Finite Group Databasebeta

← #29662254

Subgroups — #29662254

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456245C211yesmaximal, characteristic, complemented
2#21455820C211yesmaximal, characteristic, complemented
3#21455815C211yesmaximal, characteristic, complemented
4C2×#16839827C211yesmaximal, characteristic, complemented
5#21445183C212yesmaximal, complemented
6#21445173C212yesmaximal, complemented
7#21445163C212yesmaximal, complemented
8#21445159C212yesmaximal, complemented
9#21444565C211yesmaximal, characteristic, complemented
10#21444469C211yesmaximal, characteristic, complemented
11C61×#1540C211yesmaximal, characteristic, complemented
12C2×#14519797—21no—
13C2×#14519797—21no—
14C22×C8×D61—21no—
15#16840312C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17#16840144—21no—
18#16840144—21no—
19#16840141C2211yescharacteristic, complemented
20#16840138C2211yescharacteristic, complemented
21#16840136C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic, complemented
23#16840076—21no—
24#16840073—21no—
25#16840043C2212yes—
26#16840039C2212yes—
27#16839958C2212yes—
28#16839954C2212yes—
29#16839898—21no—
30#16839895—21no—
31C4×D488—22no—
32C4×#11758199—22no—
33#16839827—21no—
34#16839827—21no—
35#16839827C2212yes—
36#16839827C2212yes—
37#16839815C2211yescharacteristic
38C2×#14519689—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839509C2212yescomplemented
43#16839505C2212yescomplemented
44C61×#532C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yescomplemented
48C61×#331C2212yescomplemented
49C2×#11758222—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52#14519797—44no—
53#14519797—44no—
54C2×#11758202—21no—
55C2×#11758202—21no—
56C2×#11758200—21no—
57C2×#11758200—21no—
58C2×#11758201—21no—
59C2×#11758201—21no—
60C2×D488—22no—
61C2×#11758199—22no—
62C2×#11758199—21no—
63C2×#11758199—21no—
64C22×#92418D411yescharacteristic
65C22×#92418D411yescharacteristic
66C2×C8×D61—22no—
67C2×C8×D61—22no—
68C2×C8×D61—44no—
69#14519777C2311yescharacteristic
70#14519774C2311yescharacteristic
71#14519765—21no—
72#14519751—21no—
73#14519751—21no—
74C4×D244—21no—
75C4×D244—21no—
76C4×D244—21no—
77C4×D244—21no—
78C4×D244D412yes—
79C4×D244C2312yes—
80C2×C4⋊Dic61D411yescharacteristic
81C2×#11758198—21no—
82#14519711—22no—
83#14519710—22no—
84C4⋊#92418C2312yes—
85#14519699—22no—
86#14519699—22no—
87#14519698—22no—
88#14519697—22no—
89C22⋊#92418—21no—
90#14519689—22no—
91#14519689—22no—
92C4×#92418—21no—
93C4×#92418D411yescharacteristic
94C4×#92418D411yescharacteristic
95C4×#92418C2311yescharacteristic
96C4×#92418C2311yescharacteristic
97C2×#41×C61—21no—
98C22×C8×C61—21no—
99C4⋊Q8×C61C2311yescharacteristic
100#127×C61C2311yescharacteristic
101#124×C61—21no—
102#121×C61—21no—
103#121×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C4⋊C8×C61C2312yes—
110C22⋊C8×C61—21no—
111C4×C8×C61—21no—
112C23×D61—21no—
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—41no—
125C22×Dic61—21no—
126#11758222—42no—
127#11758222—42no—
128C2×D244—21no—
129C2×D244—24no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244C2×D412yes—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—41no—
139C2×Dic122—21no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—42no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C22⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61#4111yescharacteristic
150C42×C61C2411yescharacteristic
151#11758202—42no—
152#11758202—42no—
153#11758201—42no—
154#11758201—42no—
155#11758200—42no—
156#11758200—42no—
157#11758199—42no—
158#11758199—44no—
159#11758199—42no—
160#11758198—22no—
161#11758198—22no—
162#11758198—41no—
