Finite Group Databasebeta

← #30637755

Subgroups — #30637755

Order 16256.

236 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550209C212yesmaximal, complemented
2#22549812C211yesmaximal, characteristic
3#22549613C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic
5#543×C127C211yesmaximal, characteristic
6C2×C4⋊Dic127C2211yescharacteristic
7C2×Dic127⋊C4—21no—
8C2×C22⋊Dic127—21no—
9#16861142—22no—
10#16861133—22no—
11C22⋊(C127⋊C8)C2212yes—
12C127⋊#95C2211yescharacteristic
13C22⋊Q8×C127—22no—
14C22⋊Q8×C127—21no—
15C22⋊Q8×C127—22no—
16C22⋊Q8×C127C2212yes—
17C2×C22⋊C4×C127—21no—
18C2×C22⋊C4×C127C2211yescharacteristic
19C22×Dic127—21no—
20C22×Dic127—21no—
21C2×C127⋊C8—22no—
22C4⋊Dic127—22no—
23Dic127⋊C4—42no—
24C22⋊Dic127—42no—
25C24×C127D411yescharacteristic
26C2×Q8×C127—21no—
27C2×Q8×C127D412yes—
28C22×C4×C127—21no—
29C22×C4×C127—21no—
30C22×C4×C127D411yescharacteristic
31C22×C4×C127C2311yescharacteristic
32C4⋊C4×C127—21no—
33C4⋊C4×C127—22no—
34C4⋊C4×C127—22no—
35C4⋊C4×C127D412yes—
36C22⋊C4×C127—22no—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C2×Dic127—22no—
40C2×Dic127—41no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C2×Dic127—41no—
45C127⋊C8—42no—
46C23×C127—21no—
47C23×C127—21no—
48C23×C127—41no—
49C23×C127—41no—
50C23×C127Q1612yes—
51C23×C127C2×D411yescharacteristic
52Q8×C127—42no—
53Q8×C127—81no—
54C2×C4×C127—22no—
55C2×C4×C127—21no—
56C2×C4×C127—22no—
57C2×C4×C127—41no—
58C2×C4×C127—41no—
59C2×C4×C127—81no—
60C2×C4×C127—81no—
61C2×C4×C127C2×D412yes—
62C22×C127—21no—
63C22×C127—21no—
64C22×C127—22no—
65C22×C127—42no—
66C22×C127—41no—
67C22×C127—41no—
68C22×C127—81no—
69C22×C127#12011yescharacteristic
70C22×C127C2×Q1611yescharacteristic
71C22×C127#13711yescharacteristic
72C4×C127—22no—
73C4×C127—41no—
74C4×C127—42no—
75C4×C127—42no—
76Dic127—42no—
77Dic127—81no—
78C2×C127—21no—
79C2×C127—21no—
80C2×C127—42no—
81C2×C127#45111yescharacteristic
82C2×C127#45311yescharacteristic
83C2×C127#45811yescharacteristic
84#1614—1271nomaximal, Sylow
85C127#161411yescharacteristic, complemented, Sylow
86#543D12711yescharacteristic
87#484—1272no—
88#399—1271no—
89#328—1272no—
90#323—1271no—
91C22⋊Q8—22no—
92C22⋊Q8—21no—
93C22⋊Q8—22no—
94C22⋊Q8C2×D12712yes—
95C2×C4⋊C4—1271no—
96C2×C4⋊C4—2541no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—2541no—
99C2×C22⋊C4C2×D12711yescharacteristic
100#107—2542no—
101#103—2542no—
102C22⋊C8—1272no—
103#95—1271no—
104C24C22⋊D12711yescharacteristic
105C2×Q8—21no—
106C2×Q8C22⋊D12712yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—2541no—
110C22×C4—2541no—
111C22×C4C22⋊D12711yescharacteristic
112C22×C4C22×D12711yescharacteristic
113C2×C8—2542no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—5082no—
118C4⋊C4—2542no—
119C4⋊C4C22⋊D12712yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—5082no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23#1452323812yes—
129C23C2×C22⋊D12711yescharacteristic
130Q8—42no—
131Q8—81no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—5081no—
140C2×C4—5081no—
141C2×C4—5081no—
142C2×C4—5081no—
143C2×C4—5081no—
144C2×C4—2542no—
145C2×C4C2×C22⋊D12712yes—
146C8—5082no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1686117811yescharacteristic
155C22C2×#1452323811yescharacteristic
156C22#1686124011yescharacteristic
157C4—22no—
158C4—41no—
159C4—42no—
160C4—42no—
161C4—10161no—
162C4—5082no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2255003211yescharacteristic
167C2#2255004411yescharacteristic
168C2#2255006611yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#543×C127C2
socleC22×C127#120
prime radicalC24×C127D4
hypercenter#543D127
hypocenterC127#1614
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #22550209C2semidirect
3. #22549613C2semidirect
85. C127#1614semidirect

Maximal subgroups

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

subgroupindex
#225502092
#225498122
#225496132
#225496002
#543×C1272
#1614127