Finite Group Databasebeta

← #37582063

Subgroups — #37582063

Order 24448.

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

normal subgroups only

This group has 504 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×#20259342C211yesmaximal, characteristic, complemented
2#27133230C211yesmaximal, characteristic, complemented
3#27133146C211yesmaximal, characteristic, complemented
4#27132883C212yesmaximal, complemented
5#27132876C212yesmaximal, complemented
6#27132861C212yesmaximal, complemented
7#27132856C212yesmaximal, complemented
8#27132691C211yesmaximal, characteristic, complemented
9#27132665C211yesmaximal, characteristic, complemented
10C2×#20259243C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13#20259342—24no—
14C2×#15690277C2211yescharacteristic, complemented
15C2×#15690275C2211yescharacteristic, complemented
16C2×#15690274C2211yescharacteristic, complemented
17C2×#15690272C2211yescharacteristic, complemented
18C22×C191⋊C8C2211yescharacteristic
19#20259318C2212yes—
20#20259309C2212yes—
21#20259293—21no—
22#20259293—21no—
23#20259291—22no—
24Q8×Dic191C2212yescomplemented
25D4×Dic191—22no—
26D4×Dic191C2212yescomplemented
27#20259281—21no—
28#20259281C2211yescharacteristic
29C2×C4⋊Dic191—21no—
30C2×C4⋊Dic191C2211yescharacteristic
31Dic191⋊C8C2212yescomplemented
32Dic191⋊C8C2212yescomplemented
33#20259247C2212yescomplemented
34#20259244C2212yescomplemented
35#20259243C2212yes—
36#20259243C2212yes—
37C2×#41×C191C2211yescharacteristic
38C22×D4×C191C2211yescharacteristic
39AGL(1,ℤ/8)×C191—24no—
40C2×SD16×C191C2212yes—
41C2×D8×C191C2212yes—
42C2×M16×C191C2211yescharacteristic
43#15690287—22no—
44#15690287—22no—
45C2×Dic382C2311yescharacteristic
46C2×D764C2311yescharacteristic
47C2×C22⋊D191—22no—
48C2×C22⋊D191—22no—
49C2×C22⋊D191—21no—
50C22×Dic191—22no—
51C22×Dic191—21no—
52C2×C4×D191—21no—
53#15690277—22no—
54#15690275—22no—
55#15690274—22no—
56#15690272—22no—
57C2×C191⋊C8—22no—
58C2×C191⋊C8C2311yescharacteristic
59C2×C191⋊C8C2311yescharacteristic
60C4⋊Dic191—22no—
61C4⋊Dic191—21no—
62C4⋊Dic191D411yescharacteristic
63C4⋊Dic191D411yescharacteristic
64C4⋊Dic191C2312yes—
65C22⋊Dic191—22no—
66C22⋊Dic191—22no—
67C22⋊Dic191—22no—
68C22⋊Dic191D412yes—
69C4×Dic191—22no—
70C4×Dic191C2312yes—
71C24×C191—21no—
72#41×C191—22no—
73#41×C191D414yescomplemented
74C2×Q8×C191C2311yescharacteristic
75C2×D4×C191—21no—
76C2×D4×C191—22no—
77C2×D4×C191—21no—
78C2×D4×C191—22no—
79C2×D4×C191C2312yes—
80C2×D4×C191C2311yescharacteristic
81C22×C4×C191—21no—
82C22×C4×C191C2311yescharacteristic
83SD16×C191—24no—
84D8×C191—24no—
85M16×C191—22no—
86C2×C8×C191C2312yes—
87C22×D191—21no—
88Dic382—21no—
89Dic382—21no—
90D764—21no—
91D764—21no—
92C22⋊D191—42no—
93C22⋊D191—82no—
94C22⋊D191—82no—
95C2×Dic191—22no—
96C2×Dic191—21no—
97C2×Dic191—82no—
98C2×Dic191—81no—
99C2×Dic191C2×D412yes—
100C2×Dic191C2×D412yes—
101C4×D191—42no—
102C191⋊C8—21no—
103C191⋊C8—21no—
104C23×C191—21no—
105C23×C191—22no—
106C23×C191—42no—
107C23×C191—81no—
108C23×C191—81no—
109C23×C191C2×D411yescharacteristic
110Q8×C191—21no—
111Q8×C191C2×D412yes—
112D4×C191—24no—
113D4×C191—22no—
114D4×C191—21no—
115D4×C191—22no—
116D4×C191—42no—
117D4×C191C2×D412yes—
118C2×C4×C191—21no—
119C2×C4×C191—24no—
120C2×C4×C191C2×D411yescharacteristic
121C2×C4×C191C2×D412yes—
122C2×C4×C191#4112yes—
123C2×C4×C191C2411yescharacteristic
124C8×C191—22no—
125C22×C191—21no—
126C22×C191—21no—
127C22×C191—22no—
128C22×C191—41no—
129C22×C191—41no—
130C22×C191—42no—
131C22×C191—42no—
132C22×C191—42no—
133C22×C191—84no—
134C22×C191#12112yes—
135C22×C191C22×D411yescharacteristic
136C2×D191—41no—
137C2×D191—41no—
138C2×D191—41no—
139C4×C191—22no—
140C4×C191#12112yes—
141C4×C191C22×D411yescharacteristic
142C4×C191C2×#4111yescharacteristic
143Dic191—22no—
144Dic191—22no—
145Dic191—41no—
146Dic191—42no—
147C2×C191—22no—
148C2×C191—42no—
149C2×C191—82no—
150C2×C191C2×#12111yescharacteristic
