Finite Group Databasebeta

← #34056385

Subgroups — #34056385

Order 19072.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14558724C211yesmaximal, characteristic, complemented
2C2×SD16×D149C211yesmaximal, characteristic, complemented
3#23750288C211yesmaximal, characteristic, complemented
4D4×C22⋊D149C211yesmaximal, characteristic
5#23749996C211yesmaximal, characteristic, complemented
6#23749856C211yesmaximal, characteristic, complemented
7#23749853C211yesmaximal, characteristic, complemented
8#23749828C211yesmaximal, characteristic, complemented
9#23749824C211yesmaximal, characteristic, complemented
10#23749777C211yesmaximal, characteristic, complemented
11#23749766C211yesmaximal, characteristic, complemented
12#23749762C211yesmaximal, characteristic, complemented
13#23749716C211yesmaximal, characteristic, complemented
14#23749688C211yesmaximal, characteristic, complemented
15#450×C149C211yesmaximal, characteristic, complemented
16C22×Dic298C2211yescharacteristic, complemented
17C2×Q8×D149C2211yescharacteristic, complemented
18C22×C22⋊D149—21no—
19C2×D4×D149C2211yescharacteristic
20C2×#14558725C2211yescharacteristic, complemented
21C2×#14558725C2211yescharacteristic, complemented
22C2×#14558724—24no—
23C2×#14558724C2211yescharacteristic
24C2×#14558724C2211yescharacteristic
25C2×#14558724C2211yescharacteristic, complemented
26C2×#14558724C2211yescharacteristic, complemented
27C2×#14558723C2211yescharacteristic, complemented
28SD16×D149—24no—
29C22×#11924746C2211yescharacteristic, complemented
30C2×C8×D149C2211yescharacteristic, complemented
31#18011910—21no—
32#18011909C2211yescharacteristic, complemented
33#18011907C2211yescharacteristic
34#18011891C2211yescharacteristic, complemented
35#18011889—21no—
36#18011888—21no—
37#18011885C2211yescharacteristic
38#18011883—21no—
39#18011882C2211yescharacteristic
40#18011880—21no—
41#18011879—21no—
42#18011876C2211yescharacteristic, complemented
43D4×Dic149C2211yescharacteristic
44C4×C22⋊D149C2211yescharacteristic
45#18011871—21no—
46#18011842C2211yescharacteristic, complemented
47#18011840C2211yescharacteristic, complemented
48#18011836C2211yescharacteristic, complemented
49#18011836C2211yescharacteristic, complemented
50#18011835C2211yescharacteristic, complemented
51#18011831C2211yescharacteristic, complemented
52#18011823C2211yescharacteristic, complemented
53C8×Dic149C2211yescharacteristic, complemented
54C22×D4×C149C2211yescharacteristic
55C2×SD16×C149C2211yescharacteristic
56C2×SD16×C149C2211yescharacteristic, complemented
57C22⋊Q8×C149C2211yescharacteristic, complemented
58#102×C149C2211yescharacteristic
59#102×C149C2211yescharacteristic, complemented
60C22⋊C8×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62C2×Dic298—21no—
63C2×Dic298—22no—
64C2×Dic298—21no—
65C2×Dic298—41no—
66C2×Dic298C2311yescharacteristic
67C2×Dic298C2311yescharacteristic
68Q8×D149—22no—
69Q8×D149—41no—
70C2×D596C2311yescharacteristic
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—
81C22×Dic149—21no—
82C2×C4×D149—21no—
83C2×C4×D149C2311yescharacteristic
84#14558725—22no—
85#14558725—41no—
86#14558724—22no—
87#14558724—22no—
88#14558724—22no—
89#14558724D414yescomplemented
90#14558723—22no—
91C2×#11924746—22no—
92C2×#11924746C2311yescharacteristic
93C2×#11924746C2311yescharacteristic
94C8×D149—22no—
95C4⋊Dic149—21no—
96C4⋊Dic149C2311yescharacteristic
97#14558715—21no—
98#14558715—21no—
99#14558715—21no—
100#14558715D411yescharacteristic
101C22⋊Dic149—21no—
102C22⋊Dic149—21no—
103C22⋊Dic149D411yescharacteristic
104#14558713—21no—
105#14558713—21no—
106#14558713D411yescharacteristic
107C4×Dic149—21no—
108C4×Dic149C2311yescharacteristic
109C24×C149—21no—
110C2×Q8×C149C2311yescharacteristic
111C2×D4×C149—22no—
112C2×D4×C149—21no—
113C2×D4×C149—41no—
114C2×D4×C149C2311yescharacteristic
115C2×D4×C149C2311yescharacteristic
116C22×C4×C149C2311yescharacteristic
117SD16×C149—22no—
