Finite Group Databasebeta

← #30647079

Subgroups — #30647079

Order 16256.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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, complemented
2C2×#16861189C211yesmaximal, characteristic
3C2×#16861185C211yesmaximal, characteristic
4C2×C42×D127C211yesmaximal, characteristic
5#22549782C211yesmaximal, characteristic, complemented
6#22549773C214yesmaximal, complemented
7#22549745C211yesmaximal, characteristic, complemented
8#22549735C214yesmaximal
9#391×C127C211yesmaximal, characteristic, complemented
10C22×Dic254C2211yescharacteristic
11C22×C4×D127C2211yescharacteristic
12C22×C4×D127C2212yes—
13#16861207—24no—
14#16861189—24no—
15#16861185—24no—
16C2×C4⋊Dic127C2212yes—
17C2×C4⋊Dic127C2211yescharacteristic
18C2×Dic127⋊C4C2212yes—
19C2×Dic127⋊C4C2214yes—
20C2×Dic127⋊C4C2214yes—
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2212yes—
23C2×C4×Dic127C2212yes—
24C2×C4×Dic127C2211yescharacteristic
25C42×D127—24no—
26C127⋊#95C2214yes—
27C22×Q8×C127C2211yescharacteristic
28C2×C4⋊C4×C127C2211yescharacteristic
29C2×C42×C127C2211yescharacteristic
30#95×C127C2214yes—
31C23×D127C2311yescharacteristic
32C2×Dic254—22no—
33C2×Dic254—22no—
34C2×Dic254—22no—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C22×Dic127C2311yescharacteristic
38C22×Dic127C2312yes—
39C2×C4×D127—24no—
40C2×C4×D127—24no—
41C2×C4×D127—22no—
42C2×C4×D127—22no—
43C2×C4×D127D414yes—
44C2×C4×D127Q818yescomplemented
45C4⋊Dic127—24no—
46C4⋊Dic127—22no—
47Dic127⋊C4—24no—
48Dic127⋊C4—28no—
49Dic127⋊C4—28no—
50(C2×D127)⋊C4—24no—
51(C2×D127)⋊C4—24no—
52C4×Dic127—24no—
53C4×Dic127D414yes—
54C2×Q8×C127—22no—
55C2×Q8×C127—22no—
56C2×Q8×C127—22no—
57C22×C4×C127C2312yes—
58C22×C4×C127C2312yes—
59C22×C4×C127C2312yes—
60C22×C4×C127C2311yescharacteristic
61C4⋊C4×C127—22no—
62C42×C127—22no—
63C22×D127—21no—
64C22×D127—22no—
65C22×D127—21no—
66C22×D127C2×D412yes—
67C22×D127C2×Q814yes—
68Dic254—44no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—24no—
75C2×Dic127—22no—
76C2×Dic127C2×D412yes—
77C2×Dic127C2×D414yes—
78C2×Dic127C2×Q814yes—
79C2×Dic127#4114yes—
80C4×D127—28no—
81C4×D127—216no—
82C23×C127C2411yescharacteristic
83Q8×C127—44no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—24no—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×Q814yes—
94C2×C4×C127#4112yes—
95C22×C127C22×D411yescharacteristic
96C22×C127C22×D411yescharacteristic
97C22×C127C22×Q812yes—
98C22×C127C2×#4112yes—
99C22×C127C2×#4111yescharacteristic
100C2×D127—22no—
101C2×D127—24no—
102C2×D127—22no—
103C2×D127—24no—
104C2×D127C4⋊Q814yescomplemented
105C4×C127—24no—
106C4×C127—42no—
107C4×C127—42no—
108C4×C127#12114yes—
109Dic127—22no—
110Dic127—42no—
111Dic127—42no—
112Dic127C22⋊Q818yescomplemented
113C2×C127C2×#12111yescharacteristic
114C2×C127C2×C22⋊Q812yes—
115C2×C127C2×C4⋊Q811yescharacteristic
116C2×C127#53211yescharacteristic
117C2×C127#53312yes—
118D127—24no—
