Finite Group Databasebeta

← #30639132

Subgroups — #30639132

Order 16256.

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

normal subgroups only

This group has 344 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×#16861197C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic, complemented
3#22549882C211yesmaximal, characteristic
4#22549880C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861197—22no—
12#16861197C414yescomplemented
13C4×Dic254—22no—
14#16861174—22no—
15#16861168C2211yescharacteristic
16#16861165—22no—
17C2×(C2×D127)⋊C4—22no—
18#16861131—22no—
19#16861129—22no—
20#16861127—22no—
21C2×M16×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23C2×C42×C127C2211yescharacteristic
24C23×D127—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C2×Dic254D412yes—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508C2×C412yes—
37C22×Dic127D412yes—
38#14523232—21no—
39#14523232—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127D412yes—
42Dic127⋊C4—24no—
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—28no—
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—
57C22×D127—22no—
58C22×D127—44no—
59C22×D127—41no—
60C22×D127—41no—
61Dic254—22no—
62Dic254—22no—
63Dic254—22no—
64Dic254—22no—
65D508—22no—
66D508—22no—
67D508—41no—
68D508—41no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127C22⋊C414yes—
73C127⋊C8—41no—
74C23×C127C2×D411yescharacteristic
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—42no—
78C2×C4×C127C22×C411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C8×C127—41no—
85C22×C127—22no—
86C22×C127C2×C22⋊C411yescharacteristic
87C22×C127#12011yescharacteristic
88C22×C127#12311yescharacteristic
89C2×D127—44no—
90C2×D127—42no—
91C2×D127—82no—
92C2×D127—82no—
93C4×C127—42no—
94C4×C127—42no—
95C4×C127C4×D411yescharacteristic
96C4×C127C4×D411yescharacteristic
97C4×C127#12111yescharacteristic
98C4×C127#12111yescharacteristic
99Dic127—24no—
100Dic127—42no—
101C2×C127—21no—
102C2×C127—21no—
103C2×C127#38611yescharacteristic
104C2×C127#45412yes—
105D127—82no—
106#1888—1271nomaximal, Sylow
107C127#188811yescharacteristic, complemented, Sylow
108C2×#124—1271no—
109#519—1271no—
110#418—1271no—
111#418—1271no—
112C2×#101—1271no—
113#339—1271no—
114#339D12711yescharacteristic, complemented
115C22×Q8—1271no—
116C22×D4—1271no—
117C2×M16—1271no—
118C2×M16C2×D12711yescharacteristic
119#124—1274no—
120#124—2542no—
121C4×Q8—2542no—
122C4×Q8—2542no—
123#117—1271no—
124#117—2542no—
125#117C2×D12711yescharacteristic
126C2×C22⋊C4—2542no—
127C2×C42C2×D12711yescharacteristic
128#102—2542no—
129#102—2542no—
130#101—2542no—
131C24—2541no—
132C2×Q8—1272no—
133C2×Q8—1272no—
134C2×Q8—2541no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×D4—1272no—
139C2×D4—2541no—
140C2×D4—2541no—
141C2×D4—2541no—
142C2×D4—2541no—
143C2×D4—2541no—
144C22×C4—21no—
145C22×C4—1272no—
146C22×C4C22×D12711yescharacteristic
147M16—21no—
148M16—21no—
149M16—2541no—
150M16—2541no—
151C2×C8—21no—
152C2×C8—2541no—
153C4⋊C4—1272no—
154C4⋊C4—2544no—
155C4⋊C4D50812yes—
156C22⋊C4—21no—
157C22⋊C4—2548no—
158C22⋊C4—2541no—
159C42—21no—
160C42—21no—
161C42—2544no—
162C42—2541no—
163C42C4×D12712yes—
164C23—5084no—
165C23—5081no—
166C23—5081no—
167C23—2542no—
168C23D4×D12711yescharacteristic
169Q8—2542no—
170Q8—2542no—
171Q8—2542no—
172Q8—2542no—
173D4—5081no—
174D4—5081no—
175D4—2542no—
176D4—2542no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—1274no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4—2542no—
184C2×C4C2×C4×D12711yescharacteristic
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×D50811yescharacteristic
188C2×C4#1452324811yescharacteristic
189C2×C4#1452324911yescharacteristic
190C8—41no—
191C8—5081no—
192C22—22no—
193C22—10162no—
194C22—10162no—
195C22—5084no—
196C22—5082no—
197C22C22⋊C4×D12711yescharacteristic
198C22#1686117911yescharacteristic
199C22#1686119411yescharacteristic
200C4—42no—
201C4—42no—
202C4—5082no—
203C4—2544no—
204C4#1686117111yescharacteristic
205C4C4×D50811yescharacteristic
206C4#1686118211yescharacteristic
207C4#1686118411yescharacteristic
208C2—21no—
209C2—21no—
210C2—10162no—
211C2#2254975411yescharacteristic
212C2#2255005012yes—
subgroupG/H
centerC22#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127#120
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1888
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
2. #22550353C2semidirect
5. C2×#16861127C2semidirect
6. #22549636C2semidirect
7. #339×C127C2semidirect
12. #16861197C4semidirect
107. C127#1888semidirect
114. #339D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611972
#225503532
#225498822
#225498802
C2×#168611272
#225496362
#339×C1272
#1888127