Finite Group Databasebeta

← #34056304

Subgroups — #34056304

Order 19072.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011938C211yesmaximal, characteristic, complemented
2C2×#18011936C211yesmaximal, characteristic, complemented
3#23750294C211yesmaximal, characteristic, complemented
4#23750259C211yesmaximal, characteristic, complemented
5#23750000C211yesmaximal, characteristic, complemented
6#23749877C211yesmaximal, characteristic, complemented
7#23749876C211yesmaximal, characteristic, complemented
8#23749823C211yesmaximal, characteristic, complemented
9#23749819C211yesmaximal, characteristic, complemented
10#23749773C211yesmaximal, characteristic, complemented
11#23749761C211yesmaximal, characteristic, complemented
12#23749757C211yesmaximal, characteristic, complemented
13#23749710C211yesmaximal, characteristic, complemented
14#23749700C211yesmaximal, characteristic, complemented
15#449×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic, complemented
17C22×D596C2211yescharacteristic
18C2×D4×D149—21no—
19C2×D4×D149C2211yescharacteristic
20#18011938—24no—
21#18011936—24no—
22C2×#14558723C2211yescharacteristic
23C2×#14558723C2211yescharacteristic, complemented
24C2×D1192C2211yescharacteristic
25C2×D1192C2211yescharacteristic, complemented
26C2×#14558721C2211yescharacteristic
27C2×#14558721C2211yescharacteristic, complemented
28C2×#14558721C2211yescharacteristic, complemented
29C2×#14558720C2211yescharacteristic, complemented
30C2×#14558719C2211yescharacteristic, complemented
31#18011907—21no—
32#18011907C2211yescharacteristic
33#18011907C2211yescharacteristic, complemented
34#18011886C2211yescharacteristic
35#18011885—21no—
36#18011885C2211yescharacteristic, complemented
37#18011884—21no—
38#18011882—21no—
39#18011882C2211yescharacteristic, complemented
40#18011881—21no—
41#18011879—21no—
42Q8×Dic149C2211yescharacteristic, complemented
43#18011873C2211yescharacteristic
44C4×C22⋊D149—21no—
45C4×C22⋊D149C2211yescharacteristic
46#18011842C2211yescharacteristic, complemented
47#18011840C2211yescharacteristic, complemented
48#18011833C2211yescharacteristic
49#18011833C2211yescharacteristic, complemented
50#18011832C2211yescharacteristic, complemented
51#18011831C2211yescharacteristic, complemented
52#18011823C2211yescharacteristic, complemented
53C8×Dic149C2211yescharacteristic, complemented
54C2×#41×C149C2211yescharacteristic, complemented
55C2×SD16×C149C2211yescharacteristic, complemented
56C2×D8×C149C2211yescharacteristic, complemented
57#121×C149C2211yescharacteristic
58#103×C149C2211yescharacteristic, complemented
59#102×C149C2211yescharacteristic, complemented
60C22⋊C8×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62C23×D149—21no—
63#14558737—22no—
64#14558737—41no—
65C2×D596—22no—
66C2×D596—21no—
67C2×D596—21no—
68C2×D596—21no—
69C2×D596—41no—
70C2×D596C2311yescharacteristic
71C2×D596C2311yescharacteristic
72C2×D596C2311yescharacteristic
73C2×C22⋊D149—21no—
74C2×C22⋊D149—21no—
75C2×C22⋊D149—21no—
76C2×C22⋊D149—21no—
