Finite Group Databasebeta

← #34055516

Subgroups — #34055516

Order 19072.

414 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750135C211yesmaximal, characteristic
2C22⋊Q8×D149C211yesmaximal, characteristic, complemented
3#23749968C211yesmaximal, characteristic, complemented
4#23749964C211yesmaximal, characteristic, complemented
5#23749962C211yesmaximal, characteristic, complemented
6#23749942C211yesmaximal, characteristic, complemented
7#23749883C211yesmaximal, characteristic, complemented
8#23749881C211yesmaximal, characteristic, complemented
9#23749869C211yesmaximal, characteristic, complemented
10#23749788C211yesmaximal, characteristic, complemented
11#23749784C211yesmaximal, characteristic, complemented
12#23749782C211yesmaximal, characteristic, complemented
13#23749762C211yesmaximal, characteristic, complemented
14#23749588C211yesmaximal, characteristic, complemented
15#468×C149C211yesmaximal, characteristic, complemented
16C2×#14558735C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18C22×C4×D149C2211yescharacteristic
19C2×Dic596C2211yescharacteristic, complemented
20C2×#14558726C2211yescharacteristic, complemented
21C2×#14558724C2211yescharacteristic, complemented
22C2×#14558720C2211yescharacteristic
23#18011893—21no—
24#18011891C2211yescharacteristic
25#18011891C2211yescharacteristic
26#18011890C2211yescharacteristic
27#18011890C2211yescharacteristic
28#18011889—21no—
29#18011888—21no—
30#18011887C2211yescharacteristic
31#18011882C2211yescharacteristic
32#18011879—21no—
33D4×Dic149—21no—
34#18011871—21no—
35#18011869C2211yescharacteristic
36C4⋊C4×D149—21no—
37C4⋊C4×D149C2211yescharacteristic
38C22⋊C4×D149—21no—
39#18011844C2211yescharacteristic, complemented
40#18011843C2211yescharacteristic, complemented
41#18011842C2211yescharacteristic, complemented
42#18011836C2211yescharacteristic, complemented
43#18011836C2211yescharacteristic, complemented
44#18011835C2211yescharacteristic, complemented
45#18011834C2211yescharacteristic, complemented
46#18011834C2211yescharacteristic, complemented
47#18011832C2211yescharacteristic, complemented
48#18011823C2211yescharacteristic
49#18011823C2211yescharacteristic
50#18011823C2211yescharacteristic
51#18011823C2211yescharacteristic
52C2×SD16×C149C2211yescharacteristic, complemented
53C2×M16×C149C2211yescharacteristic
54C22⋊Q8×C149C2211yescharacteristic
55#121×C149C2211yescharacteristic
56#107×C149C2211yescharacteristic, complemented
57#103×C149C2211yescharacteristic, complemented
58#102×C149C2211yescharacteristic, complemented
59C23×D149D411yescharacteristic
60#14558735—41no—
61#14558735—41no—
62C2×Dic298—21no—
63C2×Dic298C2311yescharacteristic
64C2×Dic298C2311yescharacteristic
65Q8×D149—41no—
66Q8×D149—41no—
67C2×C22⋊D149—21no—
68C2×C22⋊D149—21no—
69C22×Dic149—21no—
70C22×Dic149D411yescharacteristic
71C2×C4×D149—21no—
72C2×C4×D149—21no—
73C2×C4×D149—41no—
74C2×C4×D149—41no—
75C2×C4×D149D411yescharacteristic
76C2×C4×D149D411yescharacteristic
77C2×C4×D149C2311yescharacteristic
78C2×C4×D149C2311yescharacteristic
79Dic596—41no—
80#14558726—41no—
81#14558724—41no—
82#14558720—41no—
83C2×#11924746C2311yescharacteristic
84C2×#11924746C2311yescharacteristic
85C4⋊Dic149—21no—
86C4⋊Dic149—21no—
87C4⋊Dic149C2311yescharacteristic
88C4⋊Dic149C2311yescharacteristic
89#14558715—21no—
90#14558715—21no—
91#14558715—21no—
92#14558715—21no—
93C22⋊Dic149—21no—
94C22⋊Dic149—21no—
95C22⋊Dic149—21no—
96#14558713—21no—
97#14558713—21no—
98#14558713—21no—
99#14558713—21no—
100C4×Dic149—21no—
