Finite Group Databasebeta

← #34056528

Subgroups — #34056528

Order 19072.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750315C211yesmaximal, characteristic, complemented
2#23750271C211yesmaximal, characteristic
3#23749980C211yesmaximal, characteristic, complemented
4#23749956C211yesmaximal, characteristic, complemented
5#23749950C211yesmaximal, characteristic, complemented
6#23749926C211yesmaximal, characteristic, complemented
7#23749866C211yesmaximal, characteristic, complemented
8#23749854C211yesmaximal, characteristic, complemented
9#23749848C211yesmaximal, characteristic, complemented
10#23749789C211yesmaximal, characteristic, complemented
11#23749778C211yesmaximal, characteristic, complemented
12C4×#14558725C211yesmaximal, characteristic, complemented
13#23749622C211yesmaximal, characteristic, complemented
14C2×#18011833C211yesmaximal, characteristic, complemented
15C8×D4×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic
17C22×D596C2211yescharacteristic
18C2×Dic596C2211yescharacteristic, complemented
19C2×#14558725C2211yescharacteristic
20C2×#14558725C2211yescharacteristic
21#18011904—21no—
22#18011903C2211yescharacteristic, complemented
23#18011901—21no—
24#18011899C2211yescharacteristic
25#18011896—21no—
26#18011892—21no—
27#18011890C2211yescharacteristic
28#18011890C2211yescharacteristic
29#18011890C2211yescharacteristic, complemented
30#18011885C2211yescharacteristic
31#18011884—21no—
32#18011879—21no—
33C4×Dic298C2211yescharacteristic, complemented
34C4×D596C2211yescharacteristic
35#18011870C2211yescharacteristic, complemented
36#18011867—21no—
37C2×C4⋊Dic149C2211yescharacteristic
38C22⋊C4×D149—21no—
39#18011845C2211yescharacteristic, complemented
40#18011843C2211yescharacteristic, complemented
41#18011843C2211yescharacteristic, complemented
42#18011836C2211yescharacteristic, complemented
43#18011836C2211yescharacteristic, complemented
44#18011836C2211yescharacteristic, complemented
45#18011836C2211yescharacteristic, complemented
46#18011833C2211yescharacteristic
47#18011833C2211yescharacteristic
48#18011833C2211yescharacteristic
49#18011833C2211yescharacteristic
50#18011833C2211yescharacteristic
51#18011833C2211yescharacteristic
52C22×C8×C149C2211yescharacteristic
53C22×C8×C149C2211yescharacteristic, complemented
54C4×D4×C149C2211yescharacteristic
55C4⋊C8×C149C2211yescharacteristic, complemented
56C22⋊C8×C149C2211yescharacteristic
57C22⋊C8×C149C2211yescharacteristic, complemented
58C4×C8×C149C2211yescharacteristic, complemented
59C23×D149—21no—
60#14558736—22no—
61#14558736—41no—
62C2×Dic298C2311yescharacteristic
63C2×Dic298C2311yescharacteristic
64C2×D596—22no—
65C2×D596—21no—
66C2×D596—41no—
67C2×D596C2311yescharacteristic
68C2×D596C2311yescharacteristic
69C2×C22⋊D149—21no—
70C2×C22⋊D149—21no—
71C22×Dic149—21no—
72C2×C4×D149—21no—
73C2×C4×D149—21no—
74Dic596—41no—
75#14558725—22no—
76#14558725—41no—
77C4⋊Dic149—21no—
78C4⋊Dic149C2311yescharacteristic
79C4⋊Dic149C2311yescharacteristic
80C4⋊Dic149C2311yescharacteristic
81C4⋊Dic149C2311yescharacteristic
82#14558715—21no—
83#14558715—21no—
84#14558715—21no—
85#14558715—21no—
86#14558715—21no—
87#14558715—21no—
88C22⋊Dic149—21no—
89C22⋊Dic149—21no—
90C22⋊Dic149—21no—
91C22⋊Dic149—21no—
92#14558713—21no—
93#14558713—21no—
94#14558713—21no—
95#14558713—21no—
96C4×Dic149—21no—
97C4×Dic149—21no—
98C2×D4×C149D411yescharacteristic
99C22×C4×C149C2311yescharacteristic
100C22×C4×C149C2311yescharacteristic
