Finite Group Databasebeta

← #30639530

Subgroups — #30639530

Order 16256.

332 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16861207C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic
5C2×C8×Dic127C211yesmaximal, characteristic
6#22549745C211yesmaximal, characteristic
7C2×#103×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861207—22no—
11#16861207C414yescomplemented
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4—22no—
14C2×Dic127⋊C4—22no—
15C2×C4×Dic127C2211yescharacteristic
16#16861134—22no—
17#16861133—22no—
18#16861132—22no—
19C8×Dic127—22no—
20C22×Q8×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#103×C127—22no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D412yes—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8D412yes—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38Dic127⋊C4—28no—
39Dic127⋊C4—44no—
40C4×Dic127C2×C412yes—
41C4×Dic127D412yes—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127C2×C412yes—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127C2311yescharacteristic
51C2×C8×C127—21no—
52C2×C8×C127—21no—
53C2×C8×C127D412yes—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56Dic254—22no—
57Dic254—22no—
58Dic254—41no—
59Dic254—41no—
60C2×Dic127—22no—
61C2×Dic127—41no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127—42no—
65C2×Dic127C22⋊C414yes—
66C2×Dic127SD1614yes—
67C2×Dic127Q1614yes—
68C127⋊C8—22no—
69C23×C127C2×D411yescharacteristic
70Q8×C127—22no—
71Q8×C127—22no—
72Q8×C127—41no—
73Q8×C127—41no—
74C2×C4×C127—22no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—42no—
79C2×C4×C127C22×C411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C8×C127—22no—
86C22×C127C2×C22⋊C411yescharacteristic
87C22×C127#12411yescharacteristic
88C22×C127#12711yescharacteristic
89C22×C127C2×SD1612yes—
90C22×C127C2×Q1612yes—
91C4×C127—42no—
92C4×C127—81no—
93C4×C127C4×D411yescharacteristic
94C4×C127C4×D411yescharacteristic
95C4×C127#12111yescharacteristic
96C4×C127#12111yescharacteristic
97Dic127—42no—
98Dic127—81no—
99Dic127#10318yescomplemented
100C2×C127#39011yescharacteristic
101C2×C127C2×#10312yes—
102C2×C127#48712yes—
103C2×C127#49211yescharacteristic
104C2×C127#49411yescharacteristic
105#1956—1271nomaximal, Sylow
106C127#195611yescharacteristic, complemented, Sylow
107C2×C4⋊Q8—1271no—
108C2×#103—1271no—
109C2×#103—1271no—
110C2×#103—1271no—
111C2×#103D12711yescharacteristic
112C2×C4×C8—1271no—
113#384—1271no—
114C22×Q8—1271no—
115C22×Q8C2×D12711yescharacteristic
116C22×C8—1271no—
117C22×C8C2×D12711yescharacteristic
118C4⋊Q8—1274no—
119C4⋊Q8—2542no—
120C2×C4⋊C4—1271no—
121C2×C4⋊C4—2542no—
122C2×C4⋊C4—2542no—
123C2×C4⋊C4C2×D12711yescharacteristic
124C2×C42—1271no—
125#103—22no—
126#103—2542no—
127#103—2542no—
128#103—2542no—
129C4×C8—2542no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—1272no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8C4×D12712yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—1272no—
145C22×C4—2541no—
146C22×C4—2541no—
147C22×C4C22×D12711yescharacteristic
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—1272no—
151C2×C8—2541no—
152C2×C8—2541no—
153C2×C8C22⋊D12712yes—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—5084no—
157C4⋊C4—2548no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C42—1272no—
161C42—1272no—
162C23D4×D12711yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—5081no—
168Q8—5081no—
169Q8—2542no—
170Q8—2542no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—1274no—
177C2×C4—1274no—
178C2×C4—1274no—
179C2×C4—5081no—
180C2×C4—5082no—
181C2×C4—5081no—
182C2×C4—5081no—
183C2×C4—2542no—
184C2×C4C2×C4×D12711yescharacteristic
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C2×C4#1452324711yescharacteristic
189C2×C4#1452324811yescharacteristic
190C8—22no—
191C8—2542no—
192C22C22⋊C4×D12711yescharacteristic
193C22#1686119511yescharacteristic
194C22#1686120511yescharacteristic
195C22SD16×D12712yes—
196C22Q16×D12712yes—
197C4—42no—
198C4—81no—
199C4—1278no—
200C4—10161no—
201C4—5082no—
202C4(C22⋊D127)⋊C411yescharacteristic
203C4C4×C22⋊D12711yescharacteristic
204C4#1686118111yescharacteristic
205C4#1686118211yescharacteristic
206C2#2254977611yescharacteristic
207C2#103×D12712yes—
208C2#2255022512yes—
209C2#2255024711yescharacteristic
210C2#2255025411yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC2×#103D127
hypocenterC127#1956
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
11. #16861207C4semidirect
99. Dic127#103semidirect
106. C127#1956semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
C2×#168611342
C2×#168611332
C2×#168611322
C2×C8×Dic1272
#225497452
C2×#103×C1272
#1956127