Finite Group Databasebeta

← #19146203

Subgroups — #19146203

Order 5504.

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×#13063353C211yesmaximal, characteristic, complemented
2C2×#13063352C211yesmaximal, characteristic, complemented
3#14583546C211yesmaximal, characteristic, complemented
4#14583540C211yesmaximal, characteristic, complemented
5#14583280C211yesmaximal, characteristic, complemented
6#14583146C211yesmaximal, characteristic, complemented
7#14583117C211yesmaximal, characteristic, complemented
8#14583113C211yesmaximal, characteristic, complemented
9#14583064C211yesmaximal, characteristic, complemented
10#14583059C211yesmaximal, characteristic, complemented
11#14583034C211yesmaximal, characteristic, complemented
12#14583032C211yesmaximal, characteristic, complemented
13#14582981C211yesmaximal, characteristic, complemented
14#14582975C211yesmaximal, characteristic, complemented
15C43×#468C211yesmaximal, characteristic, complemented
16C2×#10621902C2211yescharacteristic
17C2×#10621898C2211yescharacteristic
18C22×D172C2211yescharacteristic, complemented
19C2×D4×D43C2211yescharacteristic, complemented
20#13063353—24no—
21#13063352—24no—
22C2×#10621885C2211yescharacteristic, complemented
23C2×#10621873C2211yescharacteristic, complemented
24C2×#10621873C2211yescharacteristic, complemented
25C2×#10621884C2211yescharacteristic, complemented
26C2×D344C2211yescharacteristic
27C2×D344C2211yescharacteristic
28C2×#10621882C2211yescharacteristic
29C2×#10621882C2211yescharacteristic
30C2×#10621872C2211yescharacteristic, complemented
31#13063314C2211yescharacteristic, complemented
32#13063312C2211yescharacteristic, complemented
33#13063310—21no—
34#13063307—21no—
35#13063305C2211yescharacteristic, complemented
36#13063300—21no—
37#13063300C2211yescharacteristic
38#13063296—21no—
39#13063295—21no—
40#13063293—21no—
41#13063287C2211yescharacteristic, complemented
42#13063287C2211yescharacteristic, complemented
43C4×C22⋊D43C2211yescharacteristic
44C22⋊C4×D43—21no—
45C22⋊C4×D43—21no—
46Dic43⋊C8C2211yescharacteristic, complemented
47#13063248C2211yescharacteristic, complemented
48#13063245C2211yescharacteristic, complemented
49#13063245C2211yescharacteristic, complemented
50#13063245C2211yescharacteristic, complemented
51C22⋊(C43⋊C8)C2211yescharacteristic, complemented
52(C2×D43)⋊C8C2211yescharacteristic, complemented
53C8×Dic43C2211yescharacteristic, complemented
54C2×SD16×C43C2211yescharacteristic, complemented
55C2×M16×C43C2211yescharacteristic, complemented
56C22⋊Q8×C43C2211yescharacteristic, complemented
57#121×C43C2211yescharacteristic
58#107×C43C2211yescharacteristic, complemented
59#103×C43C2211yescharacteristic, complemented
60#102×C43C2211yescharacteristic, complemented
61C23×D43—21no—
62C23×D43—21no—
63C2×Q8×C43C2311yescharacteristic
64C2×D4×C43—21no—
