Finite Group Databasebeta

← #34056305

Subgroups — #34056305

Order 19072.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011938C211yesmaximal, characteristic, complemented
2#23750294C211yesmaximal, characteristic, complemented
3D4×C22⋊D149C211yesmaximal, characteristic, complemented
4#23750000C211yesmaximal, characteristic, complemented
5#23749878C211yesmaximal, characteristic, complemented
6#23749875C211yesmaximal, characteristic, complemented
7#23749823C211yesmaximal, characteristic, complemented
8#23749818C211yesmaximal, characteristic, complemented
9#23749770C211yesmaximal, characteristic, complemented
10#23749769C211yesmaximal, characteristic, complemented
11#23749759C211yesmaximal, characteristic, complemented
12#23749757C211yesmaximal, characteristic, complemented
13SD16×Dic149C211yesmaximal, characteristic, complemented
14#23749701C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic, complemented
17C22×C22⋊D149—21no—
18C2×D4×D149C2211yescharacteristic
19#18011938—24no—
20C2×#14558723C2211yescharacteristic, complemented
21C2×D1192C2211yescharacteristic, complemented
22C2×#14558721C2211yescharacteristic
23C2×#14558721C2211yescharacteristic, complemented
24C2×#14558721C2211yescharacteristic, complemented
25C2×#14558719C2211yescharacteristic, complemented
26#18011907—21no—
27#18011907C2211yescharacteristic
28#18011907C2211yescharacteristic, complemented
29#18011885—21no—
30#18011885C2211yescharacteristic
31#18011885C2211yescharacteristic, complemented
32#18011883—21no—
33#18011882—21no—
34#18011882C2211yescharacteristic
35#18011882C2211yescharacteristic, complemented
36#18011880—21no—
37#18011879—21no—
38Q8×Dic149C2211yescharacteristic, complemented
39D4×Dic149C2211yescharacteristic
40C4×C22⋊D149—21no—
41C4×C22⋊D149C2211yescharacteristic
42#18011843C2211yescharacteristic, complemented
43#18011840C2211yescharacteristic, complemented
44#18011835C2211yescharacteristic, complemented
45#18011833C2211yescharacteristic, 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#14558737—22no—
63#14558737—41no—
64C2×D596—21no—
65C2×D596C2311yescharacteristic
66C2×D596C2311yescharacteristic
67C2×C22⋊D149—21no—
68C2×C22⋊D149—21no—
69C2×C22⋊D149—21no—
70C2×C22⋊D149—21no—
71C2×C22⋊D149—21no—
72C2×C22⋊D149—21no—
73C2×C22⋊D149—21no—
74C2×C22⋊D149—21no—
75C2×C22⋊D149—21no—
76C2×C22⋊D149—44no—
77C2×C22⋊D149D411yescharacteristic
78D4×D149—22no—
79D4×D149—41no—
80C22×Dic149—21no—
81C2×C4×D149—21no—
82C2×C4×D149C2311yescharacteristic
83#14558723—22no—
84D1192—22no—
85#14558721—22no—
86#14558721—41no—
87#14558721—41no—
88#14558719—22no—
89C2×#11924746C2311yescharacteristic
90C2×#11924746C2311yescharacteristic
91C4⋊Dic149—21no—
92C4⋊Dic149C2311yescharacteristic
93C4⋊Dic149C2311yescharacteristic
94#14558715D411yescharacteristic
95C22⋊Dic149—21no—
96C22⋊Dic149—21no—
97C22⋊Dic149—21no—
98C22⋊Dic149D411yescharacteristic
99#14558713—21no—
100#14558713—21no—
101#14558713D411yescharacteristic
102C4×Dic149—21no—
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×C149D414yescomplemented
119D8×C149—22no—
120D8×C149—22no—
121M16×C149—22no—
122C2×C8×C149C2311yescharacteristic
123C2×C8×C149C2311yescharacteristic
124C22×D149—21no—
125C22×D149—21no—
126C22×D149—42no—
127C22×D149—42no—
128C22×D149—81no—
129C22×D149C2×D411yescharacteristic
130D596—21no—
131D596—21no—
132D596—21no—
133D596—21no—
134D596—42no—
135C22⋊D149—22no—
136C22⋊D149—42no—
137C22⋊D149—42no—
138C22⋊D149—42no—
139C22⋊D149—42no—
140C22⋊D149—42no—
141C22⋊D149—81no—
142C22⋊D149—81no—
143C22⋊D149—81no—
144C22⋊D149—81no—
145C2×Dic149—21no—
