Finite Group Databasebeta

← #34053990

Subgroups — #34053990

Order 19072.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011956C211yesmaximal, characteristic, complemented
2C2×#18011935C211yesmaximal, characteristic, complemented
3C2×#18011930C211yesmaximal, characteristic, complemented
4#23750204C211yesmaximal, characteristic, complemented
5#23749789C211yesmaximal, characteristic, complemented
6#23749788C211yesmaximal, characteristic, complemented
7#23749777C211yesmaximal, characteristic, complemented
8#23749776C211yesmaximal, characteristic, complemented
9#23749768C211yesmaximal, characteristic, complemented
10#23749767C211yesmaximal, characteristic, complemented
11#23749757C211yesmaximal, characteristic, complemented
12#23749623C211yesmaximal, characteristic, complemented
13#102×D149C211yesmaximal, characteristic, complemented
14#23749590C211yesmaximal, characteristic, complemented
15#447×C149C211yesmaximal, characteristic, complemented
16#18011956—24no—
17#18011956—24no—
18C2×#14558736C2211yescharacteristic
19C2×#14558736C2211yescharacteristic
20C2×#14558735—21no—
21C2×#14558735C2211yescharacteristic
22C2×#14558735C2211yescharacteristic
23C2×#14558735C2211yescharacteristic, complemented
24C22×C22⋊D149—21no—
25C22×C22⋊D149—21no—
26C2×D4×D149—21no—
27C2×D4×D149C2211yescharacteristic
28C2×D4×D149C2211yescharacteristic
29#18011935—24no—
30#18011930—24no—
31C2×Dic596C2211yescharacteristic, complemented
32C2×#14558725C2211yescharacteristic, complemented
33C2×#14558724C2211yescharacteristic, complemented
34C2×#14558724C2211yescharacteristic, complemented
35C2×#14558724C2211yescharacteristic, complemented
36C2×#14558721C2211yescharacteristic, complemented
37C2×#14558719C2211yescharacteristic, complemented
38C2×C8×D149C2211yescharacteristic, complemented
39#18011891C2211yescharacteristic, complemented
40#18011890C2211yescharacteristic, complemented
41#18011888—21no—
42#18011884—21no—
43#18011883—21no—
44#18011882—21no—
45#18011882C2211yescharacteristic, complemented
46C4⋊C4×D149C2211yescharacteristic, complemented
47#18011836C2211yescharacteristic, complemented
48#18011834C2211yescharacteristic, complemented
49#18011833C2211yescharacteristic, complemented
50#18011832C2211yescharacteristic, complemented
51#18011832C2211yescharacteristic, complemented
52#18011831C2211yescharacteristic, complemented
53C22⋊#11924746C2211yescharacteristic, complemented
54#18011823C2211yescharacteristic, complemented
55#18011823C2211yescharacteristic, complemented
56C22×D4×C149C2211yescharacteristic
57C2×D8×C149C2211yescharacteristic, complemented
58C2×D8×C149C2211yescharacteristic, complemented
59#121×C149C2211yescharacteristic, complemented
60#102×C149C2211yescharacteristic, complemented
61#102×C149C2211yescharacteristic, complemented
62C22⋊C8×C149C2211yescharacteristic, complemented
63C23×D149—21no—
64C23×D149—21no—
65#14558736—22no—
66#14558736—22no—
67#14558736—22no—
68#14558736—22no—
69#14558735—28no—
70#14558735—22no—
71#14558735—22no—
72#14558735—22no—
73#14558735—22no—
74#14558735—22no—
75#14558735D414yescomplemented
76C2×Dic298D411yescharacteristic
77C2×Dic298C2311yescharacteristic
78C2×Dic298C2311yescharacteristic
79C2×D596D411yescharacteristic
80C2×D596C2311yescharacteristic