65C2×D4×C43C2311yescharacteristic
66C22×C4×C43C2311yescharacteristic
67C2×C22⋊D43—21no—
68C2×C22⋊D43—21no—
69C2×C22⋊D43D411yescharacteristic
70#10621902—22no—
71#10621902—41no—
72D4×D43—22no—
73D4×D43—41no—
74#10621898—22no—
75#10621898—41no—
76C2×D172—21no—
77C2×D172—21no—
78C2×D172—21no—
79C2×D172—22no—
80C2×D172—41no—
81C2×D172C2311yescharacteristic
82C2×D172C2311yescharacteristic
83C2×D172C2311yescharacteristic
84C2×C4×D43—21no—
85C2×C4×D43—21no—
86C2×C4×D43—21no—
87C2×C4×D43—21no—
88C2×C4×D43C2311yescharacteristic
89C2×Dic86C2311yescharacteristic
90SD16×C43—22no—
91M16×C43—22no—
92C2×C8×C43C2311yescharacteristic
93C2×C8×C43C2311yescharacteristic
94C4⋊C4×C43—21no—
95C4⋊C4×C43C2311yescharacteristic
96C4⋊C4×C43C2311yescharacteristic
97C22⋊C4×C43—21no—
98C22⋊C4×C43—21no—
99C22⋊Dic43—21no—
100C22⋊Dic43—21no—
101C22⋊Dic43D411yescharacteristic
102#10621885—41no—
103#10621884—22no—
104#10621882—22no—
105#10621882—41no—
106(C2×D43)⋊C4—21no—
107(C2×D43)⋊C4—21no—
108(C2×D43)⋊C4—21no—
109(C2×D43)⋊C4—21no—
110(C2×D43)⋊C4—21no—
111(C2×D43)⋊C4—21no—
112(C2×D43)⋊C4D411yescharacteristic
113C4⋊Dic43—21no—
114Dic43⋊C4D411yescharacteristic
115C4×Dic43C2311yescharacteristic
116C2×C43⋊C8C2311yescharacteristic
117C2×C43⋊C8C2311yescharacteristic
118D344—22no—
119D344D414yescomplemented
120#10621873—22no—
121#10621873—22no—
122#10621872—22no—
123C23×C43—21no—
124C23×C43C2×D411yescharacteristic
125C22×D43—21no—
126C22×D43—21no—
127C22×D43—21no—
128C22×D43—42no—
129C22×D43—42no—
130C22×D43—42no—
131C22×D43—42no—
132C22×D43—81no—
133C22×D43—81no—
134C22×D43C2×D411yescharacteristic
135Q8×C43—21no—
136Q8×C43—21no—
137D4×C43—21no—
138D4×C43—21no—
139D4×C43—81no—
140C2×C4×C43—21no—
141C2×C4×C43—21no—
142C2×C4×C43—21no—
143C2×C4×C43—21no—
144C2×C4×C43—21no—
145C2×C4×C43C2×D411yescharacteristic
146C2×C4×C43C2411yescharacteristic
147C22⋊D43—41no—
148C22⋊D43—42no—
149C22⋊D43—42no—
150C2×Dic43—21no—
151C2×Dic43C2×D411yescharacteristic
152C2×Dic43C2×D411yescharacteristic
153C2×Dic43C2×D411yescharacteristic
154D172—21no—
155D172—21no—
156D172—21no—
157D172—21no—
158D172—42no—
159D172—42no—
160D172—81no—
161D172C2×D412yes—
162D172C2×D412yes—
163C4×D43—21no—
164C4×D43—21no—
165C4×D43—42no—
166C4×D43—42no—
167C4×D43—81no—
168C4×D43—81no—
169Dic86—21no—
170Dic86—21no—
171C8×C43—21no—
172C8×C43C2×D412yes—
173C43⋊C8—21no—
174C43⋊C8—21no—
175C22×C43—21no—
176C22×C43—21no—
177C22×C43—21no—
178C22×C43—41no—
179C22×C43—41no—
180C22×C43—41no—
181C22×C43C22×D411yescharacteristic
182C2×D43—22no—
183C2×D43—22no—
184C2×D43—21no—
185C2×D43—21no—
186C2×D43—21no—
187C2×D43—41no—
188C2×D43—41no—
189C2×D43—41no—
190C2×D43—41no—
191C2×D43—41no—
192C2×D43—41no—
193C2×D43—41no—
194C2×D43—84no—
195C2×D43—84no—
196C4×C43—21no—
197C4×C43—41no—
198C4×C43—41no—
199C4×C43—41no—
200C4×C43C22×D411yescharacteristic