101C2×Q8×C149C2311yescharacteristic
102C2×D4×C149—21no—
103C2×D4×C149C2311yescharacteristic
104C22×C4×C149C2311yescharacteristic
105SD16×C149—41no—
106M16×C149—41no—
107C2×C8×C149C2311yescharacteristic
108C2×C8×C149C2311yescharacteristic
109C4⋊C4×C149—21no—
110C4⋊C4×C149C2311yescharacteristic
111C4⋊C4×C149C2311yescharacteristic
112C22⋊C4×C149—21no—
113C22⋊C4×C149—21no—
114C22×D149—22no—
115C22×D149—41no—
116C22×D149—41no—
117C22×D149C2×D411yescharacteristic
118C22×D149C2×D411yescharacteristic
119Dic298—21no—
120Dic298—21no—
121Dic298—21no—
122Dic298—21no—
123Dic298—42no—
124C22⋊D149—42no—
125C22⋊D149—42no—
126C2×Dic149—21no—
127C2×Dic149—21no—
128C2×Dic149—21no—
129C2×Dic149—21no—
130C2×Dic149—41no—
131C2×Dic149—42no—
132C2×Dic149C2×D411yescharacteristic
133C2×Dic149C2×D411yescharacteristic
134C4×D149—21no—
135C4×D149—21no—
136C4×D149—21no—
137C4×D149—21no—
138C4×D149—41no—
139C4×D149—41no—
140C4×D149—42no—
141C4×D149—42no—
142#11924746—21no—
143#11924746—21no—
144C23×C149—21no—
145C23×C149C2×D411yescharacteristic
146Q8×C149—21no—
147Q8×C149—21no—
148D4×C149—21no—
149D4×C149—21no—
150D4×C149—42no—
151C2×C4×C149—21no—
152C2×C4×C149—21no—
153C2×C4×C149—21no—
154C2×C4×C149—21no—
155C2×C4×C149—21no—
156C2×C4×C149C2×D411yescharacteristic
157C2×C4×C149C2411yescharacteristic
158C8×C149—21no—
159C8×C149—21no—
160C22×C149—21no—
161C22×C149—21no—
162C22×C149—21no—
163C22×C149—41no—
164C22×C149—41no—
165C22×C149—41no—
166C22×C149C22×D411yescharacteristic
167C2×D149—21no—
168C2×D149—21no—
169C2×D149—21no—
170C2×D149—21no—
171C2×D149—21no—
172C2×D149—44no—
173C2×D149#13712yes—
174C4×C149—21no—
175C4×C149—41no—
176C4×C149—41no—
177C4×C149—41no—
178C4×C149H(5,𝔽2)11yescharacteristic
179C4×C149H(5,𝔽2)11yescharacteristic
180Dic149—21no—
181Dic149—21no—
182Dic149—41no—
183Dic149—41no—
184Dic149—41no—
185Dic149—41no—
186C2×C149—41no—
187C2×C149—81no—
188C2×C149#53411yescharacteristic
189C2×C149C2×#13711yescharacteristic
190C2×C149#57511yescharacteristic
191D149—22no—
192D149—41no—
193C149#319611yescharacteristic, complemented, Sylow
194#3196—1491nomaximal, Sylow
195#525—1491no—
196C2×C22⋊Q8—1491no—
197#485—1491no—
198#485—1491no—
199#484—1491no—
200#481—1491no—
201#470—1491no—
202#470—1491no—
203#468—1491no—
204#468D14911yescharacteristic, complemented
205#452—1491no—
206#452—1491no—
207#451—1491no—
208#448—1491no—
209#407—1491no—
210C2×#41—1491no—
211C22×Q8—1491no—
212C23×C4—1491no—
213C2×Q16—1491no—
214C2×Q16—1491no—
215C2×SD16—1491no—
216C2×SD16C2×D14911yescharacteristic, complemented
217C2×M16—1491no—
218C2×M16C2×D14911yescharacteristic
219#123—2981no—
220C22⋊Q8—2981no—
221C22⋊Q8—2981no—
222C22⋊Q8—1491no—
223C22⋊Q8—1491no—
224C22⋊Q8—1491no—
225C22⋊Q8—1491no—
226C22⋊Q8—1491no—
227C22⋊Q8C2×D14911yescharacteristic
228#121—1491no—
229#121C2×D14911yescharacteristic
230#120—2981no—
231C4×D4—2981no—
232C4×D4—2981no—
233#117—1491no—
234C2×C4⋊C4—2981no—
235C2×C4⋊C4—1491no—
236C2×C22⋊C4—2981no—
237#107—1491no—
238#107C2×D14911yescharacteristic, complemented
239#106—1491no—
240#106—1491no—
241#103—1491no—
242#103—1491no—
243#103—1491no—
244#103—1491no—
245#103—1491no—
246#103C2×D14911yescharacteristic, complemented
247#102—1491no—
248#102C2×D14911yescharacteristic, complemented
249C22⋊C8—1491no—
250C22⋊C8—1491no—
251C22⋊C8—1491no—
252C22⋊C8—1491no—
253C24—1491no—
254#41—5961no—
255#41—5961no—
256C2×Q8—2981no—
257C2×Q8—5961no—
258C2×Q8—5961no—
259C2×Q8—1491no—
260C2×Q8—1491no—
261C2×Q8C22×D14911yescharacteristic
262C2×D4—21no—
263C2×D4—2981no—
