Finite Group Databasebeta

← #34056361

Subgroups — #34056361

Order 19072.

620 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14558721C211yesmaximal, characteristic, complemented
2C2×D8×D149C211yesmaximal, characteristic, complemented
3#23750259C211yesmaximal, characteristic, complemented
4D4×C22⋊D149C211yesmaximal, characteristic, complemented
5#23750000C211yesmaximal, characteristic, complemented
6#23749860C211yesmaximal, characteristic, complemented
7#23749859C211yesmaximal, characteristic, complemented
8#23749819C211yesmaximal, characteristic, complemented
9#23749818C211yesmaximal, characteristic, complemented
10#23749761C211yesmaximal, characteristic, complemented
11#23749760C211yesmaximal, characteristic, complemented
12#23749757C211yesmaximal, characteristic, complemented
13#23749705C211yesmaximal, characteristic, complemented
14#23749688C211yesmaximal, characteristic, complemented
15#447×C149C211yesmaximal, characteristic, complemented
16C22×D596C2211yescharacteristic, complemented
17C22×C22⋊D149—21no—
18C2×D4×D149—21no—
19C2×D4×D149C2211yescharacteristic, complemented
20C2×D4×D149C2211yescharacteristic, complemented
21C2×D1192C2211yescharacteristic, complemented
22C2×D1192C2211yescharacteristic, complemented
23C2×#14558721—24no—
24C2×#14558721C2211yescharacteristic
25C2×#14558721C2211yescharacteristic, complemented
26C2×#14558721C2211yescharacteristic, complemented
27C2×#14558721C2211yescharacteristic, complemented
28C2×#14558721C2211yescharacteristic, complemented
29D8×D149—24no—
30C22×#11924746C2211yescharacteristic, complemented
31C2×C8×D149C2211yescharacteristic, complemented
32#18011907C2211yescharacteristic, complemented
33#18011907C2211yescharacteristic, complemented
34#18011886C2211yescharacteristic, complemented
35#18011885C2211yescharacteristic, complemented
36#18011884—21no—
37#18011883—21no—
38#18011882C2211yescharacteristic, complemented
39#18011881—21no—
40#18011880—21no—
41#18011879—21no—
42#18011879—21no—
43D4×Dic149C2211yescharacteristic, complemented
44#18011873C2211yescharacteristic, complemented
45C4×C22⋊D149C2211yescharacteristic
46#18011844C2211yescharacteristic, complemented
47#18011840C2211yescharacteristic, complemented
48#18011833C2211yescharacteristic, complemented
49#18011833C2211yescharacteristic, complemented
50#18011832C2211yescharacteristic, complemented
51#18011831C2211yescharacteristic, complemented
52#18011823C2211yescharacteristic, complemented
53C8×Dic149C2211yescharacteristic, complemented
54C22×D4×C149C2211yescharacteristic, complemented
55C2×D8×C149C2211yescharacteristic, complemented
56C2×D8×C149C2211yescharacteristic, complemented
57#121×C149C2211yescharacteristic, complemented
58#102×C149C2211yescharacteristic, complemented
59#102×C149C2211yescharacteristic, complemented
60C22⋊C8×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62C23×D149—21no—
63C23×D149—21no—
64C2×D596—22no—
65C2×D596—21no—
66C2×D596—21no—
67C2×D596—41no—
68C2×D596C2311yescharacteristic
69C2×D596C2311yescharacteristic
70C2×D596C2311yescharacteristic
71C2×C22⋊D149—21no—
72C2×C22⋊D149—21no—
73C2×C22⋊D149—21no—
74C2×C22⋊D149—21no—
75C2×C22⋊D149—21no—
76C2×C22⋊D149—21no—
77C2×C22⋊D149—21no—
78C2×C22⋊D149—21no—
79C2×C22⋊D149—44no—
80C2×C22⋊D149D411yescharacteristic
