Finite Group Databasebeta

← #34056285

Subgroups — #34056285

Order 19072.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011935C211yesmaximal, characteristic, complemented
2#23750274C211yesmaximal, characteristic, complemented
3D4×C22⋊D149C211yesmaximal, characteristic, complemented
4#23749996C211yesmaximal, characteristic, complemented
5#23749871C211yesmaximal, characteristic, complemented
6#23749868C211yesmaximal, characteristic, complemented
7#23749828C211yesmaximal, characteristic, complemented
8#23749818C211yesmaximal, characteristic, complemented
9#23749776C211yesmaximal, characteristic, complemented
10#23749774C211yesmaximal, characteristic, complemented
11#23749770C211yesmaximal, characteristic, complemented
12#23749767C211yesmaximal, characteristic, complemented
13D8×Dic149C211yesmaximal, characteristic, complemented
14#23749701C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
16C2×#14558735C2211yescharacteristic, complemented
17C22×C22⋊D149—21no—
18C2×D4×D149C2211yescharacteristic
19#18011935—24no—
20C2×#14558725C2211yescharacteristic, complemented
21C2×#14558724C2211yescharacteristic
22C2×#14558724C2211yescharacteristic, complemented
23C2×#14558723C2211yescharacteristic, complemented
24C2×#14558721C2211yescharacteristic, complemented
25C2×#14558719C2211yescharacteristic, complemented
26#18011909C2211yescharacteristic, complemented
27#18011907C2211yescharacteristic
28#18011894—21no—
29#18011892—21no—
30#18011890C2211yescharacteristic, complemented
31#18011887C2211yescharacteristic, complemented
32#18011885C2211yescharacteristic
33#18011883—21no—
34#18011883—21no—
35#18011882C2211yescharacteristic
36#18011880—21no—
37#18011879—21no—
38D4×Dic149C2211yescharacteristic
39D4×Dic149C2211yescharacteristic, complemented
40C4×C22⋊D149C2211yescharacteristic
41C2×#14558715—21no—
42#18011845C2211yescharacteristic, complemented
43#18011840C2211yescharacteristic, complemented
44#18011836C2211yescharacteristic, complemented
45#18011835C2211yescharacteristic, complemented
46#18011833C2211yescharacteristic, complemented
47#18011832C2211yescharacteristic
48#18011832C2211yescharacteristic, complemented
49#18011832C2211yescharacteristic, complemented
50C22⋊#11924746C2211yescharacteristic, complemented
51#18011823C2211yescharacteristic, complemented
52C8×Dic149C2211yescharacteristic, complemented
53C2×#41×C149C2211yescharacteristic, complemented
54C22×D4×C149C2211yescharacteristic
55AGL(1,ℤ/8)×C149—24no—
56C2×SD16×C149C2211yescharacteristic
57C2×SD16×C149C2211yescharacteristic, complemented
58C2×D8×C149C2211yescharacteristic
59C2×D8×C149C2211yescharacteristic, complemented
60C2×M16×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62#14558735—22no—
63#14558735—41no—
64C2×Dic298C2311yescharacteristic
65C2×D596C2311yescharacteristic
66C2×C22⋊D149—21no—
67C2×C22⋊D149—21no—
68C2×C22⋊D149—21no—
69C2×C22⋊D149—21no—
70C2×C22⋊D149—21no—
71C2×C22⋊D149—21no—
72C2×C22⋊D149—44no—
73C2×C22⋊D149D411yescharacteristic
