Finite Group Databasebeta

← #30642065

Subgroups — #30642065

Order 16256.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14523248C211yesmaximal, characteristic
2C2×#16861192C212yesmaximal, complemented
3#22549796C212yesmaximal, complemented
4#22549793C212yesmaximal, complemented
5#22549755C212yesmaximal, complemented
6#22549747C214yesmaximal, complemented
7#22549739C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—28no—
10C2×#14523248—24no—
11C22×Dic254C2211yescharacteristic
12C22×D508C2211yescharacteristic
13C22×C22⋊D127C2212yes—
14C22×C4×D127C2212yes—
15#16861192—28no—
16C2×Dic127⋊C4C2214yes—
17C2×C22⋊Dic127C2212yes—
18C2×(C2×D127)⋊C4C2214yes—
19C2×(C2×D127)⋊C4C2214yes—
20C127⋊#95C2212yes—
21C127⋊#95C2214yes—
22C127⋊#95C2212yes—
23C23×C4×C127C2211yescharacteristic
24C23×C4×C127C2211yescharacteristic
25C23×C4×C127C2211yescharacteristic
26C2×C4⋊C4×C127—24no—
27C2×C4⋊C4×C127C2212yes—
28C2×C4⋊C4×C127C2212yes—
29C23×D127C2312yes—
30#14523248—232no—
31C2×Dic254—24no—
32C2×Dic254—22no—
33C2×D508—24no—
34C2×D508—22no—
35C2×C22⋊D127—24no—
36C2×C22⋊D127—24no—
37C2×C22⋊D127D418yes—
38C22×Dic127C2314yes—
39C22×Dic127C2312yes—
40C2×C4×D127—24no—
41C2×C4×D127—24no—
42C2×C4×D127—24no—
43Dic127⋊C4—28no—
44C22⋊Dic127—24no—
45(C2×D127)⋊C4—28no—
46(C2×D127)⋊C4—28no—
47C24×C127C2311yescharacteristic
48C22×C4×C127—24no—
49C22×C4×C127—22no—
50C22×C4×C127—24no—
51C22×C4×C127—24no—
52C22×C4×C127—22no—
53C22×C4×C127D414yes—
54C22×C4×C127C2312yes—
55C22×C4×C127C2311yescharacteristic
56C22×C4×C127C2311yescharacteristic
57C22×C4×C127C2312yes—
58C4⋊C4×C127—24no—
59C4⋊C4×C127—24no—
60C22×D127—22no—
61C22×D127—22no—
62C22×D127—24no—
63C22×D127—24no—
64C22×D127C2×D414yes—
65Dic254—28no—
66D508—28no—
67C22⋊D127—216no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—24no—
71C2×Dic127—24no—
72C2×Dic127—24no—
73C2×Dic127C2×D414yes—
74C4×D127—216no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127C2×D414yes—
78C23×C127C2×D412yes—
79C23×C127C2411yescharacteristic
80C2×C4×C127—28no—
81C2×C4×C127—28no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—24no—
88C2×C4×C127—44no—
89C2×C4×C127C2×D412yes—
90C2×C4×C127C2×D412yes—
91C2×C4×C127#4114yescomplemented
92C22×C127—22no—
93C22×C127—22no—
94C22×C127—28no—
95C22×C127#12014yes—
96C22×C127C22×D412yes—
97C22×C127C22×D411yescharacteristic
98C22×C127C2×#4112yes—
99C22×C1272−1+412yes—
100C2×D127—24no—
101C2×D127—28no—
102C2×D127—42no—
103C2×D127—42no—
104C2×D127—44no—
105C4×C127—22no—
106C4×C127—22no—
107C4×C127—24no—
108C4×C127—42no—
109Dic127—24no—
110Dic127—44no—
111C2×C127—24no—
112C2×C127C2×#12011yescharacteristic
113C2×C127#52511yescharacteristic
114C2×C127#53611yescharacteristic
115C2×C127#54714yes—
116D127—44no—
117#2406—1271nomaximal, Sylow
118C127#240611yescharacteristic, complemented, Sylow
119C22×#41—1271no—
