Finite Group Databasebeta

← #30648902

Subgroups — #30648902

Order 16256.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#22550412C211yesmaximal, characteristic
2C2×Q8×Dic127C211yesmaximal, characteristic, complemented
3#22550209C211yesmaximal, characteristic, complemented
4#22550195C211yesmaximal, characteristic
5#22550181C212yesmaximal, complemented
6#22550100C212yesmaximal, complemented
7#22550015C211yesmaximal, characteristic
8#22549983C212yesmaximal, complemented
9#22549979C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#451×C127C211yesmaximal, characteristic, complemented
12C2×#14523235C2212yes—
13#16861204—21no—
14#16861200C2211yescharacteristic
15#16861198C2211yescharacteristic
16#16861193—21no—
17#16861183C2211yescharacteristic
18Q8×Dic127—21no—
19Q8×Dic127—21no—
20Q8×Dic127C2212yes—
21Q8×Dic127C2212yes—
22Q8×Dic127C2211yescharacteristic
23#16861171C2211yescharacteristic
24(C22⋊D127)⋊C4—21no—
25#16861167—21no—
26C2×C4⋊Dic127—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×C4×Dic127—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861142C2212yes—
31#16861141C2212yes—
32Dic127⋊C8C2212yes—
33#16861133C2212yes—
34#16861133C2212yes—
35#16861131C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yes—
38C22×Q8×C127C2211yescharacteristic
39C2×Q16×C127C2212yes—
40C22⋊Q8×C127C2211yescharacteristic
41#103×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43C2×D508C2311yescharacteristic
44C2×C22⋊D127—21no—
45C22×Dic127—21no—
46C22×Dic127D411yescharacteristic
47C22×Dic127D411yescharacteristic
48C2×C4×D127—21no—
49#14523235—24no—
50C2×C127⋊C8C2312yes—
51C4⋊Dic127—21no—
52C4⋊Dic127—21no—
53C4⋊Dic127—22no—
54C4⋊Dic127—22no—
55C4⋊Dic127—21no—
56C4⋊Dic127C2311yescharacteristic
57C4⋊Dic127C2312yes—
58Dic127⋊C4—21no—
59(C2×D127)⋊C4—21no—
60(C2×D127)⋊C4—21no—
61(C2×D127)⋊C4—21no—
62(C2×D127)⋊C4—21no—
63C4×Dic127—22no—
64C4×Dic127—22no—
65C4×Dic127—21no—
66C4×Dic127D411yescharacteristic
67C4×Dic127D411yescharacteristic
68C4×Dic127C2311yescharacteristic
69C4×Dic127C2311yescharacteristic
70C2×Q8×C127—24no—
71C2×Q8×C127—21no—
72C2×Q8×C127C2311yescharacteristic
73C2×Q8×C127C2312yes—
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76Q16×C127—24no—
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C22⋊C4×C127—21no—
81C22×D127—21no—
82D508—22no—
83C22⋊D127—42no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—41no—
88C2×Dic127—42no—
89C2×Dic127C2×D411yescharacteristic
90C2×Dic127C2×D411yescharacteristic
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C4×D127—42no—
94C127⋊C8—22no—
95C23×C127C2×D411yescharacteristic
96Q8×C127—24no—
97Q8×C127—22no—
98Q8×C127—22no—
99Q8×C127#4114yes—
100C2×C4×C127—24no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127#13712yes—
111C22×C127C22×D411yescharacteristic
112C2×D127—41no—
113C2×D127—41no—
114C2×D127—41no—
115C4×C127—24no—
116C4×C127—21no—
117C4×C127—41no—
118C4×C127—41no—
119C4×C127C2×#4112yes—
120Dic127—21no—
121Dic127—21no—
122Dic127—21no—
123Dic127—21no—
124Dic127—41no—
125Dic127—42no—
126C2×C127—22no—
127C2×C127#52511yescharacteristic
128C2×C127C2×#13711yescharacteristic
129C2×C127#57511yescharacteristic