146C2×Dic149—21no—
147C2×Dic149—42no—
148C2×Dic149C2×D411yescharacteristic
149C2×Dic149C2×D411yescharacteristic
150C2×Dic149C2×D411yescharacteristic
151C4×D149—21no—
152C4×D149—21no—
153C4×D149—42no—
154#11924746—21no—
155#11924746—21no—
156C23×C149—21no—
157C23×C149—21no—
158C23×C149—21no—
159C23×C149—42no—
160C23×C149—42no—
161C23×C149—81no—
162C23×C149C2×D411yescharacteristic
163Q8×C149—21no—
164Q8×C149C2×D412yescomplemented
165D4×C149—21no—
166D4×C149—21no—
167D4×C149—21no—
168D4×C149—21no—
169D4×C149—21no—
170D4×C149—42no—
171D4×C149—42no—
172D4×C149C2×D412yes—
173C2×C4×C149—21no—
174C2×C4×C149—21no—
175C2×C4×C149—21no—
176C2×C4×C149—42no—
177C2×C4×C149C2×D411yescharacteristic
178C2×C4×C149C2411yescharacteristic
179C8×C149—21no—
180C8×C149C2×D412yescomplemented
181C22×C149—21no—
182C22×C149—22no—
183C22×C149—21no—
184C22×C149—21no—
185C22×C149—41no—
186C22×C149—41no—
187C22×C149—41no—
188C22×C149—41no—
189C22×C149—41no—
190C22×C149—41no—
191C22×C149—41no—
192C22×C149—41no—
193C22×C149—84no—
194C22×C149C22×D411yescharacteristic
195C2×D149—21no—
196C2×D149—21no—
197C2×D149—21no—
198C2×D149—41no—
199C2×D149—41no—
200C2×D149—41no—
201C2×D149—41no—
202C2×D149—41no—
203C2×D149—41no—
204C2×D149—84no—
205C4×C149—21no—
206C4×C149—22no—
207C4×C149C22×D411yescharacteristic
208C4×C149H(5,𝔽2)11yescharacteristic
209Dic149—21no—
210Dic149—21no—
211Dic149—41no—
212Dic149—41no—
213Dic149AGL(1,ℤ/8)12yescomplemented
214C2×C149—42no—
215C2×C149—41no—
216C2×C149—81no—
217C2×C149—81no—
218C2×C149D4211yescharacteristic
219C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C149#57611yescharacteristic
221D149—41no—
222D149—81no—
223D149—81no—
224C149#327411yescharacteristic, complemented, Sylow
225#3274—1491nomaximal, Sylow
226C2×AGL(1,ℤ/8)—1491no—
227C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
228#550—1491no—
229D42—1491no—
230#493—1491no—
231#469—1491no—
232#469—1491no—
233#460—1491no—
234#459—1491no—
235#449—1491no—
236#449—1491no—
237#447—1491no—
238#447—1491no—
239C4×SD16—1491no—
240#436—1491no—
241C2×#41—1491no—
242C2×#41C2×D14911yescharacteristic, complemented
243C22×D4—2981no—
244C22×D4—1491no—
245C22×D4C2×D14911yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—2984no—
248C2×SD16—1491no—
249C2×SD16C2×D14911yescharacteristic
250C2×SD16C2×D14911yescharacteristic, complemented
251C2×D8—1491no—
252C2×D8—1491no—
253C2×D8—1491no—
254C2×D8—1491no—
255C2×D8C2×D14911yescharacteristic
256C2×D8C2×D14911yescharacteristic, complemented
257C2×M16—1491no—
258C2×M16C2×D14911yescharacteristic, complemented
259#127—2981no—
260#127—1491no—
261#127—1491no—
262#121—2981no—
263#121—2981no—
264#121—2981no—
265#121—1491no—
266#121—1491no—
267#121—1491no—
268#121—1491no—
269#120—2981no—
270#120—2981no—
271C4×Q8—1491no—
272C4×D4—2981no—
273C4×D4—1491no—
274C4×D4—1491no—
275#106—1491no—
276C4⋊C8—1491no—
277#103—1491no—
278#102—1491no—
279#102—1491no—
280#102—1491no—
281#102—1491no—
282#102—1491no—
283C22⋊C8—1491no—
284C22⋊C8—1491no—
285C4×C8—1491no—
286C24—21no—
287C24—2981no—
288#41—22no—
289#41—41no—
290#41—2982no—
291#41—5961no—
292C2×Q8C22×D14911yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—2981no—
298C2×D4—2982no—
299C2×D4—2981no—
300C2×D4—2981no—
301C2×D4—2981no—
302C2×D4—2981no—
303C2×D4—2981no—
304C2×D4—2981no—
305C2×D4—2981no—
306C2×D4—2981no—
307C2×D4—2981no—
308C2×D4—5961no—
309C2×D4—5964no—
310C2×D4—1491no—
311C2×D4—1491no—
312C2×D4—1491no—
313C2×D4C22×D14911yescharacteristic
314C2×D4C22×D14911yescharacteristic
315C2×D4C22×D14911yescharacteristic
316C22×C4—21no—
