Finite Group Databasebeta

← #34056196

Subgroups — #34056196

Order 19072.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011943C211yesmaximal, characteristic, complemented
2C2×SD16×D149C211yesmaximal, characteristic, complemented
3Q8×D596C211yesmaximal, characteristic, complemented
4#23750268C211yesmaximal, characteristic
5#23750020C211yesmaximal, characteristic, complemented
6#23749870C211yesmaximal, characteristic, complemented
7#23749865C211yesmaximal, characteristic, complemented
8#23749831C211yesmaximal, characteristic, complemented
9#23749790C211yesmaximal, characteristic, complemented
10#23749779C211yesmaximal, characteristic, complemented
11#23749777C211yesmaximal, characteristic, complemented
12#23749762C211yesmaximal, characteristic, complemented
13#23749723C211yesmaximal, characteristic, complemented
14#23749695C211yesmaximal, characteristic, complemented
15#464×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18C2×Q8×D149C2211yescharacteristic, complemented
19C2×D4×D149C2211yescharacteristic
20#18011943—24no—
21C2×#14558726C2211yescharacteristic, complemented
22C2×#14558726C2211yescharacteristic, complemented
23C2×#14558725C2211yescharacteristic
24C2×#14558725C2211yescharacteristic
25C2×#14558724C2211yescharacteristic
26C2×#14558724C2211yescharacteristic
27C2×#14558723C2211yescharacteristic
28SD16×D149—24no—
29C2×#14558719C2211yescharacteristic, complemented
30C2×C8×D149C2211yescharacteristic
31#18011911—21no—
32#18011911C2211yescharacteristic, complemented
33#18011893—21no—
34#18011891—21no—
35#18011891—21no—
36#18011891C2211yescharacteristic
37#18011891C2211yescharacteristic, complemented
38#18011887C2211yescharacteristic
39#18011884—21no—
40#18011882C2211yescharacteristic
41#18011881—21no—
42C4×D596—21no—
43C4×D596C2211yescharacteristic
44C4×D596C2211yescharacteristic
45C22⋊C4×D149—21no—
46#18011844C2211yescharacteristic, complemented
47#18011836C2211yescharacteristic, complemented
48#18011836C2211yescharacteristic, complemented
49#18011835C2211yescharacteristic, complemented
50#18011831C2211yescharacteristic
51#18011823C2211yescharacteristic
52#18011823C2211yescharacteristic, complemented
53#18011822C2211yescharacteristic, complemented
54C2×Q16×C149C2211yescharacteristic, complemented
55C2×SD16×C149C2211yescharacteristic
56#124×C149C2211yescharacteristic
57C4×Q8×C149C2211yescharacteristic, complemented
58C4⋊C8×C149C2211yescharacteristic, complemented
59#103×C149C2211yescharacteristic, complemented
60#102×C149C2211yescharacteristic
61C23×D149—21no—
62#14558737—22no—
63#14558737—41no—
64C2×Dic298—21no—
65C2×Dic298—21no—
66C2×Dic298C2311yescharacteristic
67C2×Dic298C2311yescharacteristic
68Q8×D149—22no—
69Q8×D149—22no—
70Q8×D149—41no—
71Q8×D149—41no—
72C2×D596—21no—
73C2×D596D411yescharacteristic
74C2×D596C2311yescharacteristic
75C2×C22⋊D149—21no—
76C2×C22⋊D149—21no—
77D4×D149—22no—
78D4×D149—41no—
79C2×C4×D149—21no—
80C2×C4×D149—21no—
81C2×C4×D149—21no—
82C2×C4×D149—21no—
83C2×C4×D149C2311yescharacteristic
84C2×C4×D149C2311yescharacteristic
85#14558726—22no—
86#14558726—41no—
87#14558725—22no—
88#14558725—22no—
89#14558724—22no—
90#14558724—41no—
91#14558723D414yescomplemented
92#14558719—22no—
93C2×#11924746C2311yescharacteristic
94C2×#11924746C2311yescharacteristic
95C8×D149—22no—
96C4⋊Dic149—21no—
97C4⋊Dic149—21no—
98C4⋊Dic149—21no—
99C4⋊Dic149—21no—
100C4⋊Dic149D411yescharacteristic
101C4⋊Dic149C2311yescharacteristic
102#14558715—21no—
103C22⋊Dic149—21no—
104#14558713—21no—
105#14558713—21no—
106#14558713—21no—
107#14558713—21no—
108#14558713D411yescharacteristic
109#14558713D411yescharacteristic
110C2×Q8×C149C2311yescharacteristic
