C127—42no—
96C4×C127—41no—
97C4×C127—41no—
98C4×C127C4×Q811yescharacteristic
99C4×C127C4×Q811yescharacteristic
100C4×C127C22⋊Q811yescharacteristic
101C4×C127C22⋊Q811yescharacteristic
102Dic127—21no—
103Dic127—21no—
104Dic127—21no—
105Dic127—42no—
106Dic127—41no—
107Dic127—41no—
108Dic127#10412yes—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127#39111yescharacteristic
112C2×C127C2×#10411yescharacteristic
113C2×C127#42111yescharacteristic
114#1986—1271nomaximal, Sylow
115C127#198611yescharacteristic, complemented, Sylow
116#533—1271no—
117C4×M16—1271no—
118C4×C4⋊C4—1271no—
119#339—1271no—
120#339—1271no—
121#339—1271no—
122#339D12711yescharacteristic
123C2×M16—1271no—
124C2×M16C2×D12711yescharacteristic
125C4⋊Q8—1271no—
126C4⋊Q8—1271no—
127#125—1271no—
128#125—1271no—
129C22⋊Q8—2541no—
130C22⋊Q8—2541no—
131C4×Q8—2541no—
132C4×Q8—2541no—
133#117—1271no—
134#117C2×D12711yescharacteristic
135C2×C4⋊C4—2541no—
136C2×C42—1271no—
137C2×C42—1271no—
138C2×C42C2×D12711yescharacteristic
139#97—2541no—
140C4×C8—2541no—
141#95—2541no—
142C2×Q8—2541no—
143C2×Q8—2541no—
144C22×C4—21no—
145C22×C4—1271no—
146C22×C4—1271no—
147C22×C4—2541no—
148C22×C4C22×D12711yescharacteristic
149M16—21no—
150M16—1272no—
151M16—2541no—
152M16Dic25412yescomplemented
153C2×C8—21no—
154C2×C8—2541no—
155C4⋊C4—1271no—
156C4⋊C4—1271no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C4⋊C4—2544no—
164C4⋊C4C4×D12711yescharacteristic
165C4⋊C4C4×D12711yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—2541no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—1271no—
172C42—1271no—
173C42—1271no—
174C42—1271no—
175C42—2541no—
176C42—2541no—
177C42—2541no—
178C23D4×D12711yescharacteristic
179Q8—5082no—
180Q8—5082no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1271no—
187C2×C4—1271no—
188C2×C4—1271no—
189C2×C4—1271no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4C2×C4×D12711yescharacteristic
198C2×C4D4×D12711yescharacteristic
199C2×C4Q8×D12711yescharacteristic
200C2×C4C2×Dic25411yescharacteristic
201C2×C4#1452324711yescharacteristic
202C2×C4#1452324811yescharacteristic
203C8—22no—
204C8—2542no—
205C22—21no—
206C22—21no—
207C22C22⋊C4×D12711yescharacteristic
208C22#1686119511yescharacteristic
209C22#1686120811yescharacteristic
210C4—42no—
211C4—41no—
212C4—41no—
213C4—1272no—
214C4—5082no—
215C4—5081no—
216C4—5081no—
217C4—2541no—
218C4—2541no—
219C4—2541no—
220C4#1686117411yescharacteristic
221C4C4×Dic25411yescharacteristic
222C4#1686118611yescharacteristic
223C4#1686118711yescharacteristic
224C2—21no—
225C2—21no—
226C2#2254978011yescharacteristic
227C2#104×D12711yescharacteristic
228C2#2254989111yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1986
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #16861208C4semidirect
10. #16861208C4semidirect
11. #16861200C4semidirect
12. #16861200C4semidirect
33. #14523232Q8semidirect
55. M16×C127Q8semidirect
115. C127#1986semidirect
152. M16Dic254semidirect

Maximal subgroups

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

subgroupindex
#225504292
M16×Dic1272
C4×Dic127⋊C42
#225496372
#225496362
#225496352
#339×C1272
#1986127