151C2×C191#57611yescharacteristic
152C2×C191#57711yescharacteristic
153D191—81no—
154C191#304511yescharacteristic, complemented, Sylow
155#3045—1911nomaximal, Sylow
156C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
157C2×#135—1911no—
158#535—1911no—
159#520—1911no—
160#464—1912no—
161#463—1912no—
162#460—1912no—
163#459—1912no—
164#424—1911no—
165#417—1911no—
166C2×#102—1911no—
167C2×#41—1911no—
168C2×#41C2×D19111yescharacteristic
169C22×D4C2×D19111yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—3824no—
172C2×Q16—1911no—
173C2×SD16—1911no—
174C2×SD16—1911no—
175C2×SD16C2×D19112yes—
176C2×D8—1911no—
177C2×D8C2×D19112yes—
178C2×M16C2×D19111yescharacteristic
179C22×C8—1911no—
180#127—1912no—
181#124—1912no—
182#121—3821no—
183#121—3821no—
184#120—3822no—
185C4×Q8—1912no—
186C4×D4—3822no—
187C4×D4—1912no—
188#117—3821no—
189#117—1911no—
190C2×C4⋊C4—3821no—
191C2×C4⋊C4—1911no—
192C4⋊C8—1912no—
193C4⋊C8—1912no—
194#103—1912no—
195#102—1912no—
196#102—1912no—
197#102—1912no—
198C24—21no—
199#41—22no—
200#41—3822no—
201#41—3822no—
202#41C22⋊D19114yescomplemented
203C2×Q8—1911no—
204C2×Q8C22×D19111yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—3821no—
210C2×D4—3822no—
211C2×D4—3822no—
212C2×D4—1911no—
213C2×D4C22×D19112yes—
214C2×D4C22×D19111yescharacteristic
215C22×C4—21no—
216C22×C4—3821no—
217C22×C4—3822no—
218C22×C4—3821no—
219C22×C4C22×D19111yescharacteristic
220Q16—3822no—
221SD16—24no—
222SD16—3822no—
223SD16—3822no—
224D8—24no—
225D8—3822no—
226M16—22no—
227C2×C8—3822no—
228C2×C8—1911no—
229C2×C8—1911no—
230C2×C8C22×D19112yes—
231C4⋊C4—3822no—
232C4⋊C4—3821no—
233C4⋊C4—1912no—
234C4⋊C4—1911no—
235C4⋊C4—1911no—
236C22⋊C4—3822no—
237C22⋊C4—3822no—
238C22⋊C4—3822no—
239C22⋊C4—1912no—
240C42—3822no—
241C42—1912no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—3821no—
248C23D4×D19111yescharacteristic
249Q8—21no—
250Q8—3821no—
251Q8—3821no—
252Q8C2×C22⋊D19112yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—3821no—
259D4—3821no—
260D4—15282no—
261D4—15282no—
262D4—7642no—
263D4C2×C22⋊D19112yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—3822no—
267C2×C4—3821no—
268C2×C4—1912no—
269C2×C4—1912no—
270C2×C4—15282no—
271C2×C4—15281no—
272C2×C4—7642no—
273C2×C4D4×D19111yescharacteristic
274C2×C4C2×C22⋊D19112yes—
275C2×C4#1569028612yes—
276C2×C4C23×D19111yescharacteristic
277C8—22no—
278C8—3821no—
279C8—3821no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—7641no—
290C22—7641no—
291C22—7641no—
292C22#2025929412yes—
293C22C2×D4×D19111yescharacteristic
294C4—22no—
295C4—3822no—
296C4—3822no—
297C4—7642no—
298C4—7641no—
299C4#2025929412yes—
300C4C22×C22⋊D19111yescharacteristic
301C4C2×#1569028611yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—15281no—
306C2C2×#2025929411yescharacteristic
307C2#2713346711yescharacteristic
308C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×AGL(1,ℤ/8)×C191C2
socleC22×C191C22×D4
hypercenterC2×AGL(1,ℤ/8)D191
hypocenterC191#3045
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20259342C2semidirect
2. #27133230C2semidirect
3. #27133146C2semidirect
4. #27132883C2semidirect
5. #27132876C2semidirect
6. #27132861C2semidirect
7. #27132856C2semidirect
8. #27132691C2semidirect
9. #27132665C2semidirect
10. C2×#20259243C2semidirect
11. C2×AGL(1,ℤ/8)×C191C2semidirect
14. C2×#15690277C22semidirect
15. C2×#15690275C22semidirect
16. C2×#15690274C22semidirect
17. C2×#15690272C22semidirect
24. Q8×Dic191C22semidirect
26. D4×Dic191C22semidirect
31. Dic191⋊C8C22semidirect
32. Dic191⋊C8C22semidirect
33. #20259247C22semidirect
34. #20259244C22semidirect
73. #41×C191D4semidirect
154. C191#3045semidirect
156. C2×AGL(1,ℤ/8)D191semidirect
202. #41C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593422
#271332302
#271331462
#271328832
#271328762
#271328612
#271328562
#271326912
#271326652
C2×#202592432
C2×AGL(1,ℤ/8)×C1912
#3045191