Finite Group Databasebeta

← #34056299

Subgroups — #34056299

Order 19072.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011939C211yesmaximal, characteristic, complemented
2C2×#18011938C211yesmaximal, characteristic, complemented
3#23750272C211yesmaximal, characteristic, complemented
4#23750266C211yesmaximal, characteristic, complemented
5#23750006C211yesmaximal, characteristic, complemented
6#23749872C211yesmaximal, characteristic, complemented
7#23749843C211yesmaximal, characteristic, complemented
8#23749839C211yesmaximal, characteristic, complemented
9#23749790C211yesmaximal, characteristic, complemented
10#23749785C211yesmaximal, characteristic, complemented
11#23749760C211yesmaximal, characteristic, complemented
12#23749758C211yesmaximal, characteristic, complemented
13#23749707C211yesmaximal, characteristic, complemented
14#23749701C211yesmaximal, characteristic, complemented
15#468×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×#14558736C2211yescharacteristic
18C22×D596C2211yescharacteristic, complemented
19C2×D4×D149C2211yescharacteristic, complemented
20#18011939—24no—
21#18011938—24no—
22C2×#14558726C2211yescharacteristic, complemented
23C2×#14558725C2211yescharacteristic, complemented
24C2×#14558725C2211yescharacteristic, complemented
25C2×#14558723C2211yescharacteristic, complemented
26C2×D1192C2211yescharacteristic
27C2×D1192C2211yescharacteristic
28C2×#14558721C2211yescharacteristic
29C2×#14558721C2211yescharacteristic
30C2×#14558719C2211yescharacteristic, complemented
31#18011900C2211yescharacteristic, complemented
32#18011898C2211yescharacteristic, complemented
33#18011896—21no—
34#18011893—21no—
35#18011891C2211yescharacteristic, complemented
36#18011886—21no—
37#18011886C2211yescharacteristic
38#18011882—21no—
39#18011881—21no—
40#18011879—21no—
41#18011873C2211yescharacteristic, complemented
42#18011873C2211yescharacteristic, complemented
43C4×C22⋊D149C2211yescharacteristic
44C22⋊C4×D149—21no—
45C22⋊C4×D149—21no—
46#18011840C2211yescharacteristic, complemented
47#18011834C2211yescharacteristic, complemented
48#18011831C2211yescharacteristic, complemented
49#18011831C2211yescharacteristic, complemented
50#18011831C2211yescharacteristic, complemented
51C22⋊#11924746C2211yescharacteristic, complemented
52#18011823C2211yescharacteristic, complemented
53C8×Dic149C2211yescharacteristic, complemented
54C2×SD16×C149C2211yescharacteristic, complemented
55C2×M16×C149C2211yescharacteristic, complemented
56C22⋊Q8×C149C2211yescharacteristic, complemented
57#121×C149C2211yescharacteristic
58#107×C149C2211yescharacteristic, complemented
59#103×C149C2211yescharacteristic, complemented
60#102×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62C23×D149—21no—
63#14558737—22no—
64#14558737—41no—
65#14558736—22no—
66#14558736—41no—
67C2×Dic298C2311yescharacteristic
68C2×D596—22no—
69C2×D596—21no—
70C2×D596—21no—
71C2×D596—21no—
72C2×D596—41no—
