Finite Group Databasebeta

← #30639252

Subgroups — #30639252

Order 16256.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550447C211yesmaximal, characteristic
2#22549886C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic
4#22549633C212yesmaximal, complemented
5#568×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7#16861207C412yescomplemented
8#16861196C412yescomplemented
9#16861190—22no—
10#16861188—22no—
11#16861168C2212yes—
12#16861168C2211yescharacteristic
13#16861134—24no—
14#16861124—22no—
15C2×#41×C127C2211yescharacteristic
16#131×C127—24no—
17C2×M16×C127—22no—
18C2×M16×C127C2212yes—
19C2×Dic254—21no—
20C2×Dic254—22no—
21C2×Dic254—22no—
22C2×Dic254C2×C412yes—
23C22×Dic127—21no—
24C4⋊Dic127—21no—
25C4⋊Dic127—22no—
26Dic127⋊C4—24no—
27C22⋊Dic127—24no—
28C22⋊Dic127—21no—
29C22⋊Dic127—22no—
30C4×Dic127—22no—
31C4×Dic127C2×C412yes—
32#41×C127—22no—
33#41×C127—22no—
34C2×Q8×C127C2×C411yescharacteristic
35C2×D4×C127C2×C411yescharacteristic
36C2×D4×C127D412yes—
37C22×C4×C127D412yes—
38C22×C4×C127C2311yescharacteristic
39M16×C127—24no—
40M16×C127—22no—
41M16×C127—22no—
42C2×C8×C127—24no—
43C2×C8×C127D414yes—
44Dic254—24no—
45Dic254—41no—
46Dic254—41no—
47C2×Dic127—22no—
48C2×Dic127—22no—
49C2×Dic127—42no—
50C2×Dic127—42no—
51C23×C127C22⋊C412yes—
52C23×C127C2×D411yescharacteristic
53Q8×C127—22no—
54D4×C127—24no—
55D4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127C22⋊C412yes—
59C2×C4×C127C22×C411yescharacteristic
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127#4112yes—
63C8×C127—24no—
64C22×C127—22no—
65C22×C127—41no—
66C22×C127—42no—
67C22×C127C2×C22⋊C411yescharacteristic
68C22×C127#12112yes—
69C4×C127—22no—
70C4×C127C4×D412yes—
71C4×C127#12011yescharacteristic
72C4×C127#12311yescharacteristic
73Dic127—42no—
74Dic127—42no—
75Dic127—82no—
76C2×C127—21no—
77C2×C127—22no—
78C2×C127—42no—
79C2×C127#38611yescharacteristic
80#1909—1271nomaximal, Sylow
81C127#190911yescharacteristic, complemented, Sylow
82#568D12711yescharacteristic
83#536—1271no—
84#419—1272no—
85#410—1271no—
86H(3,ℤ/4)—1272no—
87C2×#41C2×D12711yescharacteristic
88C22×Q8—1271no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D12712yes—
92C4⋊Q8—1272no—
93#124—1272no—
94#123—2542no—
95C22⋊Q8—2542no—
96#117—1271no—
97#117—1272no—
98#103—2544no—
99#99—2542no—
100#41—22no—
101#41—22no—
102C2×Q8—1272no—
103C2×Q8—2542no—
104C2×Q8—2542no—
105C2×Q8—2541no—
106C2×Q8C4×D12711yescharacteristic
107C2×D4C4×D12711yescharacteristic
108C2×D4D50812yes—
109C22×C4—2541no—
110C22×C4C22⋊D12712yes—
111C22×C4C22×D12711yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D12714yes—
117C4⋊C4—2542no—
118C4⋊C4—2544no—
119C4⋊C4—2541no—
120C22⋊C4—2542no—
121C22⋊C4—2544no—
122C22⋊C4—2541no—
123C42—1272no—
124C42—2542no—
125C23(C2×D127)⋊C412yes—
126C23C2×D50811yescharacteristic
127Q8—22no—
128Q8—5081no—
129Q8—5081no—
130Q8—2544no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—5082no—
136C2×C4—5082no—
137C2×C4—2542no—
138C2×C4—2542no—
139C2×C4(C2×D127)⋊C412yes—
140C2×C4C2×C4×D12711yescharacteristic
141C2×C4C2×C22⋊D12711yescharacteristic
142C2×C4C2×C22⋊D12712yes—
143C2×C4#1452324812yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×(C2×D127)⋊C411yescharacteristic
149C22#1686118312yes—
150C4—22no—
151C4—10162no—
152C4—5082no—
153C4—5082no—
154C4C4×C22⋊D12712yes—
155C4#1686117811yescharacteristic
156C4#1686119211yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2254975511yescharacteristic
subgroupG/H
centerC2#22549755
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#386
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1909
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22549886C2semidirect
4. #22549633C2semidirect
7. #16861207C4semidirect
8. #16861196C4semidirect
81. C127#1909semidirect

Maximal subgroups

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

subgroupindex
#225504472
#225498862
#225498502
#225496332
#568×C1272
#1909127