201C4×C43H(5,𝔽2)11yescharacteristic
202Dic43—21no—
203Dic43—21no—
204Dic43—21no—
205Dic43—41no—
206#3273—431nomaximal, Sylow
207C2×C43—41no—
208C2×C43—81no—
209C2×C43D4211yescharacteristic
210C2×C43#57511yescharacteristic
211C2×C43#57611yescharacteristic
212D43—41no—
213D43—42no—
214D43—42no—
215D43—81no—
216C2×AGL(1,ℤ/8)—431no—
217C2×AGL(1,ℤ/8)—431no—
218#546—431no—
219#546—431no—
220#495—431no—
221#468—431no—
222#468D4311yescharacteristic, complemented
223#463—431no—
224#463—431no—
225#452—431no—
226#452—431no—
227#447—431no—
228#447—431no—
229C4×D8—431no—
230#436—431no—
231C43#327311yescharacteristic, complemented, Sylow
232C2×#41—431no—
233C2×#41—431no—
234C22×D4—431no—
235C22×D4—431no—
236AGL(1,ℤ/8)—864no—
237AGL(1,ℤ/8)—864no—
238C2×Q16—431no—
239C2×SD16—431no—
240C2×SD16—431no—
241C2×SD16—431no—
242C2×SD16C2×D4311yescharacteristic, complemented
243C2×D8—431no—
244C2×D8—431no—
245C2×D8—431no—
246C2×D8—431no—
247C2×M16—431no—
248C2×M16C2×D4311yescharacteristic, complemented
249#124—431no—
250#124—431no—
251#123—861no—
252#123—861no—
253C22⋊Q8—431no—
254C22⋊Q8C2×D4311yescharacteristic, complemented
255#121—431no—
256#121—861no—
257#121—861no—
258#121C2×D4311yescharacteristic
259#120—861no—
260#120—861no—
261C4×D4—431no—
262C4×D4—431no—
263C4×D4—431no—
264C2×C22⋊C4—861no—
265C2×C22⋊C4—861no—
266#107C2×D4311yescharacteristic, complemented
267C4⋊C8—431no—
268#103—431no—
269#103C2×D4311yescharacteristic, complemented
270#102—431no—
271#102—431no—
272#102—431no—
273#102C2×D4311yescharacteristic, complemented
274C22⋊C8—431no—
275C22⋊C8—431no—
276C4×C8—431no—
277C24—861no—
278C24—861no—
279#41—862no—
280#41—862no—
281#41—1721no—
282#41—1721no—
283C2×Q8—431no—
284C2×Q8C22×D4311yescharacteristic
285C2×D4—21no—
286C2×D4—431no—
287C2×D4—431no—
288C2×D4—431no—
289C2×D4—431no—
290C2×D4—862no—
291C2×D4—861no—
292C2×D4—861no—
293C2×D4—861no—
294C2×D4—862no—
295C2×D4—861no—
296C2×D4—861no—
297C2×D4—1721no—
298C2×D4—1721no—
299C2×D4C22×D4311yescharacteristic
300C22×C4—431no—
301C22×C4—861no—
302C22×C4—861no—
303C22×C4—861no—
304C22×C4—861no—
305C22×C4C22×D4311yescharacteristic
306Q16—1721no—
307SD16—22no—
308SD16—862no—
309SD16—862no—
310SD16—862no—
311D8—434no—
312D8—862no—
313D8—862no—
314D8—1721no—
315M16—22no—
316M16—862no—
317C2×C8—431no—
318C2×C8—431no—
319C2×C8C22×D4311yescharacteristic
320C2×C8C22×D4311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—431no—
323C4⋊C4—861no—
324C4⋊C4C22×D4311yescharacteristic
325C4⋊C4C22×D4311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—431no—
329C22⋊C4—431no—
330C22⋊C4—861no—
331C22⋊C4—861no—
332C22⋊C4—861no—
333C22⋊C4—861no—
334C22⋊C4—861no—
335C22⋊C4—861no—
336C22⋊C4—861no—
337C22⋊C4—861no—
338C42—431no—
339C23—21no—
340C23—431no—
341C23—861no—
342C23—861no—
343C23—861no—
344C23—3441no—
345C23—3441no—
346C23—1722no—
347C23—1722no—
