Finite Group Databasebeta

← #34054182

Subgroups — #34054182

Order 19072.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011943C211yesmaximal, characteristic, complemented
2C2×#18011938C211yesmaximal, characteristic
3#124×D149C211yesmaximal, characteristic, complemented
4C4×#14558737C211yesmaximal, characteristic, complemented
5#23749849C211yesmaximal, characteristic, complemented
6#23749847C211yesmaximal, characteristic, complemented
7#23749845C211yesmaximal, characteristic, complemented
8#23749843C211yesmaximal, characteristic, complemented
9#23749841C211yesmaximal, characteristic, complemented
10#23749839C211yesmaximal, characteristic, complemented
11#23749837C211yesmaximal, characteristic, complemented
12#23749647C211yesmaximal, characteristic, complemented
13#23749632C211yesmaximal, characteristic, complemented
14#23749628C211yesmaximal, characteristic, complemented
15#464×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18C2×D4×D149C2211yescharacteristic
19#18011943—24no—
20#18011938—24no—
21C2×#14558726C2211yescharacteristic, complemented
22C2×#14558725C2211yescharacteristic, complemented
23C2×#14558723C2211yescharacteristic
24C2×#14558723C2211yescharacteristic, complemented
25C2×D1192C2211yescharacteristic
26C2×#14558721C2211yescharacteristic
27C2×#14558719C2211yescharacteristic
28C2×#14558719C2211yescharacteristic
29#18011900C2211yescharacteristic
30#18011899C2211yescharacteristic
31#18011898C2211yescharacteristic
32#18011897—21no—
33#18011897—21no—
34Q8×Dic149C2211yescharacteristic, complemented
35C4×D596—21no—
36C4×D596C2211yescharacteristic, complemented
37#18011873—21no—
38#18011873C2211yescharacteristic, complemented
39#18011869—21no—
40#18011869C2211yescharacteristic
41C22⋊C4×D149—21no—
42C22⋊C4×D149—21no—
43C42×D149—21no—
44C42×D149C2211yescharacteristic
45C4⋊#11924746C2211yescharacteristic, complemented
46#18011840C2211yescharacteristic, complemented
47#18011840C2211yescharacteristic, complemented
48#18011836C2211yescharacteristic, complemented
49#18011835C2211yescharacteristic, complemented
50#18011834C2211yescharacteristic, complemented
51#18011833C2211yescharacteristic
52#18011832C2211yescharacteristic
53#18011831C2211yescharacteristic
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#14558737D414yes—
64C2×Dic298—21no—
65C2×Dic298C2311yescharacteristic
66Q8×D149—22no—
67Q8×D149—22no—
68C2×D596—21no—
69C2×D596C2311yescharacteristic
70C2×D596C2311yescharacteristic
71C2×C22⋊D149—21no—
72D4×D149—22no—
73D4×D149—22no—
74C2×C4×D149—21no—
75C2×C4×D149—21no—
76C2×C4×D149—21no—
77C2×C4×D149—21no—
78C2×C4×D149D411yescharacteristic
79C2×C4×D149D411yescharacteristic
80C2×C4×D149C2311yescharacteristic
81#14558726—22no—
82#14558725—22no—
83#14558723—22no—
84#14558723—22no—
85D1192—22no—
86#14558721—22no—
87#14558719—22no—
88#14558719—22no—
89C2×#11924746C2311yescharacteristic
90C2×#11924746C2311yescharacteristic
91C4⋊Dic149—21no—
92C4⋊Dic149C2311yescharacteristic
93C22⋊Dic149—21no—
94C22⋊Dic149—21no—
95#14558713—21no—
96#14558713—21no—
97#14558713—21no—
98#14558713—21no—
99#14558713—21no—
100#14558713—21no—
101#14558713—21no—
102C4×Dic149—21no—
103C4×Dic149—21no—
104C4×Dic149D411yescharacteristic
105C4×Dic149D411yescharacteristic
106C4×Dic149C2311yescharacteristic
107C2×Q8×C149C2311yescharacteristic
108C2×Q8×C149C2311yescharacteristic
109C2×D4×C149C2311yescharacteristic
110Q16×C149—22no—
111SD16×C149—22no—
112C2×C8×C149C2311yescharacteristic
