Finite Group Databasebeta

← #30637745

Subgroups — #30637745

Order 16256.

232 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550463C211yesmaximal, characteristic
2#22550207C211yesmaximal, characteristic, complemented
3#22550187C211yesmaximal, characteristic, complemented
4#22549803C211yesmaximal, characteristic, complemented
5#22549612C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861195—21no—
10#16861193—21no—
11#16861188C2211yescharacteristic
12#16861181—21no—
13#16861177—21no—
14D4×Dic127—21no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×(C2×D127)⋊C4—21no—
17C2×(C2×D127)⋊C4C2211yescharacteristic
18#16861142—21no—
19#16861141—21no—
20#16861134—21no—
21#16861130—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic
23C127⋊#95—21no—
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C22×Dic127—21no—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D411yescharacteristic
39Dic127⋊C4—21no—
40C22⋊Dic127—21no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—42no—
47C4×Dic127—21no—
48C2×D4×C127—21no—
49C2×D4×C127D411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C2×C8×C127—21no—
53C4⋊C4×C127D411yescharacteristic
54C22⋊C4×C127—21no—
55C22×D127—21no—
56C22×D127—21no—
57C22×D127—21no—
58C22×D127—41no—
59C22×D127—41no—
60Dic254—41no—
61C22⋊D127—81no—
62C22⋊D127—81no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—81no—
71C127⋊C8—41no—
72C23×C127—21no—
73C23×C127C2×D411yescharacteristic
74D4×C127—41no—
75D4×C127—41no—
76D4×C127—41no—
77C2×C4×C127—21no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C8×C127—41no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—41no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127#12011yescharacteristic
91C22×C127#13511yescharacteristic
92C22×C127AGL(1,ℤ/8)11yescharacteristic
93C2×D127—42no—
94C2×D127—41no—
95C2×D127—41no—
96C2×D127—41no—
97C2×D127—81no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—41no—
101C4×C127—81no—
102Dic127—41no—
103Dic127—41no—
104Dic127—41no—
105Dic127—81no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—81no—
109C2×C127#44911yescharacteristic
110C2×C127#45511yescharacteristic
111C2×C127#45811yescharacteristic
112D127—81no—
113#1612—1271nomaximal, Sylow
114C127#161211yescharacteristic, complemented, Sylow
115#538—1271no—
116#483—1271no—
117#480—1271no—
118#397—1271no—
119#328—1271no—
120#327D12711yescharacteristic, complemented
121#323—1271no—
122#126—2541no—
123#124—2541no—
124#123—2541no—
125C22⋊Q8—1271no—
126#121—2541no—
127#121C2×D12711yescharacteristic
128#120—2541no—
129C4×D4—2541no—
130C2×C4⋊C4—1271no—
131C2×C22⋊C4—1271no—
132C2×C22⋊C4—2541no—
133#107—2541no—
134#106—2541no—
135#103—2541no—
136#102—2541no—
137C22⋊C8—1271no—
138C22⋊C8C2×D12711yescharacteristic
139#95—2541no—
140#95C2×D12711yescharacteristic
141C24—1271no—
142C2×Q8—1271no—
143C2×D4—21no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4D50811yescharacteristic
147C22×C4—21no—
148C22×C4—1271no—
149C22×C4—2541no—
150C22×C4—2541no—
151C22×C4C22×D12711yescharacteristic
152C2×C8—21no—
153C2×C8—2541no—
154C4⋊C4—1271no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4D50811yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—5082no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C42—2541no—
169C23—21no—
170C23—5081no—
171C23—5081no—
172C23—2541no—
173C23—2541no—
174C23—2541no—
175C23D4×D12711yescharacteristic
176Q8—5081no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—10161no—
181D4—10161no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—10161no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—2541no—
193C2×C4—2541no—
194C2×C4—2541no—
195C2×C4D4×D12711yescharacteristic
196C2×C4C2×D50811yescharacteristic
197C8—41no—
198C8—5081no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—10161no—
205C22—5082no—
206C22—5081no—
207C22—5081no—
208C22—5081no—
209C22#1686117911yescharacteristic
210C22#1686123011yescharacteristic
211C22#1686123311yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—10161no—
217C4—5081no—
218C4—5081no—
219C4—5081no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—10161no—
224C2#2255002211yescharacteristic
225C2#2255005511yescharacteristic
226C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#22550463C2
hypercenter#327D127
hypocenterC127#1612
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22550207C2semidirect
3. #22550187C2semidirect
4. #22549803C2semidirect
5. #22549612C2semidirect
6. #22549599C2semidirect
7. #327×C127C2semidirect
114. C127#1612semidirect
120. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225504632
#225502072
#225501872
#225498032
#225496122
#225495992
#327×C1272
#1612127