73C2×D596C2311yescharacteristic
74C2×D596C2311yescharacteristic
75C2×D596C2311yescharacteristic
76C2×C22⋊D149—21no—
77C2×C22⋊D149—21no—
78C2×C22⋊D149D411yescharacteristic
79D4×D149—22no—
80D4×D149—41no—
81C2×C4×D149—21no—
82C2×C4×D149—21no—
83C2×C4×D149—21no—
84C2×C4×D149—21no—
85C2×C4×D149C2311yescharacteristic
86#14558726—41no—
87#14558725—22no—
88#14558725—22no—
89#14558723—22no—
90D1192—22no—
91D1192D414yescomplemented
92#14558721—22no—
93#14558721—41no—
94#14558719—22no—
95C2×#11924746C2311yescharacteristic
96C2×#11924746C2311yescharacteristic
97C4⋊Dic149—21no—
98#14558715D411yescharacteristic
99C22⋊Dic149—21no—
100C22⋊Dic149—21no—
101C22⋊Dic149D411yescharacteristic
102#14558713—21no—
103#14558713—21no—
104#14558713—21no—
105#14558713—21no—
106#14558713—21no—
107#14558713—21no—
108#14558713D411yescharacteristic
109C4×Dic149C2311yescharacteristic
110C2×Q8×C149C2311yescharacteristic
111C2×D4×C149—21no—
112C2×D4×C149C2311yescharacteristic
113C22×C4×C149C2311yescharacteristic
114SD16×C149—22no—
115M16×C149—22no—
116C2×C8×C149C2311yescharacteristic
117C2×C8×C149C2311yescharacteristic
118C4⋊C4×C149—21no—
119C4⋊C4×C149C2311yescharacteristic
120C4⋊C4×C149C2311yescharacteristic
121C22⋊C4×C149—21no—
122C22⋊C4×C149—21no—
123C22×D149—21no—
124C22×D149—21no—
125C22×D149—21no—
126C22×D149—42no—
127C22×D149—42no—
128C22×D149—42no—
129C22×D149—42no—
130C22×D149—81no—
131C22×D149—81no—
132C22×D149C2×D411yescharacteristic
133Dic298—21no—
134Dic298—21no—
135D596—21no—
136D596—21no—
137D596—21no—
138D596—21no—
139D596—42no—
140D596—42no—
141D596—81no—
142D596C2×D412yes—
143D596C2×D412yes—
144C22⋊D149—41no—
145C22⋊D149—42no—
146C22⋊D149—42no—
147C2×Dic149—21no—
148C2×Dic149C2×D411yescharacteristic
149C2×Dic149C2×D411yescharacteristic
150C2×Dic149C2×D411yescharacteristic
151C4×D149—21no—
152C4×D149—21no—
153C4×D149—42no—
154C4×D149—42no—
155C4×D149—81no—
156C4×D149—81no—
157#11924746—21no—
158#11924746—21no—
159C23×C149—21no—
160C23×C149C2×D411yescharacteristic
161Q8×C149—21no—
162Q8×C149—21no—
163D4×C149—21no—
164D4×C149—21no—
165D4×C149—81no—
166C2×C4×C149—21no—
167C2×C4×C149—21no—
168C2×C4×C149—21no—
169C2×C4×C149—21no—
170C2×C4×C149—21no—
171C2×C4×C149C2×D411yescharacteristic
172C2×C4×C149C2411yescharacteristic
173C8×C149—21no—
174C8×C149C2×D412yes—
175C22×C149—21no—
176C22×C149—21no—
177C22×C149—21no—
178C22×C149—41no—
179C22×C149—41no—
180C22×C149—41no—
181C22×C149C22×D411yescharacteristic
182C2×D149—21no—
183C2×D149—22no—
184C2×D149—22no—
185C2×D149—21no—
186C2×D149—21no—
187C2×D149—41no—
188C2×D149—41no—
189C2×D149—41no—
190C2×D149—41no—
191C2×D149—41no—
192C2×D149—41no—
193C2×D149—41no—
194C2×D149—84no—
195C2×D149—84no—
196C4×C149—21no—
197C4×C149—41no—
198C4×C149—41no—
199C4×C149—41no—
200C4×C149C22×D411yescharacteristic
201C4×C149H(5,𝔽2)11yescharacteristic
202Dic149—21no—
203Dic149—21no—
204Dic149—21no—
205Dic149—41no—
206C2×C149—41no—
207C2×C149—81no—
208C2×C149D4211yescharacteristic
