Finite Group Databasebeta

← #30641818

Subgroups — #30641818

Order 16256.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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
2#22549783C211yesmaximal, characteristic
3#22549780C212yesmaximal
4#22549773C212yesmaximal
5#22549771C212yesmaximal
6C4×Dic127⋊C4C212yesmaximal
7#22549727C212yesmaximal
8#22549727C212yesmaximal
9#391×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861207C418yescomplemented
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4C2212yes—
14C2×Dic127⋊C4C2212yes—
15C2×Dic127⋊C4C2212yes—
16C2×Dic127⋊C4C2212yes—
17C2×Dic127⋊C4C2212yes—
18C2×C4×Dic127C2212yes—
19C2×C4×Dic127C2212yes—
20C2×C4×Dic127C2212yes—
21C2×C4×Dic127C2211yescharacteristic
22C2×C4×Dic127C2211yescharacteristic
23C127⋊#95C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2212yes—
26C127⋊#95C2212yes—
27C22×Q8×C127C2211yescharacteristic
28C2×C4⋊C4×C127C2211yescharacteristic
29C2×C42×C127C2211yescharacteristic
30#95×C127C2212yes—
31#95×C127C2212yes—
32C2×Dic254—41no—
33C2×Dic254—41no—
34C2×Dic254C2×C414yes—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C4⋊Dic127—41no—
40Dic127⋊C4—24no—
41Dic127⋊C4—42no—
42Dic127⋊C4C2×C418yes—
43Dic127⋊C4C2×C418yes—
44Dic127⋊C4Q818yescomplemented
45C4×Dic127—24no—
46C4×Dic127—24no—
47C4×Dic127—42no—
48C4×Dic127—41no—
49C4×Dic127C2×C414yes—
50C2×Q8×C127—41no—
51C2×Q8×C127—41no—
52C2×Q8×C127C2×C414yes—
53C22×C4×C127C2312yes—
54C22×C4×C127C2311yescharacteristic
55C22×C4×C127C2311yescharacteristic
56C22×C4×C127C2311yescharacteristic
57C22×C4×C127C2312yes—
58C4⋊C4×C127—41no—
59C42×C127—41no—
60Dic254—44no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127C22×C414yes—
69C2×Dic127C22×C414yes—
70C2×Dic127C2×Q814yes—
71C2×Dic127C2×Q814yes—
72C2×Dic127#4114yes—
73C23×C127C2411yescharacteristic
74Q8×C127—44no—
75C2×C4×C127—22no—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127C22×C412yes—
87C2×C4×C127C22×C412yes—
88C2×C4×C127C22×C412yes—
89C2×C4×C127C2×Q814yes—
90C22×C127C23×C411yescharacteristic
91C22×C127C22×Q812yes—
92C22×C127C2×#4112yes—
93C22×C127H(5,𝔽2)11yescharacteristic
94C22×C127H(5,𝔽2)11yescharacteristic
95C4×C127—22no—
96C4×C127—22no—
97C4×C127—22no—
98C4×C127—24no—
99C4×C127—42no—
100Dic127—24no—
101Dic127—24no—
102Dic127—42no—
103Dic127C4×Q818yes—
104C2×C127C2×C4×Q812yes—
105C2×C127#51811yescharacteristic
106C2×C127#55412yes—
107C2×C127#55511yescharacteristic
108C2×C127Q8211yescharacteristic
109#2369—1271nomaximal, Sylow
110C127#236911yescharacteristic, complemented, Sylow
111C2×C4⋊Q8—1271no—
112#391—1271no—
113#391—1272no—
114#391D12711yescharacteristic
115#389—1272no—
116#389—1272no—
117C4×C4⋊C4—1272no—
118C4×C4⋊C4—1272no—
119C4×C4⋊C4—1272no—
120C22×Q8—1271no—
121C22×Q8C2×D12711yescharacteristic
122C4⋊Q8—1278no—
123C2×C4⋊C4—1272no—
124C2×C4⋊C4—1272no—
125C2×C4⋊C4—1272no—
126C2×C4⋊C4—1272no—
127C2×C4⋊C4—1271no—
128C2×C4⋊C4—1272no—
129C2×C4⋊C4C2×D12711yescharacteristic
130C2×C42—1271no—
131C2×C42—1271no—
132C2×C42—1272no—
133C2×C42—1272no—
134C2×C42—1272no—
135C2×C42C2×D12711yescharacteristic
136#95—1272no—
137#95—1272no—
138#95—1272no—
139#95—1272no—
140#95C2×D12712yes—
141#95C2×D12712yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—1274no—
145C2×Q8—5081no—
146C2×Q8—5081no—
147C2×Q8C4×D12714yes—
148C22×C4—1272no—
149C22×C4—1272no—
150C22×C4—1272no—
151C22×C4—1272no—
152C22×C4C22×D12712yes—
153C22×C4C22×D12711yescharacteristic
154C22×C4C22×D12711yescharacteristic
155C22×C4C22×D12711yescharacteristic
156C22×C4C22×D12712yes—
157C4⋊C4—41no—
158C4⋊C4—1278no—
159C4⋊C4—1278no—
160C4⋊C4—1278no—
161C4⋊C4—5082no—
162C4⋊C4—5081no—
163C4⋊C4—2544no—
164C42—41no—
165C42—1274no—
166C42—5082no—
167C42—5081no—
168C42—2544no—
169C42—2544no—
170C23C23×D12711yescharacteristic
171Q8—44no—
172Q8—5084no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—1274no—
185C2×C4—1274no—
186C2×C4—1274no—
187C2×C4—1274no—
188C2×C4—1274no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4C2×C4×D12712yes—
197C2×C4C2×C4×D12712yes—
198C2×C4C2×C4×D12712yes—
199C2×C4Q8×D12714yes—
200C22C22×C4×D12711yescharacteristic
201C22C2×Q8×D12712yes—
202C22#41×D12712yes—
203C22#1686125411yescharacteristic
204C22#1686125411yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—1278no—
211C4—5082no—
212C4—2544no—
213C4—2544no—
214C2C4×Q8×D12712yes—
215C2#2255034911yescharacteristic
216C2#2255058012yes—
217C2#2255058811yescharacteristic
218C2#2255060611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#391×C127C2
socle, prime radical, cosocleC23×C127C24
hypercenter#391D127
hypocenterC127#2369
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #16861207C4semidirect
44. Dic127⋊C4Q8semidirect
110. C127#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
#225497832
#225497802
#225497732
#225497712
C4×Dic127⋊C42
#225497272
#391×C1272
#2369127