Finite Group Databasebeta

← #34056298

Subgroups — #34056298

Order 19072.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011937C211yesmaximal, characteristic, complemented
2C2×#18011936C211yesmaximal, characteristic, complemented
3#23750271C211yesmaximal, characteristic, complemented
4#23750007C211yesmaximal, characteristic, complemented
5#23749873C211yesmaximal, characteristic, complemented
6#23749869C211yesmaximal, characteristic, complemented
7#23749840C211yesmaximal, characteristic, complemented
8#23749837C211yesmaximal, characteristic, complemented
9#23749787C211yesmaximal, characteristic, complemented
10#23749785C211yesmaximal, characteristic, complemented
11#23749759C211yesmaximal, characteristic, complemented
12#23749758C211yesmaximal, characteristic, complemented
13C4×#14558721C211yesmaximal, characteristic, complemented
14#23749703C211yesmaximal, characteristic, complemented
15#546×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic
17C22×D596C2211yescharacteristic, complemented
18#18011937—24no—
19#18011936—24no—
20C2×#14558726C2211yescharacteristic, complemented
21C2×#14558724C2211yescharacteristic, complemented
22C2×#14558724C2211yescharacteristic, complemented
23C2×#14558723C2211yescharacteristic, complemented
24C2×#14558723C2211yescharacteristic, complemented
25C2×#14558721C2211yescharacteristic
26C2×#14558721C2211yescharacteristic
27C2×#14558721C2211yescharacteristic
28C2×#14558721C2211yescharacteristic
29C2×#14558720C2211yescharacteristic, complemented
30C2×#14558720C2211yescharacteristic, complemented
31#18011899C2211yescharacteristic, complemented
32#18011896—21no—
33#18011890C2211yescharacteristic, complemented
34#18011885C2211yescharacteristic
35#18011884—21no—
36#18011879—21no—
37C4×D596C2211yescharacteristic, complemented
38C22⋊C4×D149—21no—
39#18011844C2211yescharacteristic, complemented
40C4⋊#11924746C2211yescharacteristic, complemented
41#18011835C2211yescharacteristic, complemented
42#18011834C2211yescharacteristic, complemented
43#18011832C2211yescharacteristic, complemented
44#18011832C2211yescharacteristic, complemented
45#18011831C2211yescharacteristic, complemented
46#18011831C2211yescharacteristic, complemented
47C22⋊#11924746C2211yescharacteristic, complemented
48C22⋊#11924746C2211yescharacteristic, complemented
49C4×#11924746C2211yescharacteristic, complemented
50C2×#41×C149C2211yescharacteristic
51C22×D4×C149C2211yescharacteristic, complemented
52#124×C149C2211yescharacteristic, complemented
53#123×C149—21no—
54C22⋊Q8×C149C2211yescharacteristic, complemented
55#121×C149—21no—
56#121×C149C2211yescharacteristic
57#120×C149—21no—
58C4×D4×C149C2211yescharacteristic
59C4×D4×C149C2211yescharacteristic, complemented
60C2×C22⋊C4×C149—21no—
61C23×D149—21no—
62#14558736—22no—
63#14558736—41no—
64C2×Dic298C2311yescharacteristic
65C2×D596—22no—
66C2×D596—21no—
67C2×D596—41no—
68C2×D596C2311yescharacteristic
69C2×D596C2311yescharacteristic
70C2×C22⋊D149—21no—
71C2×C22⋊D149—21no—
72C2×C4×D149—21no—
73C2×C4×D149—21no—
74#14558726—41no—
75#14558724—22no—
76#14558724—22no—
77#14558723—22no—
78#14558723—22no—
79#14558721—22no—
80#14558721—22no—
81#14558721—41no—
82#14558721D414yescomplemented
83#14558720—22no—
84#14558720—22no—
85C2×#11924746C2311yescharacteristic
86C2×#11924746C2311yescharacteristic
87C2×#11924746C2311yescharacteristic
88C2×#11924746C2311yescharacteristic
89C4⋊Dic149C2311yescharacteristic
90#14558715—21no—
91C22⋊Dic149—21no—
92#14558713—21no—
93#14558713—21no—
94#14558713—21no—
95#14558713—21no—
96C24×C149—21no—
97#41×C149—22no—
98#41×C149—41no—
99C2×Q8×C149C2311yescharacteristic
100C2×D4×C149—22no—
101C2×D4×C149—21no—
102C2×D4×C149—21no—
103C2×D4×C149—21no—
104C2×D4×C149—41no—
105C2×D4×C149D411yescharacteristic
106C2×D4×C149C2311yescharacteristic
107C2×D4×C149C2311yescharacteristic
108C22×C4×C149—21no—