81C2×C22⋊D149—28no—
82C2×C22⋊D149—28no—
83C2×C22⋊D149—21no—
84C2×C22⋊D149—21no—
85C2×C22⋊D149—21no—
86C2×C22⋊D149—21no—
87C2×C22⋊D149—21no—
88C2×C22⋊D149—21no—
89C2×C22⋊D149—21no—
90C2×C22⋊D149D411yescharacteristic
91C2×C22⋊D149D411yescharacteristic
92D4×D149—28no—
93D4×D149—22no—
94D4×D149—22no—
95D4×D149—22no—
96D4×D149D414yescomplemented
97C22×Dic149—21no—
98C22×Dic149—21no—
99C22×Dic149—21no—
100C2×C4×D149—21no—
101C2×C4×D149—21no—
102C2×C4×D149C2311yescharacteristic
103C2×C4×D149C2311yescharacteristic
104Dic596—22no—
105#14558725—22no—
106#14558724—22no—
107#14558724—22no—
108#14558724—22no—
109#14558721—22no—
110#14558719—22no—
111C2×#11924746C2311yescharacteristic
112C2×#11924746C2311yescharacteristic
113C8×D149—22no—
114C4⋊Dic149—21no—
115C4⋊Dic149C2311yescharacteristic
116#14558715—21no—
117#14558715—21no—
118C22⋊Dic149—21no—
119C22⋊Dic149—21no—
120#14558713—21no—
121C24×C149—21no—
122C2×D4×C149—22no—
123C2×D4×C149—22no—
124C2×D4×C149—21no—
125C2×D4×C149—21no—
126C2×D4×C149C2311yescharacteristic
127C2×D4×C149C2311yescharacteristic
128C2×D4×C149C2311yescharacteristic
129C22×C4×C149C2311yescharacteristic
130D8×C149—22no—
131D8×C149—22no—
132C2×C8×C149C2311yescharacteristic
133C2×C8×C149C2311yescharacteristic
134C4⋊C4×C149C2311yescharacteristic
135C22⋊C4×C149—21no—
136C22×D149—21no—
137C22×D149—24no—
138C22×D149—24no—
139C22×D149—81no—
140C22×D149—81no—
141C22×D149—81no—
142C22×D149—81no—
143C22×D149C2×D411yescharacteristic
144C22×D149C2×D411yescharacteristic
145Dic298—21no—
146Dic298—22no—
147Dic298—21no—
148Dic298—21no—
149Dic298C2×D412yes—
150D596—22no—
151D596—21no—
152D596C2×D412yes—
153C22⋊D149—24no—
154C22⋊D149—24no—
155C22⋊D149—24no—
156C22⋊D149—24no—
157C22⋊D149—22no—
158C22⋊D149—24no—
159C22⋊D149—22no—
160C22⋊D149—42no—
161C22⋊D149—81no—
162C2×Dic149—24no—
163C2×Dic149—21no—
164C2×Dic149—24no—
165C2×Dic149—21no—
166C2×Dic149—21no—
167C2×Dic149—42no—
168C2×Dic149C2×D411yescharacteristic
169C2×Dic149C2×D411yescharacteristic
170C4×D149—24no—
171C4×D149—21no—
172C4×D149—21no—
173C4×D149—81no—
174C4×D149C2×D412yes—
175C4×D149C2×D412yes—
176#11924746—21no—
177#11924746—21no—
178C23×C149—24no—
179C23×C149—21no—
180C23×C149—21no—
181C23×C149—21no—
182C23×C149—81no—
183C23×C149—81no—
184C23×C149C2×D411yescharacteristic
185D4×C149—24no—
186D4×C149—21no—
187D4×C149—21no—
188D4×C149—21no—
189D4×C149—21no—
190D4×C149—81no—
191D4×C149C2×D412yes—
192D4×C149C2×D412yes—
193C2×C4×C149—21no—
194C2×C4×C149—21no—
195C2×C4×C149—21no—
196C2×C4×C149C2×D411yescharacteristic
197C2×C4×C149C2411yescharacteristic
198C8×C149—21no—
199C8×C149—21no—
200C22×C149—21no—
201C22×C149—22no—
202C22×C149—22no—
203C22×C149—21no—
204C22×C149—21no—
205C22×C149—41no—
206C22×C149—41no—
207C22×C149—41no—
208C22×C149—41no—
209C22×C149—41no—
210C22×C149—41no—
211C22×C149—41no—
212C22×C149—84no—
213C22×C149C22×D411yescharacteristic
214C2×D149—21no—
215C2×D149—21no—
216C2×D149—21no—
217C2×D149—22no—
218C2×D149—21no—
219C2×D149—21no—
220C2×D149—41no—
221C2×D149—41no—
222C2×D149—84no—
223C2×D149—84no—
224C2×D149#12012yes—
225C4×C149—21no—
226C4×C149—41no—
227C4×C149C22×D411yescharacteristic
228C4×C149C22×D411yescharacteristic
229Dic149—22no—
230Dic149—21no—
231Dic149—41no—