101C2×C8×C149—41no—
102C2×C8×C149—41no—
103C2×C8×C149C2311yescharacteristic
104C2×C8×C149C2311yescharacteristic
105C2×C8×C149C2311yescharacteristic
106C2×C8×C149C2311yescharacteristic
107C4⋊C4×C149D411yescharacteristic
108C22⋊C4×C149D411yescharacteristic
109C22⋊C4×C149D411yescharacteristic
110C42×C149C2311yescharacteristic
111C22×D149—21no—
112C22×D149—21no—
113C22×D149—42no—
114C22×D149—42no—
115C22×D149—81no—
116Dic298—21no—
117Dic298—21no—
118Dic298—21no—
119Dic298—21no—
120D596—21no—
121D596—21no—
122D596—21no—
123D596—42no—
124D596#4112yes—
125C22⋊D149—42no—
126C22⋊D149—81no—
127C2×Dic149—21no—
128C2×Dic149—21no—
129C2×Dic149—21no—
130C2×Dic149—21no—
131C2×Dic149—21no—
132C2×Dic149—21no—
133C2×Dic149—42no—
134C4×D149—42no—
135C4×D149—81no—
136C23×C149C2×D411yescharacteristic
137C23×C149C2×D411yescharacteristic
138D4×C149—22no—
139C2×C4×C149—21no—
140C2×C4×C149—21no—
141C2×C4×C149—21no—
142C2×C4×C149—21no—
143C2×C4×C149C2×D411yescharacteristic
144C2×C4×C149C2×D411yescharacteristic
145C2×C4×C149C2×D411yescharacteristic
146C2×C4×C149C2×D411yescharacteristic
147C2×C4×C149C2411yescharacteristic
148C8×C149—21no—
149C8×C149—21no—
150C8×C149—21no—
151C8×C149—21no—
152C22×C149—21no—
153C22×C149—21no—
154C22×C149—21no—
155C22×C149—21no—
156C22×C149—21no—
157C22×C149#13512yes—
158C22×C149C22×D411yescharacteristic
159C2×D149—22no—
160C2×D149—41no—
161C2×D149—41no—
162C2×D149—41no—
163C2×D149—41no—
164C2×D149—41no—
165C2×D149—84no—
166C4×C149—21no—
167C4×C149—21no—
168C4×C149—21no—
169C4×C149—21no—
170C4×C149C2×#4111yescharacteristic
171C4×C149H(5,𝔽2)11yescharacteristic
172Dic149—41no—
173Dic149—41no—
174Dic149—41no—
175Dic149—41no—
176Dic149—41no—
177Dic149—41no—
178C2×C149—22no—
179C2×C149—41no—
180C2×C149#54611yescharacteristic
181C2×C149C2×#13511yescharacteristic
182C2×C149#57711yescharacteristic
183D149—42no—
184D149—81no—
185C149#329611yescharacteristic, complemented, Sylow
186#3296—1491nomaximal, Sylow
187#552—1491no—
188#546—1491no—
189#488—1491no—
190#483—1491no—
191#482—1491no—
192#478—1491no—
193#467—1491no—
194#465—1491no—
195#464—1491no—
196#452—1491no—
197#450—1491no—
198C4×SD16—1491no—
199C8×D4D14911yescharacteristic, complemented
200#416—1491no—
201C2×#102—1491no—
202C2×#41—1491no—
203C22×D4—1491no—
204C2×Q16—1491no—
205C2×SD16—1491no—
206C2×SD16—1491no—
207C22×C8C2×D14911yescharacteristic
208C22×C8C2×D14911yescharacteristic, complemented
209#126—2981no—
210#125—2981no—
211#125—1491no—
212#124—1491no—
213#123—2981no—
214#123—2981no—
215C22⋊Q8—1491no—
216C22⋊Q8—1491no—
217C22⋊Q8—1491no—
218#121—2981no—
219#121—1491no—
220#120—2981no—
221C4×Q8—1491no—
222C4×D4—1491no—
223C4×D4C2×D14911yescharacteristic
224#117—2981no—
225#117—1491no—
226C2×C4⋊C4—1491no—
227C2×C22⋊C4—2981no—
228#107—1491no—
229#106—1491no—
230#106—1491no—
231C4⋊C8C2×D14911yescharacteristic, complemented
232#103—1491no—
233#103—1491no—
234#103—1491no—
235#103—1491no—
236#102—1491no—
237#102—1491no—
238#102—1491no—
239#102—1491no—
240#102—1491no—
241#102—1491no—
242C22⋊C8C2×D14911yescharacteristic
243C22⋊C8C2×D14911yescharacteristic, complemented
244C4×C8C2×D14911yescharacteristic, complemented
245C24—2981no—
246#41—2982no—
247#41—5961no—
248C2×Q8—1491no—
249C2×Q8—1491no—
250C2×D4—2982no—
251C2×D4—2981no—
252C2×D4—2981no—
253C2×D4—2981no—
254C2×D4—5961no—
255C2×D4—1491no—
256C2×D4—1491no—
257C2×D4D59611yescharacteristic
258C22×C4—2981no—
259C22×C4—2981no—
260C22×C4—2981no—
