Finite Group Databasebeta

← #37582030

Subgroups — #37582030

Order 24448.

494 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic
2C2×SD16×D191C211yesmaximal, characteristic, complemented
3#27133233C211yesmaximal, characteristic, complemented
4#27133135C211yesmaximal, characteristic, complemented
5#27132877C211yesmaximal, characteristic, complemented
6#27132876C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132869C211yesmaximal, characteristic, complemented
9#27132868C211yesmaximal, characteristic, complemented
10#27132867C211yesmaximal, characteristic, complemented
11#27132866C211yesmaximal, characteristic, complemented
12#27132688C211yesmaximal, characteristic, complemented
13#27132670C211yesmaximal, characteristic, complemented
14#27132656C211yesmaximal, characteristic, complemented
15#463×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259346—24no—
20C2×#15690276C2211yescharacteristic
21C2×#15690276C2211yescharacteristic
22C2×#15690275C2211yescharacteristic
23C2×#15690275C2211yescharacteristic
24C2×#15690274C2211yescharacteristic
25SD16×D191—24no—
26C2×#15690272C2211yescharacteristic
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259322C2211yescharacteristic, complemented
30#20259320C2211yescharacteristic, complemented
31#20259309C2211yescharacteristic, complemented
32#20259304—21no—
33#20259304—21no—
34#20259299—21no—
35#20259299—21no—
36C4×D764C2211yescharacteristic
37D4×Dic191C2211yescharacteristic
38#20259284C2211yescharacteristic
39C4×C22⋊D191—21no—
40(C22⋊D191)⋊C4—21no—
41#20259280C2211yescharacteristic
42#20259279C2211yescharacteristic
43C2×(C2×D191)⋊C4—21no—
44C22⋊C4×D191—21no—
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46Dic191⋊C8C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259247C2211yescharacteristic, complemented
50#20259246C2211yescharacteristic, complemented
51#20259245C2211yescharacteristic, complemented
52#20259245C2211yescharacteristic, complemented
53#20259243C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic
55C2×D8×C191C2211yescharacteristic
56#124×C191C2211yescharacteristic, complemented
57C4×D4×C191C2211yescharacteristic
58C4⋊C8×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690286—22no—
63#15690286—22no—
64C2×Dic382—21no—
65C2×Dic382C2311yescharacteristic
66C2×Dic382C2311yescharacteristic
67Q8×D191—22no—
68Q8×D191—22no—
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72D4×D191—22no—
73D4×D191D414yes—
74C22×Dic191—21no—
75C22×Dic191—21no—
76C2×C4×D191—21no—
77C2×C4×D191—21no—
78C2×C4×D191—21no—
79C2×C4×D191C2311yescharacteristic
80#15690276—22no—
81#15690276—22no—
82#15690275—22no—
83#15690275—22no—
84#15690274—22no—
85#15690272—22no—
86#15690270—22no—
87C2×C191⋊C8C2311yescharacteristic
88C2×C191⋊C8C2311yescharacteristic
89C8×D191—22no—
90C4⋊Dic191—21no—
91C4⋊Dic191—21no—
92C4⋊Dic191C2311yescharacteristic
93Dic191⋊C4—21no—
94Dic191⋊C4—21no—
95Dic191⋊C4D411yescharacteristic
96Dic191⋊C4D411yescharacteristic
97C22⋊Dic191—21no—
98C22⋊Dic191—21no—
99C22⋊Dic191—21no—
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4D411yescharacteristic
104(C2×D191)⋊C4D411yescharacteristic
105C4×Dic191—21no—
106C4×Dic191C2311yescharacteristic
107C2×Q8×C191C2311yescharacteristic
108C2×D4×C191C2311yescharacteristic
109C2×D4×C191C2311yescharacteristic
110C22×C4×C191—21no—
111SD16×C191—22no—
112D8×C191—22no—
113C2×C8×C191C2311yescharacteristic
114C2×C8×C191C2311yescharacteristic
