Finite Group Databasebeta

← #37582969

Subgroups — #37582969

Order 24448.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#20259349C212yesmaximal, complemented
2C2×#20259346C212yesmaximal, complemented
3#27133212C211yesmaximal, characteristic, complemented
4#127×D191C211yesmaximal, characteristic, complemented
5#27133053C211yesmaximal, characteristic, complemented
6#27133052C211yesmaximal, characteristic, complemented
7#27133050C211yesmaximal, characteristic, complemented
8#27133038C212yesmaximal, complemented
9#27133034C212yesmaximal, complemented
10#27132615C211yesmaximal, characteristic, complemented
11#496×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2211yescharacteristic
13C2×#15690286C2211yescharacteristic
14C2×D4×D191—22no—
15C2×D4×D191C2212yes—
16#20259349—28no—
17#20259346—28no—
18C2×#15690276C2212yescomplemented
19C2×#15690275C2212yescomplemented
20C2×#15690274C2212yescomplemented
21C2×D1528C2212yescomplemented
22C2×#15690272C2212yescomplemented
23C2×#15690272C2212yescomplemented
24C2×#15690270C2212yes—
25#20259320C2211yescharacteristic, complemented
26#20259319C2211yescharacteristic
27#20259318—22no—
28#20259318C2212yes—
29#20259318C2211yescharacteristic, complemented
30#20259310C2211yescharacteristic, complemented
31#20259295—22no—
32(C22⋊D191)⋊C4—22no—
33C42×D191C2211yescharacteristic
34#20259233C2211yescharacteristic, complemented
35C8×Dic191C2212yescomplemented
36C2×SD16×C191C2212yescomplemented
37C2×D8×C191C2212yescomplemented
38#127×C191C2211yescharacteristic
39#124×C191C2211yescharacteristic, complemented
40#97×C191C2211yescharacteristic, complemented
41C23×D191—22no—
42#15690288—24no—
43#15690286—24no—
44C2×Dic382C2311yescharacteristic
45C2×D764—21no—
46C2×D764—22no—
47C2×D764C2311yescharacteristic
48C2×D764C2312yes—
49C2×C22⋊D191—21no—
50C2×C22⋊D191—24no—
51C2×C22⋊D191—22no—
52C2×C22⋊D191—22no—
53D4×D191—24no—
54D4×D191—24no—
55D4×D191—48no—
56C22×Dic191—21no—
57C2×C4×D191—21no—
58C2×C4×D191D412yes—
59C2×C4×D191C2311yescharacteristic
60#15690276—24no—
61#15690275—24no—
62#15690274—24no—
63D1528—24no—
64#15690272—24no—
65#15690272—24no—
66#15690270D418yescomplemented
67C2×C191⋊C8C2312yes—
68Dic191⋊C4—22no—
69(C2×D191)⋊C4—22no—
70C4×Dic191D412yes—
71C4×Dic191C2311yescharacteristic
72C2×Q8×C191C2311yescharacteristic
73C2×D4×C191—22no—
74C2×D4×C191C2311yescharacteristic
75C2×D4×C191C2312yes—
76SD16×C191—24no—
77D8×C191—24no—
78C2×C8×C191C2312yes—
79C22⋊C4×C191—22no—
80C42×C191C2311yescharacteristic
81C22×D191—21no—
82C22×D191—22no—
83C22×D191—48no—
84C22×D191—44no—
85C22×D191C2×D411yescharacteristic
86Dic382—22no—
87D764—22no—
88D764—22no—
89D764—22no—
90D764—44no—
91D764—42no—
92C22⋊D191—42no—
93C22⋊D191—44no—
94C22⋊D191—48no—
95C22⋊D191—44no—
96C2×Dic191—21no—
97C2×Dic191—42no—
98C2×Dic191C2×D411yescharacteristic
99C2×Dic191C2×D412yes—
100C4×D191—24no—
101C4×D191—42no—
102C4×D191C2×D414yes—
103C191⋊C8C2×D414yescomplemented
104C23×C191—21no—
105C23×C191—22no—
106Q8×C191—22no—
107D4×C191—22no—
108D4×C191—22no—
109D4×C191—22no—
110D4×C191—44no—
111C2×C4×C191—21no—
112C2×C4×C191C2×D412yes—
113C2×C4×C191C2411yescharacteristic
114C8×C191C2×D414yescomplemented
115C22×C191—41no—
116C22×C191—42no—
117C22×C191—44no—
118C22×C191—42no—
119C22×C191C22×D411yescharacteristic
120C2×D191—22no—
121C2×D191—41no—
122C2×D191—42no—
123C2×D191—44no—
124C2×D191—42no—
125C2×D191—88no—
126C2×D191#12712yes—
127C4×C191—22no—
128C4×C191—41no—
129C4×C191C22×D412yes—
130Dic191—41no—
131Dic191#12712yes—
132Dic191AGL(1,ℤ/8)14yescomplemented
133C2×C191—82no—
134C2×C191—81no—
135C2×C191C2×#12711yescharacteristic
136C2×C191C2×AGL(1,ℤ/8)12yes—
137D191—22no—
138D191—81no—
139D191—82no—
140C191#315111yescharacteristic, complemented, Sylow
141#3151—1911nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1912no—
