Finite Group Databasebeta

← #30639130

Subgroups — #30639130

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×#16861196C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic, complemented
3#22549881C211yesmaximal, characteristic
4C2×#16861127C211yesmaximal, characteristic, complemented
5#22549637C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861196—22no—
11#16861196C414yescomplemented
12C4×Dic254—22no—
13#16861174—22no—
14#16861168C2211yescharacteristic
15#16861165—22no—
16C2×C22⋊Dic127—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861130—22no—
19#16861127—22no—
20C22×D4×C127C2211yescharacteristic
21C2×M16×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23#102×C127—22no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254C2×C412yes—
29C2×Dic254D412yes—
30C22×Dic127—21no—
31C22×Dic127D412yes—
32#14523232—21no—
33#14523232—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127D412yes—
36Dic127⋊C4—24no—
37C22⋊Dic127—28no—
38C22⋊Dic127—21no—
39C4×Dic127—24no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C4×Dic127C2×C412yes—
43C24×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127C2×C412yes—
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53C2×C8×C127—21no—
54C4⋊C4×C127D412yes—
55C22⋊C4×C127—21no—
56C42×C127—21no—
57Dic254—22no—
58Dic254—22no—
59Dic254—22no—
60Dic254—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—42no—
66C2×Dic127C22⋊C414yes—
67C127⋊C8—41no—
68C23×C127—22no—
69C23×C127—44no—
70C23×C127—41no—
71C23×C127—41no—
72C23×C127C2×D411yescharacteristic
73D4×C127—22no—
74D4×C127—22no—
75D4×C127—41no—
76D4×C127—41no—
77C2×C4×C127—22no—
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×C127—44no—
87C22×C127—42no—
88C22×C127—82no—
89C22×C127—82no—
90C22×C127C2×C22⋊C411yescharacteristic
91C22×C127#12011yescharacteristic
92C22×C127#12311yescharacteristic
93C4×C127—42no—
94C4×C127C4×D411yescharacteristic
95C4×C127C4×D411yescharacteristic
96C4×C127#12111yescharacteristic
97C4×C127#12111yescharacteristic
98Dic127—24no—
99Dic127—42no—
100Dic127—42no—
101C2×C127—21no—
102C2×C127—21no—
103C2×C127—82no—
104C2×C127#38611yescharacteristic
105C2×C127#45412yes—
106#1888—1271nomaximal, Sylow
107C127#188811yescharacteristic, complemented, Sylow
108C2×#124—1271no—
109#519—1271no—
110#418—1271no—
111#418D12711yescharacteristic
112C2×#101—1271no—
113#339—1271no—
114#339—1271no—
115C22×Q8—1271no—
116C22×D4C2×D12711yescharacteristic
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×C42—1271no—
128#102—22no—
129#102—2542no—
130#101—2542no—
131C24—21no—
132C2×Q8—1272no—
133C2×Q8—1272no—
134C2×Q8—2541no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C4×D12712yes—
144C22×C4—1272no—
145C22×C4—2541no—
146C22×C4C22×D12711yescharacteristic
147M16—21no—
148M16—21no—
149M16—2541no—
150M16—2541no—
151C2×C8—21no—
152C2×C8—2541no—
153C4⋊C4—1272no—
154C4⋊C4—2544no—
155C4⋊C4C22⋊D12712yes—
156C22⋊C4—21no—
157C22⋊C4—2548no—
158C22⋊C4—2541no—
159C42—21no—
160C42—1272no—
161C42—2544no—
162C42—2541no—
163C42—2541no—
164C23—22no—
165C23—44no—
166C23—41no—
167C23—41no—
168C23D4×D12711yescharacteristic
169Q8—2542no—
170Q8—2542no—
171Q8—2542no—
172Q8—2542no—
173D4—22no—
174D4—22no—
175D4—41no—
176D4—41no—
177C2×C4—22no—
178C2×C4—1274no—
179C2×C4—5082no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4—2542no—
184C2×C4C2×C4×D12711yescharacteristic
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C2×C4#1452324711yescharacteristic
189C2×C4#1452324811yescharacteristic
190C8—41no—
191C8—5081no—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—82no—
196C22—82no—
197C22C22⋊C4×D12711yescharacteristic
198C22#1686117711yescharacteristic
199C22#1686119111yescharacteristic
200C4—42no—
201C4—5082no—
202C4—5082no—
203C4—2544no—
204C4(C22⋊D127)⋊C411yescharacteristic
205C4C4×C22⋊D12711yescharacteristic
206C4#1686118011yescharacteristic
207C4#1686118211yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2254975211yescharacteristic
212C2#2255004712yes—
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC22×C127#120
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenter#418D127
hypocenterC127#1888
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #22550353C2semidirect
4. C2×#16861127C2semidirect
5. #22549637C2semidirect
6. #22549635C2semidirect
11. #16861196C4semidirect
107. C127#1888semidirect

Maximal subgroups

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

subgroupindex
C2×#168611962
#225503532
#225498812
C2×#168611272
#225496372
#225496352
#418×C1272
#1888127