130D127—81no—
131#3117—1271nomaximal, Sylow
132C127#311711yescharacteristic, complemented, Sylow
133#529—1271no—
134C2×C4×Q8—1271no—
135#484—1271no—
136#481—1271no—
137#479—1272no—
138#464—1272no—
139#451D12711yescharacteristic, complemented
140#448—1271no—
141#441—1272no—
142#440—1272no—
143#432—1271no—
144C22×Q8C2×D12711yescharacteristic
145C2×Q16C2×D12712yes—
146C2×SD16—1272no—
147#126—2541no—
148#125—1271no—
149#124—1271no—
150#123—2541no—
151C22⋊Q8C2×D12711yescharacteristic
152#121—1271no—
153C4×Q8—1272no—
154C4×Q8—1272no—
155C4×Q8—1271no—
156C4×Q8—2541no—
157C4×Q8—2541no—
158C4×D4—1271no—
159C4×D4—2541no—
160#117—2541no—
161C2×C4⋊C4—1271no—
162C2×C4⋊C4—2541no—
163C2×C42—1271no—
164C2×C42—2541no—
165#107—1272no—
166#106—1272no—
167C4⋊C8—1272no—
168#103—1272no—
169#103—1272no—
170#103C2×D12712yes—
171#102—1272no—
172C22⋊C8—1271no—
173C22⋊C8C2×D12711yescharacteristic
174#97—1272no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8C22×D12711yescharacteristic
178C2×Q8C22×D12712yes—
179C2×D4—1271no—
180C2×D4—2541no—
181C22×C4—21no—
182C22×C4—1271no—
183C22×C4—1271no—
184C22×C4—2541no—
185C22×C4—2541no—
186C22×C4C22×D12711yescharacteristic
187Q16—24no—
188SD16—2544no—
189C2×C8—1272no—
190C2×C8C22×D12712yes—
191C4⋊C4—21no—
192C4⋊C4—1272no—
193C4⋊C4—1271no—
194C4⋊C4—2541no—
195C4⋊C4—2542no—
196C4⋊C4—2542no—
197C4⋊C4—2541no—
198C4⋊C4—2541no—
199C4⋊C4—2541no—
200C4⋊C4C22×D12711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2541no—
203C22⋊C4—2541no—
204C22⋊C4—2541no—
205C22⋊C4—2541no—
206C42—1271no—
207C42—1271no—
208C42—1271no—
209C42—1271no—
210C42—2542no—
211C42—2542no—
212C42—2541no—
213C23—2541no—
214C23D4×D12711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—22no—
218Q8#1452324714yes—
219D4—5082no—
220D4—2542no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—1271no—
227C2×C4—1271no—
228C2×C4—1271no—
229C2×C4—1271no—
230C2×C4—5082no—
231C2×C4—5082no—
232C2×C4—5081no—
233C2×C4—2542no—
234C2×C4—2541no—
235C2×C4—2542no—
236C2×C4D4×D12711yescharacteristic
237C2×C4C23×D12711yescharacteristic
238C8—22no—
239C8—2542no—
240C22—21no—
241C22—21no—
242C22—5081no—
243C22—5081no—
244C22—5081no—
245C22#1686124112yes—
246C22C2×D4×D12711yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—5081no—
252C4—5082no—
253C4—2541no—
254C4—2541no—
255C4—2541no—
256C4—2541no—
257C4C2×#1452324712yes—
258C2—22no—
259C2—10161no—
260C2#2255038211yescharacteristic
261C2C2×#1686124111yescharacteristic
262C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#451×C127C2
socleC22×C127C22×D4
prime radical#16861183C22
hypercenter#451D127
hypocenterC127#3117
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic127C2semidirect
3. #22550209C2semidirect
5. #22550181C2semidirect
6. #22550100C2semidirect
8. #22549983C2semidirect
9. #22549979C2semidirect
10. #22549930C2semidirect
11. #451×C127C2semidirect
132. C127#3117semidirect
139. #451D127semidirect

Maximal subgroups

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

subgroupindex
#225504122
C2×Q8×Dic1272
#225502092
#225501952
#225501812
#225501002
#225500152
#225499832
#225499792
#225499302
#451×C1272
#3117127