74D4×D149—22no—
75D4×D149—41no—
76C22×Dic149—21no—
77C22×Dic149—21no—
78C22×Dic149—21no—
79C2×C4×D149C2311yescharacteristic
80#14558725—22no—
81#14558724—22no—
82#14558724—41no—
83#14558723—41no—
84#14558721—22no—
85#14558719—22no—
86C2×#11924746C2311yescharacteristic
87C2×#11924746C2311yescharacteristic
88C4⋊Dic149C2311yescharacteristic
89C4⋊Dic149C2311yescharacteristic
90#14558715—21no—
91#14558715—21no—
92#14558715—21no—
93#14558715—21no—
94#14558715D411yescharacteristic
95C22⋊Dic149—21no—
96C22⋊Dic149—21no—
97C22⋊Dic149—21no—
98C22⋊Dic149—21no—
99C22⋊Dic149D411yescharacteristic
100#14558713—21no—
101#14558713—21no—
102#14558713D411yescharacteristic
103C4×Dic149C2311yescharacteristic
104C24×C149—21no—
105#41×C149—22no—
106#41×C149—41no—
107C2×Q8×C149C2311yescharacteristic
108C2×D4×C149—22no—
109C2×D4×C149—21no—
110C2×D4×C149—21no—
111C2×D4×C149—41no—
112C2×D4×C149C2311yescharacteristic
113C2×D4×C149C2311yescharacteristic
114C2×D4×C149C2311yescharacteristic
115C22×C4×C149—21no—
116C22×C4×C149C2311yescharacteristic
117SD16×C149—22no—
118SD16×C149—22no—
119D8×C149—22no—
120D8×C149D414yescomplemented
121M16×C149—22no—
122C2×C8×C149C2311yescharacteristic
123C2×C8×C149C2311yescharacteristic
124C22×D149—21no—
125C22×D149—42no—
126C22×D149—42no—
127C22×D149—81no—
128C22×D149C2×D411yescharacteristic
129Dic298—21no—
130Dic298—21no—
131D596—21no—
132D596—21no—
133C22⋊D149—22no—
134C22⋊D149—42no—
135C22⋊D149—42no—
136C22⋊D149—42no—
137C22⋊D149—42no—
138C22⋊D149—42no—
139C22⋊D149—81no—
140C2×Dic149—21no—
141C2×Dic149—21no—
142C2×Dic149—21no—
143C2×Dic149—42no—
144C2×Dic149—42no—
145C2×Dic149—42no—
146C2×Dic149C2×D411yescharacteristic
147C2×Dic149C2×D411yescharacteristic
148C2×Dic149C2×D411yescharacteristic
149C4×D149—21no—
150C4×D149—21no—
151#11924746—21no—
152#11924746—21no—
153C23×C149—21no—
154C23×C149—21no—
155C23×C149—21no—
156C23×C149—42no—
157C23×C149—42no—
158C23×C149—81no—
159C23×C149C2×D411yescharacteristic
160Q8×C149—21no—
161Q8×C149—21no—
162D4×C149—21no—
163D4×C149—21no—
164D4×C149—21no—
165D4×C149—21no—
166D4×C149—42no—
167D4×C149—42no—
168D4×C149C2×D412yes—
169D4×C149C2×D412yescomplemented
170C2×C4×C149—21no—
171C2×C4×C149—21no—
172C2×C4×C149—21no—
173C2×C4×C149—42no—
174C2×C4×C149C2×D411yescharacteristic
175C2×C4×C149C2411yescharacteristic
176C8×C149—21no—
177C8×C149C2×D412yescomplemented
178C22×C149—21no—
179C22×C149—22no—
180C22×C149—22no—
181C22×C149—21no—
182C22×C149—21no—
183C22×C149—41no—
184C22×C149—41no—
185C22×C149—41no—
186C22×C149—41no—
187C22×C149—41no—
188C22×C149—41no—
189C22×C149—41no—
190C22×C149—84no—
191C22×C149C22×D411yescharacteristic
192C2×D149—21no—
193C2×D149—21no—
194C2×D149—21no—
195C2×D149—41no—
196C2×D149—41no—
197C2×D149—41no—
198C2×D149—84no—
199C4×C149—21no—
200C4×C149—41no—
201C4×C149C22×D411yescharacteristic
202C4×C149H(5,𝔽2)11yescharacteristic
203Dic149—21no—
204Dic149—21no—
205Dic149—41no—
206Dic149—41no—
207Dic149—41no—
208Dic149AGL(1,ℤ/8)12yescomplemented
209C2×C149—42no—