209C2×C149#57511yescharacteristic
210C2×C149#57611yescharacteristic
211D149—41no—
212D149—42no—
213D149—42no—
214D149—81no—
215C149#327311yescharacteristic, complemented, Sylow
216#3273—1491nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1491no—
218C2×AGL(1,ℤ/8)—1491no—
219#546—1491no—
220#546—1491no—
221#495—1491no—
222#468—1491no—
223#468D14911yescharacteristic, complemented
224#463—1491no—
225#463—1491no—
226#452—1491no—
227#452—1491no—
228#447—1491no—
229#447—1491no—
230C4×D8—1491no—
231#436—1491no—
232C2×#41—1491no—
233C2×#41—1491no—
234C22×D4—1491no—
235C22×D4—1491no—
236AGL(1,ℤ/8)—2984no—
237AGL(1,ℤ/8)—2984no—
238C2×Q16—1491no—
239C2×SD16—1491no—
240C2×SD16—1491no—
241C2×SD16—1491no—
242C2×SD16C2×D14911yescharacteristic, complemented
243C2×D8—1491no—
244C2×D8—1491no—
245C2×D8—1491no—
246C2×D8—1491no—
247C2×M16—1491no—
248C2×M16C2×D14911yescharacteristic, complemented
249#124—1491no—
250#124—1491no—
251#123—2981no—
252#123—2981no—
253C22⋊Q8—1491no—
254C22⋊Q8C2×D14911yescharacteristic, complemented
255#121—2981no—
256#121—2981no—
257#121—1491no—
258#121C2×D14911yescharacteristic
259#120—2981no—
260#120—2981no—
261C4×D4—1491no—
262C4×D4—1491no—
263C4×D4—1491no—
264C2×C22⋊C4—2981no—
265C2×C22⋊C4—2981no—
266#107C2×D14911yescharacteristic, complemented
267C4⋊C8—1491no—
268#103—1491no—
269#103C2×D14911yescharacteristic, complemented
270#102—1491no—
271#102—1491no—
272#102—1491no—
273#102C2×D14911yescharacteristic, complemented
274C22⋊C8—1491no—
275C22⋊C8—1491no—
276C4×C8—1491no—
277C24—2981no—
278C24—2981no—
279#41—2982no—
280#41—2982no—
281#41—5961no—
282#41—5961no—
283C2×Q8—1491no—
284C2×Q8C22×D14911yescharacteristic
285C2×D4—21no—
286C2×D4—2982no—
287C2×D4—2981no—
288C2×D4—2981no—
289C2×D4—2982no—
290C2×D4—2981no—
291C2×D4—2981no—
292C2×D4—2981no—
293C2×D4—5961no—
294C2×D4—5961no—
295C2×D4—1491no—
296C2×D4—1491no—
297C2×D4—1491no—
298C2×D4—1491no—
299C2×D4C22×D14911yescharacteristic
300C22×C4—2981no—
301C22×C4—2981no—
302C22×C4—2981no—
303C22×C4—2981no—
304C22×C4—1491no—
305C22×C4C22×D14911yescharacteristic
306Q16—5961no—
307SD16—22no—
308SD16—2982no—
309SD16—2982no—
310SD16—2982no—
311D8—2982no—
312D8—2982no—
313D8—5961no—
314D8—1494no—
315M16—22no—
316M16—2982no—
317C2×C8—1491no—
318C2×C8—1491no—
319C2×C8C22×D14911yescharacteristic
320C2×C8C22×D14911yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—2981no—
323C4⋊C4—1491no—
324C4⋊C4C22×D14911yescharacteristic
325C4⋊C4C22×D14911yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—2981no—
329C22⋊C4—2981no—
330C22⋊C4—2981no—
331C22⋊C4—2981no—
332C22⋊C4—2981no—
333C22⋊C4—2981no—
334C22⋊C4—2981no—
335C22⋊C4—2981no—
336C22⋊C4—1491no—
337C22⋊C4—1491no—
338C42—1491no—
339C23—21no—
340C23—2981no—
341C23—2981no—
342C23—2981no—
343C23—5962no—
344C23—5962no—
345C23—5962no—
346C23—5962no—
347C23—1491no—
348C23—11921no—
349C23—11921no—
350C23D4×D14911yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2981no—
