Finite Group Databasebeta

← #30639812

Subgroups — #30639812

Order 16256.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C22×#14523236C212yesmaximal
2C2×#16861209C211yesmaximal, characteristic, complemented
3C2×#16861132C212yesmaximal, complemented
4#22549631C211yesmaximal, characteristic, complemented
5#390×C127C211yesmaximal, characteristic
6C22×Dic254C2212yes—
7C2×#14523236—24no—
8C2×#14523236—24no—
9C2×#14523236—24no—
10C22×C127⋊C8C2212yes—
11#16861209—24no—
12C2×C4⋊Dic127—22no—
13#16861132—24no—
14C22×D4×C127C2211yescharacteristic
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C22⋊C4×C127—22no—
17C2×C42×C127C2211yescharacteristic
18C2×Dic254—22no—
19C2×Dic254—22no—
20C2×Dic254—22no—
21C2×Dic254—22no—
22C2×Dic254—22no—
23C2×Dic254D414yes—
24C22×Dic127—22no—
25#14523236—216no—
26C2×C127⋊C8—22no—
27C2×C127⋊C8—22no—
28C2×C127⋊C8D414yes—
29C4⋊Dic127—28no—
30C24×C127—21no—
31C2×D4×C127—22no—
32C2×D4×C127—22no—
33C2×D4×C127—21no—
34C2×D4×C127D412yes—
35C22×C4×C127—21no—
36C22×C4×C127D412yes—
37C22×C4×C127C2311yescharacteristic
38C4⋊C4×C127—22no—
39C22⋊C4×C127—44no—
40C42×C127—22no—
41Dic254—24no—
42Dic254—24no—
43Dic254—42no—
44Dic254—42no—
45C2×Dic127—24no—
46C2×Dic127—42no—
47C2×Dic127—42no—
48C127⋊C8—24no—
49C23×C127—22no—
50C23×C127—41no—
51C23×C127—41no—
52C23×C127—42no—
53C23×C127—42no—
54C23×C127C2×D411yescharacteristic
55D4×C127—24no—
56D4×C127—42no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—41no—
60C2×C4×C127—42no—
61C2×C4×C127SD1614yescomplemented
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127#4111yescharacteristic
66C22×C127—42no—
67C22×C127—44no—
68C22×C127—82no—
69C22×C127—81no—
70C22×C127—81no—
71C22×C127#12011yescharacteristic
72C22×C127#12111yescharacteristic
73C22×C127#12711yescharacteristic
74C22×C127C2×SD1612yes—
75C22×C127#13712yes—
76C4×C127—24no—
77C4×C127—81no—
78C4×C127#12012yes—
79C4×C127#12112yes—
80Dic127—44no—
81C2×C127—82no—
82C2×C127#39211yescharacteristic
83C2×C127#44812yes—
84C2×C127#46112yes—
85C2×C127#49211yescharacteristic
86C2×C127#49711yescharacteristic
87#2013—1271nomaximal, Sylow
88C127#201311yescharacteristic, complemented, Sylow
89C22×SD16—1272no—
90C2×C4⋊Q8—1271no—
91C2×#103—1272no—
92#390D12711yescharacteristic
93#336—1271no—
94C22×Q8—1272no—
95C22×D4C2×D12711yescharacteristic
96C2×SD16—2544no—
97C2×SD16—2544no—
98C2×SD16—2544no—
99C22×C8—1272no—
100C4⋊Q8—2544no—
101C2×C4⋊C4—2542no—
102C2×C4⋊C4C2×D12711yescharacteristic
103C2×C22⋊C4—22no—
104C2×C42C2×D12711yescharacteristic
105#103—2544no—
106C24—21no—
107C2×Q8—1274no—
108C2×Q8—2542no—
109C2×Q8—2542no—
110C2×Q8—2542no—
111C2×Q8—2542no—
112C2×Q8—2542no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4D50812yes—
117C22×C4—21no—
118C22×C4—2542no—
119C22×C4C22⋊D12712yes—
120C22×C4C22×D12711yescharacteristic
121SD16—25416no—
122C2×C8—1274no—
123C2×C8—2542no—
124C2×C8—2542no—
125C4⋊C4—22no—
126C4⋊C4—2548no—
127C22⋊C4—44no—
128C42—22no—
129C23—22no—
130C23—41no—
131C23—42no—
132C23—42no—
133C23—41no—
134C23C2×C22⋊D12711yescharacteristic
135Q8—5082no—
136Q8—5082no—
137Q8—2544no—
138Q8—2544no—
139D4—24no—
140D4—42no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—5082no—
146C2×C4—5082no—
147C2×C4—2544no—
148C2×C4#1452323614yescomplemented
149C2×C4D4×D12712yes—
150C2×C4D4×D12712yes—
151C2×C4C2×D50811yescharacteristic
152C2×C4#1452324811yescharacteristic
153C8—2544no—
154C22—42no—
155C22—44no—
156C22—81no—
157C22—82no—
158C22—81no—
159C22#1686117711yescharacteristic
160C22#1686118311yescharacteristic
161C22#1686120511yescharacteristic
162C22C2×#1452323612yes—
163C22#1686123912yes—
164C4—24no—
165C4—81no—
166C4—5084no—
167C4#1686117912yes—
168C4#1686118212yes—
169C2—82no—
170C2#2254978511yescharacteristic
171C2#2255001412yes—
172C2#2255008112yes—
173C2#2255024811yescharacteristic
174C2#2255027111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenter#390D127
hypocenterC127#2013
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16861209C2semidirect
3. C2×#16861132C2semidirect
4. #22549631C2semidirect
61. C2×C4×C127SD16semidirect
88. C127#2013semidirect
148. C2×C4#14523236semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168612092
C2×#168611322
#225496312
#390×C1272
#2013127