Finite Group Databasebeta

← #30648024

Subgroups — #30648024

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
2C22×#14523238C211yesmaximal, characteristic, complemented
3#22550450C211yesmaximal, characteristic, complemented
4#22550035C214yesmaximal, complemented
5#22550030C214yesmaximal, complemented
6#22549886C211yesmaximal, characteristic, complemented
7#22549872C211yesmaximal, characteristic, complemented
8#22549844C211yesmaximal, characteristic, complemented
9C2×#137×C127C211yesmaximal, characteristic, complemented
10#16861256—24no—
11#16861256—24no—
12C2×#14523248—21no—
13C2×#14523248C2211yescharacteristic
14C2×#14523247—22no—
15C2×#14523247C2212yes—
16C22×Dic254—21no—
17C22×Dic254C2211yescharacteristic
18C2×Q8×D127C2212yes—
19C2×#14523238—24no—
20C2×#14523238C2214yes—
21C22×C127⋊C8C2211yescharacteristic
22#16861207—21no—
23#16861198—21no—
24#16861192—22no—
25#16861185C2214yes—
26#16861168C2211yescharacteristic
27#16861134C2214yes—
28#16861133C2212yes—
29#16861130C2212yes—
30(C2×D127)⋊C8C2214yes—
31C2×#41×C127C2211yescharacteristic
32C22×Q8×C127C2211yescharacteristic
33#137×C127—24no—
34C2×Q16×C127C2212yes—
35C2×SD16×C127C2212yes—
36C2×M16×C127C2211yescharacteristic
37#14523248—28no—
38#14523248—22no—
39#14523248—22no—
40#14523247—216no—
41#14523247—24no—
42#14523247—24no—
43C2×Dic254—28no—
44C2×Dic254—22no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254D414yes—
48C2×Dic254D411yescharacteristic
49C2×Dic254C2312yes—
50Q8×D127—24no—
51Q8×D127—24no—
52C2×D508D411yescharacteristic
53C2×C22⋊D127—22no—
54C2×C22⋊D127D412yes—
55C22×Dic127—22no—
56C22×Dic127—21no—
57C2×C4×D127—22no—
58C2×C4×D127C2312yes—
59#14523238—28no—
60C2×C127⋊C8—22no—
61C2×C127⋊C8C2312yes—
62C4⋊Dic127C2312yes—
63Dic127⋊C4—24no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—24no—
66C4×Dic127—21no—
67#41×C127—22no—
68#41×C127D414yes—
69C2×Q8×C127—21no—
70C2×Q8×C127—22no—
71C2×Q8×C127—22no—
72C2×Q8×C127C2312yes—
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127C2311yescharacteristic
79Q16×C127—24no—
80SD16×C127—24no—
81M16×C127—22no—
82C2×C8×C127C2312yes—
83C22×D127C2×D412yes—
84Dic254—28no—
85Dic254—24no—
86Dic254—22no—
87Dic254—22no—
88Dic254C2×D414yes—
89D508—22no—
90C22⋊D127—28no—
91C22⋊D127—24no—
92C2×Dic127—28no—
93C2×Dic127—24no—
94C2×Dic127—22no—
95C2×Dic127—22no—
96C2×Dic127C2×D412yes—
97C4×D127—28no—
98C4×D127—22no—
99C4×D127—22no—
100C127⋊C8—22no—
101C23×C127—21no—
102C23×C127C2×D411yescharacteristic
103Q8×C127—21no—
104Q8×C127—22no—
105Q8×C127—22no—
106Q8×C127—42no—
107Q8×C127C2×D412yes—
108D4×C127—24no—
109D4×C127—21no—
110D4×C127C2×D412yes—
111C2×C4×C127—21no—
112C2×C4×C127—24no—
113C2×C4×C127—22no—
114C2×C4×C127—42no—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2411yescharacteristic
119C8×C127—22no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—42no—
123C22×C127#12012yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—24no—
127C4×C127—22no—
128C4×C127—42no—
129C4×C127#12012yes—
130C4×C127C22×D411yescharacteristic
131C4×C127C22×D411yescharacteristic
132Dic127—24no—