77C2×C22⋊D149—21no—
78C2×C22⋊D149—21no—
79C2×C22⋊D149—21no—
80C2×C22⋊D149D411yescharacteristic
81D4×D149—22no—
82D4×D149—41no—
83D4×D149—44no—
84C2×C4×D149—21no—
85C2×C4×D149—21no—
86C2×C4×D149C2311yescharacteristic
87#14558723—22no—
88#14558723D414yescomplemented
89D1192—22no—
90D1192—41no—
91#14558721—22no—
92#14558721—22no—
93#14558721—22no—
94#14558720—22no—
95#14558719—22no—
96C2×#11924746C2311yescharacteristic
97C2×#11924746C2311yescharacteristic
98C4⋊Dic149—21no—
99C4⋊Dic149C2311yescharacteristic
100#14558715D411yescharacteristic
101C22⋊Dic149—21no—
102C22⋊Dic149D411yescharacteristic
103#14558713—21no—
104#14558713—21no—
105#14558713—21no—
106#14558713D411yescharacteristic
107C4×Dic149—21no—
108C4×Dic149C2311yescharacteristic
109#41×C149—22no—
110#41×C149—41no—
111C2×Q8×C149C2311yescharacteristic
112C2×D4×C149—21no—
113C2×D4×C149—21no—
114C2×D4×C149C2311yescharacteristic
115C2×D4×C149C2311yescharacteristic
116C22×C4×C149—21no—
117C22×C4×C149C2311yescharacteristic
118SD16×C149—22no—
119D8×C149—41no—
120C2×C8×C149C2311yescharacteristic
121C2×C8×C149C2311yescharacteristic
122C4⋊C4×C149C2311yescharacteristic
123C22⋊C4×C149—21no—
124C22×D149—21no—
125C22×D149—21no—
126C22×D149—21no—
127C22×D149—42no—
128C22×D149—42no—
129C22×D149—42no—
130C22×D149—42no—
131C22×D149—81no—
132C22×D149—81no—
133C22×D149C2×D411yescharacteristic
134D596—21no—
135D596—21no—
136D596—21no—
137D596—21no—
138D596—21no—
139D596—42no—
140D596—42no—
141D596—42no—
142D596C2×D412yes—
143C22⋊D149—22no—
144C22⋊D149—42no—
145C22⋊D149—42no—
146C22⋊D149—42no—
147C22⋊D149—81no—
148C22⋊D149—81no—
149C22⋊D149—81no—
150C22⋊D149—81no—
151C2×Dic149—21no—
152C2×Dic149C2×D411yescharacteristic
153C2×Dic149C2×D411yescharacteristic
154C2×Dic149C2×D411yescharacteristic
155C4×D149—21no—
156C4×D149—21no—
157C4×D149—42no—
158C4×D149—42no—
159#11924746—21no—
160#11924746C2×D412yescomplemented
161C23×C149—21no—
162C23×C149—21no—
163C23×C149C2×D411yescharacteristic
164Q8×C149—21no—
165Q8×C149C2×D412yescomplemented
166D4×C149—21no—
167D4×C149—21no—
168D4×C149—21no—
169D4×C149—21no—
170D4×C149—42no—
171D4×C149—42no—
172C2×C4×C149—21no—
173C2×C4×C149—21no—
174C2×C4×C149—21no—
175C2×C4×C149—21no—
176C2×C4×C149—42no—
177C2×C4×C149C2×D411yescharacteristic
178C2×C4×C149C2411yescharacteristic
179C8×C149—21no—
180C8×C149—21no—
181C22×C149—21no—
182C22×C149—21no—
183C22×C149—21no—
184C22×C149—41no—
185C22×C149—41no—
186C22×C149—41no—
187C22×C149—41no—
188C22×C149—41no—
189C22×C149—41no—
190C22×C149C22×D411yescharacteristic
191C2×D149—21no—
192C2×D149—22no—
193C2×D149—21no—
194C2×D149—21no—
195C2×D149—41no—
196C2×D149—41no—
197C2×D149—41no—
198C2×D149—41no—
199C2×D149—41no—
200C2×D149—41no—
201C2×D149—41no—
202C2×D149—41no—
203C2×D149—84no—
204C2×D149—84no—
205C4×C149—21no—
206C4×C149—22no—
207C4×C149—41no—
208C4×C149C22×D411yescharacteristic
209C4×C149H(5,𝔽2)11yescharacteristic
210Dic149—21no—
211Dic149—21no—
212Dic149—41no—
213Dic149AGL(1,ℤ/8)12yescomplemented
214C2×C149—41no—
215C2×C149—81no—