81D4×D149—22no—
82D4×D149—22no—
83D4×D149—41no—
84D4×D149—41no—
85D4×D149—44no—
86C22×Dic149—21no—
87C2×C4×D149—21no—
88C2×C4×D149C2311yescharacteristic
89D1192—22no—
90D1192—41no—
91#14558721—22no—
92#14558721—22no—
93#14558721—22no—
94#14558721—22no—
95#14558721D414yescomplemented
96C2×#11924746—22no—
97C2×#11924746C2311yescharacteristic
98C2×#11924746C2311yescharacteristic
99C8×D149—22no—
100C4⋊Dic149C2311yescharacteristic
101#14558715D411yescharacteristic
102C22⋊Dic149—21no—
103C22⋊Dic149—21no—
104C22⋊Dic149D411yescharacteristic
105#14558713—21no—
106#14558713—21no—
107#14558713—21no—
108#14558713D411yescharacteristic
109C4×Dic149C2311yescharacteristic
110C24×C149—21no—
111C2×D4×C149—22no—
112C2×D4×C149—21no—
113C2×D4×C149—21no—
114C2×D4×C149—41no—
115C2×D4×C149C2311yescharacteristic
116C2×D4×C149C2311yescharacteristic
117C2×D4×C149C2311yescharacteristic
118C22×C4×C149C2311yescharacteristic
119D8×C149—22no—
120D8×C149—41no—
121C2×C8×C149C2311yescharacteristic
122C2×C8×C149C2311yescharacteristic
123C4⋊C4×C149C2311yescharacteristic
124C22⋊C4×C149—21no—
125C22×D149—21no—
126C22×D149—21no—
127C22×D149—21no—
128C22×D149—42no—
129C22×D149—42no—
130C22×D149—42no—
131C22×D149—42no—
132C22×D149—42no—
133C22×D149—42no—
134C22×D149—81no—
135C22×D149—81no—
136C22×D149—81no—
137C22×D149C2×D411yescharacteristic
138D596—21no—
139D596—21no—
140D596—21no—
141D596—21no—
142D596—21no—
143D596—42no—
144D596—42no—
145D596C2×D412yescomplemented
146C22⋊D149—42no—
147C22⋊D149—42no—
148C22⋊D149—42no—
149C22⋊D149—42no—
150C22⋊D149—42no—
151C22⋊D149—42no—
152C22⋊D149—81no—
153C22⋊D149—81no—
154C22⋊D149D814yescomplemented
155C2×Dic149—21no—
156C2×Dic149—42no—
157C2×Dic149C2×D411yescharacteristic
158C2×Dic149C2×D411yescharacteristic
159C2×Dic149C2×D411yescharacteristic
160C4×D149—21no—
161C4×D149—21no—
162C4×D149—42no—
163#11924746—21no—
164#11924746C2×D412yescomplemented
165C23×C149—21no—
166C23×C149—21no—
167C23×C149—21no—
168C23×C149—42no—
169C23×C149—42no—
170C23×C149—81no—
171C23×C149C2×D411yescharacteristic
172D4×C149—21no—
173D4×C149—21no—
174D4×C149—21no—
175D4×C149—21no—
176D4×C149—21no—
177D4×C149—42no—
178D4×C149—42no—
179D4×C149C2×D412yescomplemented
180C2×C4×C149—21no—
181C2×C4×C149—21no—
182C2×C4×C149—21no—
183C2×C4×C149C2×D411yescharacteristic
184C2×C4×C149C2411yescharacteristic
185C8×C149—21no—
186C8×C149—21no—
187C22×C149—21no—
188C22×C149—21no—
189C22×C149—22no—
190C22×C149—41no—
191C22×C149—41no—
192C22×C149—41no—
193C22×C149—41no—
194C22×C149—41no—
195C22×C149—41no—
196C22×C149—41no—
197C22×C149—41no—
198C22×C149—84no—
199C22×C149C2×D812yescomplemented
200C22×C149C22×D411yescharacteristic
201C2×D149—21no—
202C2×D149—21no—
203C2×D149—22no—
204C2×D149—41no—
205C2×D149—41no—
206C2×D149—41no—
207C2×D149—41no—
208C2×D149—41no—
209C2×D149—41no—
210C2×D149—41no—
211C2×D149—41no—
212C2×D149—84no—
213C2×D149—84no—
214C2×D149—84no—
215C2×D149C2×D812yescomplemented
216C4×C149—21no—
217C4×C149—41no—
218C4×C149C22×D411yescharacteristic
219C4×C149H(5,𝔽2)11yescharacteristic
220Dic149—21no—
221Dic149—21no—
222Dic149—41no—
223Dic149C2×D812yescomplemented
224C2×C149—22no—
225C2×C149—42no—
226C2×C149—81no—
227C2×C149—81no—
228C2×C149D4211yescharacteristic
229C2×C149C22×D811yescharacteristic