143C2×AGL(1,ℤ/8)—1912no—
144#532—1911no—
145C2×#127—1911no—
146#496—1911no—
147#496—1911no—
148#496—1911no—
149#496D19111yescharacteristic, complemented
150#493—1912no—
151#492—1912no—
152C4×M16—1911no—
153C2×#41—1911no—
154C2×#41—1911no—
155C22×D4—3822no—
156C22×D4—1912no—
157AGL(1,ℤ/8)—3828no—
158AGL(1,ℤ/8)—3828no—
159C2×SD16—1912no—
160C2×SD16—1912no—
161C2×SD16—1912no—
162C2×SD16C2×D19112yescomplemented
163C2×D8—1912no—
164C2×D8—1912no—
165C2×D8—1912no—
166C2×D8C2×D19112yescomplemented
167C2×M16—1912no—
168C4⋊Q8—1911no—
169#127—3822no—
170#127—1912no—
171#127—1911no—
172#127—1911no—
173#127C2×D19111yescharacteristic
174#124—1911no—
175#124C2×D19111yescharacteristic, complemented
176#121—3822no—
177C4×D4—3822no—
178C2×C42—1911no—
179#97—1911no—
180#97C2×D19111yescharacteristic, complemented
181C4×C8—1912no—
182C24—3822no—
183#41—3824no—
184#41—3824no—
185C2×Q8—1911no—
186C2×Q8C22×D19111yescharacteristic
187C2×D4—22no—
188C2×D4—3821no—
189C2×D4—3821no—
190C2×D4—3824no—
191C2×D4—3824no—
192C2×D4—3822no—
193C2×D4—3822no—
194C2×D4—3824no—
195C2×D4—3822no—
196C2×D4—1911no—
197C2×D4—1912no—
198C2×D4—7648no—
199C2×D4C22×D19111yescharacteristic
200C2×D4C22×D19112yes—
201C22×C4—3821no—
202C22×C4—3821no—
203C22×C4—1912no—
204C22×C4—1911no—
205SD16—24no—
206SD16—3824no—
207SD16—3824no—
208SD16—3824no—
209D8—24no—
210D8—3824no—
211D8—3824no—
212D8—3824no—
213M16—1918no—
214C2×C8—1912no—
215C2×C8C22×D19112yes—
216C4⋊C4—3822no—
217C22⋊C4—22no—
218C22⋊C4—3822no—
219C42—1912no—
220C42—1911no—
221C42C22×D19111yescharacteristic
222C23—21no—
223C23—22no—
224C23—3821no—
225C23—3822no—
226C23—1911no—
227C23—7648no—
228C23—7644no—
229Q8—22no—
230Q8—3822no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—3822no—
236D4—3822no—
237D4—3822no—
238D4—7644no—
239D4—7642no—
240D4—7644no—
241D4—7648no—
242D4—7644no—
243D4—7642no—
244C2×C4—21no—
245C2×C4—3824no—
246C2×C4—3821no—
247C2×C4—1912no—
248C2×C4—1911no—
249C2×C4—1914no—
250C2×C4—7642no—
251C2×C4—7642no—
252C2×C4D4×D19112yes—
253C2×C4C23×D19111yescharacteristic
254C8—1914no—
255C8D4×D19114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—3822no—
261C22—1912no—
262C22—15288no—
263C22—7644no—
264C22—7642no—
265C22—7641no—
266C22—7642no—
267C22C2×D4×D19111yescharacteristic
268C4—22no—
269C4—41no—
270C4—1914no—
271C4—1912no—
272C4—7641no—
273C4C2×D4×D19112yes—
274C2—82no—
275C2—81no—
276C2—3822no—
277C2—15282no—
278C2—15281no—
279C2#127×D19111yescharacteristic
280C2AGL(1,ℤ/8)×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#496×C191C2
socleC22×C191C22×D4
hypercenter#496D191
hypocenterC191#3151
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. C2×#20259346C2semidirect
3. #27133212C2semidirect
4. #127×D191C2semidirect
5. #27133053C2semidirect
6. #27133052C2semidirect
7. #27133050C2semidirect
8. #27133038C2semidirect
9. #27133034C2semidirect
10. #27132615C2semidirect
11. #496×C191C2semidirect
18. C2×#15690276C22semidirect
19. C2×#15690275C22semidirect
20. C2×#15690274C22semidirect
21. C2×D1528C22semidirect
22. C2×#15690272C22semidirect
23. C2×#15690272C22semidirect
25. #20259320C22semidirect
29. #20259318C22semidirect
30. #20259310C22semidirect
34. #20259233C22semidirect
35. C8×Dic191C22semidirect
36. C2×SD16×C191C22semidirect
37. C2×D8×C191C22semidirect
39. #124×C191C22semidirect
40. #97×C191C22semidirect
66. #15690270D4semidirect
103. C191⋊C8C2×D4semidirect
114. C8×C191C2×D4semidirect
132. Dic191AGL(1,ℤ/8)semidirect
140. C191#3151semidirect
149. #496D191semidirect
162. C2×SD16C2×D191semidirect
166. C2×D8C2×D191semidirect
175. #124C2×D191semidirect
180. #97C2×D191semidirect
255. C8D4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593492
C2×#202593462
#271332122
#127×D1912
#271330532
#271330522
#271330502
#271330382
#271330342
#271326152
#496×C1912
#3151191