216C2×C149—81no—
217C2×C149D4211yescharacteristic
218C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C149#57611yescharacteristic
220D149—41no—
221D149—42no—
222D149—81no—
223D149—81no—
224C149#327411yescharacteristic, complemented, Sylow
225#3274—1491nomaximal, Sylow
226C2×AGL(1,ℤ/8)—1491no—
227C2×AGL(1,ℤ/8)—1491no—
228#550—1491no—
229D42—1491no—
230#493—1491no—
231#469—1491no—
232#469—1491no—
233#460—1491no—
234#459—1491no—
235#449—1491no—
236#449D14911yescharacteristic, complemented
237#447—1491no—
238#447—1491no—
239C4×SD16—1491no—
240#436—1491no—
241C2×#41—1491no—
242C2×#41C2×D14911yescharacteristic, complemented
243C22×D4—2981no—
244C22×D4—1491no—
245C22×D4—1491no—
246AGL(1,ℤ/8)—2984no—
247AGL(1,ℤ/8)—2984no—
248C2×SD16—1491no—
249C2×SD16—1491no—
250C2×SD16C2×D14911yescharacteristic, complemented
251C2×D8—1491no—
252C2×D8—1491no—
253C2×D8—1491no—
254C2×D8—1491no—
255C2×D8—1491no—
256C2×D8C2×D14911yescharacteristic, complemented
257C2×M16—1491no—
258C2×M16—1491no—
259#127—2981no—
260#127—1491no—
261#127—1491no—
262#121—2981no—
263#121—2981no—
264#121—2981no—
265#121—1491no—
266#121—1491no—
267#121—1491no—
268#121C2×D14911yescharacteristic
269#120—2981no—
270#120—2981no—
271C4×Q8—1491no—
272C4×D4—2981no—
273C4×D4—1491no—
274C4×D4—1491no—
275#106—1491no—
276C4⋊C8—1491no—
277#103C2×D14911yescharacteristic, complemented
278#102—1491no—
279#102—1491no—
280#102—1491no—
281#102—1491no—
282#102C2×D14911yescharacteristic, complemented
283C22⋊C8—1491no—
284C22⋊C8C2×D14911yescharacteristic, complemented
285C4×C8—1491no—
286C24—2981no—
287C24—2981no—
288#41—22no—
289#41—41no—
290#41—2982no—
291#41—5961no—
292C2×Q8C22×D14911yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2982no—
296C2×D4—2981no—
297C2×D4—2981no—
298C2×D4—2982no—
299C2×D4—2981no—
300C2×D4—2981no—
301C2×D4—2981no—
302C2×D4—2981no—
303C2×D4—2981no—
304C2×D4—2981no—
305C2×D4—2981no—
306C2×D4—2981no—
307C2×D4—5961no—
308C2×D4—5961no—
309C2×D4—5964no—
310C2×D4—1491no—
311C2×D4—1491no—
312C2×D4—1491no—
313C2×D4—1491no—
314C2×D4C22×D14911yescharacteristic
315C2×D4C22×D14911yescharacteristic
316C22×C4—21no—
317C22×C4—2981no—
318C22×C4—2981no—
319C22×C4—1491no—
320C22×C4C22×D14911yescharacteristic
321SD16—22no—
322SD16—2982no—
323SD16—1494no—
324D8—41no—
325D8—2982no—
326D8—2982no—
327D8—2982no—
328D8—2982no—
329D8—5961no—
330M16—2982no—
331M16—2982no—
332C2×C8—1491no—
333C2×C8—1491no—
334C2×C8C22×D14911yescharacteristic
335C2×C8C22×D14911yescharacteristic
336C4⋊C4—2981no—
337C4⋊C4—1491no—
338C4⋊C4—1491no—
339C4⋊C4C22×D14911yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—2981no—
342C22⋊C4—2981no—
343C22⋊C4—2981no—
344C22⋊C4—2981no—
345C22⋊C4—1491no—
346C22⋊C4—1491no—
347C42—2981no—
348C42—1491no—
349C23—21no—
350C23—21no—
351C23—2981no—
352C23—2981no—
353C23—2981no—
354C23—5962no—
355C23—5962no—
356C23—5962no—
357C23—5962no—
358C23—1491no—
359C23—11921no—
360C23—11921no—
361C23D4×D14911yescharacteristic
362Q8—21no—
363Q8D4×D14912yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—42no—
