Finite Group Databasebeta

← #30639973

Subgroups — #30639973

Order 16256.

298 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16861133C212yesmaximal
2C2×#16861133C212yesmaximal
3#22549774C211yesmaximal, characteristic
4#22549631C211yesmaximal, characteristic
5C2×C4⋊Q8×C127C211yesmaximal, characteristic
6C22×C127⋊C8C2212yes—
7C2×C4⋊Dic127C2211yescharacteristic
8C2×C4⋊Dic127C2211yescharacteristic
9C2×Dic127⋊C4—21no—
10#16861133—24no—
11#16861133—24no—
12C127⋊#95—21no—
13C22×Q8×C127C2212yes—
14C4⋊Q8×C127—24no—
15C2×C4⋊C4×C127—21no—
16C2×C4⋊C4×C127—21no—
17C2×C42×C127C2211yescharacteristic
18C22×Dic127—21no—
19C22×Dic127—21no—
20C2×C127⋊C8—22no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23C4⋊Dic127—22no—
24C4⋊Dic127—21no—
25C4⋊Dic127D412yes—
26Dic127⋊C4—42no—
27C2×Q8×C127—22no—
28C2×Q8×C127—22no—
29C2×Q8×C127—22no—
30C2×Q8×C127—22no—
31C2×Q8×C127—22no—
32C2×Q8×C127D414yes—
33C22×C4×C127—22no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C4⋊C4×C127—24no—
38C4⋊C4×C127—24no—
39C42×C127—22no—
40C2×Dic127—22no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C127⋊C8—42no—
47C23×C127C2×D411yescharacteristic
48Q8×C127—24no—
49Q8×C127—24no—
50Q8×C127—42no—
51Q8×C127—42no—
52C2×C4×C127—21no—
53C2×C4×C127—21no—
54C2×C4×C127—21no—
55C2×C4×C127—21no—
56C2×C4×C127—24no—
57C2×C4×C127—42no—
58C2×C4×C127—42no—
59C2×C4×C127SD1612yes—
60C2×C4×C127Q1612yes—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D412yes—
63C2×C4×C127#4112yes—
64C2×C4×C127#4111yescharacteristic
65C22×C127#12011yescharacteristic
66C22×C127#12111yescharacteristic
67C22×C127#12411yescharacteristic
68C22×C127C2×SD1611yescharacteristic
69C22×C127C2×Q1611yescharacteristic
70C22×C127#13711yescharacteristic
71C22×C127#13711yescharacteristic
72C4×C127—22no—
73C4×C127—22no—
74C4×C127—44no—
75C4×C127#12112yes—
76C4×C127#12312yes—
77Dic127—42no—
78Dic127—81no—
79C2×C127#39411yescharacteristic
80C2×C127#44811yescharacteristic
81C2×C127#45111yescharacteristic
82C2×C127#46111yescharacteristic
83C2×C127#46211yescharacteristic
84C2×C127#48711yescharacteristic
85C2×C127#49111yescharacteristic
86#2043—1271nomaximal, Sylow
87C127#204311yescharacteristic, complemented, Sylow
88C2×C4⋊Q8D12711yescharacteristic
89C2×#103—1272no—
90C2×#103—1272no—
91#389—1271no—
92#336—1271no—
93C22×Q8C2×D12712yes—
94C22×C8—1272no—
95C4⋊Q8—24no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—1271no—
99C2×C4⋊C4—1271no—
100C2×C4⋊C4—2541no—
101C2×C42C2×D12711yescharacteristic
102#103—2544no—
103#103—2544no—
104#95—2541no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8C22⋊D12714yes—
111C22×C4—22no—
112C22×C4—2541no—
113C22×C4—2541no—
114C22×C4C22⋊D12711yescharacteristic
115C22×C4C22⋊D12711yescharacteristic
116C22×C4C22×D12711yescharacteristic
117C2×C8—2542no—
118C2×C8—2542no—
119C2×C8—2542no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—1272no—
123C4⋊C4—5082no—
124C4⋊C4—2542no—
125C4⋊C4—2541no—
126C42—22no—
127C23C2×C22⋊D12711yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—42no—
131Q8—42no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—24no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—5081no—
140C2×C4—5081no—
141C2×C4—5081no—
142C2×C4—5081no—
143C2×C4—5081no—
144C2×C4—2542no—
145C2×C4#1452323512yes—
146C2×C4#1452323812yes—
147C2×C4D4×D12711yescharacteristic
148C2×C4C2×C22⋊D12712yes—
149C2×C4#1452324711yescharacteristic
150C2×C4#1452324712yes—
151C8—5082no—
152C22#1686117811yescharacteristic
153C22#1686118011yescharacteristic
154C22#1686119611yescharacteristic
155C22C2×#1452323511yescharacteristic
156C22C2×#1452323811yescharacteristic
157C22#1686124011yescharacteristic
158C22#1686124011yescharacteristic
159C4—22no—
160C4—22no—
161C4—44no—
162C4—10161no—
163C4—5082no—
164C4#1686118112yes—
165C4#1686119012yes—
166C2#2254979011yescharacteristic
167C2#2255001511yescharacteristic
168C2#2255003211yescharacteristic
169C2#2255007811yescharacteristic
170C2#2255008311yescharacteristic
171C2#2255022611yescharacteristic
172C2#2255024411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×C4⋊Q8×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC2×C4⋊Q8D127
hypocenterC127#2043
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
87. C127#2043semidirect

Maximal subgroups

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

subgroupindex
C2×#168611332
#225497742
#225496312
C2×C4⋊Q8×C1272
#2043127