163#11758198—41no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61C2×D412yes—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418D814yescomplemented
171C2×#92418C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175D488—44no—
176C8×D61—44no—
177C8×D61—44no—
178C23×C61—41no—
179C23×C61#12111yescharacteristic
180C22×D61—24no—
181C22×D61—22no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61—41no—
185Q8×C61—41no—
186Q8×C61—41no—
187D4×C61—41no—
188D4×C61—42no—
189D4×C61—41no—
190D4×C61—81no—
191D4×C61—82no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—82no—
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61C2×D812yes—
200C2×C4×C61AGL(1,ℤ/8)12yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C22⋊D61—82no—
205C2×Dic61—22no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208D244—24no—
209D244—22no—
210D244—41no—
211D244—42no—
212D244—41no—
213D244—81no—
214D244#13514yescomplemented
215C4×D61—42no—
216C4×D61—42no—
217C4×D61—44no—
218C4×D61—82no—
219Dic122—41no—
220Dic122—41no—
221C8×C61—42no—
222C8×C61—42no—
223#92418—21no—
224#92418—21no—
225#92418—42no—
226#92418C2×D812yes—
227#92418C2×D812yes—
228#19101—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61#45912yes—
235C22×C61C2×#12111yescharacteristic
236C2×D61—24no—
237C2×D61—48no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—81no—
247C4×C61#45912yes—
248C4×C61C22×D811yescharacteristic
249C4×C61C2×#13512yes—
250C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
251Dic61—42no—
252Dic61—81no—
253#3147—611no—
254#3100—611no—
255#3100—611no—
256C2×C8×D4—611no—
257#1669—612no—
258#1667—612no—
259#1665—612no—
260#1665—612no—
261#1564—611no—
262#1540—611no—
263#1540D6111yescharacteristic, complemented
264C2×C61—22no—
265C2×C61—161no—
266C2×C61C2×#45911yescharacteristic
267C2×C61#305011yescharacteristic
268C2×C61#306711yescharacteristic
269D61—44no—
270D61—161no—
271C2×#135—1221no—
272C2×#135—1221no—
273C23×C8—1221no—
274#532—611no—
275#532C2×D6111yescharacteristic
276C2×C4×D4—611no—
277#495—1221no—
278#495—1221no—
279#494—611no—
280#493—611no—
281#492—611no—
282#492—611no—
283#480—1221no—
284#480—1221no—
285#474—612no—
286#474—612no—
287#459—612no—
288#459—612no—
289#447—1221no—
290#447—1221no—
291C4×D8—1222no—
292C4×D8—1222no—
293C8×D4—612no—
294C8×D4—612no—
295C8×D4—1221no—
296C8×D4—1221no—
297#431—611no—
298#431C2×D6111yescharacteristic
299C2×C4⋊C8C2×D6111yescharacteristic
300C2×C22⋊C8—1221no—
301C2×C4×C8—611no—
302#335—612no—
303#334—612no—
304#332—612no—
305#332C2×D6112yescomplemented
306#331—612no—
307#331C2×D6112yescomplemented
308C61#1910111yescharacteristic, complemented, Sylow
309C2×#41—21no—
310C2×#41—1221no—
311C22×D4—611no—
312C23×C4—1221no—
313#135—2444no—
314#135—2444no—
315C2×Q16—1221no—
316C2×Q16—1221no—
317C2×SD16—1221no—
318C2×SD16—1221no—
319C2×SD16—1221no—
320C2×SD16—1221no—
321C2×D8—1222no—
322C2×D8—1221no—
323C2×D8—1221no—
324C2×D8—1222no—
325C22×C8—21no—
326C22×C8—611no—
327C22×C8—611no—
328C22×C8—1222no—
329C22×C8—1222no—
330C22×C8—2444no—
331C4⋊Q8—611no—
332C4⋊Q8C22×D6111yescharacteristic
333#127—611no—
334#127C22×D6111yescharacteristic
335#124—21no—
336#124—1221no—
337#121—21no—
338#121—21no—
339#121—1221no—
340#121—1221no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—612no—
344C4×D4—612no—
345C4×D4—1221no—
346C4×D4—1221no—
347C4×D4—1221no—
348C4×D4—1221no—
349C2×C4⋊C4—611no—
350C2×C22⋊C4—1221no—
351C2×C42C22×D6111yescharacteristic
352#107—1222no—
353#107—1222no—
354C4⋊C8—21no—
355C4⋊C8—612no—
356C4⋊C8C22×D6112yes—
357C4⋊C8C22×D6112yes—
358#102—1222no—
359#102—1222no—
360#102—1222no—
361#102—1222no—
362C22⋊C8—21no—
363C22⋊C8—1222no—
364C22⋊C8—1222no—
365C22⋊C8—1221no—
366C4×C8—21no—
367C4×C8—611no—
368C4×C8—611no—
369C4×C8—611no—
370C4×C8—611no—