317C22×C4—2981no—
318C22×C4—2981no—
319C22×C4—1491no—
320C22×C4C22×D14911yescharacteristic
321SD16—22no—
322SD16—2982no—
323SD16C22⋊D14914yescomplemented
324D8—22no—
325D8—22no—
326D8—2982no—
327D8—2982no—
328D8—5961no—
329D8—5961no—
330M16—22no—
331M16—2982no—
332C2×C8—1491no—
333C2×C8—1491no—
334C2×C8C22×D14911yescharacteristic
335C2×C8C22×D14911yescharacteristic
336C4⋊C4—2981no—
337C4⋊C4—1491no—
338C4⋊C4—1491no—
339C4⋊C4—1491no—
340C22⋊C4—2981no—
341C22⋊C4—2981no—
342C22⋊C4—2981no—
343C22⋊C4—2981no—
344C22⋊C4—2981no—
345C22⋊C4—1491no—
346C22⋊C4—1491no—
347C42—2981no—
348C42—1491no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—2981no—
356C23—2981no—
357C23—5962no—
358C23—5962no—
359C23—1491no—
360C23—11921no—
361C23D4×D14911yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D14912yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—2982no—
372D4—2981no—
373D4—2981no—
374D4—2981no—
375D4—2981no—
376D4—5962no—
377D4—5962no—
378D4—5962no—
379D4—5962no—
380D4—5962no—
381D4—5962no—
382D4—11921no—
383D4—11921no—
384D4—11921no—
385D4—11921no—
386D4C2×C22⋊D14912yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—2981no—
392C2×C4—2981no—
393C2×C4—2981no—
394C2×C4—2981no—
395C2×C4—5962no—
396C2×C4—5962no—
397C2×C4—1491no—
398C2×C4—1491no—
399C2×C4—1491no—
400C2×C4D4×D14911yescharacteristic
401C2×C4C23×D14911yescharacteristic
402C8—21no—
403C8—2981no—
404C8—2981no—
405C8C2×C22⋊D14912yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—2981no—
420C22—2981no—
421C22—2981no—
422C22—5961no—
423C22—5961no—
424C22—5961no—
425C22—5961no—
426C22—5961no—
427C22—5961no—
428C22—11924no—
429C22C2×D4×D14911yescharacteristic
430C4—21no—
431C4—22no—
432C4—2981no—
433C4—2981no—
434C4—5961no—
435C4—5961no—
436C4—1492no—
437C4C22×C22⋊D14911yescharacteristic
438C4#1801195611yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—5961no—
444C2—11921no—
445C2—11921no—
446C2D4×C22⋊D14911yescharacteristic
447C2AGL(1,ℤ/8)×D14911yescharacteristic
448C2#2375042911yescharacteristic
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#3274
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×#18011938C2semidirect
2. #23750294C2semidirect
3. D4×C22⋊D149C2semidirect
4. #23750000C2semidirect
5. #23749878C2semidirect
6. #23749875C2semidirect
7. #23749823C2semidirect
8. #23749818C2semidirect
9. #23749770C2semidirect
10. #23749769C2semidirect
11. #23749759C2semidirect
12. #23749757C2semidirect
13. SD16×Dic149C2semidirect
14. #23749701C2semidirect
15. C2×AGL(1,ℤ/8)×C149C2semidirect
16. C2×#14558737C22semidirect
20. C2×#14558723C22semidirect
21. C2×D1192C22semidirect
23. C2×#14558721C22semidirect
24. C2×#14558721C22semidirect
25. C2×#14558719C22semidirect
28. #18011907C22semidirect
31. #18011885C22semidirect
35. #18011882C22semidirect
38. Q8×Dic149C22semidirect
42. #18011843C22semidirect
43. #18011840C22semidirect
44. #18011835C22semidirect
45. #18011833C22semidirect
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
118. SD16×C149D4semidirect
164. Q8×C149C2×D4semidirect
180. C8×C149C2×D4semidirect
213. Dic149AGL(1,ℤ/8)semidirect
224. C149#3274semidirect
227. C2×AGL(1,ℤ/8)D149semidirect
242. C2×#41C2×D149semidirect
250. C2×SD16C2×D149semidirect
256. C2×D8C2×D149semidirect
258. C2×M16C2×D149semidirect
323. SD16C22⋊D149semidirect
363. Q8C2×C22⋊D149semidirect
405. C8C2×C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119382
#237502942
D4×C22⋊D1492
#237500002
#237498782
#237498752
#237498232
#237498182
#237497702
#237497692
#237497592
#237497572
SD16×Dic1492
#237497012
C2×AGL(1,ℤ/8)×C1492
#3274149