348C23—1722no—
349C23—1722no—
350C23D4×D4311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—861no—
354Q8—861no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—432no—
359D4—432no—
360D4—861no—
361D4—861no—
362D4—861no—
363D4—861no—
364D4—3441no—
365D4—1722no—
366D4—1722no—
367D4—1721no—
368D4—1722no—
369D4—1722no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—431no—
376C2×C4—431no—
377C2×C4—431no—
378C2×C4—861no—
379C2×C4—861no—
380C2×C4—861no—
381C2×C4—3441no—
382C2×C4—3441no—
383C2×C4—1722no—
384C2×C4—1722no—
385C2×C4D4×D4311yescharacteristic
386C2×C4C23×D4311yescharacteristic
387C8—21no—
388C8—861no—
389C8—861no—
390C8D4×D4312yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—861no—
398C22—861no—
399C22—861no—
400C22—862no—
401C22—862no—
402C22—3444no—
403C22—3444no—
404C22—1721no—
405C22—1721no—
406C22—1721no—
407C22—1721no—
408C22—1721no—
409C22—1721no—
410C22—1721no—
411C22C2×D4×D4311yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—861no—
417C4—861no—
418C4—861no—
419C4—1721no—
420C4C2×D4×D4311yescharacteristic
421C4#1306337111yescharacteristic
422C2—41no—
423C2—81no—
424C2—3441no—
425C2—1721no—
426C2—1722no—
427C2—1722no—
428C2#1458353311yescharacteristic
429C2#1458369811yescharacteristic
430C2#1458370311yescharacteristic
subgroupG/H
centerC22C2×D4×D43
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#468C2
socleC22×C43C22×D4
hypercenter#468D43
hypocenterC43#3273
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#13063353C2semidirect
2. C2×#13063352C2semidirect
3. #14583546C2semidirect
4. #14583540C2semidirect
5. #14583280C2semidirect
6. #14583146C2semidirect
7. #14583117C2semidirect
8. #14583113C2semidirect
9. #14583064C2semidirect
10. #14583059C2semidirect
11. #14583034C2semidirect
12. #14583032C2semidirect
13. #14582981C2semidirect
14. #14582975C2semidirect
15. C43×#468C2semidirect
18. C22×D172C22semidirect
19. C2×D4×D43C22semidirect
22. C2×#10621885C22semidirect
23. C2×#10621873C22semidirect
24. C2×#10621873C22semidirect
25. C2×#10621884C22semidirect
30. C2×#10621872C22semidirect
31. #13063314C22semidirect
32. #13063312C22semidirect
35. #13063305C22semidirect
41. #13063287C22semidirect
42. #13063287C22semidirect
46. Dic43⋊C8C22semidirect
47. #13063248C22semidirect
48. #13063245C22semidirect
49. #13063245C22semidirect
50. #13063245C22semidirect
51. C22⋊(C43⋊C8)C22semidirect
52. (C2×D43)⋊C8C22semidirect
53. C8×Dic43C22semidirect
54. C2×SD16×C43C22semidirect
55. C2×M16×C43C22semidirect
56. C22⋊Q8×C43C22semidirect
58. #107×C43C22semidirect
59. #103×C43C22semidirect
60. #102×C43C22semidirect
119. D344D4semidirect
222. #468D43semidirect
231. C43#3273semidirect
242. C2×SD16C2×D43semidirect
248. C2×M16C2×D43semidirect
254. C22⋊Q8C2×D43semidirect
266. #107C2×D43semidirect
269. #103C2×D43semidirect
273. #102C2×D43semidirect