210C2×C149—41no—
211C2×C149—42no—
212C2×C149—81no—
213C2×C149D4211yescharacteristic
214C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C149#57711yescharacteristic
216D149—41no—
217D149—81no—
218C149#327211yescharacteristic, complemented, Sylow
219#3272—1491nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1491no—
221C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
222#547—1491no—
223D42—1491no—
224#492—1491no—
225#468—1491no—
226#468—1491no—
227#461—1491no—
228#459—1491no—
229#450—1491no—
230#450—1491no—
231#449—1491no—
232#449—1491no—
233C4×D8—1491no—
234#436—1491no—
235C2×#41—1491no—
236C2×#41C2×D14911yescharacteristic, complemented
237C22×D4—2981no—
238C22×D4—1491no—
239C22×D4C2×D14911yescharacteristic
240AGL(1,ℤ/8)—24no—
241AGL(1,ℤ/8)—2984no—
242C2×SD16—1491no—
243C2×SD16—1491no—
244C2×SD16—1491no—
245C2×SD16—1491no—
246C2×SD16C2×D14911yescharacteristic
247C2×SD16C2×D14911yescharacteristic, complemented
248C2×D8—1491no—
249C2×D8C2×D14911yescharacteristic
250C2×D8C2×D14911yescharacteristic, complemented
251C2×M16—1491no—
252C2×M16C2×D14911yescharacteristic, complemented
253C4⋊Q8—1491no—
254#127—1491no—
255#123—2981no—
256#123—2981no—
257C22⋊Q8—1491no—
258C22⋊Q8—1491no—
259#121—2981no—
260#121—2981no—
261#121—1491no—
262#121—1491no—
263#120—2981no—
264#120—2981no—
265C4×D4—1491no—
266C4×D4—1491no—
267C4×D4—1491no—
268C2×C4⋊C4—2981no—
269#107—1491no—
270C4⋊C8—1491no—
271#103—1491no—
272#103—1491no—
273#102—1491no—
274#102—1491no—
275#102—1491no—
276#102—1491no—
277C22⋊C8—1491no—
278C22⋊C8—1491no—
279C4×C8—1491no—
280C24—21no—
281C24—2981no—
282#41—22no—
283#41—41no—
284#41—2982no—
285#41—5961no—
286C2×Q8—1491no—
287C2×Q8C22×D14911yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—2982no—
293C2×D4—2981no—
294C2×D4—2981no—
295C2×D4—2981no—
296C2×D4—2981no—
297C2×D4—2981no—
298C2×D4—2981no—
299C2×D4—5961no—
300C2×D4—5964no—
301C2×D4—1491no—
302C2×D4—1491no—
303C2×D4C22×D14911yescharacteristic
304C2×D4C22×D14911yescharacteristic
305C2×D4C22×D14911yescharacteristic
306C22×C4—21no—
307C22×C4—2981no—
308C22×C4—2981no—
309C22×C4—2981no—
310C22×C4—1491no—
311C22×C4C22×D14911yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—2982no—
315SD16—2982no—
316SD16—5961no—
317SD16—5961no—
318D8—22no—
319D8—2982no—
320D8C22⋊D14914yescomplemented
321M16—22no—
322M16—2982no—
323C2×C8—1491no—
324C2×C8—1491no—
325C2×C8C22×D14911yescharacteristic
326C2×C8C22×D14911yescharacteristic
327C4⋊C4—2981no—
328C4⋊C4—2981no—
329C4⋊C4—2981no—
330C4⋊C4—2981no—
331C4⋊C4—1491no—
332C4⋊C4—1491no—
333C4⋊C4—1491no—
334C22⋊C4—2981no—
335C22⋊C4—2981no—
336C22⋊C4—2981no—
337C22⋊C4—2981no—
338C22⋊C4—2981no—
339C22⋊C4—2981no—
340C22⋊C4—1491no—
341C22⋊C4—1491no—
342C42—1491no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—2981no—
350C23—5962no—
351C23—5962no—
352C23—1491no—
353C23—11921no—
354C23D4×D14911yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2981no—
358Q8—2981no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—2982no—
