Finite Group Databasebeta

← #30647078

Subgroups — #30647078

Order 16256.

644 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 201 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×#16861188C211yesmaximal, characteristic
3C22×C4×Dic127C211yesmaximal, characteristic
4#22549783C211yesmaximal, characteristic, complemented
5#22549781C211yesmaximal, characteristic, complemented
6#22549771C214yesmaximal, complemented
7#22549745C211yesmaximal, characteristic, complemented
8#22549734C214yesmaximal
9C2×C22⋊Q8×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11C23×Dic127C2212yes—
12#16861207—24no—
13#16861188—24no—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4C2212yes—
16C2×Dic127⋊C4C2214yes—
17C2×Dic127⋊C4C2214yes—
18C2×C22⋊Dic127C2212yes—
19C2×C4×Dic127—24no—
20C2×C4×Dic127C2212yes—
21C2×C4×Dic127C2211yescharacteristic
22C2×C4×Dic127C2211yescharacteristic
23C127⋊#95C2214yes—
24C127⋊#95C2214yes—
25C22×Q8×C127C2211yescharacteristic
26C23×C4×C127C2211yescharacteristic
27C22⋊Q8×C127—24no—
28C2×C4⋊C4×C127C2212yes—
29C2×C4⋊C4×C127C2211yescharacteristic
30C2×C22⋊C4×C127C2212yes—
31C2×Dic254—22no—
32C2×Dic254—22no—
33C2×Dic254—22no—
34C22×Dic127—24no—
35C22×Dic127—24no—
36C22×Dic127Q818yescomplemented
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C22×Dic127C2312yes—
40C22×Dic127C2312yes—
41C4⋊Dic127—22no—
42Dic127⋊C4—28no—
43Dic127⋊C4—28no—
44Dic127⋊C4—24no—
45C22⋊Dic127—24no—
46C4×Dic127—24no—
47C4×Dic127—22no—
48C4×Dic127D414yes—
49C24×C127C2311yescharacteristic
50C2×Q8×C127—22no—
51C2×Q8×C127—22no—
52C2×Q8×C127—22no—
53C22×C4×C127—22no—
54C22×C4×C127—22no—
55C22×C4×C127D414yes—
56C22×C4×C127C2312yes—
57C22×C4×C127C2312yes—
58C22×C4×C127C2311yescharacteristic
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127—22no—
61C4⋊C4×C127—24no—
62C22⋊C4×C127—24no—
63Dic254—44no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—216no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—24no—
71C2×Dic127—22no—
72C2×Dic127C2×D414yes—
73C2×Dic127C2×Q814yes—
74C2×Dic127C2×Q814yes—
75C2×Dic127#4114yes—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—22no—
79C23×C127C2×D412yes—
80C23×C127C2×Q814yes—
81C23×C127C2411yescharacteristic
82Q8×C127—44no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—28no—
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×D412yes—
94C2×C4×C127#4112yes—
95C22×C127—22no—
96C22×C127—24no—
97C22×C127—22no—
98C22×C127—24no—
99C22×C127C4⋊Q814yescomplemented
100C22×C127C22×D411yescharacteristic
101C22×C127C22×D411yescharacteristic
102C22×C127C22×Q812yes—
103C22×C127C2×#4112yes—
104C22×C127C2×#4111yescharacteristic
105C4×C127—22no—
106C4×C127—42no—
107C4×C127—42no—
108C4×C127#12114yes—
109Dic127—24no—
110Dic127—42no—
111Dic127—42no—
112Dic127C22⋊Q818yescomplemented
113C2×C127—24no—
114C2×C127C2×#12111yescharacteristic
115C2×C127C2×C22⋊Q812yes—
116C2×C127C2×C4⋊Q811yescharacteristic
117C2×C127#53211yescharacteristic
118C2×C127#53312yes—