118SD16×C149—41no—
119C2×C8×C149C2311yescharacteristic
120C2×C8×C149C2311yescharacteristic
121C4⋊C4×C149—21no—
122C4⋊C4×C149C2311yescharacteristic
123C22⋊C4×C149—21no—
124C22×D149—21no—
125C22×D149—42no—
126C22×D149—42no—
127C22×D149—81no—
128C22×D149C2×D411yescharacteristic
129Dic298—21no—
130Dic298—21no—
131Dic298—21no—
132Dic298—42no—
133Dic298—42no—
134Dic298C2×D412yescomplemented
135D596—21no—
136D596—21no—
137C22⋊D149—42no—
138C22⋊D149—42no—
139C22⋊D149—42no—
140C22⋊D149—42no—
141C22⋊D149—81no—
142C22⋊D149SD1614yescomplemented
143C2×Dic149—21no—
144C2×Dic149—21no—
145C2×Dic149—21no—
146C2×Dic149—42no—
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#11924746C2×D412yescomplemented
156C23×C149—21no—
157C23×C149—21no—
158C23×C149—42no—
159C23×C149—42no—
160C23×C149—81no—
161C23×C149C2×D411yescharacteristic
162Q8×C149—21no—
163Q8×C149—21no—
164D4×C149—21no—
165D4×C149—21no—
166D4×C149—21no—
167D4×C149—42no—
168D4×C149C2×D412yes—
169C2×C4×C149—21no—
170C2×C4×C149—21no—
171C2×C4×C149—21no—
172C2×C4×C149—21no—
173C2×C4×C149C2×D411yescharacteristic
174C2×C4×C149C2411yescharacteristic
175C8×C149—21no—
176C8×C149—21no—
177C22×C149—21no—
178C22×C149—21no—
179C22×C149—22no—
180C22×C149—41no—
181C22×C149—41no—
182C22×C149—41no—
183C22×C149—41no—
184C22×C149—41no—
185C22×C149—84no—
186C22×C149C2×SD1612yescomplemented
187C22×C149C22×D411yescharacteristic
188C2×D149—21no—
189C2×D149—21no—
190C2×D149—41no—
191C2×D149—41no—
192C2×D149—41no—
193C2×D149—84no—
194C2×D149C2×SD1612yescomplemented
195C4×C149—21no—
196C4×C149—41no—
197C4×C149—41no—
198C4×C149C22×D411yescharacteristic
199C4×C149H(5,𝔽2)11yescharacteristic
200Dic149—21no—
201Dic149—22no—
202Dic149—21no—
203Dic149—41no—
204Dic149—41no—
205Dic149C2×SD1612yescomplemented
206C2×C149—22no—
207C2×C149—42no—
208C2×C149—81no—
209C2×C149D4211yescharacteristic
210C2×C149C22×SD1611yescharacteristic
211C2×C149#57711yescharacteristic
212D149—22no—
213D149—81no—
214C149D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—1491nomaximal, Sylow
216C22×SD16—1491no—
217C22×SD16—1491no—
218D4×Q8—1491no—
219D42—1491no—
220#492—1491no—
221#465—1491no—
222#465—1491no—
223#461—1491no—
224#460—1491no—
225#450—1491no—
226#450D14911yescharacteristic, complemented
227#448—1491no—
228#448—1491no—
229C4×SD16—1491no—
230C8×D4—1491no—
231C22×Q8—1491no—
232C22×Q8—1491no—
233C22×D4—2981no—
234C22×D4—1491no—
235C22×D4C2×D14911yescharacteristic
236C2×SD16—2984no—
237C2×SD16—2984no—
238C2×SD16—1491no—
239C2×SD16—1491no—
240C2×SD16—1491no—
241C2×SD16—1491no—
242C2×SD16—1491no—
243C2×SD16—1491no—
244C2×SD16—1491no—
245C2×SD16C2×D14911yescharacteristic
246C2×SD16C2×D14911yescharacteristic, complemented
247C22×C8—1491no—
248C22×C8—1491no—
249C4⋊Q8—2981no—
250C4⋊Q8—1491no—
251#127—1491no—
252C22⋊Q8—2981no—
253C22⋊Q8—2981no—
254C22⋊Q8—1491no—
255C22⋊Q8C2×D14911yescharacteristic, complemented
256#121—2981no—
257#121—1491no—
258#121—1491no—
259#120—2981no—
260#120—2981no—
261C4×Q8—1491no—
262C4×D4—2981no—
263C4×D4—1491no—
264C4×D4—1491no—
265#106—1491no—
266C4⋊C8—1491no—
267#103—1491no—
268#103—1491no—
269#103—1491no—
270#102—1491no—
271#102C2×D14911yescharacteristic
272#102C2×D14911yescharacteristic, complemented
273C22⋊C8—1491no—
274C22⋊C8C2×D14911yescharacteristic, complemented
275C4×C8—1491no—
276C24—21no—
277C24—2981no—
278C2×Q8—2981no—
279C2×Q8—2982no—
280C2×Q8—2981no—
281C2×Q8—2982no—
282C2×Q8—5961no—
283C2×Q8—5961no—
284C2×Q8—1491no—
285C2×Q8—1491no—
286C2×Q8C22×D14911yescharacteristic
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2982no—
291C2×D4—2981no—
292C2×D4—2981no—
293C2×D4—2981no—
294C2×D4—2981no—
295C2×D4—2981no—
296C2×D4—5961no—
297C2×D4—5964no—
298C2×D4—1491no—
299C2×D4—1491no—
300C2×D4C22×D14911yescharacteristic
301C2×D4C22×D14911yescharacteristic
302C22×C4—2981no—
303C22×C4—2981no—
304C22×C4—2981no—
305C22×C4—1491no—