366D4—2981no—
367D4—2981no—
368D4—5962no—
369D4—5962no—
370D4—5962no—
371D4—5962no—
372D4—5962no—
373D4—11921no—
374D4C2×C22⋊D14912yes—
375D4C2×C22⋊D14912yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—2981no—
381C2×C4—2981no—
382C2×C4—2981no—
383C2×C4—2981no—
384C2×C4—2981no—
385C2×C4—5962no—
386C2×C4—5962no—
387C2×C4—5962no—
388C2×C4—1491no—
389C2×C4—1491no—
390C2×C4—1491no—
391C2×C4D4×D14911yescharacteristic
392C2×C4C23×D14911yescharacteristic
393C8—21no—
394C8—2981no—
395C8—2981no—
396C8C2×C22⋊D14912yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—2981no—
411C22—2981no—
412C22—2981no—
413C22—5961no—
414C22—5961no—
415C22—5961no—
416C22—11924no—
417C22C2×D4×D14911yescharacteristic
418C4—21no—
419C4—41no—
420C4—2981no—
421C4—2981no—
422C4—5961no—
423C4—5961no—
424C4—5961no—
425C4—1492no—
426C4C22×C22⋊D14911yescharacteristic
427C4#1801195611yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—5961no—
433C2—11921no—
434C2D4×C22⋊D14911yescharacteristic
435C2AGL(1,ℤ/8)×D14911yescharacteristic
436C2#2375043511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×AGL(1,ℤ/8)×C149C2
socleC22×C149C22×D4
hypercenterC2×AGL(1,ℤ/8)D149
hypocenterC149#3272
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
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×#18011935C2semidirect
2. #23750274C2semidirect
3. D4×C22⋊D149C2semidirect
4. #23749996C2semidirect
5. #23749871C2semidirect
6. #23749868C2semidirect
7. #23749828C2semidirect
8. #23749818C2semidirect
9. #23749776C2semidirect
10. #23749774C2semidirect
11. #23749770C2semidirect
12. #23749767C2semidirect
13. D8×Dic149C2semidirect
14. #23749701C2semidirect
15. C2×AGL(1,ℤ/8)×C149C2semidirect
16. C2×#14558735C22semidirect
20. C2×#14558725C22semidirect
22. C2×#14558724C22semidirect
23. C2×#14558723C22semidirect
24. C2×#14558721C22semidirect
25. C2×#14558719C22semidirect
26. #18011909C22semidirect
30. #18011890C22semidirect
31. #18011887C22semidirect
39. D4×Dic149C22semidirect
42. #18011845C22semidirect
43. #18011840C22semidirect
44. #18011836C22semidirect
45. #18011835C22semidirect
46. #18011833C22semidirect
48. #18011832C22semidirect
49. #18011832C22semidirect
50. C22⋊#11924746C22semidirect
51. #18011823C22semidirect
52. C8×Dic149C22semidirect
53. C2×#41×C149C22semidirect
57. C2×SD16×C149C22semidirect
59. C2×D8×C149C22semidirect
60. C2×M16×C149C22semidirect
120. D8×C149D4semidirect
169. D4×C149C2×D4semidirect
177. C8×C149C2×D4semidirect
208. Dic149AGL(1,ℤ/8)semidirect
218. C149#3272semidirect
221. C2×AGL(1,ℤ/8)D149semidirect
236. C2×#41C2×D149semidirect
247. C2×SD16C2×D149semidirect
250. C2×D8C2×D149semidirect
252. C2×M16C2×D149semidirect
320. D8C22⋊D149semidirect
375. D4C2×C22⋊D149semidirect
396. C8C2×C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119352
#237502742
D4×C22⋊D1492
#237499962
#237498712
#237498682
#237498282
#237498182
#237497762
#237497742
#237497702
#237497672
D8×Dic1492
#237497012
C2×AGL(1,ℤ/8)×C1492
#3272149