119#2857—1271nomaximal, Sylow
120C127#285711yescharacteristic, complemented, Sylow
121C2×C4⋊Q8—1271no—
122C2×C22⋊Q8—1271no—
123C2×C22⋊Q8D12711yescharacteristic
124C22×C42—1271no—
125#391—1271no—
126#391—1271no—
127#389—1274no—
128#384—1271no—
129#380—1274no—
130C22×Q8—1271no—
131C22×Q8C2×D12711yescharacteristic
132C23×C4—1272no—
133C23×C4C2×D12711yescharacteristic
134C4⋊Q8—2544no—
135C22⋊Q8—24no—
136C22⋊Q8—2544no—
137C2×C4⋊C4—1271no—
138C2×C4⋊C4—1272no—
139C2×C4⋊C4—1274no—
140C2×C4⋊C4—1274no—
141C2×C4⋊C4C2×D12711yescharacteristic
142C2×C4⋊C4C2×D12712yes—
143C2×C22⋊C4—1272no—
144C2×C22⋊C4C2×D12712yes—
145C2×C42—1271no—
146C2×C42—1272no—
147C2×C42—1271no—
148C2×C42—2544no—
149#95—1274no—
150#95—1274no—
151C24C22×D12711yescharacteristic
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—2542no—
156C2×Q8—2542no—
157C2×Q8—2542no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—1272no—
161C22×C4—1278no—
162C22×C4—1272no—
163C22×C4—1272no—
164C22×C4—1272no—
165C22×C4—2544no—
166C22×C4—2544no—
167C22×C4C22⋊D12714yes—
168C22×C4C22×D12712yes—
169C22×C4C22×D12712yes—
170C22×C4C22×D12711yescharacteristic
171C22×C4C22×D12711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—2548no—
175C4⋊C4—2548no—
176C4⋊C4—2544no—
177C4⋊C4—2542no—
178C22⋊C4—24no—
179C22⋊C4—2544no—
180C42—1274no—
181C42—2544no—
182C42—2542no—
183C23—21no—
184C23—22no—
185C23—21no—
186C23C2×C22⋊D12712yes—
187C23Q8×D12714yes—
188C23C23×D12711yescharacteristic
189Q8—44no—
190Q8—5084no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—1274no—
200C2×C4—1274no—
201C2×C4—1274no—
202C2×C4—1274no—
203C2×C4—25416no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4—2542no—
207C2×C4—2544no—
208C2×C4—2542no—
209C2×C4—2542no—
210C2×C4—2542no—
211C2×C4D4×D12712yes—
212C2×C4C2×C22⋊D12712yes—
213C2×C4C2×C22⋊D12712yes—
214C2×C4#1452324712yes—
215C22—22no—
216C22—24no—
217C22—22no—
218C22—24no—
219C22#1686120714yescomplemented
220C22C2×D4×D12711yescharacteristic
221C22C22×C22⋊D12711yescharacteristic
222C22C2×Q8×D12712yes—
223C22#41×D12712yes—
224C22C2×#1452324711yescharacteristic
225C4—22no—
226C4—42no—
227C4—42no—
228C4—1278no—
229C4—5082no—
230C4—5082no—
231C4—2544no—
232C4#1686118114yes—
233C2—24no—
234C2C2×#1686118111yescharacteristic
235C2C22⋊Q8×D12712yes—
236C2C2×#1686120711yescharacteristic
237C2#2255042211yescharacteristic
238C2#2255042812yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×C22⋊Q8×C127C2
prime radicalC24×C127C23
hypercenterC2×C22⋊Q8D127
hypocenterC127#2857
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861207C2semidirect
4. #22549783C2semidirect
5. #22549781C2semidirect
6. #22549771C2semidirect
7. #22549745C2semidirect
36. C22×Dic127Q8semidirect
99. C22×C127C4⋊Q8semidirect
112. Dic127C22⋊Q8semidirect
120. C127#2857semidirect
219. C22#16861207semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
C2×#168611882
C22×C4×Dic1272
#225497832
#225497812
#225497712
#225497452
#225497342
C2×C22⋊Q8×C1272
#2857127