370D4—2982no—
371D4—2981no—
372D4—2981no—
373D4—2981no—
374D4—2981no—
375D4—2981no—
376D4—5962no—
377D4—5962no—
378D4—5962no—
379D4—5962no—
380D4—5962no—
381D4—5962no—
382D4—1492no—
383D4—11921no—
384D4—11921no—
385D4—11921no—
386D4—11921no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—2981no—
393C2×C4—2981no—
394C2×C4—2981no—
395C2×C4—5962no—
396C2×C4—5962no—
397C2×C4—1491no—
398C2×C4—1491no—
399C2×C4—1491no—
400C2×C4D4×D14911yescharacteristic
401C2×C4C23×D14911yescharacteristic
402C8—21no—
403C8—21no—
404C8—2981no—
405C8—1492no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—2981no—
416C22—2981no—
417C22—2981no—
418C22—2982no—
419C22—5961no—
420C22—5961no—
421C22—5961no—
422C22—5961no—
423C22—5961no—
424C22—5961no—
425C22—5961no—
426C22—5961no—
427C22—11924no—
428C22—11924no—
429C22C2×D4×D14911yescharacteristic
430C4—21no—
431C4—22no—
432C4—41no—
433C4—2981no—
434C4—2981no—
435C4—5961no—
436C4—1492no—
437C4C2×D4×D14911yescharacteristic
438C4#1801195611yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—5961no—
443C2—5962no—
444C2—11921no—
445C2—11921no—
446C2#2375025711yescharacteristic
447C2AGL(1,ℤ/8)×D14911yescharacteristic
448C2#2375042711yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#449×C149C2
socleC22×C149C22×D4
hypercenter#449D149
hypocenterC149#3274
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#18011938C2semidirect
2. C2×#18011936C2semidirect
3. #23750294C2semidirect
4. #23750259C2semidirect
5. #23750000C2semidirect
6. #23749877C2semidirect
7. #23749876C2semidirect
8. #23749823C2semidirect
9. #23749819C2semidirect
10. #23749773C2semidirect
11. #23749761C2semidirect
12. #23749757C2semidirect
13. #23749710C2semidirect
14. #23749700C2semidirect
15. #449×C149C2semidirect
16. C2×#14558737C22semidirect
23. C2×#14558723C22semidirect
25. C2×D1192C22semidirect
27. C2×#14558721C22semidirect
28. C2×#14558721C22semidirect
29. C2×#14558720C22semidirect
30. C2×#14558719C22semidirect
33. #18011907C22semidirect
36. #18011885C22semidirect
39. #18011882C22semidirect
42. Q8×Dic149C22semidirect
46. #18011842C22semidirect
47. #18011840C22semidirect
49. #18011833C22semidirect
50. #18011832C22semidirect
51. #18011831C22semidirect
52. #18011823C22semidirect
53. C8×Dic149C22semidirect
54. C2×#41×C149C22semidirect
55. C2×SD16×C149C22semidirect
56. C2×D8×C149C22semidirect
58. #103×C149C22semidirect
59. #102×C149C22semidirect
60. C22⋊C8×C149C22semidirect
88. #14558723D4semidirect
160. #11924746C2×D4semidirect
165. Q8×C149C2×D4semidirect
213. Dic149AGL(1,ℤ/8)semidirect
224. C149#3274semidirect
236. #449D149semidirect
242. C2×#41C2×D149semidirect
250. C2×SD16C2×D149semidirect
256. C2×D8C2×D149semidirect
277. #103C2×D149semidirect
282. #102C2×D149semidirect
284. C22⋊C8C2×D149semidirect
363. Q8D4×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119382
C2×#180119362
#237502942
#237502592
#237500002
#237498772
#237498762
#237498232
#237498192
#237497732
#237497612
#237497572
#237497102
#237497002
#449×C1492
#3274149