Finite Group Databasebeta

← #30639283

Subgroups — #30639283

Order 16256.

308 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 93 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#22550643C211yesmaximal, characteristic
2#22549860C211yesmaximal, characteristic
3C2×#16861128C212yesmaximal, complemented
4#22549646C212yesmaximal, complemented
5#536×C127C211yesmaximal, characteristic
6#16861216—24no—
7C2×#14523232—22no—
8C2×#14523232C2212yes—
9C2×#14523232C2212yes—
10#16861128—24no—
11#16861128—21no—
12#16861126—21no—
13C2×#41×C127C2211yescharacteristic
14C22×Q8×C127C2211yescharacteristic
15C4⋊Q8×C127—21no—
16#124×C127—21no—
17#123×C127C414yes—
18C22⋊Q8×C127—22no—
19#117×C127C2211yescharacteristic
20#14523232—24no—
21#14523232—22no—
22#14523232—22no—
23#14523232—22no—
24#14523232D414yes—
25C2×C127⋊C8—24no—
26C2×C127⋊C8—22no—
27C2×C127⋊C8—22no—
28#41×C127—22no—
29#41×C127—22no—
30C2×Q8×C127—21no—
31C2×Q8×C127—21no—
32C2×Q8×C127—22no—
33C2×Q8×C127—22no—
34C2×Q8×C127D411yescharacteristic
35C2×D4×C127C2×C412yes—
36C2×D4×C127D411yescharacteristic
37C22×C4×C127C2×C412yes—
38C22×C4×C127D412yes—
39C22×C4×C127C2311yescharacteristic
40C4⋊C4×C127—24no—
41C4⋊C4×C127—21no—
42C22⋊C4×C127—24no—
43C22⋊C4×C127C2×C412yes—
44C42×C127—21no—
45C127⋊C8—24no—
46C127⋊C8—42no—
47C23×C127C22⋊C412yes—
48C23×C127C22×C411yescharacteristic
49Q8×C127—22no—
50Q8×C127—41no—
51Q8×C127—42no—
52Q8×C127—41no—
53D4×C127—24no—
54D4×C127—22no—
55C2×C4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—24no—
60C2×C4×C127C22⋊C412yes—
61C2×C4×C127C2×D412yes—
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127#4112yes—
65C22×C127—22no—
66C22×C127—41no—
67C22×C127—42no—
68C22×C127C2×C22⋊C411yescharacteristic
69C22×C127C4×D412yes—
70C4×C127—22no—
71C4×C127—42no—
72C4×C127—42no—
73C4×C127#12011yescharacteristic
74C4×C127#12112yes—
75C4×C127#12311yescharacteristic
76C2×C127—21no—
77C2×C127—22no—
78C2×C127—42no—
79C2×C127#38611yescharacteristic
80#1915—1271nomaximal, Sylow
81C127#191511yescharacteristic, complemented, Sylow
82#568—1271no—
83#536D12711yescharacteristic
84#413—1271no—
85C2×#101—1272no—
86#344—1272no—
87C2×#41C2×D12711yescharacteristic
88C22×Q8C2×D12711yescharacteristic
89#131—2544no—
90C2×M16—1272no—
91C2×M16—1272no—
92C2×M16—2542no—
93C4⋊Q8—21no—
94#124—21no—
95#123Dic12714yes—
96C22⋊Q8—22no—
97#117C2×D12711yescharacteristic
98#101—2544no—
99#101—2541no—
100#100—2541no—
101#41—22no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—22no—
107C2×Q8C22⋊D12711yescharacteristic
108C2×D4C2×Dic12712yes—
109C2×D4C22⋊D12711yescharacteristic
110C22×C4C2×Dic12712yes—
111C22×C4C22⋊D12712yes—
112C22×C4C22×D12711yescharacteristic
113M16—1274no—
114M16—2542no—
115M16—2542no—
116M16—2544no—
117M16—2542no—
118C2×C8—2542no—
119C2×C8—2544no—
120C2×C8—2542no—
121C4⋊C4—24no—
122C4⋊C4—21no—
123C22⋊C4—24no—
124C22⋊C4C2×Dic12712yes—
125C42—21no—
126C23C22⋊Dic12712yes—
127C23C22×Dic12711yescharacteristic
128Q8—22no—
129Q8—41no—
130Q8—41no—
131Q8—42no—
132D4—24no—
133D4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—24no—
139C2×C4C22⋊Dic12712yes—
140C2×C4D4×D12712yes—
141C2×C4C2×C22⋊D12711yescharacteristic
142C2×C4C2×C22⋊D12711yescharacteristic
143C2×C4#1452324712yes—
144C8—5082no—
145C8—2544no—
146C22—22no—
147C22—41no—
148C22—42no—
149C22C2×C22⋊Dic12711yescharacteristic
150C22D4×Dic12712yes—
151C4—22no—
152C4—42no—
153C4—42no—
154C4#1686117711yescharacteristic
155C4#1686118112yes—
156C4#1686119011yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#2254975611yescharacteristic
subgroupG/H
centerC2#22549756
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#536×C127C2
socleC2×C127#386
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#536D127
hypocenterC127#1915
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
3. C2×#16861128C2semidirect
4. #22549646C2semidirect
81. C127#1915semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225498602
C2×#168611282
#225496462
#536×C1272
#1915127