354Q8—2981no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—2981no—
359D4—2981no—
360D4—2981no—
361D4—2981no—
362D4—5962no—
363D4—5962no—
364D4—5961no—
365D4—5962no—
366D4—5962no—
367D4—1492no—
368D4—1492no—
369D4—11921no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—2981no—
376C2×C4—2981no—
377C2×C4—2981no—
378C2×C4—5962no—
379C2×C4—5962no—
380C2×C4—1491no—
381C2×C4—1491no—
382C2×C4—1491no—
383C2×C4—11921no—
384C2×C4—11921no—
385C2×C4D4×D14911yescharacteristic
386C2×C4C23×D14911yescharacteristic
387C8—21no—
388C8—2981no—
389C8—2981no—
390C8D4×D14912yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—2981no—
398C22—2981no—
399C22—2981no—
400C22—2982no—
401C22—2982no—
402C22—5961no—
403C22—5961no—
404C22—5961no—
405C22—5961no—
406C22—5961no—
407C22—5961no—
408C22—5961no—
409C22—11924no—
410C22—11924no—
411C22C2×D4×D14911yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—2981no—
417C4—2981no—
418C4—2981no—
419C4—5961no—
420C4C2×D4×D14911yescharacteristic
421C4#1801195711yescharacteristic
422C2—41no—
423C2—81no—
424C2—5961no—
425C2—5962no—
426C2—5962no—
427C2—11921no—
428C2#2375025911yescharacteristic
429C2#2375042411yescharacteristic
430C2#2375042911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#468×C149C2
socleC22×C149C22×D4
hypercenter#468D149
hypocenterC149#3273
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#18011939C2semidirect
2. C2×#18011938C2semidirect
3. #23750272C2semidirect
4. #23750266C2semidirect
5. #23750006C2semidirect
6. #23749872C2semidirect
7. #23749843C2semidirect
8. #23749839C2semidirect
9. #23749790C2semidirect
10. #23749785C2semidirect
11. #23749760C2semidirect
12. #23749758C2semidirect
13. #23749707C2semidirect
14. #23749701C2semidirect
15. #468×C149C2semidirect
18. C22×D596C22semidirect
19. C2×D4×D149C22semidirect
22. C2×#14558726C22semidirect
23. C2×#14558725C22semidirect
24. C2×#14558725C22semidirect
25. C2×#14558723C22semidirect
30. C2×#14558719C22semidirect
31. #18011900C22semidirect
32. #18011898C22semidirect
35. #18011891C22semidirect
41. #18011873C22semidirect
42. #18011873C22semidirect
46. #18011840C22semidirect
47. #18011834C22semidirect
48. #18011831C22semidirect
49. #18011831C22semidirect
50. #18011831C22semidirect
51. C22⋊#11924746C22semidirect
52. #18011823C22semidirect
53. C8×Dic149C22semidirect
54. C2×SD16×C149C22semidirect
55. C2×M16×C149C22semidirect
56. C22⋊Q8×C149C22semidirect
58. #107×C149C22semidirect
59. #103×C149C22semidirect
60. #102×C149C22semidirect
91. D1192D4semidirect
215. C149#3273semidirect
223. #468D149semidirect
242. C2×SD16C2×D149semidirect
248. C2×M16C2×D149semidirect
254. C22⋊Q8C2×D149semidirect
266. #107C2×D149semidirect
269. #103C2×D149semidirect
273. #102C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119392
C2×#180119382
#237502722
#237502662
#237500062
#237498722
#237498432
#237498392
#237497902
#237497852
#237497602
#237497582
#237497072
#237497012
#468×C1492
#3273149