109C22×C4×C149—21no—
110C22×C4×C149C2311yescharacteristic
111C22×C4×C149C2311yescharacteristic
112C4⋊C4×C149—21no—
113C4⋊C4×C149D411yescharacteristic
114C4⋊C4×C149C2311yescharacteristic
115C22⋊C4×C149—21no—
116C22⋊C4×C149—21no—
117C22⋊C4×C149—21no—
118C22⋊C4×C149—21no—
119C22⋊C4×C149—21no—
120C22⋊C4×C149D411yescharacteristic
121C22⋊C4×C149D411yescharacteristic
122C42×C149C2311yescharacteristic
123C22×D149—21no—
124C22×D149—21no—
125C22×D149—42no—
126C22×D149—42no—
127C22×D149—81no—
128Dic298—21no—
129Dic298—21no—
130D596—21no—
131D596—21no—
132D596—21no—
133D596—42no—
134D596C2×D412yes—
135C22⋊D149—42no—
136C22⋊D149—81no—
137C2×Dic149—21no—
138C2×Dic149—21no—
139C4×D149—42no—
140C4×D149—81no—
141#11924746—21no—
142#11924746—21no—
143#11924746—21no—
144#11924746C2×D412yes—
145C23×C149—21no—
146C23×C149—21no—
147C23×C149—42no—
148C23×C149—42no—
149C23×C149—81no—
150C23×C149C2×D411yescharacteristic
151C23×C149C2×D411yescharacteristic
152Q8×C149—21no—
153Q8×C149—21no—
154D4×C149—21no—
155D4×C149—21no—
156D4×C149—21no—
157D4×C149—41no—
158D4×C149—42no—
159D4×C149—42no—
160D4×C149—81no—
161D4×C149C2×D412yes—
162C2×C4×C149—21no—
163C2×C4×C149—21no—
164C2×C4×C149—21no—
165C2×C4×C149—21no—
166C2×C4×C149—21no—
167C2×C4×C149—21no—
168C2×C4×C149—42no—
169C2×C4×C149—81no—
170C2×C4×C149C2×D411yescharacteristic
171C2×C4×C149C2×D411yescharacteristic
172C2×C4×C149C2×D411yescharacteristic
173C2×C4×C149C2×D411yescharacteristic
174C2×C4×C149C2411yescharacteristic
175C22×C149—21no—
176C22×C149—21no—
177C22×C149—22no—
178C22×C149—21no—
179C22×C149—21no—
180C22×C149—21no—
181C22×C149—21no—
182C22×C149—41no—
183C22×C149—41no—
184C22×C149—41no—
185C22×C149—41no—
186C22×C149—41no—
187C22×C149—84no—
188C22×C149C22×D411yescharacteristic
189C2×D149—22no—
190C2×D149—41no—
191C2×D149—41no—
192C2×D149—41no—
193C2×D149—41no—
194C2×D149—41no—
195C2×D149—84no—
196C4×C149—21no—
197C4×C149—21no—
198C4×C149—21no—
199C4×C149—21no—
200C4×C149—41no—
201C4×C149—41no—
202C4×C149C22×D411yescharacteristic
203C4×C149H(5,𝔽2)11yescharacteristic
204Dic149—41no—
205Dic149—41no—
206C2×C149—41no—
207C2×C149—41no—
208C2×C149—42no—
209C2×C149—81no—
210C2×C149D4211yescharacteristic
211C2×C149#57511yescharacteristic
212C2×C149#57611yescharacteristic
213D149—42no—
214D149—81no—
215C149#327311yescharacteristic, complemented, Sylow
216#3273—1491nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1491no—
218C2×AGL(1,ℤ/8)—1491no—
219#546—1491no—
220#546D14911yescharacteristic, complemented
221#495—1491no—
222#468—1491no—
223#468—1491no—
224#463—1491no—
225#463—1491no—
226#452—1491no—
227#452—1491no—
228#447—1491no—
229#447—1491no—
230C4×D8—1491no—
231#436—1491no—
232C2×#41—1491no—
233C2×#41C2×D14911yescharacteristic
234C22×D4—1491no—
235C22×D4C2×D14911yescharacteristic, complemented
236AGL(1,ℤ/8)—2984no—
237AGL(1,ℤ/8)—2984no—
238C2×Q16—1491no—
239C2×SD16—1491no—
240C2×SD16—1491no—
241C2×SD16—1491no—
242C2×SD16—1491no—
243C2×D8—1491no—
244C2×D8—1491no—
245C2×D8—1491no—
246C2×D8—1491no—
247C2×M16—1491no—
248C2×M16—1491no—
249#124—1491no—
250#124C2×D14911yescharacteristic, complemented
251#123—21no—
252#123—2981no—
253C22⋊Q8—1491no—
254C22⋊Q8C2×D14911yescharacteristic, complemented
255#121—21no—
256#121—2981no—
257#121—1491no—
258#121C2×D14911yescharacteristic
259#120—21no—
260#120—2981no—
261C4×D4—1491no—
262C4×D4C2×D14911yescharacteristic
263C4×D4C2×D14911yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—2981no—
266#107—1491no—
267C4⋊C8—1491no—
268#103—1491no—
269#103—1491no—
270#102—1491no—
271#102—1491no—
272#102—1491no—
273#102—1491no—
274C22⋊C8—1491no—
275C22⋊C8—1491no—
276C4×C8—1491no—
277C24—21no—
278C24—2981no—
279#41—22no—
280#41—41no—
281#41—2982no—
282#41—5961no—
283C2×Q8—1491no—
284C2×Q8C22×D14911yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2982no—
291C2×D4—2981no—
292C2×D4—2981no—