111C2×Q8×C149C2311yescharacteristic
112C2×D4×C149C2311yescharacteristic
113Q16×C149—22no—
114SD16×C149—22no—
115C2×C8×C149C2311yescharacteristic
116C2×C8×C149C2311yescharacteristic
117C4⋊C4×C149—21no—
118C4⋊C4×C149C2311yescharacteristic
119C22⋊C4×C149—21no—
120C22⋊C4×C149—21no—
121C42×C149—21no—
122C42×C149C2311yescharacteristic
123C22×D149—21no—
124C22×D149—42no—
125C22×D149—42no—
126C22×D149—81no—
127C22×D149C2×D411yescharacteristic
128C22×D149C2×D411yescharacteristic
129Dic298—21no—
130Dic298—21no—
131Dic298—21no—
132Dic298—21no—
133Dic298—42no—
134Dic298—42no—
135D596—22no—
136D596—21no—
137D596—42no—
138D596C2×D412yes—
139C22⋊D149—42no—
140C22⋊D149—81no—
141C2×Dic149—21no—
142C2×Dic149—21no—
143C2×Dic149—21no—
144C2×Dic149C2×D411yescharacteristic
145C2×Dic149C2×D411yescharacteristic
146C4×D149—21no—
147C4×D149—21no—
148C4×D149—21no—
149C4×D149—21no—
150C4×D149—42no—
151C4×D149—42no—
152C4×D149—42no—
153C4×D149—81no—
154#11924746—21no—
155#11924746C2×D412yes—
156C23×C149—21no—
157Q8×C149—21no—
158Q8×C149—21no—
159Q8×C149—21no—
160Q8×C149C2×D412yes—
161D4×C149—21no—
162D4×C149—21no—
163C2×C4×C149—21no—
164C2×C4×C149—21no—
165C2×C4×C149—21no—
166C2×C4×C149C2×D411yescharacteristic
167C2×C4×C149C2×D411yescharacteristic
168C2×C4×C149C2411yescharacteristic
169C8×C149—21no—
170C8×C149—21no—
171C22×C149—41no—
172C22×C149—41no—
173C22×C149—41no—
174C22×C149C22×D411yescharacteristic
175C2×D149—21no—
176C2×D149—21no—
177C2×D149—21no—
178C2×D149—22no—
179C2×D149—21no—
180C2×D149—21no—
181C2×D149—41no—
182C2×D149—41no—
183C2×D149—84no—
184C2×D149#13712yescomplemented
185C4×C149—22no—
186C4×C149—21no—
187C4×C149—21no—
188C4×C149—41no—
189C4×C149—41no—
190C4×C149C22×D411yescharacteristic
191C4×C149H(5,𝔽2)11yescharacteristic
192Dic149—21no—
193Dic149—21no—
194Dic149—41no—
195Dic149—41no—
196Dic149—41no—
197C2×C149—81no—
198C2×C149D4211yescharacteristic
199C2×C149C2×#13711yescharacteristic
200C2×C149#57711yescharacteristic
201D149—22no—
202D149—41no—
203D149—42no—
204C149#326611yescharacteristic, complemented, Sylow
205#3266—1491nomaximal, Sylow
206C2×#137—1491no—
207C22×SD16—1491no—
208D4×Q8—1491no—
209#546—1491no—
210#497—1491no—
211#468—1491no—
212#467—1491no—
213#464D14911yescharacteristic, complemented
214#461—1491no—
215#452—1491no—
216#451—1491no—
217#450—1491no—
218#448—1491no—
219#440—1491no—
220#435—1491no—
221C2×#41—1491no—
222C22×Q8—1491no—
223C22×Q8—1491no—
224C22×D4—1491no—
225#137—2984no—
226C2×Q16—1491no—
227C2×Q16—1491no—
228C2×Q16C2×D14911yescharacteristic, complemented
229C2×SD16—2984no—
230C2×SD16—1491no—
231C2×SD16—1491no—
232C2×SD16—1491no—
233C2×SD16—1491no—
234C2×SD16—1491no—
235C2×SD16C2×D14911yescharacteristic
236C2×M16—1491no—
237C22×C8—1491no—
238C4⋊Q8—2981no—
239C4⋊Q8—1491no—
240#124C2×D14911yescharacteristic
241#123—2981no—
242C22⋊Q8—2981no—
243C22⋊Q8—2981no—
244C22⋊Q8—1491no—
245C22⋊Q8—1491no—
246C22⋊Q8—1491no—
247#121—2981no—
248#121—1491no—
249#120—2981no—
250C4×Q8C2×D14911yescharacteristic, complemented
251C4×D4—2981no—
252C4×D4—1491no—
253C4×D4—1491no—
254C2×C22⋊C4—2981no—
255#107—1491no—
256C4⋊C8C2×D14911yescharacteristic, complemented
257#103—1491no—
258#103—1491no—
259#103—1491no—
260#103C2×D14911yescharacteristic, complemented
261#102—1491no—
262#102C2×D14911yescharacteristic
263C22⋊C8—1491no—
264C22⋊C8—1491no—
265#97—1491no—
266C24—2981no—
267#41—2982no—
268#41—5961no—
269C2×Q8—2981no—
270C2×Q8—2981no—
271C2×Q8—2982no—
272C2×Q8—2982no—
273C2×Q8—5961no—
274C2×Q8—5961no—
275C2×Q8—1491no—
276C2×Q8—1491no—
277C2×Q8C22×D14911yescharacteristic
278C2×Q8C22×D14911yescharacteristic
279C2×D4—2981no—
280C2×D4—2982no—
281C2×D4—2981no—