120C2×#123—1272no—
121C22×C4⋊C4D12711yescharacteristic, complemented
122#395—1272no—
123#394—1272no—
124#386—1272no—
125#385—1274no—
126#381—1271no—
127C2×#41—2548no—
128C2×#41—2544no—
129C22×Q8—1271no—
130C22×D4—1272no—
131C22×D4—1271no—
132C23×C4—1272no—
133C23×C4C2×D12711yescharacteristic
134C23×C4C2×D12711yescharacteristic
135C23×C4C2×D12711yescharacteristic
136#123—2548no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—1274no—
139C2×C4⋊C4C2×D12712yes—
140C2×C4⋊C4C2×D12712yes—
141C2×C22⋊C4—1272no—
142C2×C22⋊C4—1274no—
143C2×C22⋊C4—1274no—
144#95—1272no—
145#95—1272no—
146#95—1274no—
147C24—1272no—
148C24C22×D12711yescharacteristic
149#41—25432no—
150C2×Q8—2544no—
151C2×Q8—2542no—
152C2×D4—1278no—
153C2×D4—2544no—
154C2×D4—2544no—
155C2×D4—2544no—
156C2×D4—2542no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—1272no—
163C22×C4—1274no—
164C22×C4—2544no—
165C22×C4—2544no—
166C22×C4—2544no—
167C22×C4C22⋊D12714yes—
168C22×C4C22×D12712yes—
169C22×C4C22×D12711yescharacteristic
170C22×C4C22×D12711yescharacteristic
171C22×C4C22×D12712yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—2548no—
175C22⋊C4—2548no—
176C22⋊C4—2544no—
177C22⋊C4—2548no—
178C23—22no—
179C23—22no—
180C23—1274no—
181C23—2542no—
182C23—2542no—
183C23—2544no—
184C23—2544no—
185C23D4×D12714yes—
186C23C2×C22⋊D12712yes—
187C23C23×D12711yescharacteristic
188Q8—2548no—
189D4—2548no—
190D4—25416no—
191C2×C4—28no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—28no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—1274no—
201C2×C4—25416no—
202C2×C4—2544no—
203C2×C4—2544no—
204C2×C4—2544no—
205C2×C4—2542no—
206C2×C4—2542no—
207C2×C4C2×C22⋊D12712yes—
208C2×C4C2×C22⋊D12712yes—
209C2×C4#1452324814yescomplemented
210C22—22no—
211C22—22no—
212C22—28no—
213C22—5082no—
214C22—5082no—
215C22—5084no—
216C22—2544no—
217C22—2548no—
218C22#1686117714yes—
219C22C2×D4×D12712yes—
220C22C22×C22⋊D12711yescharacteristic
221C22C2×#1452324812yes—
222C22#1686125712yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—5084no—
228C4—2544no—
229C2—24no—
230C2—5084no—
231C2C2×#1686117711yescharacteristic
232C2#2255038711yescharacteristic
233C2#2255045011yescharacteristic
234C2#2255052714yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC22×C4⋊C4×C127C2
prime radicalC22×#14523248C2
hypercenterC22×C4⋊C4D127
hypocenterC127#2406
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16861192C2semidirect
3. #22549796C2semidirect
4. #22549793C2semidirect
5. #22549755C2semidirect
6. #22549747C2semidirect
7. #22549739C2semidirect
8. C22×C4⋊C4×C127C2semidirect
91. C2×C4×C127#41semidirect
118. C127#2406semidirect
121. C22×C4⋊C4D127semidirect
209. C2×C4#14523248semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168611922
#225497962
#225497932
#225497552
#225497472
#225497392
C22×C4⋊C4×C1272
#2406127