115C4⋊C4×C191C2311yescharacteristic
116C22⋊C4×C191—21no—
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C42×C191C2311yescharacteristic
120C22×D191—21no—
121C22×D191—24no—
122C22×D191—81no—
123C22×D191—81no—
124C22×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128Dic382—21no—
129Dic382—42no—
130D764—21no—
131D764C2×D412yes—
132C22⋊D191—24no—
133C22⋊D191—42no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191—21no—
137C2×Dic191—42no—
138C2×Dic191—81no—
139C2×Dic191C2×D411yescharacteristic
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C4×D191—42no—
143C4×D191—81no—
144C4×D191—81no—
145C4×D191C2×D412yes—
146C4×D191#4112yes—
147C191⋊C8—21no—
148C191⋊C8—21no—
149C23×C191—21no—
150C23×C191—21no—
151Q8×C191—21no—
152Q8×C191—21no—
153D4×C191—21no—
154D4×C191—21no—
155D4×C191—21no—
156D4×C191C2×D412yes—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—81no—
160C2×C4×C191C2×D411yescharacteristic
161C2×C4×C191C2×D411yescharacteristic
162C2×C4×C191C2411yescharacteristic
163C8×C191—21no—
164C8×C191—21no—
165C22×C191—22no—
166C22×C191—41no—
167C22×C191—41no—
168C22×C191—41no—
169C22×C191—41no—
170C22×C191—41no—
171C22×C191C22×D411yescharacteristic
172C2×D191—21no—
173C2×D191—21no—
174C2×D191—22no—
175C2×D191—41no—
176C2×D191—41no—
177C2×D191—84no—
178C2×D191#12112yes—
179C4×C191—21no—
180C4×C191—21no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191C22×D411yescharacteristic
184C4×C191C2×#4111yescharacteristic
185Dic191—21no—
186Dic191—21no—
187Dic191—41no—
188Dic191—41no—
189Dic191—41no—
190Dic191#12112yes—
191C2×C191—42no—
192C2×C191—81no—
193C2×C191C2×#12111yescharacteristic
194C2×C191#57711yescharacteristic
195C2×C191#57711yescharacteristic
196D191—22no—
197D191—42no—
198C191#304211yescharacteristic, complemented, Sylow
199#3042—1911nomaximal, Sylow
200C2×AGL(1,ℤ/8)—1911no—
201C22×SD16—1911no—
202#536—1911no—
203#518—1911no—
204#463—1911no—
205#463—1911no—
206#463—1911no—
207#463D19111yescharacteristic, complemented
208#461—1911no—
209#461—1911no—
210#461—1911no—
211#461—1911no—
212#424—1911no—
213#419—1911no—
214#416—1911no—
215C2×#41—1911no—
216C22×Q8—1911no—
217C22×D4—1911no—
218AGL(1,ℤ/8)—3824no—
219C2×SD16—3824no—
220C2×SD16—1911no—
221C2×SD16—1911no—
222C2×SD16—1911no—
223C2×SD16—1911no—
224C2×SD16—1911no—
225C2×SD16C2×D19111yescharacteristic
226C2×D8—1911no—
227C2×D8C2×D19111yescharacteristic
228C2×M16—1911no—
229C22×C8—1911no—
230C4⋊Q8—1911no—
231C4⋊Q8—1911no—
232#124—1911no—
233#124C2×D19111yescharacteristic, complemented
234#123—3821no—
235#123—3821no—
236C22⋊Q8—3821no—
237C22⋊Q8—3821no—
238C4×D4—3821no—
239C4×D4—3821no—
240C4×D4—1911no—
241C4×D4—1911no—
242C4×D4—1911no—
243C4×D4C2×D19111yescharacteristic
244#117—1911no—
245#117—1911no—
246C2×C22⋊C4—3821no—
247C2×C22⋊C4—3821no—
248C4⋊C8—1911no—
249C4⋊C8—1911no—
250C4⋊C8—1911no—
251C4⋊C8C2×D19111yescharacteristic, complemented
252#103—1911no—
253#103—1911no—
254#103—1911no—
255#103—1911no—
256#103—1911no—
257#103C2×D19111yescharacteristic, complemented
258#102—1911no—
259#102C2×D19111yescharacteristic, complemented
260C24—3821no—
261#41—3822no—
262#41—3822no—
263C2×Q8—3821no—
264C2×Q8—3822no—
265C2×Q8—3822no—
266C2×Q8—1911no—
267C2×Q8—1911no—
268C2×Q8C22×D19111yescharacteristic
269C2×D4—3821no—
270C2×D4—3822no—
271C2×D4—3821no—
272C2×D4—1914no—
273C2×D4—1911no—
274C2×D4C22×D19111yescharacteristic
275C2×D4C22×D19111yescharacteristic
