Finite Group Databasebeta

← #30637757

Subgroups — #30637757

Order 16256.

236 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550212C212yesmaximal, complemented
3#22549812C211yesmaximal, characteristic
4#22549614C212yesmaximal, complemented
5#323×C127C211yesmaximal, characteristic
6#16861188—21no—
7#16861188C2212yes—
8#16861186—22no—
9#16861186—22no—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—21no—
12C2×C22⋊Dic127—21no—
13C2×C22⋊Dic127—21no—
14#16861143—22no—
15#16861134—22no—
16C127⋊#95C2211yescharacteristic
17C2×C22⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C2×Dic254—21no—
20C2×Dic254D412yes—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C22×Dic127—21no—
24C22×Dic127—21no—
25C4⋊Dic127—21no—
26C4⋊Dic127—22no—
27C4⋊Dic127D412yes—
28Dic127⋊C4—22no—
29Dic127⋊C4—22no—
30Dic127⋊C4—42no—
31C22⋊Dic127—22no—
32C22⋊Dic127—22no—
33C22⋊Dic127—42no—
34C24×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C2×C8×C127—22no—
38C22⋊C4×C127—22no—
39Dic254—42no—
40Dic254—81no—
41C2×Dic127—22no—
42C2×Dic127—22no—
43C2×Dic127—22no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C2×Dic127—41no—
47C2×Dic127—41no—
48C2×Dic127—41no—
49C2×Dic127—81no—
50C2×Dic127—81no—
51C23×C127—21no—
52C23×C127—21no—
53C23×C127—41no—
54C23×C127—41no—
55C23×C127Q1612yes—
56C23×C127C2×D411yescharacteristic
57C2×C4×C127—21no—
58C2×C4×C127—41no—
59C2×C4×C127—41no—
60C2×C4×C127C2×D412yes—
61C8×C127—42no—
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—
74Dic127—42no—
75Dic127—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#543—1271no—
87#484—1272no—
88#399—1271no—
89#328—1272no—
90#323D12711yescharacteristic
91C22⋊Q8—1272no—
92C22⋊Q8—2542no—
93C22⋊Q8—2542no—
94C22⋊Q8—2541no—
95C2×C4⋊C4—1271no—
96C2×C4⋊C4—2541no—
97C2×C22⋊C4—2541no—
98C2×C22⋊C4—2541no—
99C2×C22⋊C4C2×D12711yescharacteristic
100#107—2542no—
101#103—2542no—
102C22⋊C8C2×D12712yes—
103#95—1271no—
104C24D50811yescharacteristic
105C2×Q8—1272no—
106C2×Q8—2541no—
107C22×C4—2541no—
108C22×C4—2541no—
109C22×C4—2541no—
110C22×C4—2541no—
111C22×C4D50811yescharacteristic
112C22×C4C22×D12711yescharacteristic
113C2×C8—22no—
114C4⋊C4—1272no—
115C4⋊C4—5082no—
116C4⋊C4—2542no—
117C4⋊C4—2541no—
118C4⋊C4—2542no—
119C4⋊C4—2542no—
120C22⋊C4—22no—
121C22⋊C4—5082no—
122C22⋊C4—2542no—
123C22⋊C4—2542no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23Dic50812yes—
129C23C2×D50811yescharacteristic
130Q8—10161no—
131Q8—5082no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—10161no—
136C2×C4—10161no—
137C2×C4—5081no—
138C2×C4—5081no—
139C2×C4—5081no—
140C2×C4—5081no—
141C2×C4—5081no—
142C2×C4—2542no—
143C2×C4—2542no—
144C2×C4—2542no—
145C2×C4D4×D12712yes—
146C8—42no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1686117911yescharacteristic
155C22C2×Dic50811yescharacteristic
156C22#1686124211yescharacteristic
157C4—22no—
158C4—41no—
159C4—10161no—
160C4—5082no—
161C4—5082no—
162C4—5082no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2255003411yescharacteristic
167C2#2255004611yescharacteristic
168C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127#120
prime radicalC24×C127D4
hypercenter#323D127
hypocenterC127#1614
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22550212C2semidirect
4. #22549614C2semidirect
85. C127#1614semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225502122
#225498122
#225496142
#323×C1272
#1614127