Finite Group Databasebeta

← #30648023

Subgroups — #30648023

Order 16256.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861256C211yesmaximal, characteristic
2C2×#16861240C211yesmaximal, characteristic, complemented
3C22×#14523238C211yesmaximal, characteristic, complemented
4#22550036C214yesmaximal, complemented
5#22550031C214yesmaximal, complemented
6#22549884C211yesmaximal, characteristic, complemented
7#22549870C211yesmaximal, characteristic, complemented
8#22549843C211yesmaximal, characteristic, complemented
9#536×C127C211yesmaximal, characteristic, complemented
10#16861256—24no—
11#16861256—24no—
12C2×#14523248C2212yes—
13C2×#14523248C2211yescharacteristic
14C2×#14523247—22no—
15C2×#14523247—21no—
16C22×Dic254C2212yes—
17C22×Dic254C2211yescharacteristic
18C2×Q8×D127—21no—
19#16861240—24no—
20C2×#14523238—24no—
21C2×#14523238C2212yes—
22C2×#14523238C2214yes—
23C2×#14523235C2212yes—
24C2×#14523232C2211yescharacteristic
25C22×C127⋊C8C2211yescharacteristic
26#16861132C2212yes—
27#16861132C2214yes—
28#16861129C2212yes—
29C22⋊(C127⋊C8)C2214yes—
30C2×#41×C127C2211yescharacteristic
31C22×Q8×C127C2211yescharacteristic
32C4⋊Q8×C127—21no—
33#124×C127—21no—
34#123×C127—22no—
35C22⋊Q8×C127C2214yes—
36#117×C127C2211yescharacteristic
37#14523248—24no—
38#14523248—24no—
39#14523248—22no—
40#14523248D414yes—
41#14523247—216no—
42#14523247—28no—
43C2×Dic254—24no—
44C2×Dic254—24no—
45C2×Dic254—22no—
46C2×Dic254—21no—
47C2×Dic254—22no—
48C2×Dic254D414yes—
49C2×Dic254C2311yescharacteristic
50C2×Dic254C2312yes—
51Q8×D127—28no—
52C2×D508C2311yescharacteristic
53C2×C22⋊D127—22no—
54C2×C22⋊D127—21no—
55C22×Dic127—22no—
56C22×Dic127—21no—
57C2×C4×D127—22no—
58C2×C4×D127—21no—
59#14523238—28no—
60#14523238—24no—
61#14523235—24no—
62#14523232—22no—
63C2×C127⋊C8—22no—
64C2×C127⋊C8C2312yes—
65C2×C127⋊C8C2312yes—
66#41×C127—22no—
67#41×C127—22no—
68C2×Q8×C127—21no—
69C2×Q8×C127—22no—
70C2×Q8×C127—22no—
71C2×Q8×C127D411yescharacteristic
72C2×Q8×C127C2312yes—
73C2×D4×C127D412yes—
74C2×D4×C127D411yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127C2312yes—
77C22×C4×C127C2311yescharacteristic
78C4⋊C4×C127—24no—
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C22⋊C4×C127—24no—
82C42×C127—21no—
83C22×D127—21no—
84Dic254—28no—
85Dic254—24no—
86Dic254—21no—
87Dic254—22no—
88Dic254C2×D412yes—
89Dic254C2×D414yes—
90D508—21no—
91D508C2×D412yes—
92C22⋊D127—28no—
93C22⋊D127—24no—
94C2×Dic127—28no—
95C2×Dic127—24no—
96C2×Dic127—22no—
97C2×Dic127—21no—
98C4×D127—28no—
99C4×D127—24no—
100C127⋊C8—22no—
101C127⋊C8—22no—
102C23×C127C2×D412yes—
103C23×C127C2×D411yescharacteristic
104Q8×C127—22no—
105Q8×C127—22no—
106Q8×C127—22no—
107Q8×C127—42no—
108D4×C127—24no—
109D4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—42no—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2411yescharacteristic
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—24no—
123C22×C127#12012yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—42no—
127C4×C127—22no—
128C4×C127—42no—
129C4×C127—42no—
130C4×C127#12012yes—
131C4×C127C22×D411yescharacteristic
132C4×C127C22×D411yescharacteristic
133Dic127—24no—
134Dic127—22no—
135C2×C127—22no—
136C2×C127—42no—