261C22×C4C22×D14911yescharacteristic
262C22×C4C22×D14911yescharacteristic
263Q16—5961no—
264SD16—2982no—
265SD16—5961no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D14911yescharacteristic
269C2×C8C22×D14911yescharacteristic
270C2×C8C22×D14911yescharacteristic
271C2×C8C22×D14911yescharacteristic
272C4⋊C4—2981no—
273C4⋊C4—2981no—
274C4⋊C4—2981no—
275C4⋊C4—2981no—
276C4⋊C4—2981no—
277C4⋊C4—2981no—
278C4⋊C4—2981no—
279C4⋊C4—1491no—
280C4⋊C4—1491no—
281C4⋊C4—1491no—
282C4⋊C4—1491no—
283C4⋊C4D59611yescharacteristic
284C22⋊C4—2981no—
285C22⋊C4—2981no—
286C22⋊C4—2981no—
287C22⋊C4—2981no—
288C22⋊C4—2981no—
289C22⋊C4—2981no—
290C22⋊C4—2981no—
291C22⋊C4—2981no—
292C22⋊C4D59611yescharacteristic
293C22⋊C4D59611yescharacteristic
294C42—2981no—
295C42—2981no—
296C42C22×D14911yescharacteristic
297C23—2981no—
298C23—2981no—
299C23—5962no—
300C23—5962no—
301C23—11921no—
302C23C2×D59611yescharacteristic
303C23C2×D59611yescharacteristic
304Q8—2981no—
305Q8—2981no—
306Q8—2981no—
307Q8—2981no—
308D4—22no—
309D4—2981no—
310D4—2981no—
311D4—2981no—
312D4—5962no—
313D4—5962no—
314D4—1492no—
315D4—11921no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2981no—
321C2×C4—2981no—
322C2×C4—2981no—
323C2×C4—2981no—
324C2×C4—2981no—
325C2×C4—2981no—
326C2×C4—5962no—
327C2×C4—5962no—
328C2×C4—11921no—
329C2×C4C2×D59611yescharacteristic
330C2×C4C2×D59611yescharacteristic
331C2×C4C2×D59611yescharacteristic
332C2×C4C2×D59611yescharacteristic
333C2×C4C23×D14911yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—2982no—
344C22—5961no—
345C22—5961no—
346C22—5961no—
347C22—5961no—
348C22—5961no—
349C22—11924no—
350C22#1801193312yes—
351C22C22×D59611yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—5961no—
357C4—5961no—
358C4—5961no—
359C4—5961no—
360C4—5961no—
361C4—5961no—
362C4#41×D14911yescharacteristic
363C4#1801195611yescharacteristic
364C2—22no—
365C2—41no—
366C2—5962no—
367C2—11921no—
368C2#2375027011yescharacteristic
369C2C2×#1801193311yescharacteristic
370C2#2375043711yescharacteristic
subgroupG/H
centerC22C22×D596
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC8×D4×C149C2
socleC22×C149C22×D4
prime radical#23750271C2
hypercenterC8×D4D149
hypocenterC149#3296
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #23750315C2semidirect
3. #23749980C2semidirect
4. #23749956C2semidirect
5. #23749950C2semidirect
6. #23749926C2semidirect
7. #23749866C2semidirect
8. #23749854C2semidirect
9. #23749848C2semidirect
10. #23749789C2semidirect
11. #23749778C2semidirect
12. C4×#14558725C2semidirect
13. #23749622C2semidirect
14. C2×#18011833C2semidirect
15. C8×D4×C149C2semidirect
18. C2×Dic596C22semidirect
22. #18011903C22semidirect
29. #18011890C22semidirect
33. C4×Dic298C22semidirect
35. #18011870C22semidirect
39. #18011845C22semidirect
40. #18011843C22semidirect
41. #18011843C22semidirect
42. #18011836C22semidirect
43. #18011836C22semidirect
44. #18011836C22semidirect
45. #18011836C22semidirect
53. C22×C8×C149C22semidirect
55. C4⋊C8×C149C22semidirect
57. C22⋊C8×C149C22semidirect
58. C4×C8×C149C22semidirect
185. C149#3296semidirect
199. C8×D4D149semidirect
208. C22×C8C2×D149semidirect
231. C4⋊C8C2×D149semidirect
243. C22⋊C8C2×D149semidirect
244. C4×C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237503152
#237502712
#237499802
#237499562
#237499502
#237499262
#237498662
#237498542
#237498482
#237497892
#237497782
C4×#145587252
#237496222
C2×#180118332
C8×D4×C1492
#3296149