Finite Group Databasebeta

← #30639888

Subgroups — #30639888

Order 16256.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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#22549884C212yesmaximal
2#22549781C211yesmaximal, characteristic
3#22549645C211yesmaximal, characteristic
4#22549635C212yesmaximal
5#544×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7C2×#14523232C2212yes—
8C2×C4×Dic127C2211yescharacteristic
9#16861132—22no—
10#16861132—22no—
11C127⋊#95—22no—
12C4⋊Q8×C127—22no—
13#125×C127—22no—
14C22⋊Q8×C127—22no—
15#117×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×Dic254—22no—
18C2×Dic254—21no—
19C2×Dic254—22no—
20C2×Dic254Q812yescomplemented
21C22×Dic127—21no—
22C22×Dic127—21no—
23#14523232—22no—
24#14523232—22no—
25C2×C127⋊C8—22no—
26C4×Dic127—22no—
27C2×Q8×C127—22no—
28C22×C4×C127D412yes—
29C22×C4×C127C2311yescharacteristic
30C4⋊C4×C127—22no—
31C4⋊C4×C127—24no—
32C4⋊C4×C127—22no—
33C4⋊C4×C127D412yes—
34C4⋊C4×C127D412yes—
35C22⋊C4×C127—22no—
36C42×C127—22no—
37Dic254—24no—
38Dic254—42no—
39C2×Dic127—22no—
40C2×Dic127—22no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C2×Dic127—41no—
45C127⋊C8—42no—
46C23×C127C2×D411yescharacteristic
47Q8×C127—82no—
48C2×C4×C127—22no—
49C2×C4×C127—22no—
50C2×C4×C127—22no—
51C2×C4×C127—42no—
52C2×C4×C127C2×D412yes—
53C2×C4×C127C2×Q811yescharacteristic
54C2×C4×C127#4112yes—
55C2×C4×C127#4111yescharacteristic
56C22×C127—21no—
57C22×C127—21no—
58C22×C127#12011yescharacteristic
59C22×C127C22⋊Q811yescharacteristic
60C22×C127#12411yescharacteristic
61C4×C127—42no—
62C4×C127—42no—
63C4×C127—42no—
64C4×C127C22⋊Q812yes—
65C4×C127#12412yes—
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#544D12711yescharacteristic
76#419—1272no—
77#391—1271no—
78#343—1271no—
79#339—1272no—
80C22×Q8—1271no—
81C2×M16—1272no—
82C4⋊Q8—22no—
83#125—22no—
84C22⋊Q8—22no—
85#117C2×D12712yes—
86C2×C4⋊C4C2×D12711yescharacteristic
87C2×C42—1271no—
88#103—2542no—
89#103—2542no—
90#95—2542no—
91C2×Q8—22no—
92C2×Q8—1272no—
93C2×Q8—2542no—
94C2×Q8—2542no—
95C2×Q8—2541no—
96C22×C4—2541no—
97C22×C4—2541no—
98C22×C4C22⋊D12712yes—
99C22×C4C22×D12711yescharacteristic
100M16—2542no—
101M16—2542no—
102C2×C8—2542no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—22no—
106C4⋊C4C22⋊D12712yes—
107C4⋊C4C22⋊D12712yes—
108C22⋊C4—22no—
109C42—22no—
110C42—2542no—
111C23C2×C22⋊D12711yescharacteristic
112Q8—82no—
113Q8—5082no—
114Q8—2544no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—5081no—
120C2×C4—5081no—
121C2×C4—5081no—
122C2×C4—5081no—
123C2×C4—2542no—
124C2×C4—2542no—
125C2×C4C2×C22⋊D12712yes—
126C2×C4Q8×D12711yescharacteristic
127C2×C4#1452324912yes—
128C2×C4#1452324911yescharacteristic
129C8—5082no—
130C22—21no—
131C22—21no—
132C22#1686117811yescharacteristic
133C22#1686118511yescharacteristic
134C22#1686119811yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—5082no—
139C4—5082no—
140C4#1686118512yes—
141C4#1686119812yes—
142C2—21no—
143C2—21no—
144C2#2254978711yescharacteristic
145C2#2255005411yescharacteristic
146C2#2255005911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#544×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#544D127
hypocenterC127#2028
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. 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
#225498842
#225497812
#225496452
#225496352
#544×C1272
#2028127