Finite Group Databasebeta

← #30638835

Subgroups — #30638835

Order 16256.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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#41×Dic127C211yesmaximal, characteristic
2#22549850C211yesmaximal, characteristic
3#22549637C212yesmaximal, complemented
4#22549633C212yesmaximal, complemented
5#413×C127C211yesmaximal, characteristic
6Q8×Dic127—21no—
7D4×Dic127—21no—
8D4×Dic127—22no—
9#16861168C412yescomplemented
10#16861168C2212yes—
11#16861168C2211yescharacteristic
12C2×C4×Dic127C412yescomplemented
13C2×C4×Dic127C2211yescharacteristic
14#16861124—22no—
15C2×#41×C127C2211yescharacteristic
16C2×M16×C127C2212yes—
17#101×C127—21no—
18#100×C127—21no—
19C22×Dic127—22no—
20C22×Dic127—21no—
21C4⋊Dic127—22no—
22C4⋊Dic127—21no—
23C4⋊Dic127—22no—
24C22⋊Dic127—22no—
25C22⋊Dic127—21no—
26C22⋊Dic127—22no—
27C4×Dic127—22no—
28C4×Dic127—21no—
29C4×Dic127—22no—
30C4×Dic127C2×C411yescharacteristic
31C4×Dic127C2×C412yes—
32C4×Dic127C2×C411yescharacteristic
33#41×C127—22no—
34#41×C127—22no—
35C2×Q8×C127D411yescharacteristic
36C2×D4×C127D411yescharacteristic
37C2×D4×C127D412yes—
38C22×C4×C127C2×C412yes—
39C22×C4×C127C2311yescharacteristic
40M16×C127—22no—
41M16×C127—22no—
42C2×C8×C127—22no—
43C2×Dic127—21no—
44C2×Dic127—22no—
45C2×Dic127—21no—
46C2×Dic127—42no—
47C2×Dic127—44no—
48C2×Dic127—42no—
49C23×C127C22⋊C412yes—
50C23×C127C2×D411yescharacteristic
51Q8×C127—22no—
52D4×C127—24no—
53D4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—22no—
56C2×C4×C127C22⋊C412yes—
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127#4112yes—
60C2×C4×C127#4112yes—
61C8×C127—42no—
62C22×C127—22no—
63C22×C127—41no—
64C22×C127—42no—
65C22×C127C2×C22⋊C411yescharacteristic
66C22×C127#12312yes—
67C4×C127—22no—
68C4×C127#11711yescharacteristic
69C4×C127#11711yescharacteristic
70C4×C127#12312yes—
71Dic127—22no—
72Dic127—42no—
73Dic127—41no—
74Dic127—82no—
75C2×C127—21no—
76C2×C127—22no—
77C2×C127—42no—
78C2×C127#38111yescharacteristic
79#1827—1271nomaximal, Sylow
80C127#182711yescharacteristic, complemented, Sylow
81C4×#41—1271no—
82#413D12711yescharacteristic
83#410—1271no—
84#339—1272no—
85H(3,ℤ/4)—1272no—
86C2×#41C2×D12711yescharacteristic
87C2×M16C2×D12712yes—
88C4×Q8—2541no—
89C4×D4—2541no—
90C4×D4—2542no—
91#117—1272no—
92#117—1271no—
93#117—1272no—
94C2×C42—1272no—
95C2×C42—1271no—
96#101—21no—
97#100—21no—
98#99—2542no—
99#41—22no—
100#41—22no—
101C2×Q8C22⋊D12711yescharacteristic
102C2×D4C22⋊D12711yescharacteristic
103C2×D4D50812yes—
104C22×C4—2542no—
105C22×C4—2541no—
106C22×C4C4×D12712yes—
107C22×C4C22×D12711yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—2542no—
112C4⋊C4—2542no—
113C4⋊C4—2541no—
114C22⋊C4—2542no—
115C22⋊C4—2542no—
116C22⋊C4—2541no—
117C42—1271no—
118C42—1272no—
119C42—1271no—
120C42—2542no—
121C42—2542no—
122C42—2541no—
123C23(C2×D127)⋊C412yes—
124C23C2×D50811yescharacteristic
125Q8—22no—
126D4—24no—
127D4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—5082no—
131C2×C4—5082no—
132C2×C4—5084no—
133C2×C4—2542no—
134C2×C4—2541no—
135C2×C4—2541no—
136C2×C4(C2×D127)⋊C412yes—
137C2×C4C2×C4×D12711yescharacteristic
138C2×C4C2×C22⋊D12711yescharacteristic
139C2×C4#1452324712yes—
140C2×C4#1452324712yes—
141C8—42no—
142C22—22no—
143C22—41no—
144C22—42no—
145C22C2×(C2×D127)⋊C411yescharacteristic
146C22#1686119312yes—
147C4—22no—
148C4—10162no—
149C4—5081no—
150C4—5082no—
151C4—2542no—
152C4#1686116511yescharacteristic
153C4#1686116511yescharacteristic
154C4#1686119012yes—
155C2—22no—
156C2—21no—
157C2—42no—
158C2#2254973711yescharacteristic
subgroupG/H
centerC4#16861165
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#413×C127C2
socleC2×C127#381
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#413D127
hypocenterC127#1827
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #22549637C2semidirect
4. #22549633C2semidirect
9. #16861168C4semidirect
12. C2×C4×Dic127C4semidirect
80. C127#1827semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1272
#225498502
#225496372
#225496332
#413×C1272
#1827127