Finite Group Databasebeta

← #30639709

Subgroups — #30639709

Order 16256.

244 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 83 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#22550429C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic
3#22549722C211yesmaximal, characteristic
4#22549637C211yesmaximal, characteristic
5#22549636C211yesmaximal, characteristic
6#22549635C211yesmaximal, characteristic
7#339×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9#16861208C412yescomplemented
10#16861200C412yescomplemented
11#16861186—22no—
12C4×Dic254—22no—
13#16861168C2211yescharacteristic
14C2×C4×Dic127C2211yescharacteristic
15C2×C4×Dic127C2211yescharacteristic
16Dic127⋊C8—22no—
17C127⋊#95—22no—
18C2×M16×C127C2211yescharacteristic
19#117×C127C2211yescharacteristic
20C2×C42×C127C2211yescharacteristic
21C2×Dic254—22no—
22C22×Dic127—21no—
23C22×Dic127D412yes—
24#14523232—21no—
25#14523232—21no—
26C2×C127⋊C8—21no—
27C4⋊Dic127—22no—
28C4⋊Dic127C2×C412yes—
29Dic127⋊C4—22no—
30Dic127⋊C4—22no—
31C22⋊Dic127—21no—
32C4×Dic127—21no—
33C4×Dic127—21no—
34C4×Dic127C2×C411yescharacteristic
35C4×Dic127C2×C411yescharacteristic
36C4×Dic127D412yes—
37C4×Dic127Q812yescomplemented
38C22×C4×C127—21no—
39C22×C4×C127C2311yescharacteristic
40M16×C127—21no—
41M16×C127—21no—
42C2×C8×C127—21no—
43C4⋊C4×C127C2×C412yes—
44C22⋊C4×C127—21no—
45C42×C127—21no—
46C42×C127—21no—
47C42×C127Q812yescomplemented
48Dic254—44no—
49C2×Dic127—22no—
50C2×Dic127—22no—
51C2×Dic127—42no—
52C2×Dic127—42no—
53C2×Dic127C22⋊C412yes—
54C2×Dic127C22⋊C412yes—
55C127⋊C8—41no—
56C23×C127C2×D411yescharacteristic
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—42no—
60C2×C4×C127C22×C411yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×Q811yescharacteristic
63C2×C4×C127C2×Q811yescharacteristic
64C2×C4×C127#4111yescharacteristic
65C2×C4×C127#4111yescharacteristic
66C8×C127—41no—
67C22×C127—22no—
68C22×C127C2×C22⋊C411yescharacteristic
69C22×C127C22⋊Q811yescharacteristic
70C22×C127C22⋊Q811yescharacteristic
71C4×C127—42no—
72C4×C127—42no—
73C4×C127C4×Q811yescharacteristic
74C4×C127C4×Q811yescharacteristic
75C4×C127#12411yescharacteristic
76C4×C127C4⋊Q811yescharacteristic
77Dic127—22no—
78Dic127—22no—
79Dic127—42no—
80Dic127—42no—
81C2×C127—21no—
82C2×C127—21no—
83C2×C127#39111yescharacteristic
84C2×C127#42112yes—
85#1995—1271nomaximal, Sylow
86C127#199511yescharacteristic, complemented, Sylow
87#533—1271no—
88#423—1271no—
89#376—1271no—
90#339—1271no—
91#339—1271no—
92#339—1271no—
93#339D12711yescharacteristic
94C2×M16—1271no—
95C2×M16C2×D12711yescharacteristic
96C4⋊Q8—1272no—
97#125—1272no—
98C22⋊Q8—2542no—
99C4×Q8—2542no—
100#117—1271no—
101#117C2×D12711yescharacteristic
102C2×C42—1271no—
103C2×C42—1271no—
104C2×C42C2×D12711yescharacteristic
105C4⋊C8—2542no—
106#95—2542no—
107C2×Q8—2542no—
108C22×C4—21no—
109C22×C4—1272no—
110C22×C4—2541no—
111C22×C4C22×D12711yescharacteristic
112M16—21no—
113M16—21no—
114M16—2541no—
115M16—2541no—
116C2×C8—21no—
117C2×C8—2541no—
118C4⋊C4—1272no—
119C4⋊C4—2542no—
120C4⋊C4—2542no—
121C4⋊C4—2542no—
122C4⋊C4C4×D12712yes—
123C22⋊C4—21no—
124C22⋊C4—2541no—
125C42—21no—
126C42—21no—
127C42—1271no—
128C42—1271no—
129C42—1272no—
130C42—1272no—
131C42—2541no—
132C42—2541no—
133C42Dic25412yescomplemented
134C23D4×D12711yescharacteristic
135Q8—5084no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—1272no—
140C2×C4—1272no—
141C2×C4—5082no—
142C2×C4—5082no—
143C2×C4—2542no—
144C2×C4—2542no—
145C2×C4C2×C4×D12711yescharacteristic
146C2×C4D4×D12711yescharacteristic
147C2×C4Q8×D12711yescharacteristic
148C2×C4C2×Dic25411yescharacteristic
149C2×C4#1452324711yescharacteristic
150C2×C4#1452324811yescharacteristic
151C8—41no—
152C8—5081no—
153C22—22no—
154C22C22⋊C4×D12711yescharacteristic
155C22#1686118611yescharacteristic
156C22#1686118711yescharacteristic
157C4—42no—
158C4—42no—
159C4—5082no—
160C4—5082no—
161C4—2542no—
162C4—2542no—
163C4#1686117411yescharacteristic
164C4C4×Dic25411yescharacteristic
165C4#1686119511yescharacteristic
166C4#1686120811yescharacteristic
167C2—21no—
168C2—21no—
169C2#2254978011yescharacteristic
170C2#2254989112yes—
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1995
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #16861208C4semidirect
10. #16861200C4semidirect
37. C4×Dic127Q8semidirect
47. C42×C127Q8semidirect
86. C127#1995semidirect
133. C42Dic254semidirect

Maximal subgroups

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

subgroupindex
#225504292
#225498992
#225497222
#225496372
#225496362
#225496352
#339×C1272
#1995127