293C2×D4—2981no—
294C2×D4—5961no—
295C2×D4—1491no—
296C2×D4—1491no—
297C2×D4C22⋊D14911yescharacteristic
298C2×D4C22×D14911yescharacteristic
299C2×D4C22×D14911yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—2981no—
303C22×C4—2981no—
304C22×C4C22×D14911yescharacteristic
305C22×C4C22×D14911yescharacteristic
306Q16—5961no—
307SD16—2982no—
308SD16—2982no—
309SD16—2982no—
310SD16—2982no—
311D8—2982no—
312D8—2982no—
313D8—5961no—
314D8—1494no—
315M16—2982no—
316M16—2982no—
317C2×C8—1491no—
318C2×C8—1491no—
319C2×C8—1491no—
320C2×C8—1491no—
321C4⋊C4—21no—
322C4⋊C4—2981no—
323C4⋊C4—1491no—
324C4⋊C4C22⋊D14911yescharacteristic
325C4⋊C4C22×D14911yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—2981no—
332C22⋊C4—2981no—
333C22⋊C4—2981no—
334C22⋊C4—2981no—
335C22⋊C4—2981no—
336C22⋊C4C22⋊D14911yescharacteristic
337C22⋊C4C22⋊D14911yescharacteristic
338C42C22×D14911yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—2981no—
345C23—2981no—
346C23—5962no—
347C23—5962no—
348C23—11921no—
349C23C2×C22⋊D14911yescharacteristic
350C23C2×C22⋊D14911yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2981no—
354Q8—2981no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—2981no—
363D4—2981no—
364D4—2981no—
365D4—5962no—
366D4—5962no—
367D4—1492no—
368D4—11921no—
369D4D4×D14912yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—2981no—
379C2×C4—2981no—
380C2×C4—5962no—
381C2×C4—11921no—
382C2×C4C2×C22⋊D14911yescharacteristic
383C2×C4C2×C22⋊D14911yescharacteristic
384C2×C4C2×C22⋊D14911yescharacteristic
385C2×C4C2×C22⋊D14911yescharacteristic
386C2×C4C23×D14911yescharacteristic
387C8—2981no—
388C8—2981no—
389C8—2981no—
390C8—1492no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—2982no—
405C22—5961no—
406C22—5961no—
407C22—5961no—
408C22—5961no—
409C22—5961no—
410C22—11924no—
411C22C22×C22⋊D14911yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—5961no—
419C4—5961no—
420C4C2×D4×D14911yescharacteristic
421C4#1801195611yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—5962no—
427C2—11921no—
428C2D4×C22⋊D14911yescharacteristic
429C2#2375042311yescharacteristic
430C2#2375042811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#546×C149C2
socleC22×C149C22×D4
hypercenter#546D149
hypocenterC149#3273
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#18011937C2semidirect
2. C2×#18011936C2semidirect
3. #23750271C2semidirect
4. #23750007C2semidirect
5. #23749873C2semidirect
6. #23749869C2semidirect
7. #23749840C2semidirect
8. #23749837C2semidirect
9. #23749787C2semidirect
10. #23749785C2semidirect
11. #23749759C2semidirect
12. #23749758C2semidirect
13. C4×#14558721C2semidirect
14. #23749703C2semidirect
15. #546×C149C2semidirect
17. C22×D596C22semidirect
20. C2×#14558726C22semidirect
21. C2×#14558724C22semidirect
22. C2×#14558724C22semidirect
23. C2×#14558723C22semidirect
24. C2×#14558723C22semidirect
29. C2×#14558720C22semidirect
30. C2×#14558720C22semidirect
31. #18011899C22semidirect
33. #18011890C22semidirect
37. C4×D596C22semidirect
39. #18011844C22semidirect
40. C4⋊#11924746C22semidirect
41. #18011835C22semidirect
42. #18011834C22semidirect
43. #18011832C22semidirect
44. #18011832C22semidirect
45. #18011831C22semidirect
46. #18011831C22semidirect
47. C22⋊#11924746C22semidirect
48. C22⋊#11924746C22semidirect
49. C4×#11924746C22semidirect
51. C22×D4×C149C22semidirect
52. #124×C149C22semidirect
54. C22⋊Q8×C149C22semidirect
59. C4×D4×C149C22semidirect
82. #14558721D4semidirect
215. C149#3273semidirect
220. #546D149semidirect
235. C22×D4C2×D149semidirect
250. #124C2×D149semidirect
254. C22⋊Q8C2×D149semidirect
263. C4×D4C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119372
C2×#180119362
#237502712
#237500072
#237498732
#237498692
#237498402
#237498372
#237497872
#237497852
#237497592
#237497582
C4×#145587212
#237497032
#546×C1492
#3273149