Finite Group Databasebeta

← #30637254

Subgroups — #30637254

Order 16256.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#22550412C211yesmaximal, characteristic
2#22549930C211yesmaximal, characteristic, complemented
3#22549727C211yesmaximal, characteristic, complemented
4#22549613C211yesmaximal, characteristic, complemented
5#22549611C211yesmaximal, characteristic
6#22549609C211yesmaximal, characteristic
7#328×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861200C411yescharacteristic, complemented
10#16861198C411yescharacteristic, complemented
11#16861193—21no—
12#16861183C2211yescharacteristic
13Q8×Dic127C411yescharacteristic, complemented
14#16861171C411yescharacteristic, complemented
15(C22⋊D127)⋊C4—21no—
16#16861167—21no—
17C2×Dic127⋊C4—21no—
18C2×C4×Dic127—21no—
19C2×C4×Dic127C2211yescharacteristic
20Dic127⋊C8—21no—
21C22⋊(C127⋊C8)C2211yescharacteristic
22#16861119—21no—
23C127⋊#95C2211yescharacteristic
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C2×D508C2×C411yescharacteristic
28C2×C22⋊D127—21no—
29C22×Dic127—21no—
30C22×Dic127D411yescharacteristic
31C22×Dic127D411yescharacteristic
32C2×C4×D127—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127C2×C411yescharacteristic
37Dic127⋊C4—24no—
38Dic127⋊C4—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43C4×Dic127—21no—
44C4×Dic127—42no—
45C4×Dic127C2×C411yescharacteristic
46C4×Dic127C2×C411yescharacteristic
47C4×Dic127D411yescharacteristic
48C4×Dic127D411yescharacteristic
49C2×Q8×C127C2×C411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C2×C8×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127C2×C411yescharacteristic
55C22⋊C4×C127—21no—
56C22×D127—21no—
57D508—41no—
58C22⋊D127—41no—
59C22⋊D127—41no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—21no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127C22⋊C411yescharacteristic
68C2×Dic127C22⋊C411yescharacteristic
69C2×Dic127C22⋊C411yescharacteristic
70C2×Dic127C22⋊C411yescharacteristic
71C4×D127—42no—
72C127⋊C8—41no—
73C23×C127C2×D411yescharacteristic
74Q8×C127—41no—
75C2×C4×C127—22no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C8×C127—41no—
84C22×C127—21no—
85C22×C127—21no—
86C22×C127C2×C22⋊C411yescharacteristic
87C22×C127AGL(1,ℤ/8)11yescharacteristic
88C22×C127#13711yescharacteristic
89C2×D127—41no—
90C2×D127—41no—
91C2×D127—41no—
92C4×C127—21no—
93C4×C127—21no—
94C4×C127—42no—
95C4×C127—41no—
96C4×C127—41no—
97Dic127—21no—
98Dic127—21no—
99Dic127—21no—
100Dic127—42no—
101Dic127—41no—
102Dic127#10412yes—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127#41011yescharacteristic
106C2×C127#41711yescharacteristic
107C2×C127C2×#10411yescharacteristic
108D127—81no—
109#1509—1271nomaximal, Sylow
110C127#150911yescharacteristic, complemented, Sylow
111#529—1271no—
112#432—1271no—
113C4×C4⋊C4—1271no—
114#328—1271no—
115#328D12711yescharacteristic, complemented
116#327—1271no—
117#327—1271no—
118#126—2541no—
119#125—1271no—
120#124—1271no—
121#123—2541no—
122C22⋊Q8C2×D12711yescharacteristic
123#121—1271no—
124C4×Q8—1271no—
125C4×D4—1271no—
126C4×D4—2541no—
127#117—2541no—
128C2×C4⋊C4—2541no—
129C2×C42—1271no—
130C2×C42—2541no—
131C4⋊C8—2541no—
132C22⋊C8—1271no—
133C22⋊C8C2×D12711yescharacteristic
134#97—2541no—
135#95—1271no—
136#95C2×D12711yescharacteristic
137C2×Q8C4×D12711yescharacteristic
138C2×D4—1271no—
139C2×D4—2541no—
140C22×C4—21no—
141C22×C4—1271no—
142C22×C4—1271no—
143C22×C4—2541no—
144C22×C4—2541no—
145C22×C4C22×D12711yescharacteristic
146C2×C8—21no—
147C2×C8—2541no—
148C4⋊C4—21no—
149C4⋊C4—1271no—
150C4⋊C4—2541no—
151C4⋊C4—2541no—
152C4⋊C4—2541no—
153C4⋊C4—2544no—
154C4⋊C4C4×D12711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—2541no—
157C22⋊C4—2541no—
158C22⋊C4—2541no—
159C22⋊C4—2541no—
160C42—1271no—
161C42—1271no—
162C42—1271no—
163C42—1271no—
164C42—5082no—
165C42—2541no—
166C23—2541no—
167C23D4×D12711yescharacteristic
168Q8—41no—
169D4—5081no—
170D4—5081no—
171D4—5081no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—1271no—
179C2×C4—1271no—
180C2×C4—1271no—
181C2×C4—1271no—
182C2×C4—5082no—
183C2×C4—5081no—
184C2×C4—5081no—
185C2×C4—2542no—
186C2×C4—2542no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4C2×C4×D12711yescharacteristic
191C2×C4D4×D12711yescharacteristic
192C8—41no—
193C8—5081no—
194C22—21no—
195C22—21no—
196C22—5081no—
197C22—5081no—
198C22—5081no—
199C22C22⋊C4×D12711yescharacteristic
200C22#1686123611yescharacteristic
201C22#1686124111yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—1272no—
208C4—5082no—
209C4—5081no—
210C4—2541no—
211C4—2541no—
212C4—2541no—
213C2—21no—
214C2—21no—
215C2—10161no—
216C2#2254984811yescharacteristic
217C2#2254987611yescharacteristic
218C2#104×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861183C22
cosocleC22×C4×C127C23
hypercenter#328D127
hypocenterC127#1509
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #22549930C2semidirect
3. #22549727C2semidirect
4. #22549613C2semidirect
7. #328×C127C2semidirect
9. #16861200C4semidirect
10. #16861198C4semidirect
13. Q8×Dic127C4semidirect
14. #16861171C4semidirect
110. C127#1509semidirect
115. #328D127semidirect

Maximal subgroups

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

subgroupindex
#225504122
#225499302
#225497272
#225496132
#225496112
#225496092
#328×C1272
#1509127