119#2857—1271nomaximal, Sylow
120C127#285711yescharacteristic, complemented, Sylow
121C2×C4⋊Q8—1271no—
122C2×C22⋊Q8—1271no—
123C2×C22⋊Q8—1271no—
124C22×C42—1271no—
125#391—1271no—
126#391D12711yescharacteristic, complemented
127#389—1274no—
128#384—1271no—
129#380—1274no—
130C22×Q8—1271no—
131C22×Q8C2×D12711yescharacteristic
132C23×C4—1271no—
133C23×C4—1272no—
134C4⋊Q8—2544no—
135C22⋊Q8—2544no—
136C22⋊Q8—2544no—
137C2×C4⋊C4—1271no—
138C2×C4⋊C4—1272no—
139C2×C4⋊C4—1272no—
140C2×C4⋊C4—1274no—
141C2×C4⋊C4—1274no—
142C2×C4⋊C4C2×D12711yescharacteristic
143C2×C22⋊C4—1272no—
144C2×C22⋊C4—1272no—
145C2×C42—1271no—
146C2×C42—1272no—
147C2×C42—2544no—
148C2×C42C2×D12711yescharacteristic
149#95—1274no—
150#95C2×D12714yes—
151C24—1271no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—2542no—
156C2×Q8—2542no—
157C2×Q8—2542no—
158C22×C4—1272no—
159C22×C4—1271no—
160C22×C4—1274no—
161C22×C4—1278no—
162C22×C4—1272no—
163C22×C4—1272no—
164C22×C4—2544no—
165C22×C4—2544no—
166C22×C4—2542no—
167C22×C4—2542no—
168C22×C4C22×D12712yes—
169C22×C4C22×D12712yes—
170C22×C4C22×D12712yes—
171C22×C4C22×D12711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—2548no—
174C4⋊C4—2544no—
175C4⋊C4—2548no—
176C4⋊C4—2544no—
177C4⋊C4—2542no—
178C22⋊C4—2544no—
179C22⋊C4—2544no—
180C42—22no—
181C42—1274no—
182C42—2544no—
183C23—1272no—
184C23—1274no—
185C23—2541no—
186C23—2542no—
187C23—2541no—
188C23C23×D12711yescharacteristic
189Q8—44no—
190Q8—5084no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—1274no—
199C2×C4—1274no—
200C2×C4—1274no—
201C2×C4—1272no—
202C2×C4—25416no—
203C2×C4—2542no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4—2544no—
207C2×C4—2542no—
208C2×C4—2541no—
209C2×C4—2541no—
210C2×C4—2548no—
211C2×C4D4×D12712yes—
212C2×C4D4×D12712yes—
213C2×C4Q8×D12714yes—
214C2×C4#1452324812yes—
215C22—1274no—
216C22—2542no—
217C22—2544no—
218C22—2542no—
219C22—2544no—
220C22C2×D4×D12711yescharacteristic
221C22C2×D4×D12711yescharacteristic
222C22C2×Q8×D12712yes—
223C22#41×D12712yes—
224C22C2×#1452324811yescharacteristic
225C4—24no—
226C4—42no—
227C4—42no—
228C4—1278no—
229C4—5082no—
230C4—5082no—
231C4—2542no—
232C4#1686118214yes—
233C2—2544no—
234C2C2×#1686118211yescharacteristic
235C2C22⋊Q8×D12712yes—
236C2C4⋊Q8×D12711yescharacteristic
237C2#2255042511yescharacteristic
238C2#2255043012yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#391×C127C2
prime radicalC23×D127C23
hypercenter#391D127
hypocenterC127#2857
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#16861207C2semidirect
5. #22549782C2semidirect
6. #22549773C2semidirect
7. #22549745C2semidirect
9. #391×C127C2semidirect
44. C2×C4×D127Q8semidirect
104. C2×D127C4⋊Q8semidirect
112. Dic127C22⋊Q8semidirect
120. C127#2857semidirect
126. #391D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
C2×#168611892
C2×#168611852
C2×C42×D1272
#225497822
#225497732
#225497452
#225497352
#391×C1272
#2857127