Finite Group Databasebeta

← #30647084

Subgroups — #30647084

Order 16256.

752 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 177 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×#16861208C211yesmaximal, characteristic, complemented
2C2×#16861205C211yesmaximal, characteristic
3C2×#16861195C214yesmaximal, complemented
4#22549779C211yesmaximal, characteristic, complemented
5#22549778C211yesmaximal, characteristic, complemented
6#22549777C211yesmaximal, characteristic, complemented
7#22549776C214yesmaximal, complemented
8C42×Dic127C211yesmaximal, characteristic, complemented
9#390×C127C211yesmaximal, characteristic, complemented
10C22×Dic254C2212yes—
11C22×D508C2211yescharacteristic
12C22×C22⋊D127C2212yes—
13C22×C22⋊D127C2212yes—
14#16861208—24no—
15#16861205—24no—
16#16861195—216no—
17C2×C4⋊Dic127C2211yescharacteristic
18C2×Dic127⋊C4C2212yes—
19C2×C22⋊Dic127C2214yes—
20C2×(C2×D127)⋊C4C2214yes—
21C2×(C2×D127)⋊C4C2214yes—
22C2×C4×Dic127C2212yes—
23C2×C4×Dic127C2211yescharacteristic
24C2×C4×Dic127C2211yescharacteristic
25C2×C4×Dic127C2212yes—
26C22×D4×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2214yes—
29C2×C42×C127C2211yescharacteristic
30C23×D127C2312yes—
31C2×Dic254—28no—
32C2×Dic254—24no—
33C2×D508—22no—
34C2×D508—22no—
35C2×D508—22no—
36C2×C22⋊D127—28no—
37C2×C22⋊D127—24no—
38C2×C22⋊D127—24no—
39C2×C22⋊D127—28no—
40C22×Dic127C2312yes—
41C22×Dic127C2312yes—
42C22×Dic127C2311yescharacteristic
43C22×Dic127C2311yescharacteristic
44C4⋊Dic127—22no—
45Dic127⋊C4—24no—
46C22⋊Dic127—28no—
47(C2×D127)⋊C4—28no—
48(C2×D127)⋊C4—28no—
49C4×Dic127—24no—
50C4×Dic127—22no—
51C4×Dic127D414yes—
52C4×Dic127D418yescomplemented
53C24×C127C2312yes—
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127—22no—
57C22×C4×C127C2312yes—
58C22×C4×C127C2312yes—
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127—22no—
61C22⋊C4×C127—28no—
62C42×C127—22no—
63C22×D127—24no—
64C22×D127—22no—
65C22×D127—24no—
66C22×D127—22no—
67C22×D127—22no—
68Dic254—48no—
69D508—44no—
70C22⋊D127—48no—
71C22⋊D127—48no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—24no—
75C2×Dic127—22no—
76C2×Dic127C2×D414yes—
77C2×Dic127C2×D412yes—
78C2×Dic127C2×D412yes—
79C2×Dic127C2×D414yes—
80C2×Dic127#4114yes—
81C23×C127—24no—
82C23×C127—22no—
83C23×C127—22no—
84C23×C127—22no—
85C23×C127—24no—
86C23×C127C2411yescharacteristic
87D4×C127—44no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—24no—
91C2×C4×C127—22no—
92C2×C4×C127C2×D414yes—
93C2×C4×C127C2×D412yes—
94C2×C4×C127#4112yes—
95C2×C4×C127#4112yes—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127—42no—
99C22×C127—44no—
100C22×C127—44no—
101C22×C127C22×D411yescharacteristic
102C22×C127C22×D412yes—
103C22×C127C2×#4112yes—
104C22×C127C2×#4111yescharacteristic
105C22×C127C2×#4111yescharacteristic
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—42no—
109C2×D127—44no—
110C2×D127—44no—
111C4×C127—24no—
112C4×C127—42no—
113C4×C127#12414yescomplemented
114Dic127—24no—
115Dic127—42no—
116Dic127#12414yescomplemented
117Dic127#12714yes—
118C2×C127—82no—
119C2×C127C2×#12411yescharacteristic
120C2×C127C2×#12411yescharacteristic
121C2×C127C2×#12711yescharacteristic
122C2×C127#53212yes—
123C2×C127#53212yes—
