Finite Group Databasebeta

← #30639892

Subgroups — #30639892

Order 16256.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550507C211yesmaximal, characteristic
2#22549886C212yesmaximal
3#22549781C211yesmaximal, characteristic
4#22549637C212yesmaximal
5#343×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7#16861208—22no—
8#16861200—22no—
9#16861188—22no—
10#16861168C2212yes—
11C2×C4×Dic127C2211yescharacteristic
12#16861134—22no—
13#16861134—22no—
14C127⋊#95—22no—
15C2×M16×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×Dic254—22no—
18C2×Dic254—22no—
19C2×Dic254—21no—
20C2×Dic254—22no—
21C2×Dic254Q812yescomplemented
22C22×Dic127—21no—
23C22×Dic127—21no—
24C4⋊Dic127—22no—
25C4⋊Dic127D412yes—
26C4⋊Dic127D412yes—
27Dic127⋊C4—24no—
28C22⋊Dic127—22no—
29C4×Dic127—22no—
30C4×Dic127—22no—
31C22×C4×C127D412yes—
32C22×C4×C127C2311yescharacteristic
33M16×C127—22no—
34M16×C127—22no—
35C2×C8×C127—22no—
36C4⋊C4×C127—22no—
37Dic254—24no—
38Dic254—42no—
39Dic254—82no—
40C2×Dic127—22no—
41C2×Dic127—22no—
42C2×Dic127—22no—
43C2×Dic127—22no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C2×Dic127—41no—
47C2×Dic127—41no—
48C23×C127C2×D411yescharacteristic
49C2×C4×C127—22no—
50C2×C4×C127—42no—
51C2×C4×C127C2×D412yes—
52C2×C4×C127C2×Q811yescharacteristic
53C2×C4×C127#4111yescharacteristic
54C2×C4×C127#4112yes—
55C8×C127—42no—
56C22×C127—21no—
57C22×C127—21no—
58C22×C127#12011yescharacteristic
59C22×C127C22⋊Q811yescharacteristic
60C22×C127#12411yescharacteristic
61C4×C127—42no—
62C4×C127C22⋊Q812yes—
63C4×C127#12412yes—
64Dic127—42no—
65Dic127—42no—
66Dic127—42no—
67Dic127—42no—
68C2×C127—21no—
69C2×C127—21no—
70C2×C127#39311yescharacteristic
71C2×C127#45511yescharacteristic
72C2×C127#45611yescharacteristic
73#2028—1271nomaximal, Sylow
74C127#202811yescharacteristic, complemented, Sylow
75#544—1271no—
76#419—1272no—
77#391—1271no—
78#343D12711yescharacteristic
79#339—1272no—
80C22×Q8—1271no—
81C2×M16C2×D12712yes—
82C4⋊Q8—2542no—
83#125—2542no—
84C22⋊Q8—2542no—
85#117—1272no—
86C2×C4⋊C4C2×D12711yescharacteristic
87C2×C42—1271no—
88#103—2542no—
89#103—2542no—
90#95—2542no—
91C2×Q8—1272no—
92C2×Q8—2542no—
93C2×Q8—2542no—
94C2×Q8—2542no—
95C2×Q8—2541no—
96C22×C4—2541no—
97C22×C4—2541no—
98C22×C4D50812yes—
99C22×C4C22×D12711yescharacteristic
100M16—22no—
101M16—22no—
102C2×C8—22no—
103C4⋊C4—22no—
104C4⋊C4—1272no—
105C4⋊C4—1272no—
106C4⋊C4—2544no—
107C4⋊C4—2542no—
108C22⋊C4—2542no—
109C42—2542no—
110C42—2542no—
111C23C2×D50811yescharacteristic
112Q8—10162no—
113Q8—5082no—
114Q8—2544no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—5081no—
118C2×C4—5081no—
119C2×C4—5081no—
120C2×C4—5081no—
121C2×C4—2542no—
122C2×C4—2542no—
123C2×C4—2542no—
124C2×C4—2542no—
125C2×C4D4×D12712yes—
126C2×C4Q8×D12711yescharacteristic
127C2×C4#1452324711yescharacteristic
128C2×C4#1452324812yes—
129C8—42no—
130C22—21no—
131C22—21no—
132C22#1686117911yescharacteristic
133C22#1686118911yescharacteristic
134C22#1686119711yescharacteristic
135C4—42no—
136C4—5082no—
137C4—5082no—
138C4—5082no—
139C4—5082no—
140C4#1686118712yes—
141C4#1686119512yes—
142C2—21no—
143C2—21no—
144C2#2254978911yescharacteristic
145C2#2255005611yescharacteristic
146C2#2255006111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#343D127
hypocenterC127#2028
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
21. C2×Dic254Q8semidirect
74. C127#2028semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225498862
#225497812
#225496372
#343×C1272
#2028127