Finite Group Databasebeta

← #30639194

Subgroups — #30639194

Order 16256.

390 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14523232C211yesmaximal, characteristic, complemented
2C2×#16861126C212yesmaximal, complemented
3#22549840C211yesmaximal, characteristic, complemented
4#22549645C212yesmaximal, complemented
5C2×#121×C127C211yesmaximal, characteristic
6C2×#14523232—24no—
7C2×#14523232—21no—
8C2×#14523232C2212yes—
9C2×#14523232C2212yes—
10C22×C127⋊C8—21no—
11#16861126—24no—
12C22⋊(C127⋊C8)—21no—
13C22⋊(C127⋊C8)—21no—
14C22×D4×C127C412yes—
15C22×D4×C127C2211yescharacteristic
16C23×C4×C127C2211yescharacteristic
17#121×C127—21no—
18#121×C127—21no—
19#121×C127—21no—
20#121×C127—21no—
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C22⋊C4×C127C412yes—
23#14523232—24no—
24#14523232—22no—
25#14523232—22no—
26#14523232—22no—
27#14523232D414yescomplemented
28C2×C127⋊C8—24no—
29C2×C127⋊C8—22no—
30C2×C127⋊C8—22no—
31C24×C127C2×C411yescharacteristic
32C24×C127C2×C412yes—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127—21no—
36C2×D4×C127—21no—
37C2×D4×C127—22no—
38C2×D4×C127—22no—
39C2×D4×C127—44no—
40C22×C4×C127—22no—
41C22×C4×C127—22no—
42C22×C4×C127C2×C411yescharacteristic
43C22×C4×C127C2×C412yes—
44C22×C4×C127D411yescharacteristic
45C22×C4×C127D411yescharacteristic
46C22×C4×C127D411yescharacteristic
47C22×C4×C127D411yescharacteristic
48C22×C4×C127C2311yescharacteristic
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C22⋊C4×C127—22no—
52C22⋊C4×C127—22no—
53C127⋊C8—24no—
54C127⋊C8—42no—
55C23×C127—22no—
56C23×C127—22no—
57C23×C127—22no—
58C23×C127—24no—
59C23×C127—41no—
60C23×C127—42no—
61C23×C127—42no—
62C23×C127C22⋊C411yescharacteristic
63C23×C127C22⋊C411yescharacteristic
64C23×C127C22×C411yescharacteristic
65D4×C127—44no—
66D4×C127—44no—
67D4×C127—41no—
68D4×C127—42no—
69D4×C127—41no—
70C2×C4×C127—22no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—42no—
79C2×C4×C127C22⋊C411yescharacteristic
80C2×C4×C127C22⋊C411yescharacteristic
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4112yes—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—44no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—84no—
97C22×C127#10012yescomplemented
98C22×C127C2×C22⋊C411yescharacteristic
99C22×C127C4×D412yes—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—42no—
103C4×C127#12011yescharacteristic
104C4×C127#12112yes—
105C4×C127#12311yescharacteristic
106C2×C127—22no—
107C2×C127—22no—
108C2×C127—41no—
109C2×C127—82no—
110C2×C127#38611yescharacteristic
111C2×C127C2×#10011yescharacteristic
112C2×C127#41311yescharacteristic
113#1900—1271nomaximal, Sylow
114C127#190011yescharacteristic, complemented, Sylow
115C22×M16—1271no—
116C2×#121D12711yescharacteristic
117C2×#100—1272no—
118#407—1271no—
119#343—1272no—
120C22×D4Dic12712yes—
121C22×D4C2×D12711yescharacteristic
122C23×C4C2×D12711yescharacteristic
123C2×M16—1272no—
124C2×M16—1272no—
125C2×M16—2544no—
126C2×M16—2541no—
127C22×C8—2541no—
128#121—21no—
129#121—21no—
130#121—21no—
131#121—21no—
132C2×C4⋊C4C2×D12711yescharacteristic
133C2×C22⋊C4Dic12712yes—
134#100—2544no—
135C22⋊C8—2541no—
136C22⋊C8—2541no—
137C24C2×Dic12711yescharacteristic
138C24C2×Dic12712yes—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4—44no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4C2×Dic12711yescharacteristic
149C22×C4C2×Dic12712yes—
150C22×C4C22⋊D12711yescharacteristic
151C22×C4C22⋊D12711yescharacteristic
152C22×C4C22⋊D12711yescharacteristic
153C22×C4C22⋊D12711yescharacteristic
154C22×C4C22×D12711yescharacteristic
155M16—1274no—
156M16—2542no—
157M16—2542no—
158M16—2544no—
159M16—2542no—
160C2×C8—2542no—
161C2×C8—2544no—
162C2×C8—2542no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—24no—
171C23—41no—
172C23—42no—
173C23—42no—
174C23C22⋊Dic12711yescharacteristic
175C23C22⋊Dic12711yescharacteristic
176C23C22×Dic12711yescharacteristic
177D4—44no—
178D4—44no—
179D4—41no—
180D4—41no—
181D4—42no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4C22⋊Dic12711yescharacteristic
192C2×C4C22⋊Dic12711yescharacteristic
193C2×C4D4×D12712yes—
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4C2×C22⋊D12711yescharacteristic
196C2×C4#1452324712yes—
197C8—5082no—
198C8—2544no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—44no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—84no—
211C22#1686112612yescomplemented
212C22C2×C22⋊Dic12711yescharacteristic
213C22D4×Dic12712yes—
214C4—21no—
215C4—21no—
216C4—42no—
217C4#1686117711yescharacteristic
218C4#1686118112yes—
219C4#1686119011yescharacteristic
220C2—22no—
221C2—22no—
222C2—41no—
223C2—82no—
224C2#2254975611yescharacteristic
225C2C2×#1686112611yescharacteristic
226C2#2254986011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×#121×C127C2
socleC22×C127C2×C22⋊C4
cosocleC22×C4×C127C23
hypercenterC2×#121D127
hypocenterC127#1900
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14523232C2semidirect
2. C2×#16861126C2semidirect
3. #22549840C2semidirect
4. #22549645C2semidirect
27. #14523232D4semidirect
97. C22×C127#100semidirect
114. C127#1900semidirect
211. C22#16861126semidirect

Maximal subgroups

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

subgroupindex
C22×#145232322
C2×#168611262
#225498402
#225496452
C2×#121×C1272
#1900127