Finite Group Databasebeta

← #30637744

Subgroups — #30637744

Order 16256.

232 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550204C211yesmaximal, characteristic, complemented
2#22550187C211yesmaximal, characteristic, complemented
3#22549801C211yesmaximal, characteristic, complemented
4#22549613C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549600C211yesmaximal, characteristic, complemented
7#538×C127C211yesmaximal, characteristic
8C2×C4⋊Dic127C2211yescharacteristic
9C2×C22⋊Dic127—21no—
10#16861142—21no—
11#16861140—21no—
12#16861133—21no—
13#16861130—21no—
14C22⋊(C127⋊C8)C2211yescharacteristic
15C22⋊(C127⋊C8)C2211yescharacteristic
16C127⋊#95—21no—
17C127⋊#95C2211yescharacteristic
18#126×C127—21no—
19#124×C127—21no—
20#123×C127—21no—
21C22⋊Q8×C127C2211yescharacteristic
22#121×C127—21no—
23#121×C127C2211yescharacteristic
24#120×C127—21no—
25C4×D4×C127—21no—
26C2×C22⋊C4×C127C2211yescharacteristic
27C22×Dic127—21no—
28C22×Dic127—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33C22⋊Dic127—42no—
34C24×C127D411yescharacteristic
35C2×Q8×C127D411yescharacteristic
36C2×D4×C127—21no—
37C2×D4×C127—21no—
38C2×D4×C127—21no—
39C2×D4×C127D411yescharacteristic
40C22×C4×C127—21no—
41C22×C4×C127D411yescharacteristic
42C22×C4×C127C2311yescharacteristic
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C4⋊C4×C127D411yescharacteristic
47C22⋊C4×C127—21no—
48C22⋊C4×C127—21no—
49C22⋊C4×C127—21no—
50C22⋊C4×C127—21no—
51C22⋊C4×C127—21no—
52C22⋊C4×C127—21no—
53C22⋊C4×C127—21no—
54C42×C127—21no—
55C2×Dic127—41no—
56C2×Dic127—41no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C127⋊C8—41no—
62C127⋊C8—41no—
63C23×C127—21no—
64C23×C127—21no—
65C23×C127—21no—
66C23×C127—21no—
67C23×C127—41no—
68C23×C127—41no—
69C23×C127C2×D411yescharacteristic
70Q8×C127—41no—
71D4×C127—41no—
72D4×C127—41no—
73D4×C127—41no—
74D4×C127—81no—
75D4×C127—81no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—81no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—42no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—41no—
94C22×C127—81no—
95C22×C127#12011yescharacteristic
96C22×C127#13511yescharacteristic
97C22×C127AGL(1,ℤ/8)11yescharacteristic
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—41no—
101C4×C127—41no—
102C4×C127—41no—
103C4×C127—41no—
104Dic127—81no—
105Dic127—81no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—81no—
109C2×C127—81no—
110C2×C127#44911yescharacteristic
111C2×C127#45511yescharacteristic
112C2×C127#45811yescharacteristic
113#1612—1271nomaximal, Sylow
114C127#161211yescharacteristic, complemented, Sylow
115#538D12711yescharacteristic
116#483—1271no—
117#480—1271no—
118#397—1271no—
119#328—1271no—
120#327—1271no—
121#323—1271no—
122#126—21no—
123#124—21no—
124#123—21no—
125C22⋊Q8C2×D12711yescharacteristic
126#121—21no—
127#121C2×D12711yescharacteristic
128#120—21no—
129C4×D4—21no—
130C2×C4⋊C4—1271no—
131C2×C22⋊C4—2541no—
132C2×C22⋊C4C2×D12711yescharacteristic
133#107—2541no—
134#106—2541no—
135#103—2541no—
136#102—2541no—
137C22⋊C8—1271no—
138C22⋊C8—1271no—
139#95—1271no—
140#95—2541no—
141C24C22⋊D12711yescharacteristic
142C2×Q8C22⋊D12711yescharacteristic
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C22⋊D12711yescharacteristic
147C22×C4—21no—
148C22×C4—2541no—
149C22×C4—2541no—
150C22×C4C22⋊D12711yescharacteristic
151C22×C4C22×D12711yescharacteristic
152C2×C8—2541no—
153C2×C8—2541no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4C22⋊D12711yescharacteristic
159C4⋊C4C22⋊D12711yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—5082no—
168C42—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23C2×C22⋊D12711yescharacteristic
176Q8—41no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—81no—
181D4—81no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—81no—
189C2×C4—5081no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—5081no—
195C2×C4C2×C22⋊D12711yescharacteristic
196C2×C4C2×C22⋊D12711yescharacteristic
197C8—5081no—
198C8—5081no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22#1686117811yescharacteristic
210C22#1686122911yescharacteristic
211C22#1686123411yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—10161no—
219C4—10161no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—81no—
224C2#2255002111yescharacteristic
225C2#2255005411yescharacteristic
226C2#2255006611yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#538×C127C2
socleC22×C127#120
hypercenter#538D127
hypocenterC127#1612
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #22550204C2semidirect
2. #22550187C2semidirect
3. #22549801C2semidirect
4. #22549613C2semidirect
5. #22549610C2semidirect
6. #22549600C2semidirect
114. C127#1612semidirect

Maximal subgroups

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

subgroupindex
#225502042
#225501872
#225498012
#225496132
#225496102
#225496002
#538×C1272
#1612127