Finite Group Databasebeta

← #34056296

Subgroups — #34056296

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×#18011935C211yesmaximal, characteristic, complemented
2#23750269C211yesmaximal, characteristic, complemented
3#23750263C211yesmaximal, characteristic, complemented
4#23750006C211yesmaximal, characteristic, complemented
5#23749871C211yesmaximal, characteristic, complemented
6#23749868C211yesmaximal, characteristic, complemented
7#23749842C211yesmaximal, characteristic, complemented
8#23749838C211yesmaximal, characteristic, complemented
9#23749787C211yesmaximal, characteristic, complemented
10#23749784C211yesmaximal, characteristic, complemented
11#23749759C211yesmaximal, characteristic, complemented
12#23749757C211yesmaximal, characteristic, complemented
13D8×Dic149C211yesmaximal, characteristic, complemented
14#23749701C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
16C2×#14558735C2211yescharacteristic
17C2×D4×D149C2211yescharacteristic, complemented
18#18011935—24no—
19C2×#14558726C2211yescharacteristic, complemented
20C2×#14558725C2211yescharacteristic, complemented
21C2×#14558724C2211yescharacteristic, complemented
22C2×#14558721C2211yescharacteristic
23C2×#14558721C2211yescharacteristic
24C2×#14558719C2211yescharacteristic, complemented
25#18011900C2211yescharacteristic, complemented
26#18011898C2211yescharacteristic, complemented
27#18011894—21no—
28#18011892—21no—
29#18011890C2211yescharacteristic, complemented
30#18011887C2211yescharacteristic, complemented
31#18011885C2211yescharacteristic
32#18011883—21no—
33#18011883—21no—
34#18011882C2211yescharacteristic
35#18011880—21no—
36#18011879—21no—
37D4×Dic149C2211yescharacteristic, complemented
38D4×Dic149C2211yescharacteristic, complemented
39C4×C22⋊D149C2211yescharacteristic
40C2×C22⋊Dic149—21no—
41C2×#14558713—21no—
42#18011845C2211yescharacteristic, complemented
43#18011840C2211yescharacteristic, complemented
44#18011836C2211yescharacteristic, complemented
45#18011835C2211yescharacteristic, complemented
46#18011833C2211yescharacteristic, complemented
47#18011832C2211yescharacteristic, complemented
48#18011832C2211yescharacteristic, complemented
49#18011832C2211yescharacteristic, complemented
50C22⋊#11924746C2211yescharacteristic, complemented
51#18011823C2211yescharacteristic, complemented
52C8×Dic149C2211yescharacteristic, complemented
53C2×#41×C149C2211yescharacteristic
54C22×D4×C149C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C149—24no—
56C2×SD16×C149C2211yescharacteristic, complemented
57C2×SD16×C149C2211yescharacteristic, complemented
58C2×D8×C149C2211yescharacteristic
59C2×D8×C149C2211yescharacteristic
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⋊D149D411yescharacteristic
71D4×D149—22no—
72D4×D149—41no—
73C22×Dic149—21no—
74C22×Dic149—21no—
75C22×Dic149—21no—
76C2×C4×D149C2311yescharacteristic
77#14558726—41no—
78#14558725—22no—
79#14558724—22no—
80#14558721—22no—
81#14558721—41no—
82#14558719—22no—
83C2×#11924746C2311yescharacteristic
84C2×#11924746C2311yescharacteristic
85C4⋊Dic149C2311yescharacteristic
86C4⋊Dic149C2311yescharacteristic
87#14558715—21no—
88#14558715—21no—
89#14558715D411yescharacteristic
90C22⋊Dic149—21no—
91C22⋊Dic149—21no—
92C22⋊Dic149—21no—
93C22⋊Dic149—21no—
94C22⋊Dic149—21no—
95C22⋊Dic149—21no—
96C22⋊Dic149D411yescharacteristic
97#14558713—21no—
98#14558713—21no—
99#14558713—21no—
100#14558713—21no—
101#14558713D411yescharacteristic
102C4×Dic149C2311yescharacteristic
103C24×C149—21no—
104#41×C149—22no—
105#41×C149—41no—
106C2×Q8×C149C2311yescharacteristic
107C2×D4×C149—22no—
108C2×D4×C149—21no—
109C2×D4×C149—21no—
