Finite Group Databasebeta

← #30639725

Subgroups — #30639725

Order 16256.

496 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#16861180C212yesmaximal, complemented
2C2×#16861130C212yesmaximal, complemented
3C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
4#22549639C211yesmaximal, characteristic, complemented
5C22×D8×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127—22no—
7C22×C4×D127C2211yescharacteristic
8C22×C127⋊C8C2211yescharacteristic
9#16861180—24no—
10#16861180—24no—
11C2×C4⋊Dic127C2212yes—
12C2×C22⋊Dic127—22no—
13#16861130—24no—
14(C2×D127)⋊C8—22no—
15C22×D4×C127C2212yes—
16C2×D8×C127—24no—
17C2×D8×C127—22no—
18C22×C8×C127C2211yescharacteristic
19C23×D127D411yescharacteristic
20C2×C22⋊D127—28no—
21C2×C22⋊D127—44no—
22C2×C22⋊D127—44no—
23C22×Dic127—22no—
24C22×Dic127D411yescharacteristic
25C2×C4×D127—21no—
26C2×C4×D127—22no—
27C2×C4×D127—22no—
28C2×C4×D127—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—22no—
31C4⋊Dic127—22no—
32C4⋊Dic127D414yescomplemented
33C22⋊Dic127—28no—
34C24×C127—22no—
35C2×D4×C127—22no—
36C2×D4×C127—22no—
37C2×D4×C127—22no—
38C2×D4×C127—22no—
39C2×D4×C127—22no—
40C2×D4×C127D414yescomplemented
41C22×C4×C127C2311yescharacteristic
42D8×C127—28no—
43C2×C8×C127—22no—
44C2×C8×C127D412yescomplemented
45C22×D127—22no—
46C22×D127—21no—
47C22×D127—21no—
48C22×D127—22no—
49C22×D127D812yescomplemented
50C22⋊D127—416no—
51C2×Dic127—22no—
52C2×Dic127—24no—
53C2×Dic127—21no—
54C2×Dic127—42no—
55C2×Dic127—42no—
56C4×D127—41no—
57C4×D127—42no—
58C4×D127—41no—
59C127⋊C8—41no—
60C23×C127—24no—
61C23×C127—42no—
62C23×C127—42no—
63C23×C127—42no—
64C23×C127—42no—
65C23×C127—42no—
66C23×C127—42no—
67C23×C127C2×D411yescharacteristic
68D4×C127—24no—
69D4×C127—24no—
70D4×C127—42no—
71D4×C127—42no—
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D412yes—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127#4111yescharacteristic
76C8×C127—22no—
77C22×C127—44no—
78C22×C127—44no—
79C22×C127—44no—
80C22×C127—82no—
81C22×C127—82no—
82C22×C127—82no—
83C22×C127—82no—
84C22×C127#12012yes—
85C22×C127#12111yescharacteristic
86C22×C127C2×D811yescharacteristic
87C22×C127#13511yescharacteristic
88C22×C127AGL(1,ℤ/8)12yes—
89C2×D127—24no—
90C2×D127—22no—
91C2×D127—41no—
92C2×D127—42no—
93C2×D127—41no—
94C4×C127#12011yescharacteristic
95C4×C127#12112yes—
96C4×C127#12711yescharacteristic
97Dic127—41no—
98Dic127—44no—
99C2×C127—84no—
100C2×C127#39211yescharacteristic
101C2×C127#44712yes—
102C2×C127#44912yes—
103C2×C127#46611yescharacteristic
104C2×C127#46911yescharacteristic
105D127—42no—
106#1999—1271nomaximal, Sylow
107C127#199911yescharacteristic, complemented, Sylow
108C22×D8D12711yescharacteristic, complemented
109C2×#121—1272no—
110C2×#102—1272no—
111C2×C22⋊C8—1271no—
112#340—1271no—
113C22×D4—2542no—
114C22×D4C2×D12712yes—
115C23×C4—1271no—
116C2×D8—24no—
117C2×D8—22no—
118C22×C8—1271no—
119C22×C8C2×D12711yescharacteristic
120#121—2544no—
121#121—2544no—
122C2×C4⋊C4—1272no—
123C2×C22⋊C4—2542no—
124#102—2544no—
125C22⋊C8—2542no—
126C24—22no—
127C24—1271no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—5084no—
134C2×D4—5084no—
135C2×D4—2548no—
136C2×D4C22⋊D12714yescomplemented
137C22×C4—1271no—
138C22×C4—2542no—
139C22×C4—2542no—
140C22×C4—2541no—
141C22×C4—2542no—
142C22×C4—2541no—
143C22×C4C22×D12711yescharacteristic
144D8—28no—
145C2×C8—22no—
146C2×C8—2542no—
147C2×C8—2541no—
148C2×C8C22⋊D12712yescomplemented
149C4⋊C4—1274no—
150C4⋊C4—2542no—
151C22⋊C4—2548no—
152C23—24no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—1272no—
160C23—2541no—
161C23—2542no—
162C23—2542no—
163C23—2541no—
164C23D4×D12711yescharacteristic
165D4—24no—
166D4—24no—
167D4—42no—
168D4—42no—
169D4—50816no—
170C2×C4—5082no—
171C2×C4—5082no—
172C2×C4—5081no—
173C2×C4—5081no—
174C2×C4—5082no—
175C2×C4—2542no—
176C2×C4—2544no—
177C2×C4—2541no—
178C2×C4D4×D12712yes—
179C2×C4C2×C22⋊D12711yescharacteristic
180C2×C4C2×C22⋊D12712yes—
181C2×C4#1452324711yescharacteristic
182C8—22no—
183C8—5081no—
184C22—44no—
185C22—44no—
186C22—44no—
187C22—82no—
188C22—82no—
189C22—82no—
190C22—82no—
191C22—5082no—
192C22—5081no—
193C22—5081no—
194C22—2542no—
195C22—2544no—
196C22#1686117712yes—
197C22#1686118011yescharacteristic
198C22D8×D12711yescharacteristic
199C22#1686122811yescharacteristic
200C22#1686123312yes—
201C4—5081no—
202C4—5084no—
203C4#1686117811yescharacteristic
204C4#1686118112yes—
205C4#1686120511yescharacteristic
206C2—84no—
207C2—5082no—
208C2#2254978411yescharacteristic
209C2#2255000812yes—
210C2#2255001812yes—
211C2#2255010911yescharacteristic
212C2#2255012611yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC22×D8×C127C2
socleC23×C127C2×D4
hypercenterC22×D8D127
hypocenterC127#1999
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861180C2semidirect
2. C2×#16861130C2semidirect
3. C2×(C2×D127)⋊C8C2semidirect
4. #22549639C2semidirect
5. C22×D8×C127C2semidirect
32. C4⋊Dic127D4semidirect
40. C2×D4×C127D4semidirect
44. C2×C8×C127D4semidirect
49. C22×D127D8semidirect
107. C127#1999semidirect
108. C22×D8D127semidirect
136. C2×D4C22⋊D127semidirect
148. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611802
C2×#168611302
C2×(C2×D127)⋊C82
#225496392
C22×D8×C1272
#1999127