232Dic149—41no—
233Dic149#12012yes—
234C2×C149—42no—
235C2×C149—42no—
236C2×C149—41no—
237C2×C149—81no—
238C2×C149C2×#12011yescharacteristic
239C2×C149#57611yescharacteristic
240C2×C149#57711yescharacteristic
241D149—22no—
242D149—41no—
243D149—42no—
244C149#301211yescharacteristic, complemented, Sylow
245#3012—1491nomaximal, Sylow
246C2×H(5,𝔽2)—1491no—
247C2×AGL(1,ℤ/8)—1491no—
248C2×#135—1491no—
249#537—1491no—
250#452—1491no—
251#452—1491no—
252#450—1491no—
253#450—1491no—
254#449—1491no—
255#449—1491no—
256#447—1491no—
257#447D14911yescharacteristic, complemented
258#417—1491no—
259C2×#102—1491no—
260#408—1491no—
261H(5,𝔽2)—2984no—
262H(5,𝔽2)—2984no—
263C2×#41—2981no—
264C2×#41—1491no—
265C2×#41—1491no—
266C2×#41—1491no—
267C2×#41—1491no—
268C2×#41—1491no—
269C22×D4—2981no—
270C22×D4—2981no—
271C22×D4—2981no—
272C22×D4—1491no—
273C22×D4—1491no—
274C22×D4C2×D14911yescharacteristic
275AGL(1,ℤ/8)—2984no—
276#135—2984no—
277C2×Q16—1491no—
278C2×SD16—1491no—
279C2×SD16—1491no—
280C2×SD16—1491no—
281C2×SD16—1491no—
282C2×D8—1491no—
283C2×D8C2×D14911yescharacteristic, complemented
284C2×D8C2×D14911yescharacteristic, complemented
285C2×M16—1491no—
286C22×C8—1491no—
287C22⋊Q8—2981no—
288C22⋊Q8—1491no—
289C22⋊Q8—1491no—
290#121—2981no—
291#121—2981no—
292#121—2981no—
293#121—1491no—
294#121C2×D14911yescharacteristic, complemented
295C2×C4⋊C4—1491no—
296#103—1491no—
297#103—1491no—
298#102—1491no—
299#102—1491no—
300#102—1491no—
301#102—1491no—
302#102C2×D14911yescharacteristic, complemented
303#102C2×D14911yescharacteristic, complemented
304C22⋊C8—1491no—
305C22⋊C8—1491no—
306C22⋊C8—1491no—
307C22⋊C8C2×D14911yescharacteristic, complemented
308C24—21no—
309C24—2981no—
310C24—2981no—
311#41—2982no—
312#41—2982no—
313#41—2982no—
314#41—2982no—
315#41—2988no—
316#41—2982no—
317#41—2982no—
318#41—2982no—
319#41—2982no—
320#41—2982no—
321#41—1494no—
322C2×Q8—1491no—
323C2×Q8—1491no—
324C2×Q8—1491no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—2988no—
330C2×D4—2982no—
331C2×D4—2988no—
332C2×D4—2988no—
333C2×D4—2981no—
334C2×D4—2981no—
335C2×D4—2981no—
336C2×D4—2982no—
337C2×D4—2981no—
338C2×D4—2981no—
339C2×D4—2982no—
340C2×D4—2981no—
341C2×D4—2981no—
342C2×D4—1491no—
343C2×D4—1491no—
344C2×D4—1494no—
345C2×D4—1491no—
346C2×D4—1491no—
347C2×D4C22×D14911yescharacteristic
348C2×D4C22×D14911yescharacteristic
349C2×D4C22×D14911yescharacteristic
350C22×C4—2981no—
351C22×C4—2981no—
352C22×C4—2981no—
353C22×C4—2981no—
354C22×C4—2981no—
355C22×C4—1491no—
356C22×C4—1491no—
357C22×C4C22×D14911yescharacteristic
358Q16—2982no—
359SD16—2982no—
360SD16—2982no—
361SD16—2982no—
362SD16—2982no—
363D8—22no—
364D8—22no—
365D8—2982no—
366M16—2982no—
367C2×C8—2982no—
368C2×C8—1491no—
369C2×C8—1491no—
370C2×C8C22×D14911yescharacteristic
371C2×C8C22×D14911yescharacteristic
372C4⋊C4—2981no—
373C4⋊C4—2981no—
374C4⋊C4—2981no—
375C4⋊C4—1491no—
376C4⋊C4C22×D14911yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—2981no—
379C22⋊C4—2981no—
380C22⋊C4—2981no—
381C23—24no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—81no—
386C23—81no—
387C23—2984no—
388C23—2984no—
389C23—2981no—
390C23—1491no—
391C23—1491no—
392C23—11921no—
393C23—11921no—
394C23—11921no—
395C23—11921no—
