Finite Group Databasebeta

← #30641299

Subgroups — #30641299

Order 16256.

588 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q8×Dic127C211yesmaximal, characteristic
3C2×#16861174C211yesmaximal, characteristic
4#22549782C211yesmaximal, characteristic
5#22549773C214yesmaximal
6#22549745C211yesmaximal, characteristic
7#22549727C214yesmaximal
8C42×Dic127C211yesmaximal, characteristic
9#391×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861207C418yescomplemented
12Q8×Dic127—24no—
13#16861174—24no—
14C2×C4⋊Dic127C2212yes—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4C2212yes—
17C2×Dic127⋊C4C2214yes—
18C2×Dic127⋊C4C2214yes—
19C2×C4×Dic127C2212yes—
20C2×C4×Dic127C2212yes—
21C2×C4×Dic127C2212yes—
22C2×C4×Dic127C2211yescharacteristic
23C2×C4×Dic127C2211yescharacteristic
24C2×C4×Dic127C2212yes—
25C127⋊#95C2214yes—
26C22×Q8×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28C2×C42×C127C2211yescharacteristic
29#95×C127C2214yes—
30C2×Dic254—22no—
31C2×Dic254—22no—
32C2×Dic254C2×C414yes—
33C22×Dic127C2312yes—
34C22×Dic127C2312yes—
35C22×Dic127C2311yescharacteristic
36C22×Dic127C2312yes—
37C22×Dic127C2311yescharacteristic
38C4⋊Dic127—24no—
39C4⋊Dic127—22no—
40Dic127⋊C4—24no—
41Dic127⋊C4—28no—
42Dic127⋊C4C2×C4116yes—
43C4×Dic127—24no—
44C4×Dic127—24no—
45C4×Dic127—24no—
46C4×Dic127C2×C414yes—
47C4×Dic127D414yes—
48C4×Dic127Q818yescomplemented
49C2×Q8×C127—22no—
50C2×Q8×C127—22no—
51C2×Q8×C127C2×C414yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2312yes—
54C22×C4×C127C2312yes—
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127—22no—
57C42×C127—22no—
58Dic254—28no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—24no—
64C2×Dic127C22×C414yes—
65C2×Dic127C22×C414yes—
66C2×Dic127C2×D412yes—
67C2×Dic127C2×D412yes—
68C2×Dic127C2×Q814yes—
69C2×Dic127C2×Q814yes—
70C2×Dic127#4114yes—
71C23×C127C2411yescharacteristic
72Q8×C127—28no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—24no—
79C2×C4×C127C22×C414yes—
80C2×C4×C127C22×C412yes—
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×Q814yes—
83C2×C4×C127#4112yes—
84C22×C127C23×C411yescharacteristic
85C22×C127C22×D411yescharacteristic
86C22×C127C22×Q812yes—
87C22×C127C2×#4112yes—
88C22×C127C2×#4111yescharacteristic
89C4×C127—24no—
90C4×C127—24no—
91C4×C127—42no—
92C4×C127C4×D414yes—
93Dic127—24no—
94Dic127—22no—
95Dic127—42no—
96Dic127C4×Q818yescomplemented
97Dic127C4⋊Q814yescomplemented
98C2×C127C2×C4×D411yescharacteristic
99C2×C127C2×C4×Q812yes—
100C2×C127C2×C4⋊Q811yescharacteristic
101C2×C127#53211yescharacteristic
102C2×C127#53312yes—
103C4×C4⋊Q8—1271nomaximal, Sylow
104C127C4×C4⋊Q811yescharacteristic, complemented, Sylow
105C2×C4⋊Q8—1271no—
106C2×C4×Q8—1271no—
107C2×C4×Q8—1271no—
108#391—1271no—
109#391D12711yescharacteristic
110#389—1274no—
111#384—1271no—
112C4×C4⋊C4—1274no—
113C43—1271no—
114C22×Q8—1271no—
115C22×Q8C2×D12711yescharacteristic
116C4⋊Q8—1278no—
117C4×Q8—2544no—
118C4×Q8—2544no—
119C2×C4⋊C4—1274no—
120C2×C4⋊C4—1271no—
121C2×C4⋊C4—1272no—
122C2×C4⋊C4—1272no—
123C2×C4⋊C4—1274no—
124C2×C4⋊C4C2×D12711yescharacteristic
125C2×C42—1271no—
126C2×C42—1271no—
127C2×C42—1272no—
128C2×C42—1272no—
129C2×C42—1272no—
130C2×C42—1272no—
131C2×C42C2×D12711yescharacteristic
132#95—1274no—
133#95C2×D12714yes—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—1274no—
137C2×Q8—2542no—
138C2×Q8—2542no—
139C2×Q8C4×D12714yes—
140C22×C4—1271no—
141C22×C4—1272no—
142C22×C4—1272no—
143C22×C4—1271no—
144C22×C4—1272no—
145C22×C4C22×D12712yes—
146C22×C4C22×D12712yes—
147C22×C4C22×D12712yes—
148C22×C4C22×D12711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—12716no—
151C4⋊C4—2548no—
152C4⋊C4—2544no—
153C4⋊C4—2544no—
154C4⋊C4—2542no—
155C42—22no—
156C42—1274no—
157C42—1274no—
158C42—1278no—
159C42—2544no—
160C42—2544no—
161C42—2544no—
162C23C23×D12711yescharacteristic
163Q8—28no—
164Q8—2548no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—1274no—
172C2×C4—1272no—
173C2×C4—1274no—
174C2×C4—1274no—
175C2×C4—1274no—
176C2×C4—1274no—
177C2×C4—1272no—
178C2×C4—2542no—
179C2×C4—2544no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4C2×C4×D12714yes—
184C2×C4C2×C4×D12712yes—
185C2×C4D4×D12712yes—
186C2×C4Q8×D12714yes—
187C2×C4#1452324712yes—
188C22C22×C4×D12711yescharacteristic
189C22C2×D4×D12711yescharacteristic
190C22C2×Q8×D12712yes—
191C22#41×D12712yes—
192C22C2×#1452324711yescharacteristic
193C4—24no—
194C4—24no—
195C4—42no—
196C4—1278no—
197C4—1274no—
198C4—5082no—
199C4—2544no—
200C4—2542no—
201C4(C22⋊D127)⋊C414yes—
202C2C2×(C22⋊D127)⋊C411yescharacteristic
203C2C4×Q8×D12712yes—
204C2C4⋊Q8×D12711yescharacteristic
205C2#2255042511yescharacteristic
206C2#2255042812yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#391×C127C2
socle, prime radical, cosocleC23×C127C24
hypercenter#391D127
hypocenterC127C4×C4⋊Q8
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #16861207C4semidirect
48. C4×Dic127Q8semidirect
96. Dic127C4×Q8semidirect
97. Dic127C4⋊Q8semidirect
104. C127C4×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
C2×Q8×Dic1272
C2×#168611742
#225497822
#225497732
#225497452
#225497272
C42×Dic1272
#391×C1272
C4×C4⋊Q8127