306C22×C4C22×D14911yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—2982no—
310SD16—2982no—
311SD16—2982no—
312SD16—2982no—
313SD16—2982no—
314SD16—5961no—
315SD16—1494no—
316C2×C8—2982no—
317C2×C8—2982no—
318C2×C8—1491no—
319C2×C8—1491no—
320C2×C8C22×D14911yescharacteristic
321C2×C8C22×D14911yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—2981no—
324C4⋊C4—2981no—
325C4⋊C4—2981no—
326C4⋊C4—2981no—
327C4⋊C4—1491no—
328C4⋊C4—1491no—
329C4⋊C4C22×D14911yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—2981no—
332C22⋊C4—2981no—
333C22⋊C4—2981no—
334C22⋊C4—2981no—
335C22⋊C4—1491no—
336C22⋊C4—1491no—
337C42—2981no—
338C42—1491no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—2981no—
345C23—5962no—
346C23—5962no—
347C23—1491no—
348C23—11921no—
349C23D4×D14911yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—2981no—
353Q8—2981no—
354Q8—2981no—
355Q8—5962no—
356Q8—5962no—
357Q8—1492no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—2981no—
363D4—2981no—
364D4—5962no—
365D4—5962no—
366D4—5962no—
367D4—5962no—
368D4—1494no—
369D4—11921no—
370D4D4×D14912yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—2981no—
376C2×C4—2981no—
377C2×C4—2981no—
378C2×C4—2981no—
379C2×C4—2981no—
380C2×C4—5962no—
381C2×C4—5962no—
382C2×C4—5962no—
383C2×C4—1491no—
384C2×C4—1491no—
385C2×C4—1491no—
386C2×C4D4×D14911yescharacteristic
387C2×C4C23×D14911yescharacteristic
388C8—21no—
389C8—21no—
390C8—2981no—
391C8—1492no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—2981no—
402C22—2981no—
403C22—5961no—
404C22—5961no—
405C22—5961no—
406C22—1492no—
407C22—11924no—
408C22SD16×D14912yescomplemented
409C22C2×D4×D14911yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—2981no—
414C4—2981no—
415C4—2982no—
416C4—5961no—
417C4—5961no—
418C4—1492no—
419C4C2×D4×D14911yescharacteristic
420C4#1801195611yescharacteristic
421C2—22no—
422C2—42no—
423C2—81no—
424C2—2982no—
425C2—11921no—
426C2#2375025711yescharacteristic
427C2C2×SD16×D14911yescharacteristic
428C2#2375043511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#450×C149C2
socleC22×C149C22×D4
prime radicalD4×C22⋊D149C2
hypercenter#450D149
hypocenterC149D4×SD16
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14558724C2semidirect
2. C2×SD16×D149C2semidirect
3. #23750288C2semidirect
5. #23749996C2semidirect
6. #23749856C2semidirect
7. #23749853C2semidirect
8. #23749828C2semidirect
9. #23749824C2semidirect
10. #23749777C2semidirect
11. #23749766C2semidirect
12. #23749762C2semidirect
13. #23749716C2semidirect
14. #23749688C2semidirect
15. #450×C149C2semidirect
16. C22×Dic298C22semidirect
17. C2×Q8×D149C22semidirect
20. C2×#14558725C22semidirect
21. C2×#14558725C22semidirect
25. C2×#14558724C22semidirect
26. C2×#14558724C22semidirect
27. C2×#14558723C22semidirect
29. C22×#11924746C22semidirect
30. C2×C8×D149C22semidirect
32. #18011909C22semidirect
34. #18011891C22semidirect
42. #18011876C22semidirect
46. #18011842C22semidirect
47. #18011840C22semidirect
48. #18011836C22semidirect
49. #18011836C22semidirect
50. #18011835C22semidirect
51. #18011831C22semidirect
52. #18011823C22semidirect
53. C8×Dic149C22semidirect
56. C2×SD16×C149C22semidirect
57. C22⋊Q8×C149C22semidirect
59. #102×C149C22semidirect
60. C22⋊C8×C149C22semidirect
89. #14558724D4semidirect
134. Dic298C2×D4semidirect
142. C22⋊D149SD16semidirect
155. #11924746C2×D4semidirect
186. C22×C149C2×SD16semidirect
194. C2×D149C2×SD16semidirect
205. Dic149C2×SD16semidirect
214. C149D4×SD16semidirect
226. #450D149semidirect
246. C2×SD16C2×D149semidirect
255. C22⋊Q8C2×D149semidirect
272. #102C2×D149semidirect
274. C22⋊C8C2×D149semidirect
408. C22SD16×D149semidirect

Maximal subgroups

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

subgroupindex
C22×#145587242
C2×SD16×D1492
#237502882
D4×C22⋊D1492
#237499962
#237498562
#237498532
#237498282
#237498242
#237497772
#237497662
#237497622
#237497162
#237496882
#450×C1492
D4×SD16149