396C23D4×D14911yescharacteristic
397Q8—2982no—
398Q8—2981no—
399Q8—2981no—
400Q8—2981no—
401Q8—1492no—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—21no—
407D4—81no—
408D4—2982no—
409D4—2984no—
410D4—2984no—
411D4—2984no—
412D4—2984no—
413D4—2982no—
414D4—2982no—
415D4—2984no—
416D4—2981no—
417D4—5962no—
418D4—1492no—
419D4—11921no—
420D4D4×D14912yes—
421D4D4×D14912yes—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—2984no—
426C2×C4—2984no—
427C2×C4—2981no—
428C2×C4—2984no—
429C2×C4—2981no—
430C2×C4—2981no—
431C2×C4—2981no—
432C2×C4—2981no—
433C2×C4—5962no—
434C2×C4—1491no—
435C2×C4—1491no—
436C2×C4—1492no—
437C2×C4—1492no—
438C2×C4—11921no—
439C2×C4D4×D14911yescharacteristic
440C2×C4C23×D14911yescharacteristic
441C8—21no—
442C8—21no—
443C8—2981no—
444C8—2981no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—84no—
458C22—2981no—
459C22—2981no—
460C22—2981no—
461C22—2981no—
462C22—2981no—
463C22—2982no—
464C22—5961no—
465C22—5961no—
466C22—1492no—
467C22—11924no—
468C22—11924no—
469C22C2×D4×D14911yescharacteristic
470C4—21no—
471C4—41no—
472C4—2982no—
473C4—2981no—
474C4—5961no—
475C4—5961no—
476C4—1492no—
477C4C2×D4×D14911yescharacteristic
478C4C2×D4×D14911yescharacteristic
479C2—41no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—2982no—
484C2—5961no—
485C2—5962no—
486C2#120×D14911yescharacteristic
487C2#2375042611yescharacteristic
488C2#2375043511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#447×C149C2
socleC22×C149C22×D4
hypercenter#447D149
hypocenterC149#3012
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. C2×#18011956C2semidirect
2. C2×#18011935C2semidirect
3. C2×#18011930C2semidirect
4. #23750204C2semidirect
5. #23749789C2semidirect
6. #23749788C2semidirect
7. #23749777C2semidirect
8. #23749776C2semidirect
9. #23749768C2semidirect
10. #23749767C2semidirect
11. #23749757C2semidirect
12. #23749623C2semidirect
13. #102×D149C2semidirect
14. #23749590C2semidirect
15. #447×C149C2semidirect
23. C2×#14558735C22semidirect
31. C2×Dic596C22semidirect
32. C2×#14558725C22semidirect
33. C2×#14558724C22semidirect
34. C2×#14558724C22semidirect
35. C2×#14558724C22semidirect
36. C2×#14558721C22semidirect
37. C2×#14558719C22semidirect
38. C2×C8×D149C22semidirect
39. #18011891C22semidirect
40. #18011890C22semidirect
45. #18011882C22semidirect
46. C4⋊C4×D149C22semidirect
47. #18011836C22semidirect
48. #18011834C22semidirect
49. #18011833C22semidirect
50. #18011832C22semidirect
51. #18011832C22semidirect
52. #18011831C22semidirect
53. C22⋊#11924746C22semidirect
54. #18011823C22semidirect
55. #18011823C22semidirect
57. C2×D8×C149C22semidirect
58. C2×D8×C149C22semidirect
59. #121×C149C22semidirect
60. #102×C149C22semidirect
61. #102×C149C22semidirect
62. C22⋊C8×C149C22semidirect
75. #14558735D4semidirect
96. D4×D149D4semidirect
244. C149#3012semidirect
257. #447D149semidirect
283. C2×D8C2×D149semidirect
284. C2×D8C2×D149semidirect
294. #121C2×D149semidirect
302. #102C2×D149semidirect
303. #102C2×D149semidirect
307. C22⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119562
C2×#180119352
C2×#180119302
#237502042
#237497892
#237497882
#237497772
#237497762
#237497682
#237497672
#237497572
#237496232
#102×D1492
#237495902
#447×C1492
#3012149