Finite Group Databasebeta

← #30641320

Subgroups — #30641320

Order 16256.

880 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 229 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×#14523247C211yesmaximal, characteristic
2C2×#16861166C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549782C211yesmaximal, characteristic, complemented
5#22549781C211yesmaximal, characteristic, complemented
6#22549779C211yesmaximal, characteristic, complemented
7#22549752C214yesmaximal, complemented
8#22549734C214yesmaximal, complemented
9#390×C127C211yesmaximal, characteristic, complemented
10C2×#14523247—24no—
11C2×#14523247—24no—
12C2×#14523247C418yescomplemented
13C22×Dic254C2211yescharacteristic
14C22×C22⋊D127C2212yes—
15C23×Dic127C2212yes—
16C22×C4×D127C2212yes—
17C22×C4×D127C2211yescharacteristic
18#16861166—24no—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4C2212yes—
21C2×Dic127⋊C4C2212yes—
22C4⋊C4×D127—24no—
23C2×C22⋊Dic127C2214yes—
24C2×(C2×D127)⋊C4C2212yes—
25C2×C4×Dic127C2211yescharacteristic
26C127⋊#95C2214yes—
27C127⋊#95C2214yes—
28C22×D4×C127C2211yescharacteristic
29C2×C4⋊C4×C127C2211yescharacteristic
30C2×C22⋊C4×C127C2214yes—
31C2×C42×C127C2211yescharacteristic
32C23×D127C2311yescharacteristic
33#14523247—232no—
34C2×Dic254—22no—
35C2×Dic254—22no—
36C2×Dic254C2×C414yescomplemented
37C2×C22⋊D127—24no—
38C2×C22⋊D127—24no—
39C2×C22⋊D127C2×C418yes—
40C22×Dic127—24no—
41C22×Dic127—24no—
42C22×Dic127C2×C418yescomplemented
43C22×Dic127C2312yes—
44C22×Dic127C2312yes—
45C22×Dic127C2312yes—
46C22×Dic127C2311yescharacteristic
47C2×C4×D127—24no—
48C2×C4×D127—44no—
49C2×C4×D127C2×C414yes—
50C2×C4×D127D414yes—
51C2×C4×D127D414yes—
52C4⋊Dic127—22no—
53Dic127⋊C4—24no—
54Dic127⋊C4—24no—
55C22⋊Dic127—44no—
56(C2×D127)⋊C4—24no—
57C4×Dic127—22no—
58C24×C127C2312yes—
59C2×D4×C127—22no—
60C2×D4×C127—22no—
61C2×D4×C127C2×C414yescomplemented
62C22×C4×C127C2312yes—
63C22×C4×C127C2312yes—
64C22×C4×C127C2311yescharacteristic
65C4⋊C4×C127—22no—
66C22⋊C4×C127—44no—
67C42×C127—22no—
68C22×D127—22no—
69C22×D127—22no—
70C22×D127C22×C412yes—
71C22×D127C2×D412yes—
72C22×D127C2×D412yes—
73Dic254—28no—
74C22⋊D127—216no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—216no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—24no—
82C2×Dic127—22no—
83C2×Dic127C22×C414yes—
84C2×Dic127C22×C412yes—
85C2×Dic127C2×D412yes—
86C2×Dic127C2×D412yes—
87C4×D127—48no—
88C4×D127C22⋊C4116yescomplemented
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—24no—
92C23×C127—24no—
93C23×C127C22×C414yes—
94C23×C127C2411yescharacteristic
95D4×C127—28no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—24no—
100C2×C4×C127—22no—
101C2×C4×C127C22×C412yes—
102C2×C4×C127C2×D412yes—
103C2×C4×C127C2×D412yes—
104C22×C127—24no—
105C22×C127—28no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127—44no—
109C22×C127C23×C411yescharacteristic
110C22×C127C22×D411yescharacteristic
111C22×C127C22×D411yescharacteristic
112C22×C127H(5,𝔽2)12yes—
113C22×C1272−1+412yes—
114C2×D127—24no—
115C2×D127—22no—
116C2×D127—22no—
117C2×D127—24no—
118C2×D127C2×C22⋊C414yes—
119C4×C127—42no—
120C4×C127—42no—
121C4×C127C2×C22⋊C414yes—
122Dic127—24no—
123Dic127—42no—
124Dic127—42no—
125Dic127C2×C22⋊C414yes—
126C2×C127—44no—
127C2×C127C22×C22⋊C411yescharacteristic
128C2×C127#52012yes—
129C2×C127#53511yescharacteristic
130C2×C127#53611yescharacteristic
131C2×C127#53712yes—
