Finite Group Databasebeta

← #30639282

Subgroups — #30639282

Order 16256.

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

normal subgroups only

This group has 308 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#22549860C211yesmaximal, characteristic
3C2×#16861127C212yesmaximal, complemented
4#22549648C212yesmaximal, complemented
5#568×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7C2×#14523232C2212yes—
8#16861207—21no—
9#16861196—21no—
10#16861190C414yes—
11#16861188—22no—
12#16861168C2211yescharacteristic
13#16861128—21no—
14#16861127—24no—
15#16861126—21no—
16C2×#41×C127C2211yescharacteristic
17#131×C127—24no—
18C2×M16×C127—22no—
19C2×M16×C127C2212yes—
20C2×Dic254—21no—
21C2×Dic254—21no—
22C2×Dic254—22no—
23C2×Dic254—22no—
24C22×Dic127C2×C412yes—
25#14523232—22no—
26#14523232—22no—
27C2×C127⋊C8—22no—
28C4⋊Dic127—21no—
29Dic127⋊C4—24no—
30C22⋊Dic127—24no—
31C22⋊Dic127C2×C412yes—
32C4×Dic127—21no—
33#41×C127—22no—
34#41×C127—22no—
35C2×Q8×C127D411yescharacteristic
36C2×D4×C127C2×C412yes—
37C2×D4×C127D411yescharacteristic
38C22×C4×C127D412yes—
39C22×C4×C127C2311yescharacteristic
40M16×C127—24no—
41M16×C127—22no—
42M16×C127D414yes—
43C2×C8×C127—24no—
44C2×C8×C127—22no—
45Dic254—41no—
46Dic254—42no—
47Dic254—41no—
48C2×Dic127—22no—
49C2×Dic127—22no—
50C2×Dic127—24no—
51C127⋊C8—42no—
52C23×C127C22⋊C412yes—
53C23×C127C22×C411yescharacteristic
54Q8×C127—22no—
55D4×C127—24no—
56D4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127C22⋊C412yes—
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127#4112yes—
64C8×C127—24no—
65C22×C127—22no—
66C22×C127—41no—
67C22×C127—42no—
68C22×C127C2×C22⋊C411yescharacteristic
69C22×C127C4×D412yes—
70C4×C127—22no—
71C4×C127#12011yescharacteristic
72C4×C127#12112yes—
73C4×C127#12311yescharacteristic
74Dic127—42no—
75Dic127—42no—
76C2×C127—21no—
77C2×C127—22no—
78C2×C127—42no—
79C2×C127#38611yescharacteristic
80#1915—1271nomaximal, Sylow
81C127#191511yescharacteristic, complemented, Sylow
82#568D12711yescharacteristic
83#536—1271no—
84#413—1271no—
85C2×#101—1272no—
86#344—1272no—
87C2×#41C2×D12711yescharacteristic
88C22×Q8—1271no—
89#131—24no—
90C2×M16—22no—
91C2×M16—1272no—
92C2×M16C2×D12712yes—
93C4⋊Q8—2541no—
94#124—2541no—
95#123—1274no—
96C22⋊Q8—2542no—
97#117—1271no—
98#101—2541no—
99#101—2544no—
100#100—2541no—
101#41—22no—
102#41—22no—
103C2×Q8—2542no—
104C2×Q8—2541no—
105C2×Q8—2541no—
106C2×Q8—2542no—
107C2×Q8D50811yescharacteristic
108C2×D4C4×D12712yes—
109C2×D4D50811yescharacteristic
110C22×C4—1272no—
111C22×C4C22⋊D12712yes—
112C22×C4C22×D12711yescharacteristic
113M16—24no—
114M16—22no—
115M16—2542no—
116M16—2542no—
117M16C22⋊D12714yes—
118C2×C8—24no—
119C2×C8—22no—
120C2×C8—2542no—
121C4⋊C4—2544no—
122C4⋊C4—2541no—
123C22⋊C4—1272no—
124C22⋊C4—2544no—
125C42—2541no—
126C23(C2×D127)⋊C412yes—
127C23C2×C4×D12711yescharacteristic
128Q8—22no—
129Q8—5081no—
130Q8—5081no—
131Q8—5082no—
132D4—24no—
133D4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—2542no—
137C2×C4—2542no—
138C2×C4—2544no—
139C2×C4(C2×D127)⋊C412yes—
140C2×C4C2×C22⋊D12711yescharacteristic
141C2×C4C2×C22⋊D12712yes—
142C2×C4C2×D50811yescharacteristic
143C2×C4#1452324812yes—
144C8—24no—
145C8—5082no—
146C22—22no—
147C22—41no—
148C22—42no—
149C22C2×(C2×D127)⋊C411yescharacteristic
150C22C4×C22⋊D12712yes—
151C4—22no—
152C4—5082no—
153C4—5082no—
154C4#1686117811yescharacteristic
155C4#1686118312yes—
156C4#1686119211yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#2254975511yescharacteristic
subgroupG/H
centerC2#22549755
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#386
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1915
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
3. C2×#16861127C2semidirect
4. #22549648C2semidirect
81. C127#1915semidirect

Maximal subgroups

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

subgroupindex
#225504472
#225498602
C2×#168611272
#225496482
#568×C1272
#1915127