Finite Group Databasebeta

← #30639225

Subgroups — #30639225

Order 16256.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16861242C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic, complemented
3#22549894C211yesmaximal, characteristic
4#22549884C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic
6#22549647C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10#16861242—22no—
11#16861242—22no—
12C2×Dic508C412yescomplemented
13C2×#14523237C412yescomplemented
14C2×#14523232C2211yescharacteristic
15C4×Dic254—22no—
16#16861174—22no—
17#16861168C2211yescharacteristic
18#16861165—22no—
19#16861137—22no—
20#16861132—22no—
21#16861123—22no—
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254C2×C411yescharacteristic
33C2×Dic254C2×C412yes—
34C2×Dic254D412yes—
35C2×Dic254D412yes—
36C2×D508C2×C411yescharacteristic
37C2×C22⋊D127—21no—
38C22×Dic127—21no—
39C2×C4×D127—21no—
40Dic508—24no—
41#14523237—24no—
42#14523232—21no—
43#14523232—21no—
44C2×C127⋊C8—21no—
45C4⋊Dic127—21no—
46Dic127⋊C4—24no—
47C22⋊Dic127D412yes—
48C4×Dic127—24no—
49C4×Dic127—21no—
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53C2×C8×C127C2×C412yes—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—21no—
58C42×C127—21no—
59C42×C127D412yes—
60C22×D127—21no—
61Dic254—22no—
62Dic254—22no—
63Dic254—22no—
64Dic254—21no—
65Dic254—21no—
66Dic254C22⋊C414yes—
67D508—21no—
68D508—21no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127—22no—
76C4×D127—41no—
77C4×D127—41no—
78C127⋊C8—41no—
79C23×C127C2×D411yescharacteristic
80C2×C4×C127—22no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—22no—
89C22×C127—41no—
90C22×C127C4×D411yescharacteristic
91C22×C127#12011yescharacteristic
92C22×C127#12111yescharacteristic
93C2×D127—41no—
94C2×D127—81no—
95C4×C127—42no—
96C4×C127—81no—
97C4×C127C2×C22⋊C411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127#12111yescharacteristic
100C4×C127#12311yescharacteristic
101Dic127—24no—
102Dic127—42no—
103Dic127—41no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127#38611yescharacteristic
108D127—81no—
109#1905—1271nomaximal, Sylow
110C127#190511yescharacteristic, complemented, Sylow
111C2×#137—1271no—
112#519—1271no—
113#421—1271no—
114#419—1271no—
115#410—1271no—
116#344—1271no—
117H(3,ℤ/4)D12711yescharacteristic, complemented
118C2×#41—1271no—
119C22×Q8—1271no—
120#137—2542no—
121#137—2542no—
122C2×Q16—1272no—
123C2×SD16—1272no—
124C2×M16—1271no—
125C2×M16C2×D12711yescharacteristic
126C4×Q8—2542no—
127C4×Q8—2542no—
128#117—1271no—
129#117—2542no—
130#117C2×D12711yescharacteristic
131#117C2×D12711yescharacteristic
132#104—2542no—
133#103—2542no—
134#99—2542no—
135#41—2541no—
136#41—2541no—
137#41—2541no—
138#41—2541no—
139C2×Q8—1272no—
140C2×Q8—1271no—
141C2×Q8—1272no—
142C2×Q8—1272no—
143C2×Q8—2541no—
144C2×Q8—2541no—
145C2×Q8—2541no—
146C2×D4—1271no—
147C2×D4—2541no—
148C22×C4—2541no—
149C22×C4—2541no—
150C22×C4C22×D12711yescharacteristic
151Q16—2544no—
152SD16—2544no—
153M16—21no—
154M16—21no—
155M16—2541no—
156M16—2541no—
157C2×C8—2541no—
158C2×C8C4×D12712yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—2544no—
162C4⋊C4—2541no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—1272no—
166C42—21no—
167C42—2544no—
168C42—2541no—
169C42D50812yes—
170C23—2541no—
171C23D4×D12711yescharacteristic
172Q8—1274no—
173Q8—2541no—
174Q8—2541no—
175Q8—2542no—
176Q8—2542no—
177Q8—2542no—
178D4—5081no—
179D4—5081no—
180D4—2541no—
181D4—2541no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—5081no—
185C2×C4—5081no—
186C2×C4—2541no—
187C2×C4—2542no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4D4×D12711yescharacteristic
194C2×C4C2×D50811yescharacteristic
195C2×C4#1452324811yescharacteristic
196C2×C4#1452324911yescharacteristic
197C8—22no—
198C8—5081no—
199C22—41no—
200C22—10161no—
201C22—5081no—
202C22#1686117111yescharacteristic
203C22#1686117911yescharacteristic
204C22#1686118211yescharacteristic
205C4—42no—
206C4—81no—
207C4—5082no—
208C4—5081no—
209C4—2544no—
210C4C22⋊C4×D12711yescharacteristic
211C4C4×D50811yescharacteristic
212C4#1686118411yescharacteristic
213C4#1686119411yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—10161no—
218C2#2254975411yescharacteristic
subgroupG/H
centerC2#22549754
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingH(3,ℤ/4)×C127C2
socleC2×C127#386
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenterH(3,ℤ/4)D127
hypocenterC127#1905
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #22550353C2semidirect
4. #22549884C2semidirect
6. #22549647C2semidirect
7. H(3,ℤ/4)×C127C2semidirect
12. C2×Dic508C4semidirect
13. C2×#14523237C4semidirect
110. C127#1905semidirect
117. H(3,ℤ/4)D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
#225503532
#225498942
#225498842
#225498492
#225496472
H(3,ℤ/4)×C1272
#1905127