113C2×C8×C149C2311yescharacteristic
114C4⋊C4×C149—21no—
115C4⋊C4×C149C2311yescharacteristic
116C22⋊C4×C149—21no—
117C22⋊C4×C149—21no—
118C42×C149—21no—
119C42×C149C2311yescharacteristic
120C22×D149—21no—
121C22×D149—21no—
122C22×D149—42no—
123C22×D149—42no—
124C22×D149—42no—
125C22×D149C2×D411yescharacteristic
126Dic298—21no—
127Dic298—21no—
128Dic298—42no—
129D596—24no—
130D596—21no—
131D596—21no—
132D596—21no—
133D596C2×D412yes—
134C22⋊D149—42no—
135C2×Dic149—21no—
136C2×Dic149—21no—
137C2×Dic149C2×D411yescharacteristic
138C2×Dic149C2×D411yescharacteristic
139C2×Dic149C2×D411yescharacteristic
140C4×D149—24no—
141C4×D149—41no—
142C4×D149—41no—
143C4×D149—42no—
144C4×D149—42no—
145C4×D149—42no—
146C4×D149C2×D412yes—
147C4×D149#4112yes—
148#11924746—21no—
149#11924746—21no—
150C23×C149—21no—
151Q8×C149—21no—
152Q8×C149—21no—
153Q8×C149—21no—
154Q8×C149C2×D412yes—
155D4×C149—21no—
156D4×C149—21no—
157C2×C4×C149—21no—
158C2×C4×C149—21no—
159C2×C4×C149—21no—
160C2×C4×C149C2×D411yescharacteristic
161C2×C4×C149C2×D411yescharacteristic
162C2×C4×C149C2411yescharacteristic
163C8×C149—21no—
164C8×C149—21no—
165C22×C149—41no—
166C22×C149—41no—
167C22×C149—41no—
168C22×C149C22×D411yescharacteristic
169C2×D149—21no—
170C2×D149—21no—
171C2×D149—22no—
172C2×D149—41no—
173C2×D149—41no—
174C2×D149—41no—
175C2×D149—41no—
176C2×D149—41no—
177C2×D149—84no—
178C2×D149#12112yes—
179C4×C149—22no—
180C4×C149—21no—
181C4×C149—21no—
182C4×C149—41no—
183C4×C149—41no—
184C4×C149C22×D411yescharacteristic
185C4×C149C2×#4111yescharacteristic
186Dic149—21no—
187Dic149—21no—
188Dic149—41no—
189Dic149—41no—
190Dic149#12112yes—
191C2×C149—81no—
192C2×C149C2×#12111yescharacteristic
193C2×C149#57511yescharacteristic
194C2×C149#57511yescharacteristic
195D149—22no—
196D149—42no—
197D149—81no—
198C149#304011yescharacteristic, complemented, Sylow
199#3040—1491nomaximal, Sylow
200C2×#137—1491no—
201C2×AGL(1,ℤ/8)—1491no—
202C2×#124—1491no—
203C4×#41—1491no—
204#464—1491no—
205#464—1491no—
206#464—1491no—
207#464D14911yescharacteristic, complemented
208#463—1491no—
209#463—1491no—
210#463—1491no—
211#463—1491no—
212#423—1491no—
213#419—1491no—
214#418—1491no—
215C2×#41—1491no—
216C22×Q8—1491no—
217C22×D4—1491no—
218#137—2984no—
219AGL(1,ℤ/8)—2984no—
220C2×Q16—1491no—
221C2×Q16C2×D14911yescharacteristic, complemented
222C2×SD16—1491no—
223C2×SD16—1491no—
224C2×SD16—1491no—
225C2×SD16C2×D14911yescharacteristic
226C2×D8—1491no—
227C2×D8—1491no—
228C2×M16—1491no—
229C2×M16—1491no—
230#124—2981no—
231#124—2981no—
232#124—1491no—
233#124—1491no—
234#124—1491no—
235#124C2×D14911yescharacteristic
236C4×Q8—1491no—
237C4×Q8C2×D14911yescharacteristic, complemented
238C4×D4—2981no—
239C4×D4—2981no—
240C4×D4—1491no—
241C4×D4—1491no—
242#117—2981no—
243#117—1491no—
244C2×C22⋊C4—2981no—
245C2×C22⋊C4—2981no—
246C2×C42—2981no—
247C2×C42—1491no—
248C4⋊C8—1491no—
249C4⋊C8—1491no—
250C4⋊C8—1491no—
251C4⋊C8C2×D14911yescharacteristic, complemented
252#103—1491no—
253#103—1491no—
254#103—1491no—
255#103C2×D14911yescharacteristic, complemented
256#102—1491no—
257#102—1491no—
258#102—1491no—
259#102C2×D14911yescharacteristic
260C24—2981no—
261#41—2982no—
262#41—1494no—
263C2×Q8—2981no—
264C2×Q8—2982no—
265C2×Q8—2982no—
266C2×Q8—1491no—
267C2×Q8C22×D14911yescharacteristic
268C2×Q8C22×D14911yescharacteristic
269C2×D4—2981no—
270C2×D4—2982no—
271C2×D4—2982no—
272C2×D4—2981no—
273C2×D4—1491no—
274C2×D4—1491no—
275C2×D4C22×D14911yescharacteristic
