Finite Group Databasebeta

← #30640083

Subgroups — #30640083

Order 16256.

314 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550502C211yesmaximal, characteristic
2#22549749C212yesmaximal
3#22549643C211yesmaximal, characteristic
4#22549642C212yesmaximal
5#385×C127C211yesmaximal, characteristic
6#16861188—21no—
7#16861188—21no—
8#16861186—22no—
9#16861186—22no—
10C2×Dic127⋊C4—22no—
11C2×C22⋊Dic127C2212yes—
12C2×C22⋊Dic127C2212yes—
13C127⋊#95—22no—
14C23×C4×C127C2211yescharacteristic
15C2×C4⋊C4×C127—21no—
16C2×C22⋊C4×C127C2211yescharacteristic
17C2×C22⋊C4×C127C2211yescharacteristic
18#95×C127—21no—
19C2×Dic254—21no—
20C2×Dic254—21no—
21C22×Dic127—22no—
22C22×Dic127D412yes—
23C22×Dic127D412yes—
24C4⋊Dic127—21no—
25C4⋊Dic127—21no—
26Dic127⋊C4—22no—
27Dic127⋊C4—22no—
28Dic127⋊C4—44no—
29C22⋊Dic127—22no—
30C22⋊Dic127—22no—
31C22⋊Dic127—22no—
32C22⋊Dic127Q814yescomplemented
33C24×C127C2311yescharacteristic
34C22×C4×C127—21no—
35C22×C4×C127—21no—
36C22×C4×C127—22no—
37C22×C4×C127—42no—
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C4⋊C4×C127—42no—
41C22⋊C4×C127—22no—
42C22⋊C4×C127—21no—
43C22⋊C4×C127Q812yescomplemented
44Dic254—81no—
45Dic254—81no—
46C2×Dic127—22no—
47C2×Dic127—22no—
48C2×Dic127—24no—
49C2×Dic127—42no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C2×Dic127—42no—
53C23×C127—41no—
54C23×C127—41no—
55C23×C127C2×D412yes—
56C23×C127C2×D411yescharacteristic
57C23×C127C2×Q812yes—
58C23×C127C2×Q811yescharacteristic
59C23×C127#4111yescharacteristic
60C2×C4×C127—22no—
61C2×C4×C127—24no—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—22no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127—42no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C22×C127—21no—
73C22×C127—22no—
74C22×C127—21no—
75C22×C127—22no—
76C22×C127—42no—
77C22×C127—41no—
78C22×C127—41no—
79C22×C127#12011yescharacteristic
80C22×C127C22⋊Q812yes—
81C22×C127C22⋊Q811yescharacteristic
82C22×C127C4⋊Q811yescharacteristic
83C22×C127C4⋊Q812yes—
84C4×C127—42no—
85C4×C127—42no—
86C4×C127—41no—
87C4×C127—41no—
88Dic127—44no—
89Dic127—42no—
90Dic127—42no—
91C2×C127—22no—
92C2×C127—21no—
93C2×C127—21no—
94C2×C127—22no—
95C2×C127#39511yescharacteristic
96C2×C127B(4,𝔽2)11yescharacteristic
97C2×C127#45811yescharacteristic
98#2062—1271nomaximal, Sylow
99C127#206211yescharacteristic, complemented, Sylow
100#543—1271no—
101#385—1272no—
102#385D12711yescharacteristic
103#342—1272no—
104#342—1271no—
105C23×C4C2×D12711yescharacteristic
106C22⋊Q8—2542no—
107C22⋊Q8—2542no—
108C22⋊Q8—2541no—
109C22⋊Q8—2541no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—2542no—
112C2×C22⋊C4—1272no—
113C2×C22⋊C4—1272no—
114C2×C22⋊C4C2×D12711yescharacteristic
115C2×C22⋊C4C2×D12711yescharacteristic
116#95—21no—
117#95—2542no—
118C24C22×D12711yescharacteristic
119C2×Q8—2541no—
120C2×Q8—2541no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—1272no—
126C22×C4—1272no—
127C22×C4—2542no—
128C22×C4C22⋊D12711yescharacteristic
129C22×C4C22⋊D12711yescharacteristic
130C4⋊C4—42no—
131C4⋊C4—5084no—
132C4⋊C4—2542no—
133C4⋊C4—2542no—
134C4⋊C4—2541no—
135C4⋊C4—2541no—
136C22⋊C4—22no—
137C22⋊C4—21no—
138C22⋊C4—1274no—
139C22⋊C4—2542no—
140C22⋊C4—2542no—
141C22⋊C4—2542no—
142C22⋊C4Dic25412yescomplemented
143C23—41no—
144C23—41no—
145C23D4×D12712yes—
146C23C2×C22⋊D12711yescharacteristic
147C23Q8×D12712yes—
148C23C2×Dic25411yescharacteristic
149C23#1452324811yescharacteristic
150Q8—10161no—
151Q8—10161no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—5082no—
165C2×C4—5082no—
166C2×C4—5082no—
167C2×C4—5082no—
168C2×C4—2544no—
169C2×C4—2542no—
170C2×C4—2542no—
171C22—21no—
172C22—22no—
173C22—21no—
174C22—22no—
175C22—42no—
176C22—41no—
177C22—41no—
178C22#1686117711yescharacteristic
179C22#1686118712yes—
180C22#1686118811yescharacteristic
181C22#1686120711yescharacteristic
182C22#1686120812yes—
183C4—42no—
184C4—42no—
185C4—41no—
186C4—41no—
187C4—5084no—
188C4—5082no—
189C4—5082no—
190C2—21no—
191C2—22no—
192C2—21no—
193C2—22no—
194C2#2254979611yescharacteristic
195C2#2255006211yescharacteristic
196C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, prime radical, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#385×C127C2
socleC22×C127#120
hypercenter#385D127
hypocenterC127#2062
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

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

NH ≅ G/Ntype
32. C22⋊Dic127Q8semidirect
43. C22⋊C4×C127Q8semidirect
99. C127#2062semidirect
142. C22⋊C4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225497492
#225496432
#225496422
#385×C1272
#2062127