110C2×D4×C149—41no—
111C2×D4×C149C2311yescharacteristic
112C2×D4×C149C2311yescharacteristic
113C2×D4×C149C2311yescharacteristic
114C22×C4×C149—21no—
115C22×C4×C149C2311yescharacteristic
116SD16×C149—22no—
117SD16×C149—22no—
118D8×C149—22no—
119D8×C149D414yescomplemented
120M16×C149—22no—
121C2×C8×C149C2311yescharacteristic
122C2×C8×C149C2311yescharacteristic
123C22×D149—21no—
124C22×D149—42no—
125C22×D149—42no—
126C22×D149—81no—
127C22×D149C2×D411yescharacteristic
128Dic298—21no—
129Dic298—21no—
130D596—21no—
131D596—21no—
132C22⋊D149—41no—
133C22⋊D149—42no—
134C22⋊D149—42no—
135C22⋊D149—81no—
136C22⋊D149—81no—
137C2×Dic149—21no—
138C2×Dic149—21no—
139C2×Dic149—21no—
140C2×Dic149—42no—
141C2×Dic149—81no—
142C2×Dic149—81no—
143C2×Dic149C2×D411yescharacteristic
144C2×Dic149C2×D411yescharacteristic
145C2×Dic149C2×D411yescharacteristic
146C4×D149—21no—
147C4×D149—21no—
148#11924746—21no—
149#11924746—21no—
150C23×C149—21no—
151C23×C149—21no—
152C23×C149—21no—
153C23×C149—42no—
154C23×C149—42no—
155C23×C149—81no—
156C23×C149C2×D411yescharacteristic
157Q8×C149—21no—
158Q8×C149—21no—
159D4×C149—21no—
160D4×C149—21no—
161D4×C149—21no—
162D4×C149—21no—
163D4×C149—42no—
164D4×C149—42no—
165D4×C149C2×D412yes—
166D4×C149C2×D412yes—
167C2×C4×C149—21no—
168C2×C4×C149—21no—
169C2×C4×C149—21no—
170C2×C4×C149—42no—
171C2×C4×C149C2×D411yescharacteristic
172C2×C4×C149C2411yescharacteristic
173C8×C149—21no—
174C8×C149C2×D412yes—
175C22×C149—21no—
176C22×C149—22no—
177C22×C149—22no—
178C22×C149—21no—
179C22×C149—21no—
180C22×C149—41no—
181C22×C149—41no—
182C22×C149—41no—
183C22×C149—41no—
184C22×C149—41no—
185C22×C149—41no—
186C22×C149—41no—
187C22×C149—84no—
188C22×C149C22×D411yescharacteristic
189C2×D149—21no—
190C2×D149—21no—
191C2×D149—21no—
192C2×D149—41no—
193C2×D149—41no—
194C2×D149—41no—
195C2×D149—84no—
196C4×C149—21no—
197C4×C149—41no—
198C4×C149C22×D411yescharacteristic
199C4×C149H(5,𝔽2)11yescharacteristic
200Dic149—21no—
201Dic149—21no—
202Dic149—21no—
203Dic149—41no—
204Dic149—41no—
205Dic149—41no—
206C2×C149—42no—
207C2×C149—41no—
208C2×C149—42no—
209C2×C149—81no—
210C2×C149D4211yescharacteristic
211C2×C149#57511yescharacteristic
212C2×C149#57611yescharacteristic
213D149—41no—
214D149—81no—
215C149#327311yescharacteristic, complemented, Sylow
216#3273—1491nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1491no—
218C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
219#546—1491no—
220#546—1491no—
221#495—1491no—
222#468—1491no—
223#468—1491no—
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×#41C2×D14911yescharacteristic
234C22×D4—1491no—
235C22×D4C2×D14911yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—2984no—
238C2×Q16—1491no—
239C2×SD16—1491no—
240C2×SD16—1491no—
241C2×SD16C2×D14911yescharacteristic, complemented
242C2×SD16C2×D14911yescharacteristic, complemented
243C2×D8—1491no—
244C2×D8—1491no—
245C2×D8C2×D14911yescharacteristic
246C2×D8C2×D14911yescharacteristic
247C2×M16—1491no—
248C2×M16C2×D14911yescharacteristic, complemented
249#124—1491no—
250#124—1491no—
251#123—2981no—
252#123—2981no—
253C22⋊Q8—1491no—
254C22⋊Q8—1491no—
255#121—2981no—
256#121—2981no—
257#121—1491no—
258#121—1491no—
259#120—2981no—
260#120—2981no—
261C4×D4—1491no—
262C4×D4—1491no—
263C4×D4—1491no—
264C2×C22⋊C4—2981no—
265C2×C22⋊C4—2981no—
266#107—1491no—
267C4⋊C8—1491no—
268#103—1491no—
269#103—1491no—
270#102—1491no—
271#102—1491no—
272#102—1491no—
273#102—1491no—
274C22⋊C8—1491no—
275C22⋊C8—1491no—
276C4×C8—1491no—
277C24—21no—
278C24—2981no—
279#41—22no—
280#41—41no—
281#41—2982no—
282#41—5961no—
283C2×Q8—1491no—