371C4×C8—1221no—
372C24—1221no—
373#41—42no—
374#41—42no—
375#41—2442no—
376#41—2442no—
377C2×Q8—21no—
378C2×Q8—1221no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—612no—
384C2×D4—1224no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—2441no—
390C22×C4—41no—
391C22×C4—1221no—
392C22×C4—1222no—
393C22×C4—1222no—
394C22×C4—2441no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4#1175822211yescharacteristic
398C22×C4D4×D6111yescharacteristic
399C22×C4D4×D6111yescharacteristic
400Q16—2442no—
401Q16—2442no—
402SD16—2442no—
403SD16—2442no—
404SD16—2442no—
405SD16—2442no—
406D8—2444no—
407D8—2442no—
408D8—2442no—
409D8—2444no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—42no—
413C2×C8—611no—
414C2×C8—611no—
415C2×C8—614no—
416C2×C8—611no—
417C2×C8—611no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—2444no—
421C2×C8—2444no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—612no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—1222no—
431C22⋊C4—1222no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C42#1175822211yescharacteristic
435C42D4×D6111yescharacteristic
436C42D4×D6111yescharacteristic
437C42C23×D6111yescharacteristic
438C23—41no—
439C23—1224no—
440C23—1222no—
441C23—2442no—
442C23—2441no—
443C23—2442no—
444C23#1451975011yescharacteristic
445Q8—41no—
446Q8—41no—
447Q8—2441no—
448Q8—2441no—
449D4—41no—
450D4—42no—
451D4—41no—
452D4—81no—
453D4—82no—
454D4—614no—
455D4—1224no—
456D4—1222no—
457D4—4881no—
458D4—4882no—
459D4—2441no—
460D4—2442no—
461D4—2441no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—82no—
466C2×C4—1222no—
467C2×C4—4882no—
468C2×C4—2441no—
469C2×C4—2442no—
470C2×C4—2444no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4#1451975011yescharacteristic
474C2×C4#1451975011yescharacteristic
475C2×C4#1451975011yescharacteristic
476C2×C4D8×D6112yes—
477C2×C4#1451980412yes—
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C2×#1175822211yescharacteristic
481C8—42no—
482C8—42no—
483C8—612no—
484C8—612no—
485C8—1221no—
486C8—1221no—
487C8—2442no—
488C22—21no—
489C22—21no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—1224no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22—2448no—
498C22—2442no—
499C22—2442no—
500C22#1683995612yes—
501C22C2×#1451975011yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—81no—
506C4—4881no—
507C4—2442no—
508C4#1683995612yes—
509C4C2×D8×D6111yescharacteristic
510C4#135×D6112yes—
511C4C2×#1451980411yescharacteristic
512C2—22no—
513C2—161no—
514C2—9761no—
515C2—2444no—
516C2C2×#1683995611yescharacteristic
517C2#2145539911yescharacteristic
518C2#2145553811yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1540C2
socleC22×C61C2×#121
hypercenter#1540D61
hypocenterC61#19101
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
1. #21456245C2semidirect
2. #21455820C2semidirect
3. #21455815C2semidirect
4. C2×#16839827C2semidirect
5. #21445183C2semidirect
6. #21445173C2semidirect
7. #21445163C2semidirect
8. #21445159C2semidirect
9. #21444565C2semidirect
10. #21444469C2semidirect
11. C61×#1540C2semidirect
19. #16840141C22semidirect
20. #16840138C22semidirect
21. #16840136C22semidirect
22. #16840134C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839509C22semidirect
43. #16839505C22semidirect
47. C61×#332C22semidirect
48. C61×#331C22semidirect
170. C2×#92418D8semidirect
214. D244#135semidirect
263. #1540D61semidirect
305. #332C2×D61semidirect
307. #331C2×D61semidirect
308. C61#19101semidirect

Maximal subgroups

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

subgroupindex
#214562452
#214558202
#214558152
C2×#168398272
#214451832
#214451732
#214451632
#214451592
#214445652
#214444692
C61×#15402
#1910161