230C2×C149#57611yescharacteristic
231D149—22no—
232D149—42no—
233D149—81no—
234D149—81no—
235C149D4×D811yescharacteristic, complemented, Sylow
236D4×D8—1491nomaximal, Sylow
237C22×D8—1491no—
238C22×D8—1491no—
239D42—1491no—
240D42—1491no—
241#493—1491no—
242#466—1491no—
243#466—1491no—
244#459—1491no—
245#459—1491no—
246#447—1491no—
247#447—1491no—
248#447—1491no—
249#447D14911yescharacteristic, complemented
250C4×D8—1491no—
251C8×D4—1491no—
252C22×D4—2981no—
253C22×D4—2981no—
254C22×D4—1491no—
255C22×D4—1491no—
256C22×D4—1491no—
257C22×D4C2×D14911yescharacteristic, complemented
258C2×D8—2984no—
259C2×D8—2984no—
260C2×D8—1491no—
261C2×D8—1491no—
262C2×D8—1491no—
263C2×D8—1491no—
264C2×D8—1491no—
265C2×D8—1491no—
266C2×D8—1491no—
267C2×D8C2×D14911yescharacteristic, complemented
268C2×D8C2×D14911yescharacteristic, complemented
269C22×C8—1491no—
270C22×C8—1491no—
271#127—1491no—
272#127—1491no—
273#121—2981no—
274#121—2981no—
275#121—1491no—
276#121—1491no—
277#121—1491no—
278#121C2×D14911yescharacteristic, complemented
279#120—2981no—
280#120—2981no—
281#120—2981no—
282#120—2981no—
283C4×D4—1491no—
284C4×D4—1491no—
285C4×D4—1491no—
286#107—1491no—
287C4⋊C8—1491no—
288#102—1491no—
289#102—1491no—
290#102—1491no—
291#102—1491no—
292#102C2×D14911yescharacteristic, complemented
293#102C2×D14911yescharacteristic, complemented
294C22⋊C8—1491no—
295C22⋊C8C2×D14911yescharacteristic, complemented
296C4×C8—1491no—
297C24—21no—
298C24—2981no—
299C24—2981no—
300C24—2981no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—2982no—
306C2×D4—2981no—
307C2×D4—2982no—
308C2×D4—2982no—
309C2×D4—2981no—
310C2×D4—2981no—
311C2×D4—2981no—
312C2×D4—2981no—
313C2×D4—2981no—
314C2×D4—2981no—
315C2×D4—2981no—
316C2×D4—2981no—
317C2×D4—2981no—
318C2×D4—5961no—
319C2×D4—5961no—
320C2×D4—5961no—
321C2×D4—5964no—
322C2×D4—5964no—
323C2×D4—1491no—
324C2×D4—1491no—
325C2×D4—1491no—
326C2×D4—1491no—
327C2×D4C22×D14911yescharacteristic
328C2×D4C22×D14911yescharacteristic
329C2×D4C22×D14911yescharacteristic
330C22×C4—2981no—
331C22×C4—2981no—
332C22×C4—1491no—
333C22×C4C22×D14911yescharacteristic
334D8—22no—
335D8—41no—
336D8—2982no—
337D8—2982no—
338D8—2982no—
339D8—2982no—
340D8—2982no—
341D8—5961no—
342D8—1494no—
343C2×C8—2982no—
344C2×C8—2982no—
345C2×C8—1491no—
346C2×C8—1491no—
347C2×C8C22×D14911yescharacteristic
348C2×C8C22×D14911yescharacteristic
349C4⋊C4—1491no—
350C4⋊C4—1491no—
351C4⋊C4C22×D14911yescharacteristic
352C22⋊C4—21no—
353C22⋊C4—2981no—
354C22⋊C4—2981no—
355C22⋊C4—2981no—
356C22⋊C4—2981no—
357C22⋊C4—2981no—
358C22⋊C4—1491no—
359C22⋊C4—1491no—
360C42—1491no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—42no—
365C23—42no—
366C23—81no—
367C23—2981no—
368C23—2981no—
369C23—2981no—
370C23—5962no—
371C23—5962no—
372C23—5962no—
373C23—5962no—
374C23—5962no—
375C23—5962no—
376C23—1491no—
377C23—11921no—
378C23—11921no—
379C23—11921no—
380C23D4×D14911yescharacteristic
381D4—21no—
382D4—21no—
383D4—21no—
384D4—21no—
385D4—21no—
386D4—42no—
387D4—42no—
388D4—2981no—
389D4—2981no—
390D4—2981no—
391D4—2981no—
392D4—2981no—
393D4—5962no—
394D4—5962no—
395D4—5962no—
396D4—5962no—
397D4—5962no—
398D4—5962no—
399D4—5962no—
400D4—5962no—
401D4—1494no—
402D4—1492no—
403D4—11921no—
404D4—11921no—
405D4D4×D14912yescomplemented