276C22×C4—21no—
277C22×C4—3821no—
278C22×C4—3821no—
279C22×C4—3821no—
280C22×C4—3821no—
281C22×C4—3821no—
282C22×C4—1911no—
283SD16—22no—
284SD16—3822no—
285SD16—3822no—
286SD16—3822no—
287SD16—3822no—
288SD16—3822no—
289D8—22no—
290D8—3822no—
291M16—3822no—
292C2×C8—3822no—
293C2×C8—1911no—
294C2×C8—1911no—
295C2×C8C22×D19111yescharacteristic
296C2×C8C22×D19111yescharacteristic
297C4⋊C4—3821no—
298C4⋊C4—3821no—
299C4⋊C4—3821no—
300C4⋊C4—3821no—
301C4⋊C4—1911no—
302C4⋊C4—1911no—
303C4⋊C4—1911no—
304C4⋊C4C22×D19111yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—3821no—
309C22⋊C4—3821no—
310C22⋊C4—3821no—
311C22⋊C4—3821no—
312C22⋊C4—3821no—
313C22⋊C4—3821no—
314C22⋊C4—1911no—
315C22⋊C4—1911no—
316C42—3821no—
317C42—1911no—
318C42C22×D19111yescharacteristic
319C23—21no—
320C23—21no—
321C23—3824no—
322C23—3821no—
323C23—1911no—
324C23—15281no—
325C23—15281no—
326Q8—21no—
327Q8—21no—
328Q8—3821no—
329Q8—3821no—
330Q8—3821no—
331Q8—3821no—
332Q8—7642no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—3824no—
337D4—3821no—
338D4—1912no—
339D4—7642no—
340D4D4×D19112yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—81no—
344C2×C4—3821no—
345C2×C4—3821no—
346C2×C4—3821no—
347C2×C4—1911no—
348C2×C4—1911no—
349C2×C4—1911no—
350C2×C4—1912no—
351C2×C4—1912no—
352C2×C4—15281no—
353C2×C4—15281no—
354C2×C4—15281no—
355C2×C4—7642no—
356C2×C4—7642no—
357C2×C4D4×D19111yescharacteristic
358C2×C4D4×D19111yescharacteristic
359C2×C4C23×D19111yescharacteristic
360C8—21no—
361C8—21no—
362C8—3821no—
363C8—3821no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—3821no—
371C22—3821no—
372C22—3822no—
373C22—1912no—
374C22—15284no—
375C22—7641no—
376C22—7641no—
377C22C2×D4×D19111yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—3821no—
383C4—3821no—
384C4—1912no—
385C4—7641no—
386C4—7641no—
387C4—7641no—
388C4C2×D4×D19111yescharacteristic
389C4#41×D19111yescharacteristic
390C2—42no—
391C2—81no—
392C2—3822no—
393C2—7642no—
394C2#121×D19111yescharacteristic
395C2#2713347011yescharacteristic
396C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#463×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259346C2
hypercenter#463D191
hypocenterC191#3042
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D191C2semidirect
3. #27133233C2semidirect
4. #27133135C2semidirect
5. #27132877C2semidirect
6. #27132876C2semidirect
7. #27132875C2semidirect
8. #27132869C2semidirect
9. #27132868C2semidirect
10. #27132867C2semidirect
11. #27132866C2semidirect
12. #27132688C2semidirect
13. #27132670C2semidirect
14. #27132656C2semidirect
15. #463×C191C2semidirect
29. #20259322C22semidirect
30. #20259320C22semidirect
31. #20259309C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. Dic191⋊C8C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259247C22semidirect
49. #20259247C22semidirect
50. #20259246C22semidirect
51. #20259245C22semidirect
52. #20259245C22semidirect
53. #20259243C22semidirect
56. #124×C191C22semidirect
58. C4⋊C8×C191C22semidirect
59. #103×C191C22semidirect
60. #102×C191C22semidirect
198. C191#3042semidirect
207. #463D191semidirect
233. #124C2×D191semidirect
251. C4⋊C8C2×D191semidirect
257. #103C2×D191semidirect
259. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×SD16×D1912
#271332332
#271331352
#271328772
#271328762
#271328752
#271328692
#271328682
#271328672
#271328662
#271326882
#271326702
#271326562
#463×C1912
#3042191