284C2×Q8C22×D14911yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—2982no—
290C2×D4—2981no—
291C2×D4—2981no—
292C2×D4—2981no—
293C2×D4—2981no—
294C2×D4—5961no—
295C2×D4—1491no—
296C2×D4—1491no—
297C2×D4C22×D14911yescharacteristic
298C2×D4C22×D14911yescharacteristic
299C2×D4C22×D14911yescharacteristic
300C22×C4—21no—
301C22×C4—2981no—
302C22×C4—2981no—
303C22×C4—2981no—
304C22×C4—1491no—
305C22×C4C22×D14911yescharacteristic
306Q16—5961no—
307SD16—22no—
308SD16—22no—
309SD16—2982no—
310SD16—2982no—
311D8—22no—
312D8—2982no—
313D8—5961no—
314D8C22⋊D14914yescomplemented
315M16—22no—
316M16—2982no—
317C2×C8—1491no—
318C2×C8—1491no—
319C2×C8C22×D14911yescharacteristic
320C2×C8C22×D14911yescharacteristic
321C4⋊C4—2981no—
322C4⋊C4—2981no—
323C4⋊C4—1491no—
324C4⋊C4—1491no—
325C4⋊C4—1491no—
326C22⋊C4—2981no—
327C22⋊C4—2981no—
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—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—2981no—
346C23—5962no—
347C23—5962no—
348C23—1491no—
349C23—11921no—
350C23D4×D14911yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2981no—
354Q8—2981no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—2981no—
362D4—2981no—
363D4—5961no—
364D4—5962no—
365D4—5962no—
366D4—11921no—
367D4—11921no—
368D4C2×C22⋊D14912yes—
369D4C2×C22⋊D14912yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—2981no—
375C2×C4—2981no—
376C2×C4—2981no—
377C2×C4—2981no—
378C2×C4—2981no—
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—
390C8C2×C22⋊D14912yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—2981no—
405C22—2981no—
406C22—2981no—
407C22—5961no—
408C22—5961no—
409C22—5961no—
410C22—11924no—
411C22C2×D4×D14911yescharacteristic
412C4—21no—
413C4—41no—
414C4—2981no—
415C4—2981no—
416C4—2981no—
417C4—5961no—
418C4—5961no—
419C4—5961no—
420C4C22×C22⋊D14911yescharacteristic
421C4#1801195611yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—5961no—
427C2—11921no—
428C2D4×C22⋊D14911yescharacteristic
429C2#2375042111yescharacteristic
430C2#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#3273
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. #23750269C2semidirect
3. #23750263C2semidirect
4. #23750006C2semidirect
5. #23749871C2semidirect
6. #23749868C2semidirect
7. #23749842C2semidirect
8. #23749838C2semidirect
9. #23749787C2semidirect
10. #23749784C2semidirect
11. #23749759C2semidirect
12. #23749757C2semidirect
13. D8×Dic149C2semidirect
14. #23749701C2semidirect
15. C2×AGL(1,ℤ/8)×C149C2semidirect
17. C2×D4×D149C22semidirect
19. C2×#14558726C22semidirect
20. C2×#14558725C22semidirect
21. C2×#14558724C22semidirect
24. C2×#14558719C22semidirect
25. #18011900C22semidirect
26. #18011898C22semidirect
29. #18011890C22semidirect
30. #18011887C22semidirect
37. D4×Dic149C22semidirect
38. D4×Dic149C22semidirect
42. #18011845C22semidirect
43. #18011840C22semidirect
44. #18011836C22semidirect
45. #18011835C22semidirect
46. #18011833C22semidirect
47. #18011832C22semidirect
48. #18011832C22semidirect
49. #18011832C22semidirect
50. C22⋊#11924746C22semidirect
51. #18011823C22semidirect
52. C8×Dic149C22semidirect
54. C22×D4×C149C22semidirect
56. C2×SD16×C149C22semidirect
57. C2×SD16×C149C22semidirect
60. C2×M16×C149C22semidirect
119. D8×C149D4semidirect
215. C149#3273semidirect
218. C2×AGL(1,ℤ/8)D149semidirect
235. C22×D4C2×D149semidirect
241. C2×SD16C2×D149semidirect
242. C2×SD16C2×D149semidirect
248. C2×M16C2×D149semidirect
314. D8C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119352
#237502692
#237502632
#237500062
#237498712
#237498682
#237498422
#237498382
#237497872
#237497842
#237497592
#237497572
D8×Dic1492
#237497012
C2×AGL(1,ℤ/8)×C1492
#3273149