282C2×D4—2981no—
283C2×D4—5961no—
284C2×D4—1491no—
285C2×D4—1491no—
286C2×D4C22×D14911yescharacteristic
287C22×C4—2981no—
288C22×C4—2981no—
289C22×C4—2981no—
290C22×C4—2981no—
291C22×C4—1491no—
292C22×C4—1491no—
293Q16—22no—
294Q16—2982no—
295Q16—5961no—
296SD16—22no—
297SD16—2982no—
298SD16—2982no—
299SD16—2982no—
300SD16—5961no—
301SD16—1494no—
302M16—2982no—
303C2×C8—2982no—
304C2×C8—1491no—
305C2×C8—1491no—
306C2×C8C22×D14911yescharacteristic
307C2×C8C22×D14911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2981no—
310C4⋊C4—2981no—
311C4⋊C4—2981no—
312C4⋊C4—2981no—
313C4⋊C4—2981no—
314C4⋊C4—1491no—
315C4⋊C4—1491no—
316C4⋊C4C22×D14911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2981no—
320C22⋊C4—2981no—
321C22⋊C4—2981no—
322C22⋊C4—2981no—
323C22⋊C4—2981no—
324C22⋊C4—1491no—
325C22⋊C4—1491no—
326C42—21no—
327C42C22×D14911yescharacteristic
328C23—21no—
329C23—2981no—
330C23—5962no—
331C23—5962no—
332C23—1491no—
333C23—1491no—
334C23—11921no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—2981no—
339Q8—2981no—
340Q8—2981no—
341Q8—2981no—
342Q8—5962no—
343Q8—5962no—
344Q8D4×D14912yes—
345D4—21no—
346D4—21no—
347D4—2982no—
348D4—2981no—
349D4—5962no—
350D4—5962no—
351D4—1492no—
352D4—11921no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—2981no—
357C2×C4—2981no—
358C2×C4—2981no—
359C2×C4—2981no—
360C2×C4—2981no—
361C2×C4—2981no—
362C2×C4—2981no—
363C2×C4—5962no—
364C2×C4—5962no—
365C2×C4—5962no—
366C2×C4—1491no—
367C2×C4—1491no—
368C2×C4—11921no—
369C2×C4D4×D14911yescharacteristic
370C2×C4D4×D14911yescharacteristic
371C2×C4C23×D14911yescharacteristic
372C8—21no—
373C8—21no—
374C8—2981no—
375C8—1492no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—2981no—
380C22—2981no—
381C22—2981no—
382C22—2981no—
383C22—2981no—
384C22—2982no—
385C22—5961no—
386C22—5961no—
387C22—1492no—
388C22—11924no—
389C22C2×D4×D14911yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—2981no—
396C4—2981no—
397C4—5961no—
398C4—5961no—
399C4—5961no—
400C4C2×D4×D14911yescharacteristic
401C4#1801195711yescharacteristic
402C2—81no—
403C2—2982no—
404C2—5961no—
405C2—5962no—
406C2#2375025911yescharacteristic
407C2#137×D14911yescharacteristic
408C2#2375043211yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#464×C149C2
socleC22×C149C22×D4
prime radical#23750268C2
hypercenter#464D149
hypocenterC149#3266
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#18011943C2semidirect
2. C2×SD16×D149C2semidirect
3. Q8×D596C2semidirect
5. #23750020C2semidirect
6. #23749870C2semidirect
7. #23749865C2semidirect
8. #23749831C2semidirect
9. #23749790C2semidirect
10. #23749779C2semidirect
11. #23749777C2semidirect
12. #23749762C2semidirect
13. #23749723C2semidirect
14. #23749695C2semidirect
15. #464×C149C2semidirect
18. C2×Q8×D149C22semidirect
21. C2×#14558726C22semidirect
22. C2×#14558726C22semidirect
29. C2×#14558719C22semidirect
32. #18011911C22semidirect
37. #18011891C22semidirect
46. #18011844C22semidirect
47. #18011836C22semidirect
48. #18011836C22semidirect
49. #18011835C22semidirect
52. #18011823C22semidirect
53. #18011822C22semidirect
54. C2×Q16×C149C22semidirect
57. C4×Q8×C149C22semidirect
58. C4⋊C8×C149C22semidirect
59. #103×C149C22semidirect
91. #14558723D4semidirect
184. C2×D149#137semidirect
204. C149#3266semidirect
213. #464D149semidirect
228. C2×Q16C2×D149semidirect
250. C4×Q8C2×D149semidirect
256. C4⋊C8C2×D149semidirect
260. #103C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119432
C2×SD16×D1492
Q8×D5962
#237502682
#237500202
#237498702
#237498652
#237498312
#237497902
#237497792
#237497772
#237497622
#237497232
#237496952
#464×C1492
#3266149