137C2×C127C2×#12011yescharacteristic
138C2×C127#57812yes—
139D127—42no—
140#3011—1271nomaximal, Sylow
141C127#301111yescharacteristic, complemented, Sylow
142C2×2−1+4—1271no—
143C2×#137—1271no—
144C22×Q16—1271no—
145#536D12711yescharacteristic, complemented
146#452—1274no—
147#451—1274no—
148#419—1271no—
149#416—1271no—
150#408—1271no—
1512−1+4—2544no—
1522−1+4—2544no—
153C2×#41—1271no—
154C2×#41—1272no—
155C2×#41—2542no—
156C2×#41—2541no—
157C2×#41C2×D12711yescharacteristic
158C22×Q8—1271no—
159C22×Q8—1272no—
160C22×Q8—2541no—
161C22×Q8C2×D12711yescharacteristic
162#137—2544no—
163C2×Q16—1274no—
164C2×Q16—1272no—
165C2×Q16—2544no—
166C2×SD16—1272no—
167C2×M16—1271no—
168C22×C8—1271no—
169C4⋊Q8—21no—
170#124—21no—
171#123—22no—
172C22⋊Q8C2×D12714yes—
173#117C2×D12711yescharacteristic
174#103—1272no—
175#103—1274no—
176#102—1272no—
177C22⋊C8—1274no—
178#41—22no—
179#41—22no—
180#41—1274no—
181#41—2544no—
182#41—2544no—
183#41—25416no—
184#41—2548no—
185#41—2542no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—1274no—
190C2×Q8—1272no—
191C2×Q8—1271no—
192C2×Q8—2544no—
193C2×Q8—2544no—
194C2×Q8—2548no—
195C2×Q8—2542no—
196C2×Q8—2542no—
197C2×Q8—2541no—
198C2×Q8C22⋊D12711yescharacteristic
199C2×Q8C22×D12712yes—
200C2×D4—1271no—
201C2×D4—2542no—
202C2×D4—2541no—
203C2×D4C22⋊D12712yes—
204C2×D4C22⋊D12711yescharacteristic
205C22×C4—21no—
206C22×C4—2542no—
207C22×C4—2541no—
208C22×C4—2542no—
209C22×C4—2541no—
210C22×C4C22×D12712yes—
211C22×C4C22×D12711yescharacteristic
212Q16—2548no—
213Q16—2544no—
214SD16—2544no—
215M16—2542no—
216C2×C8—1272no—
217C2×C8—1272no—
218C2×C8—2542no—
219C4⋊C4—24no—
220C4⋊C4C22×D12712yes—
221C22⋊C4—21no—
222C22⋊C4—24no—
223C42—21no—
224C23—2541no—
225C23C2×C22⋊D12712yes—
226C23C2×C22⋊D12711yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—22no—
230Q8—42no—
231Q8—1274no—
232Q8—1272no—
233Q8—2548no—
234Q8—2542no—
235Q8—2541no—
236Q8—2544no—
237D4—24no—
238D4—22no—
239D4—1272no—
240D4—2548no—
241D4—2544no—
242D4—2541no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—2548no—
249C2×C4—2548no—
250C2×C4—2542no—
251C2×C4—2541no—
252C2×C4—2544no—
253C2×C4—2544no—
254C2×C4D4×D12712yes—
255C2×C4D4×D12712yes—
256C2×C4C2×C22⋊D12712yes—
257C2×C4C2×C22⋊D12711yescharacteristic
258C2×C4C23×D12711yescharacteristic
259C8—2542no—
260C8—2542no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—5082no—
265C22—2542no—
266C22#1686117712yes—
267C22C22×C22⋊D12711yescharacteristic
268C4—22no—
269C4—42no—
270C4—42no—
271C4—2544no—
272C4—2542no—
273C4#1686117712yes—
274C4C2×D4×D12711yescharacteristic
275C4C2×D4×D12711yescharacteristic
276C2—22no—
277C2—42no—
278C2—5082no—
279C2C2×#1686117711yescharacteristic
280C2#2255069112yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861256C2
hypercenter#536D127
hypocenterC127#3011
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. C22×#14523238C2semidirect
4. #22550036C2semidirect
5. #22550031C2semidirect
6. #22549884C2semidirect
7. #22549870C2semidirect
8. #22549843C2semidirect
9. #536×C127C2semidirect
141. C127#3011semidirect
145. #536D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#168612402
C22×#145232382
#225500362
#225500312
#225498842
#225498702
#225498432
#536×C1272
#3011127