133Dic127—22no—
134Dic127—42no—
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×#137D12711yescharacteristic, complemented
144C22×Q16—1271no—
145#536—1271no—
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—2541no—
156C2×#41—2542no—
157C2×#41C2×D12711yescharacteristic
158C22×Q8—1272no—
159C22×Q8—1271no—
160C22×Q8—2541no—
161C22×Q8C2×D12711yescharacteristic
162#137—24no—
163C2×Q16—1274no—
164C2×Q16—2544no—
165C2×Q16C2×D12712yes—
166C2×SD16C2×D12712yes—
167C2×M16C2×D12711yescharacteristic
168C22×C8—1271no—
169C4⋊Q8—2541no—
170#124—2541no—
171#123—2542no—
172C22⋊Q8—1274no—
173#117—1271no—
174#103—1272no—
175#103—1274no—
176#102—1272no—
177C22⋊C8—1274no—
178#41—22no—
179#41—2548no—
180#41—2544no—
181#41—25416no—
182#41—2544no—
183#41—2542no—
184#41—2542no—
185#41C22⋊D12714yes—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—1271no—
190C2×Q8—1274no—
191C2×Q8—1272no—
192C2×Q8—2542no—
193C2×Q8—2548no—
194C2×Q8—2544no—
195C2×Q8—2544no—
196C2×Q8—2541no—
197C2×Q8—2542no—
198C2×Q8C22×D12712yes—
199C2×Q8C22×D12711yescharacteristic
200C2×D4—21no—
201C2×D4—1272no—
202C2×D4—1271no—
203C2×D4—2542no—
204C2×D4C22×D12711yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—1272no—
208C22×C4—2542no—
209C22×C4—2542no—
210C22×C4—2541no—
211C22×C4C22×D12711yescharacteristic
212Q16—24no—
213Q16—2548no—
214SD16—24no—
215M16—22no—
216C2×C8—1272no—
217C2×C8—2542no—
218C2×C8C22×D12712yes—
219C4⋊C4—1272no—
220C4⋊C4—2544no—
221C22⋊C4—2544no—
222C22⋊C4—2541no—
223C42—2541no—
224C23—21no—
225C23—1272no—
226C23D4×D12711yescharacteristic
227Q8—21no—
228Q8—22no—
229Q8—22no—
230Q8—42no—
231Q8—1274no—
232Q8—2548no—
233Q8—2542no—
234Q8—2542no—
235Q8—2544no—
236Q8C2×C22⋊D12712yes—
237D4—24no—
238D4—21no—
239D4—2548no—
240D4—2544no—
241D4—2542no—
242D4C2×C22⋊D12712yes—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—1272no—
248C2×C4—2548no—
249C2×C4—2542no—
250C2×C4—2548no—
251C2×C4—2542no—
252C2×C4—2544no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4D4×D12712yes—
256C2×C4D4×D12711yescharacteristic
257C2×C4C2×C22⋊D12712yes—
258C2×C4C23×D12711yescharacteristic
259C8—22no—
260C8—2542no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—2542no—
265C22—2544no—
266C22#1686117712yes—
267C22C2×D4×D12711yescharacteristic
268C4—22no—
269C4—42no—
270C4—5082no—
271C4—2542no—
272C4—2544no—
273C4#1686117712yes—
274C4C2×D4×D12711yescharacteristic
275C4C22×C22⋊D12711yescharacteristic
276C2—22no—
277C2—42no—
278C2—5082no—
279C2C2×#1686117711yescharacteristic
280C2#2255069212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861256C2
hypercenterC2×#137D127
hypocenterC127#3011
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14523238C2semidirect
3. #22550450C2semidirect
4. #22550035C2semidirect
5. #22550030C2semidirect
6. #22549886C2semidirect
7. #22549872C2semidirect
8. #22549844C2semidirect
9. C2×#137×C127C2semidirect
141. C127#3011semidirect
143. C2×#137D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C22×#145232382
#225504502
#225500352
#225500302
#225498862
#225498722
#225498442
C2×#137×C1272
#3011127