124D127—82no—
125#2858—1271nomaximal, Sylow
126C127#285811yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—1271no—
128C2×#127—1271no—
129C2×#124—1274no—
130#390—1271no—
131#390—1271no—
132#390—1274no—
133#390—1271no—
134#390D12711yescharacteristic, complemented
135C43—1271no—
136C22×Q8—1272no—
137C22×D4—1272no—
138C22×D4—1272no—
139C22×D4—1271no—
140C22×D4C2×D12711yescharacteristic
141C4⋊Q8—2544no—
142#127—2544no—
143#124—25416no—
144C2×C4⋊C4—1272no—
145C2×C4⋊C4—1271no—
146C2×C4⋊C4C2×D12711yescharacteristic
147C2×C22⋊C4—1274no—
148C2×C22⋊C4—1274no—
149C2×C22⋊C4—1274no—
150C2×C22⋊C4C2×D12714yes—
151C2×C42—1272no—
152C2×C42—1271no—
153C2×C42—1272no—
154C2×C42—1271no—
155C2×C42C2×D12711yescharacteristic
156C24—1272no—
157C24C22×D12712yes—
158C2×Q8—2548no—
159C2×Q8—2544no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—2548no—
164C2×D4—2544no—
165C2×D4—2544no—
166C2×D4—2548no—
167C2×D4—2542no—
168C2×D4—2542no—
169C2×D4—2542no—
170C22×C4—1271no—
171C22×C4—1271no—
172C22×C4—1272no—
173C22×C4—1272no—
174C22×C4C22×D12712yes—
175C22×C4C22×D12712yes—
176C22×C4C22×D12711yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—2544no—
179C4⋊C4—2542no—
180C22⋊C4—28no—
181C22⋊C4—2548no—
182C22⋊C4—2548no—
183C22⋊C4—2548no—
184C42—22no—
185C42—1274no—
186C42—1278no—
187C42—2544no—
188C42—2542no—
189C23—24no—
190C23—22no—
191C23—24no—
192C23—22no—
193C23—22no—
194C23—2544no—
195C23—2542no—
196C23—2544no—
197C23—2542no—
198C23—2542no—
199C23C23×D12711yescharacteristic
200Q8—5088no—
201D4—44no—
202D4—5088no—
203D4—5088no—
204D4—5084no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—1272no—
210C2×C4—1272no—
211C2×C4—1274no—
212C2×C4—1274no—
213C2×C4—1274no—
214C2×C4—2544no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4—2542no—
218C2×C4D4×D12714yes—
219C2×C4D4×D12712yes—
220C2×C4#1452324712yes—
221C2×C4#1452324812yes—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—44no—
226C22—44no—
227C22—5082no—
228C22—5082no—
229C22—5082no—
230C22—5084no—
231C22—5084no—
232C22C2×D4×D12711yescharacteristic
233C22C2×D4×D12712yes—
234C22#41×D12712yes—
235C22C2×#1452324711yescharacteristic
236C22C2×#1452324811yescharacteristic
237C4—24no—
238C4—42no—
239C4—1274no—
240C4—1274no—
241C4—5082no—
242C4—2544no—
243C4#1686119514yescomplemented
244C2—82no—
245C2—10162no—
246C2#124×D12711yescharacteristic
247C2C2×#1686119511yescharacteristic
248C2#127×D12711yescharacteristic
249C2#2255042112yes—
250C2#2255042312yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#390×C127C2
prime radicalC2×#16861205C2
hypercenter#390D127
hypocenterC127#2858
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16861208C2semidirect
3. C2×#16861195C2semidirect
4. #22549779C2semidirect
5. #22549778C2semidirect
6. #22549777C2semidirect
7. #22549776C2semidirect
8. C42×Dic127C2semidirect
9. #390×C127C2semidirect
52. C4×Dic127D4semidirect
113. C4×C127#124semidirect
116. Dic127#124semidirect
126. C127#2858semidirect
134. #390D127semidirect
243. C4#16861195semidirect

Maximal subgroups

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

subgroupindex
C2×#168612082
C2×#168612052
C2×#168611952
#225497792
#225497782
#225497772
#225497762
C42×Dic1272
#390×C1272
#2858127