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—2981no—
410C2×C4—2981no—
411C2×C4—2981no—
412C2×C4—5962no—
413C2×C4—5962no—
414C2×C4—1491no—
415C2×C4—1491no—
416C2×C4—1491no—
417C2×C4D4×D14911yescharacteristic
418C2×C4C23×D14911yescharacteristic
419C8—21no—
420C8—21no—
421C8—2981no—
422C8—1492no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—84no—
435C22—2981no—
436C22—2981no—
437C22—2982no—
438C22—5961no—
439C22—5961no—
440C22—5961no—
441C22—5961no—
442C22—5961no—
443C22—5961no—
444C22—5961no—
445C22—5961no—
446C22—1492no—
447C22—11924no—
448C22—11924no—
449C22—11924no—
450C22D8×D14912yescomplemented
451C22C2×D4×D14911yescharacteristic
452C4—21no—
453C4—41no—
454C4—2981no—
455C4—2981no—
456C4—5961no—
457C4—1492no—
458C4C2×D4×D14911yescharacteristic
459C4#1801195611yescharacteristic
460C2—22no—
461C2—42no—
462C2—81no—
463C2—81no—
464C2—2982no—
465C2—5962no—
466C2—11921no—
467C2—11921no—
468C2#2375025711yescharacteristic
469C2C2×D8×D14911yescharacteristic
470C2#2375042911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#447×C149C2
socleC22×C149C22×D4
hypercenter#447D149
hypocenterC149D4×D8
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#14558721C2semidirect
2. C2×D8×D149C2semidirect
3. #23750259C2semidirect
4. D4×C22⋊D149C2semidirect
5. #23750000C2semidirect
6. #23749860C2semidirect
7. #23749859C2semidirect
8. #23749819C2semidirect
9. #23749818C2semidirect
10. #23749761C2semidirect
11. #23749760C2semidirect
12. #23749757C2semidirect
13. #23749705C2semidirect
14. #23749688C2semidirect
15. #447×C149C2semidirect
16. C22×D596C22semidirect
19. C2×D4×D149C22semidirect
20. C2×D4×D149C22semidirect
21. C2×D1192C22semidirect
22. C2×D1192C22semidirect
25. C2×#14558721C22semidirect
26. C2×#14558721C22semidirect
27. C2×#14558721C22semidirect
28. C2×#14558721C22semidirect
30. C22×#11924746C22semidirect
31. C2×C8×D149C22semidirect
32. #18011907C22semidirect
33. #18011907C22semidirect
34. #18011886C22semidirect
35. #18011885C22semidirect
38. #18011882C22semidirect
43. D4×Dic149C22semidirect
44. #18011873C22semidirect
46. #18011844C22semidirect
47. #18011840C22semidirect
48. #18011833C22semidirect
49. #18011833C22semidirect
50. #18011832C22semidirect
51. #18011831C22semidirect
52. #18011823C22semidirect
53. C8×Dic149C22semidirect
54. C22×D4×C149C22semidirect
55. C2×D8×C149C22semidirect
56. C2×D8×C149C22semidirect
57. #121×C149C22semidirect
58. #102×C149C22semidirect
59. #102×C149C22semidirect
60. C22⋊C8×C149C22semidirect
95. #14558721D4semidirect
145. D596C2×D4semidirect
154. C22⋊D149D8semidirect
164. #11924746C2×D4semidirect
179. D4×C149C2×D4semidirect
199. C22×C149C2×D8semidirect
215. C2×D149C2×D8semidirect
223. Dic149C2×D8semidirect
235. C149D4×D8semidirect
249. #447D149semidirect
257. C22×D4C2×D149semidirect
267. C2×D8C2×D149semidirect
268. C2×D8C2×D149semidirect
278. #121C2×D149semidirect
292. #102C2×D149semidirect
293. #102C2×D149semidirect
295. C22⋊C8C2×D149semidirect
405. D4D4×D149semidirect
450. C22D8×D149semidirect

Maximal subgroups

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

subgroupindex
C22×#145587212
C2×D8×D1492
#237502592
D4×C22⋊D1492
#237500002
#237498602
#237498592
#237498192
#237498182
#237497612
#237497602
#237497572
#237497052
#237496882
#447×C1492
D4×D8149