Finite Group Databasebeta

← #30647964

Subgroups — #30647964

Order 16256.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16861240C212yesmaximal, complemented
2#22550387C211yesmaximal, characteristic
3#22550032C214yesmaximal, complemented
4#22550015C214yesmaximal, complemented
5#22549885C212yesmaximal, complemented
6#22549840C211yesmaximal, characteristic, complemented
7C23×Q8×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—28no—
10C2×#14523238C2214yescomplemented
11C2×#14523235C2214yescomplemented
12C2×#14523232C2212yes—
13#16861192—21no—
14#16861188C2212yes—
15#16861183C2212yes—
16#16861178—21no—
17C4×C22⋊D127—22no—
18#16861168C2211yescharacteristic
19#16861133C2218yescomplemented
20C22⋊(C127⋊C8)C2214yes—
21C22×Q8×C127—28no—
22C22×Q8×C127—21no—
23C22×Q8×C127—22no—
24C22×Q8×C127C2214yes—
25C22×Q8×C127C2212yes—
26C23×C4×C127—21no—
27C23×C4×C127C2211yescharacteristic
28#14523248—24no—
29C2×Dic254C2311yescharacteristic
30C2×D508C2311yescharacteristic
31C2×C22⋊D127—21no—
32C2×C22⋊D127—22no—
33C2×C4×D127—21no—
34#14523238—28no—
35#14523235—28no—
36#14523232—24no—
37C2×C127⋊C8C2314yes—
38C4⋊Dic127C2312yes—
39Dic127⋊C4—22no—
40C22⋊Dic127—22no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—22no—
43C4×Dic127—21no—
44C24×C127D411yescharacteristic
45C2×Q8×C127—216no—
46C2×Q8×C127—28no—
47C2×Q8×C127—216no—
48C2×Q8×C127—24no—
49C2×Q8×C127—24no—
50C2×Q8×C127—22no—
51C2×Q8×C127D418yescomplemented
52C2×Q8×C127C2314yes—
53C22×C4×C127—28no—
54C22×C4×C127—24no—
55C22×C4×C127—22no—
56C22×C4×C127—24no—
57C22×C4×C127D411yescharacteristic
58C22×C4×C127D412yes—
59C22×C4×C127C2312yes—
60C22×C4×C127C2311yescharacteristic
61C22×D127—21no—
62Dic254—22no—
63D508—22no—
64C22⋊D127—42no—
65C22⋊D127—44no—
66C2×Dic127—22no—
67C2×Dic127—21no—
68C4×D127—42no—
69C127⋊C8—24no—
70C23×C127—22no—
71C23×C127—24no—
72C23×C127C2×D411yescharacteristic
73C23×C127C2×D412yes—
74Q8×C127—216no—
75Q8×C127—28no—
76Q8×C127—24no—
77Q8×C127C2×D418yes—
78C2×C4×C127—216no—
79C2×C4×C127—28no—
80C2×C4×C127—22no—
81C2×C4×C127—24no—
82C2×C4×C127—24no—
83C2×C4×C127—24no—
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D414yes—
87C2×C4×C127C2411yescharacteristic
88C22×C127—28no—
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—22no—
92C22×C127#12012yes—
93C22×C127#13714yescomplemented
94C22×C127C22×D411yescharacteristic
95C2×D127—41no—
96C2×D127—42no—
97C4×C127—28no—
98C4×C127—22no—
99C4×C127#12012yes—
100C4×C127C22×D412yes—
101Dic127—41no—
102Dic127—42no—
103C2×C127—22no—
104C2×C127—24no—
105C2×C127C2×#12011yescharacteristic
106C2×C127C2×#13712yes—
107D127—81no—
108#3004—1271nomaximal, Sylow
109C127#300411yescharacteristic, complemented, Sylow
110C23×Q8D12711yescharacteristic, complemented
111C2×#137—1272no—
112#525—1271no—
113#451—1274no—
114#448—1274no—
115#419—1272no—
116#407—1271no—
117C2×#41—1271no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D12714yes—
122C22×Q8C2×D12712yes—
123C23×C4—21no—
124C23×C4C2×D12711yescharacteristic
125#137—2548no—
126C2×Q16—1274no—
127C2×SD16—1274no—
128C2×M16—1272no—
129#123—2541no—
130C22⋊Q8—1272no—
131#121—1272no—
132#120—2541no—
133C4×D4—2542no—
134#117—1271no—
135#103—1278no—
136C22⋊C8—1274no—
137C24C22⋊D12711yescharacteristic
138#41—2544no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—24no—
145C2×Q8—1271no—
146C2×Q8C22⋊D12718yescomplemented
147C2×Q8C22×D12714yes—
148C2×D4—1271no—
149C2×D4—2542no—
150C2×D4—2541no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—2541no—
156C22×C4C22⋊D12712yes—
157C22×C4C22⋊D12711yescharacteristic
158C22×C4C22×D12712yes—
159C22×C4C22×D12711yescharacteristic
160Q16—2548no—
161SD16—2548no—
162M16—2544no—
163C2×C8—1274no—
164C4⋊C4—1272no—
165C4⋊C4—2542no—
166C22⋊C4—2541no—
167C22⋊C4—2542no—
168C22⋊C4—2542no—
169C42—2541no—
170C23—24no—
171C23—22no—
172C23—2541no—
173C23C2×C22⋊D12712yes—
174C23C2×C22⋊D12711yescharacteristic
175Q8—216no—
176Q8—24no—
177Q8—28no—
178Q8—2542no—
179Q8C2×C22⋊D12718yes—
180D4—5084no—
181D4—5082no—
182D4—2542no—
183C2×C4—216no—
184C2×C4—24no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—5082no—
190C2×C4—2542no—
191C2×C4—2541no—
192C2×C4C2×C22⋊D12712yes—
193C2×C4C2×C22⋊D12714yes—
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4C23×D12711yescharacteristic
196C8—2544no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—5081no—
202C22—5082no—
203C22#1686117812yes—
204C22#1686124014yescomplemented
205C22C22×C22⋊D12711yescharacteristic
206C4—28no—
207C4—22no—
208C4—5081no—
209C4—5082no—
210C4#1686117812yes—
211C4C22×C22⋊D12712yes—
212C2—24no—
213C2—22no—
214C2—10161no—
215C2C2×#1686117811yescharacteristic
216C2C2×#1686124012yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC23×Q8×C127C2
socleC22×C127C22×D4
prime radical#22550387C2
hypercenterC23×Q8D127
hypocenterC127#3004
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. #22550032C2semidirect
4. #22550015C2semidirect
5. #22549885C2semidirect
6. #22549840C2semidirect
7. C23×Q8×C127C2semidirect
10. C2×#14523238C22semidirect
11. C2×#14523235C22semidirect
19. #16861133C22semidirect
51. C2×Q8×C127D4semidirect
93. C22×C127#137semidirect
109. C127#3004semidirect
110. C23×Q8D127semidirect
146. C2×Q8C22⋊D127semidirect
204. C22#16861240semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225503872
#225500322
#225500152
#225498852
#225498402
C23×Q8×C1272
#3004127