Finite Group Databasebeta

← #30647105

Subgroups — #30647105

Order 16256.

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

normal subgroups only

This group has 504 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
1C2×#16861208C211yesmaximal, characteristic
2C2×#16861207C211yesmaximal, characteristic
3#22549783C211yesmaximal, characteristic
4#22549782C211yesmaximal, characteristic
5#22549781C211yesmaximal, characteristic
6#22549780C214yesmaximal
7#22549740C214yesmaximal
8C42×Dic127C211yesmaximal, characteristic
9#391×C127C211yesmaximal, characteristic
10C22×Dic254C2212yes—
11C22×Dic254C2211yescharacteristic
12#16861208—24no—
13#16861207—24no—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4C2212yes—
16C2×Dic127⋊C4C2212yes—
17C2×Dic127⋊C4C2212yes—
18C2×C4×Dic127C2212yes—
19C2×C4×Dic127C2211yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21C2×C4×Dic127C2212yes—
22C127⋊#95C2214yes—
23C127⋊#95C2214yes—
24C127⋊#95C2214yes—
25C22×Q8×C127C2211yescharacteristic
26C2×C4⋊C4×C127C2211yescharacteristic
27C2×C42×C127C2211yescharacteristic
28#95×C127C2214yes—
29C2×Dic254—24no—
30C2×Dic254—28no—
31C2×Dic254—22no—
32C2×Dic254—22no—
33C2×Dic254—22no—
34C22×Dic127C2312yes—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C22×Dic127C2311yescharacteristic
38C22×Dic127C2311yescharacteristic
39C4⋊Dic127—22no—
40Dic127⋊C4—24no—
41Dic127⋊C4—24no—
42Dic127⋊C4—24no—
43C4×Dic127—24no—
44C4×Dic127—22no—
45C4×Dic127D414yes—
46C4×Dic127Q818yescomplemented
47C2×Q8×C127—22no—
48C2×Q8×C127—22no—
49C2×Q8×C127—22no—
50C22×C4×C127C2312yes—
51C22×C4×C127C2312yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—22no—
55C42×C127—22no—
56Dic254—44no—
57Dic254—48no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—24no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127C2×D412yes—
66C2×Dic127C2×D412yes—
67C2×Dic127C2×Q814yes—
68C2×Dic127C2×Q814yes—
69C2×Dic127#4114yes—
70C23×C127C2411yescharacteristic
71Q8×C127—44no—
72C2×C4×C127—22no—
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×C127C2×D412yes—
80C2×C4×C127C2×Q814yes—
81C2×C4×C127#4112yes—
82C2×C4×C127#4112yes—
83C22×C127C22×D411yescharacteristic
84C22×C127C22×Q812yes—
85C22×C127C2×#4112yes—
86C22×C127C2×#4111yescharacteristic
87C22×C127C2×#4111yescharacteristic
88C4×C127—24no—
89C4×C127—42no—
90C4×C127—42no—
91C4×C127#12414yes—
92Dic127—24no—
93Dic127—42no—
94Dic127—42no—
95Dic127#12414yes—
96Dic127C4⋊Q814yes—
97C2×C127C2×#12411yescharacteristic
98C2×C127C2×#12411yescharacteristic
99C2×C127C2×C4⋊Q811yescharacteristic
100C2×C127#53312yes—
101C2×C127#53312yes—
102#2862—1271nomaximal, Sylow
103C127#286211yescharacteristic, complemented, Sylow
104C2×C4⋊Q8—1271no—
105C2×C4⋊Q8—1271no—
106#391—1271no—
107#391—1271no—
108#391—1271no—
109#391—1274no—
110#391D12711yescharacteristic
111#382—1274no—
112C43—1271no—
113C22×Q8—1272no—
114C22×Q8—1271no—
115C22×Q8C2×D12711yescharacteristic
116C4⋊Q8—2544no—
117C4⋊Q8—2544no—
118C2×C4⋊C4—1272no—
119C2×C4⋊C4—1272no—
120C2×C4⋊C4—1272no—
121C2×C4⋊C4—1271no—
122C2×C4⋊C4C2×D12711yescharacteristic
123C2×C42—1271no—
124C2×C42—1272no—
125C2×C42—1272no—
126C2×C42—1271no—
127C2×C42C2×D12711yescharacteristic
128#95—1274no—
129#95—1274no—
130#95—1274no—
131#95C2×D12714yes—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—2544no—
136C2×Q8—2548no—
137C2×Q8—2542no—
138C2×Q8—2542no—
139C2×Q8—2542no—
140C22×C4—1271no—
141C22×C4—1271no—
142C22×C4—1272no—
143C22×C4—1272no—
144C22×C4—1272no—
145C22×C4C22×D12712yes—
146C22×C4C22×D12712yes—
147C22×C4C22×D12712yes—
148C22×C4C22×D12711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—2544no—
151C4⋊C4—2544no—
152C4⋊C4—2544no—
153C4⋊C4—2542no—
154C42—22no—
155C42—1274no—
156C42—1278no—
157C42—2544no—
158C42—2542no—
159C23C23×D12711yescharacteristic
160Q8—44no—
161Q8—5088no—
162Q8—5084no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—1272no—
171C2×C4—1272no—
172C2×C4—1274no—
173C2×C4—1274no—
174C2×C4—1274no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4—2544no—
178C2×C4—2542no—
179C2×C4—2542no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4D4×D12712yes—
183C2×C4Q8×D12714yes—
184C2×C4#1452324712yes—
185C2×C4#1452324812yes—
186C22C2×D4×D12711yescharacteristic
187C22C2×Q8×D12712yes—
188C22#41×D12712yes—
189C22C2×#1452324711yescharacteristic
190C22C2×#1452324811yescharacteristic
191C4—24no—
192C4—42no—
193C4—42no—
194C4—1274no—
195C4—1274no—
196C4—5082no—
197C4—5082no—
198C4—2544no—
199C4#1686119514yes—
200C2#124×D12711yescharacteristic
201C2C2×#1686119511yescharacteristic
202C2C4⋊Q8×D12711yescharacteristic
203C2#2255042812yes—
204C2#2255043012yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, prime radical, cosocleC23×C127C24
Fitting#391×C127C2
hypercenter#391D127
hypocenterC127#2862
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
46. C4×Dic127Q8semidirect
103. C127#2862semidirect

Maximal subgroups

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

subgroupindex
C2×#168612082
C2×#168612072
#225497832
#225497822
#225497812
#225497802
#225497402
C42×Dic1272
#391×C1272
#2862127