Finite Group Databasebeta

← #30637747

Subgroups — #30637747

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#22550466C211yesmaximal, characteristic
2#22550204C211yesmaximal, characteristic, complemented
3#22550190C211yesmaximal, characteristic, complemented
4#22549803C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#328×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861198—21no—
10#16861194—21no—
11#16861184—21no—
12#16861183C2211yescharacteristic
13#16861179—21no—
14(C22⋊D127)⋊C4—21no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×(C2×D127)⋊C4—21no—
17C2×(C2×D127)⋊C4C2211yescharacteristic
18#16861143—21no—
19#16861140—21no—
20#16861133—21no—
21#16861131—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic
23C127⋊#95—21no—
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×D508—21no—
29C2×D508—21no—
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C2×C4×D127—21no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D411yescharacteristic
39Dic127⋊C4—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—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×Q8×C127D411yescharacteristic
49C22×C4×C127—21no—
50C22×C4×C127C2311yescharacteristic
51C2×C8×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127D411yescharacteristic
54C22⋊C4×C127—21no—
55C22×D127—21no—
56C22×D127—21no—
57C22×D127—21no—
58C22×D127—21no—
59C22×D127—41no—
60C22×D127—41no—
61D508—41no—
62D508—81no—
63D508—81no—
64C22⋊D127—41no—
65C22⋊D127—41no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C4×D127—81no—
73C127⋊C8—41no—
74C23×C127C2×D411yescharacteristic
75Q8×C127—41no—
76C2×C4×C127—21no—
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#12011yescharacteristic
88C22×C127#13511yescharacteristic
89C22×C127AGL(1,ℤ/8)11yescharacteristic
90C2×D127—42no—
91C2×D127—41no—
92C2×D127—41no—
93C2×D127—41no—
94C2×D127—41no—
95C2×D127—41no—
96C2×D127—41no—
97C2×D127—81no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—41no—
101C4×C127—41no—
102C4×C127—81no—
103Dic127—41no—
104Dic127—41no—
105Dic127—81no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127#44911yescharacteristic
109C2×C127#45511yescharacteristic
110C2×C127#45811yescharacteristic
111D127—81no—
112D127—81no—
113#1612—1271nomaximal, Sylow
114C127#161211yescharacteristic, complemented, Sylow
115#538—1271no—
116#483—1271no—
117#480—1271no—
118#397—1271no—
119#328D12711yescharacteristic, complemented
120#327—1271no—
121#323—1271no—
122#126—2541no—
123#124—2541no—
124#123—2541no—
125C22⋊Q8C2×D12711yescharacteristic
126#121—1271no—
127#121—2541no—
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×Q8D50811yescharacteristic
143C2×D4—1271no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C22×C4—21no—
148C22×C4—1271no—
149C22×C4—2541no—
150C22×C4—2541no—
151C22×C4C22×D12711yescharacteristic
152C2×C8—21no—
153C2×C8—2541no—
154C4⋊C4—21no—
155C4⋊C4—1271no—
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—5081no—
170C23—5081no—
171C23—2541no—
172C23—2541no—
173C23—2541no—
174C23—2541no—
175C23D4×D12711yescharacteristic
176Q8—41no—
177D4—10161no—
178D4—10161no—
179D4—5081no—
180D4—5081no—
181D4—5081no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—10161no—
189C2×C4—5081no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—2541no—
194C2×C4—2541no—
195C2×C4D4×D12711yescharacteristic
196C2×C4C2×D50811yescharacteristic
197C8—41no—
198C8—5081no—
199C22—21no—
200C22—21no—
201C22—10161no—
202C22—5082no—
203C22—5081no—
204C22—5081no—
205C22—5081no—
206C22—5081no—
207C22—5081no—
208C22—5081no—
209C22#1686117911yescharacteristic
210C22#1686123211yescharacteristic
211C22#1686123611yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—81no—
217C4—10161no—
218C4—5081no—
219C4—5081no—
220C2—21no—
221C2—21no—
222C2—10161no—
223C2—10161no—
224C2#2255002411yescharacteristic
225C2#2255005511yescharacteristic
226C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127#120
prime radical#22550466C2
hypercenter#328D127
hypocenterC127#1612
LCSG ▷ C22×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. #22550204C2semidirect
3. #22550190C2semidirect
4. #22549803C2semidirect
5. #22549609C2semidirect
6. #22549599C2semidirect
7. #328×C127C2semidirect
114. C127#1612semidirect
119. #328D127semidirect

Maximal subgroups

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

subgroupindex
#225504662
#225502042
#225501902
#225498032
#225496092
#225495992
#328×C1272
#1612127