132D127—24no—
133#2311—1271nomaximal, Sylow
134C127#231111yescharacteristic, complemented, Sylow
135C22×#41—1271no—
136C2×#117—1271no—
137C22×C4⋊C4—1271no—
138#391—1271no—
139#391—1271no—
140#390—1271no—
141#390D12711yescharacteristic, complemented
142#386—1274no—
143#380—1274no—
144C2×#41—1278no—
145C2×#41—2544no—
146C2×#41—2544no—
147C22×Q8—1271no—
148C22×D4—1272no—
149C22×D4C2×D12711yescharacteristic
150C23×C4—1271no—
151C23×C4—1272no—
152C23×C4—1272no—
153#117—2544no—
154C2×C4⋊C4—1272no—
155C2×C4⋊C4—1271no—
156C2×C4⋊C4—1272no—
157C2×C4⋊C4—2544no—
158C2×C4⋊C4C2×D12711yescharacteristic
159C2×C22⋊C4—1274no—
160C2×C22⋊C4—1272no—
161C2×C22⋊C4C2×D12714yes—
162C2×C42—1271no—
163C2×C42C2×D12711yescharacteristic
164#95—1274no—
165#95—1274no—
166C24—1271no—
167C24C22×D12712yes—
168#41—25432no—
169C2×Q8—1274no—
170C2×Q8—2542no—
171C2×Q8—2542no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—1278no—
175C2×D4—2544no—
176C2×D4—2544no—
177C2×D4C4×D12714yescomplemented
178C22×C4—1271no—
179C22×C4—1274no—
180C22×C4—1274no—
181C22×C4—1274no—
182C22×C4—1278no—
183C22×C4—1272no—
184C22×C4—1272no—
185C22×C4—1272no—
186C22×C4—5084no—
187C22×C4—2544no—
188C22×C4—2544no—
189C22×C4—2544no—
190C22×C4C22×D12712yes—
191C22×C4C22×D12712yes—
192C22×C4C22×D12711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—2544no—
195C4⋊C4—2544no—
196C4⋊C4—2542no—
197C22⋊C4—44no—
198C22⋊C4—5084no—
199C22⋊C4—2544no—
200C42—22no—
201C42—2542no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—1272no—
207C23—1272no—
208C23—1272no—
209C23—2542no—
210C23—2542no—
211C23C2×C4×D12714yes—
212C23C23×D12711yescharacteristic
213Q8—2548no—
214D4—28no—
215D4—25416no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—1272no—
222C2×C4—1272no—
223C2×C4—1272no—
224C2×C4—12716no—
225C2×C4—1274no—
226C2×C4—5088no—
227C2×C4—25416no—
228C2×C4—2542no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4—2544no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4—2542no—
235C2×C4C2×C4×D12712yes—
236C2×C4D4×D12712yes—
237C2×C4D4×D12712yes—
238C22—24no—
239C22—28no—
240C22—42no—
241C22—42no—
242C22—44no—
243C22—1274no—
244C22—2544no—
245C22—2542no—
246C22—2542no—
247C22—2544no—
248C22C22×C4×D12711yescharacteristic
249C22C2×D4×D12711yescharacteristic
250C22C2×D4×D12711yescharacteristic
251C22#1686125412yes—
252C22#1686125612yes—
253C4—42no—
254C4—42no—
255C4—1274no—
256C4—5082no—
257C4—5082no—
258C4—2544no—
259C4C22⋊C4×D12714yes—
260C2—44no—
261C2—2544no—
262C2C2×C22⋊C4×D12711yescharacteristic
263C2#2255035512yes—
264C2#2255043911yescharacteristic
265C2#2255044611yescharacteristic
266C2#2255045512yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#390×C127C2
socle, cosocleC23×C127C24
prime radicalC22×#14523247C2
hypercenter#390D127
hypocenterC127#2311
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16861166C2semidirect
3. C2×C4⋊C4×D127C2semidirect
4. #22549782C2semidirect
5. #22549781C2semidirect
6. #22549779C2semidirect
7. #22549752C2semidirect
8. #22549734C2semidirect
9. #390×C127C2semidirect
12. C2×#14523247C4semidirect
36. C2×Dic254C2×C4semidirect
42. C22×Dic127C2×C4semidirect
61. C2×D4×C127C2×C4semidirect
88. C4×D127C22⋊C4semidirect
134. C127#2311semidirect
141. #390D127semidirect
177. C2×D4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168611662
C2×C4⋊C4×D1272
#225497822
#225497812
#225497792
#225497522
#225497342
#390×C1272
#2311127