Finite Group Databasebeta

← #30639122

Subgroups — #30639122

Order 16256.

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

normal subgroups only

This group has 324 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
1C2×#16861209C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic
3#22549886C211yesmaximal, characteristic
4#22549884C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic
6#22549636C211yesmaximal, characteristic
7#339×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861209—22no—
12#16861209C414yescomplemented
13C4×Dic254—22no—
14#16861174—22no—
15#16861168C2211yescharacteristic
16#16861165—22no—
17C2×C4⋊Dic127—22no—
18#16861134—22no—
19#16861132—22no—
20#16861127—22no—
21C2×M16×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23C2×C42×C127C2211yescharacteristic
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254C2×C412yes—
34C2×Dic254C2×C412yes—
35C2×Dic254D412yes—
36C22×Dic127—21no—
37C22×Dic127D412yes—
38#14523232—21no—
39#14523232—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127—28no—
42C4⋊Dic127D412yes—
43Dic127⋊C4—24no—
44C22⋊Dic127—21no—
45C4×Dic127—24no—
46C4×Dic127—21no—
47C22×C4×C127—21no—
48C22×C4×C127C2311yescharacteristic
49M16×C127—21no—
50M16×C127—21no—
51C2×C8×C127—21no—
52C4⋊C4×C127D412yes—
53C22⋊C4×C127—21no—
54C42×C127—21no—
55C42×C127—21no—
56C42×C127C2×C412yes—
57Dic254—22no—
58Dic254—22no—
59Dic254—22no—
60Dic254—22no—
61Dic254—22no—
62Dic254—22no—
63Dic254—41no—
64Dic254—41no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127C22⋊C414yes—
72C127⋊C8—41no—
73C23×C127C2×D411yescharacteristic
74C2×C4×C127—24no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127C22×C411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C8×C127—41no—
84C22×C127—22no—
85C22×C127C2×C22⋊C411yescharacteristic
86C22×C127#12011yescharacteristic
87C22×C127#12311yescharacteristic
88C4×C127—24no—
89C4×C127—42no—
90C4×C127C4×D411yescharacteristic
91C4×C127C4×D411yescharacteristic
92C4×C127#12111yescharacteristic
93C4×C127#12111yescharacteristic
94Dic127—24no—
95Dic127—42no—
96Dic127—42no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127#38611yescharacteristic
100C2×C127#45612yes—
101#1886—1271nomaximal, Sylow
102C127#188611yescharacteristic, complemented, Sylow
103C2×C4⋊Q8—1271no—
104#519—1271no—
105#419—1271no—
106#419—1271no—
107C2×#101—1271no—
108#339—1271no—
109#339D12711yescharacteristic
110C22×Q8—1271no—
111C22×Q8—1271no—
112C2×M16—1271no—
113C2×M16C2×D12711yescharacteristic
114C4⋊Q8—1274no—
115C4⋊Q8—2542no—
116C4×Q8—2542no—
117C4×Q8—2542no—
118#117—1271no—
119#117—2542no—
120#117C2×D12711yescharacteristic
121C2×C4⋊C4—2542no—
122C2×C42C2×D12711yescharacteristic
123#103—2542no—
124#103—2542no—
125#101—2542no—
126C2×Q8—1272no—
127C2×Q8—1272no—
128C2×Q8—1272no—
129C2×Q8—2541no—
130C2×Q8—2541no—
131C2×Q8—2541no—
132C2×Q8—2541no—
133C2×Q8—2541no—
134C2×Q8—2541no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C22×C4—21no—
139C22×C4—1272no—
140C22×C4—2541no—
141C22×C4C22×D12711yescharacteristic
142M16—21no—
143M16—21no—
144M16—2541no—
145M16—2541no—
146C2×C8—21no—
147C2×C8—2541no—
148C4⋊C4—1272no—
149C4⋊C4—2548no—
150C4⋊C4—2544no—
151C4⋊C4D50812yes—
152C22⋊C4—21no—
153C22⋊C4—2541no—
154C42—21no—
155C42—21no—
156C42—2544no—
157C42—2541no—
158C42C4×D12712yes—
159C23D4×D12711yescharacteristic
160Q8—5081no—
161Q8—5081no—
162Q8—2542no—
163Q8—2542no—
164Q8—2542no—
165Q8—2542no—
166Q8—2542no—
167Q8—2542no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—1274no—
172C2×C4—5081no—
173C2×C4—5081no—
174C2×C4—2542no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4—2542no—
178C2×C4C2×C4×D12711yescharacteristic
179C2×C4D4×D12711yescharacteristic
180C2×C4D4×D12711yescharacteristic
181C2×C4C2×D50811yescharacteristic
182C2×C4#1452324811yescharacteristic
183C2×C4#1452324911yescharacteristic
184C8—41no—
185C8—5081no—
186C22—22no—
187C22C22⋊C4×D12711yescharacteristic
188C22#1686117911yescharacteristic
189C22#1686119411yescharacteristic
190C4—24no—
191C4—42no—
192C4—5082no—
193C4—5082no—
194C4—2544no—
195C4#1686117111yescharacteristic
196C4C4×D50811yescharacteristic
197C4#1686118211yescharacteristic
198C4#1686118411yescharacteristic
199C2—21no—
200C2—21no—
201C2#2254975411yescharacteristic
202C2#2255006012yes—
subgroupG/H
centerC22#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1886
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
12. #16861209C4semidirect
102. C127#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#168612092
#225503532
#225498862
#225498842
C2×#168611272
#225496362
#339×C1272
#1886127