276C22×C4—2981no—
277C22×C4—2981no—
278C22×C4—2981no—
279C22×C4—2981no—
280C22×C4—1491no—
281C22×C4—1491no—
282C22×C4—1491no—
283Q16—22no—
284Q16—2982no—
285SD16—22no—
286SD16—2982no—
287SD16—2982no—
288SD16—2982no—
289D8—2982no—
290D8—2982no—
291M16—2982no—
292M16—2982no—
293C2×C8—1491no—
294C2×C8—1491no—
295C2×C8C22×D14911yescharacteristic
296C2×C8C22×D14911yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—2981no—
299C4⋊C4—1491no—
300C4⋊C4C22×D14911yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2981no—
304C22⋊C4—2981no—
305C22⋊C4—2981no—
306C22⋊C4—2981no—
307C22⋊C4—2981no—
308C22⋊C4—2981no—
309C22⋊C4—2981no—
310C22⋊C4—2981no—
311C22⋊C4—2981no—
312C42—21no—
313C42—2981no—
314C42—2981no—
315C42—1491no—
316C42—1491no—
317C42—1491no—
318C42C22×D14911yescharacteristic
319C23—21no—
320C23—2981no—
321C23—2981no—
322C23—5962no—
323C23—5962no—
324C23—5962no—
325C23—1491no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—2981no—
330Q8—2981no—
331Q8—5962no—
332Q8D4×D14912yes—
333D4—21no—
334D4—21no—
335D4—2984no—
336D4—2981no—
337D4—2981no—
338D4—2981no—
339D4—5962no—
340D4—1492no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2984no—
345C2×C4—2981no—
346C2×C4—2981no—
347C2×C4—5961no—
348C2×C4—5961no—
349C2×C4—5962no—
350C2×C4—5962no—
351C2×C4—5962no—
352C2×C4—1491no—
353C2×C4—1491no—
354C2×C4—1491no—
355C2×C4—1492no—
356C2×C4—1492no—
357C2×C4D4×D14911yescharacteristic
358C2×C4D4×D14911yescharacteristic
359C2×C4C23×D14911yescharacteristic
360C8—21no—
361C8—21no—
362C8—2981no—
363C8—2981no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—2981no—
368C22—2981no—
369C22—2982no—
370C22—5961no—
371C22—5961no—
372C22—5961no—
373C22—5961no—
374C22—5961no—
375C22—1492no—
376C22—11924no—
377C22C2×D4×D14911yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—2981no—
384C4—2981no—
385C4—5961no—
386C4—5961no—
387C4—1492no—
388C4C2×D4×D14911yescharacteristic
389C4#41×D14911yescharacteristic
390C2—81no—
391C2—2982no—
392C2—5962no—
393C2—11921no—
394C2#121×D14911yescharacteristic
395C2#2375042211yescharacteristic
396C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#464×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011938C2
hypercenter#464D149
hypocenterC149#3040
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
3. #124×D149C2semidirect
4. C4×#14558737C2semidirect
5. #23749849C2semidirect
6. #23749847C2semidirect
7. #23749845C2semidirect
8. #23749843C2semidirect
9. #23749841C2semidirect
10. #23749839C2semidirect
11. #23749837C2semidirect
12. #23749647C2semidirect
13. #23749632C2semidirect
14. #23749628C2semidirect
15. #464×C149C2semidirect
21. C2×#14558726C22semidirect
22. C2×#14558725C22semidirect
24. C2×#14558723C22semidirect
34. Q8×Dic149C22semidirect
36. C4×D596C22semidirect
38. #18011873C22semidirect
45. C4⋊#11924746C22semidirect
46. #18011840C22semidirect
47. #18011840C22semidirect
48. #18011836C22semidirect
49. #18011835C22semidirect
50. #18011834C22semidirect
54. C2×Q16×C149C22semidirect
57. C4×Q8×C149C22semidirect
58. C4⋊C8×C149C22semidirect
59. #103×C149C22semidirect
198. C149#3040semidirect
207. #464D149semidirect
221. C2×Q16C2×D149semidirect
237. C4×Q8C2×D149semidirect
251. C4⋊C8C2×D149semidirect
255. #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×#180119382
#124×D1492
C4×#145587372
#237498492
#237498472
#237498452
#237498432
#237498412
#237498392
#237498372
#237496472
#237496322
#237496282
#464×C1492
#3040149