264C2×D4—2981no—
265C2×D4C22×D14911yescharacteristic
266C22×C4—2981no—
267C22×C4—2981no—
268C22×C4—2981no—
269C22×C4—5961no—
270C22×C4—5961no—
271C22×C4—1491no—
272C22×C4—1491no—
273C22×C4—1491no—
274C22×C4—1491no—
275C22×C4—1491no—
276C22×C4C22×D14911yescharacteristic
277Q16—5961no—
278Q16—5961no—
279SD16—41no—
280SD16—5961no—
281M16—41no—
282M16—5961no—
283C2×C8—1491no—
284C2×C8—1491no—
285C2×C8C22×D14911yescharacteristic
286C2×C8C22×D14911yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—2981no—
289C4⋊C4—2981no—
290C4⋊C4—2981no—
291C4⋊C4—2981no—
292C4⋊C4—2981no—
293C4⋊C4—2981no—
294C4⋊C4—1491no—
295C4⋊C4—1491no—
296C4⋊C4C22×D14911yescharacteristic
297C4⋊C4C22×D14911yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—2981no—
301C22⋊C4—2981no—
302C22⋊C4—2981no—
303C22⋊C4—2981no—
304C22⋊C4—2981no—
305C22⋊C4—2981no—
306C22⋊C4—2981no—
307C42—2981no—
308C23—21no—
309C23—2982no—
310C23—5961no—
311C23—5961no—
312C23—1491no—
313C23—1491no—
314C23D4×D14911yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—2981no—
318Q8—2981no—
319Q8—2981no—
320Q8—2981no—
321Q8—5962no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—5962no—
326D4—5962no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—2981no—
333C2×C4—2981no—
334C2×C4—2981no—
335C2×C4—2981no—
336C2×C4—2981no—
337C2×C4—2981no—
338C2×C4—2981no—
339C2×C4—2981no—
340C2×C4—5961no—
341C2×C4—5961no—
342C2×C4—5962no—
343C2×C4—5962no—
344C2×C4—5961no—
345C2×C4—5962no—
346C2×C4—1491no—
347C2×C4—1491no—
348C2×C4D4×D14911yescharacteristic
349C2×C4C23×D14911yescharacteristic
350C8—21no—
351C8—21no—
352C8—2981no—
353C8—2981no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2981no—
361C22—2981no—
362C22—2981no—
363C22—2981no—
364C22—2981no—
365C22—5964no—
366C22—1492no—
367C22C2×D4×D14911yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—2981no—
373C4—2981no—
374C4—5961no—
375C4—5961no—
376C4—5961no—
377C4—5961no—
378C4#1801195611yescharacteristic
379C4#1801195711yescharacteristic
380C2—41no—
381C2—81no—
382C2—2982no—
383C2—5961no—
384C2#2375018311yescharacteristic
385C2#137×D14911yescharacteristic
386C2#2375042111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#468×C149C2
socleC22×C149C22×D4
prime radical#23750135C2
hypercenter#468D149
hypocenterC149#3196
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
2. C22⋊Q8×D149C2semidirect
3. #23749968C2semidirect
4. #23749964C2semidirect
5. #23749962C2semidirect
6. #23749942C2semidirect
7. #23749883C2semidirect
8. #23749881C2semidirect
9. #23749869C2semidirect
10. #23749788C2semidirect
11. #23749784C2semidirect
12. #23749782C2semidirect
13. #23749762C2semidirect
14. #23749588C2semidirect
15. #468×C149C2semidirect
19. C2×Dic596C22semidirect
20. C2×#14558726C22semidirect
21. C2×#14558724C22semidirect
39. #18011844C22semidirect
40. #18011843C22semidirect
41. #18011842C22semidirect
42. #18011836C22semidirect
43. #18011836C22semidirect
44. #18011835C22semidirect
45. #18011834C22semidirect
46. #18011834C22semidirect
47. #18011832C22semidirect
52. C2×SD16×C149C22semidirect
56. #107×C149C22semidirect
57. #103×C149C22semidirect
58. #102×C149C22semidirect
193. C149#3196semidirect
204. #468D149semidirect
216. C2×SD16C2×D149semidirect
238. #107C2×D149semidirect
246. #103C2×D149semidirect
248. #102C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237501352
C22⋊Q8×D1492
#237499682
#237499642
#237499622
#237499422
#237498832
#237498812
#237498692
#237497882
#237497842
#237497822
#237497622
#237495882
#468×C1492
#3196149