Finite Group Databasebeta

← #30647064

Subgroups — #30647064

Order 16256.

564 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 177 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×C4×Dic127C211yesmaximal, characteristic
2#22549783C212yesmaximal, complemented
3#22549772C212yesmaximal, complemented
4#22549757C214yesmaximal, complemented
5#22549748C212yesmaximal
6#22549737C212yesmaximal
7#22549736C211yesmaximal, characteristic
8C2×C22⋊Q8×C127C211yesmaximal, characteristic
9C23×Dic127C2211yescharacteristic
10C23×Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2212yes—
12C2×Dic127⋊C4C2214yes—
13C2×C22⋊Dic127C2212yes—
14C2×C4×Dic127—24no—
15C2×C4×Dic127C2212yes—
16C2×C4×Dic127C2212yes—
17C127⋊#95C2214yes—
18C127⋊#95C2214yes—
19C127⋊#95C2214yes—
20C127⋊#95C2212yes—
21C22×Q8×C127C2211yescharacteristic
22C23×C4×C127C2211yescharacteristic
23C22⋊Q8×C127—24no—
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2212yes—
27C22×Dic127—24no—
28C22×Dic127—22no—
29C22×Dic127—22no—
30C22×Dic127—22no—
31C22×Dic127Q814yescomplemented
32C22×Dic127C2314yes—
33C22×Dic127C2312yes—
34C22×Dic127C2312yes—
35C4⋊Dic127—24no—
36Dic127⋊C4—28no—
37C22⋊Dic127—24no—
38C4×Dic127—24no—
39C4×Dic127—24no—
40C24×C127C2311yescharacteristic
41C2×Q8×C127—22no—
42C2×Q8×C127—24no—
43C22×C4×C127—24no—
44C22×C4×C127D414yes—
45C22×C4×C127C2312yes—
46C22×C4×C127C2312yes—
47C22×C4×C127C2311yescharacteristic
48C22×C4×C127C2311yescharacteristic
49C4⋊C4×C127—22no—
50C4⋊C4×C127—24no—
51C22⋊C4×C127—24no—
52C2×Dic127—28no—
53C2×Dic127—28no—
54C2×Dic127—22no—
55C2×Dic127—22no—
56C2×Dic127—22no—
57C2×Dic127—22no—
58C2×Dic127—24no—
59C2×Dic127—24no—
60C2×Dic127—24no—
61C2×Dic127C2×Q814yes—
62C2×Dic127#4114yes—
63C23×C127—22no—
64C23×C127—22no—
65C23×C127C2×D412yes—
66C23×C127C2×Q812yes—
67C23×C127#4112yes—
68C23×C127C2411yescharacteristic
69Q8×C127—44no—
70C2×C4×C127—22no—
71C2×C4×C127—28no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—24no—
75C2×C4×C127—24no—
76C2×C4×C127C2×D412yes—
77C2×C4×C127C2×D412yes—
78C2×C4×C127#4114yes—
79C22×C127—22no—
80C22×C127—22no—
81C22×C127—24no—
82C22×C127—24no—
83C22×C127C22⋊Q814yes—
84C22×C127C22×D411yescharacteristic
85C22×C127C22×Q811yescharacteristic
86C22×C127C2×#4112yes—
87C22×C127C2×#4112yes—
88C22×C127C2×#4111yescharacteristic
89C4×C127—22no—
90C4×C127—42no—
91C4×C127—42no—
92C4×C127#12314yes—
93Dic127—24no—
94Dic127—24no—
95Dic127—44no—
96C2×C127—24no—
97C2×C127C2×C22⋊Q811yescharacteristic
98C2×C127C2×#12311yescharacteristic
99C2×C127#52511yescharacteristic
100C2×C127#52912yes—
101C2×C127#53312yes—
102#2855—1271nomaximal, Sylow
103C127#285511yescharacteristic, complemented, Sylow
104C2×C22⋊Q8D12711yescharacteristic
105C22×C42—1271no—
106#391—1272no—
107#389—1272no—
108#387—1274no—
109#385—1272no—
110#381—1272no—
111#380—1271no—
112C22×Q8C2×D12711yescharacteristic
113C23×C4—1271no—
114C23×C4—1271no—
115C23×C4C2×D12711yescharacteristic
116C22⋊Q8—24no—
117C2×C4⋊C4—1272no—
118C2×C4⋊C4—1274no—
119C2×C4⋊C4C2×D12712yes—
120C2×C4⋊C4C2×D12711yescharacteristic
121C2×C22⋊C4—1272no—
122C2×C22⋊C4C2×D12712yes—
123C2×C42—1272no—
124C2×C42—1272no—
125C2×C42—2544no—
126#95—1272no—
127#95—1274no—
128#95—1274no—
129#95—1274no—
130C24C22×D12711yescharacteristic
131C2×Q8—22no—
132C2×Q8—24no—
133C22×C4—24no—
134C22×C4—1274no—
135C22×C4—1272no—
136C22×C4—1272no—
137C22×C4—1274no—
138C22×C4—2544no—
139C22×C4—2542no—
140C22×C4—2542no—
141C22×C4—2542no—
142C22×C4C22⋊D12714yes—
143C22×C4C22×D12712yes—
144C22×C4C22×D12712yes—
145C22×C4C22×D12711yescharacteristic
146C22×C4C22×D12711yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—2548no—
150C4⋊C4—2544no—
151C22⋊C4—24no—
152C22⋊C4—2544no—
153C42—2544no—
154C42—2544no—
155C23—22no—
156C23—22no—
157C23C2×C22⋊D12712yes—
158C23Q8×D12712yes—
159C23#1452324912yes—
160C23C23×D12711yescharacteristic
161Q8—44no—
162C2×C4—22no—
163C2×C4—28no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—1274no—
169C2×C4—1274no—
170C2×C4—2544no—
171C2×C4—2544no—
172C2×C4—2544no—
173C2×C4—2548no—
174C2×C4—2548no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4—2542no—
178C2×C4—2542no—
179C2×C4C2×C22⋊D12712yes—
180C2×C4C2×C22⋊D12712yes—
181C2×C4#1452324714yes—
182C22—22no—
183C22—22no—
184C22—24no—
185C22—24no—
186C22#1686118514yes—
187C22C22×C22⋊D12711yescharacteristic
188C22C2×Q8×D12711yescharacteristic
189C22#41×D12712yes—
190C22C2×#1452324712yes—
191C22C2×#1452324911yescharacteristic
192C4—22no—
193C4—42no—
194C4—42no—
195C4—5084no—
196C4—2544no—
197C4—2544no—
198C4#1686119014yes—
199C2—24no—
200C2C2×#1686118511yescharacteristic
201C2C2×#1686119011yescharacteristic
202C2#2255038311yescharacteristic
203C2#2255041212yes—
204C2#2255042812yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×C22⋊Q8×C127C2
prime radicalC24×C127C23
hypercenterC2×C22⋊Q8D127
hypocenterC127#2855
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22549783C2semidirect
3. #22549772C2semidirect
4. #22549757C2semidirect
31. C22×Dic127Q8semidirect
103. C127#2855semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1272
#225497832
#225497722
#225497572
#225497482
#225497372
#225497362
C2×C22⋊Q8×C1272
#2855127