Finite Group Databasebeta

← #29730395

Subgroups — #29730395

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840039C211yesmaximal, characteristic, complemented
2#21455279C211yesmaximal, characteristic
3#21454563C211yesmaximal, characteristic, complemented
4#21447262C212yesmaximal, complemented
5#21447109C212yesmaximal, complemented
6#21446809C211yesmaximal, characteristic, complemented
7#21446594C211yesmaximal, characteristic, complemented
8#21446325C212yesmaximal, complemented
9#21445881C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic, complemented
11C61×#3141C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C2×#14519784C2212yes—
14C2×C4×D244C2211yescharacteristic
15#16840039—21no—
16#16840039—22no—
17#16840039—21no—
18#16839974—22no—
19#16839964—21no—
20#16839954—21no—
21#16839840—21no—
22#16839834—22no—
23D4×#92418—21no—
24C2×#14519712C2212yes—
25C2×#14519710C2212yes—
26#16839786C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28#16839763C2212yes—
29C2×#14519697C2212yes—
30C2×#14519693C2211yescharacteristic
31C2×#14519693C2211yescharacteristic
32#16839729C2212yes—
33C4×#11758194C2211yescharacteristic
34#16839531C2212yes—
35#16839531C2212yes—
36#16839530C2211yescharacteristic
37#16839530C2211yescharacteristic
38#16839526C2212yes—
39C61×#532C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C61×#489—22no—
42C61×#487—21no—
43C61×#486—21no—
44C61×#417C2212yes—
45C61×#417C2212yes—
46C4×M16×C61C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49#14519802—24no—
50#14519802—42no—
51C2×#11758201—22no—
52C2×#11758199—22no—
53#14519784—24no—
54#14519784—42no—
55C2×#11758194—22no—
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2312yes—
58C2×#11758194C2311yescharacteristic
59C22×#92418—22no—
60C22×#92418C2312yes—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244—21no—
64C4×D244—21no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×#11758198—21no—
67#14519712—42no—
68#14519710—22no—
69#14519710—22no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418D411yescharacteristic
73C4⋊#92418D412yes—
74C4⋊#92418D411yescharacteristic
75#14519701—22no—
76#14519698—22no—
77#14519697—22no—
78#14519697—22no—
79#14519693—22no—
80#14519693—22no—
81C22⋊#92418—22no—
82C22⋊#92418—22no—
83#14519688—21no—
84C4×#92418—21no—
85C2×#41×C61C2312yes—
86C22×Q8×C61—21no—
87C2×M16×C61C2311yescharacteristic
88C2×M16×C61C2311yescharacteristic
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61D411yescharacteristic
94#127×C61D411yescharacteristic
95#124×C61D412yes—
96#121×C61—21no—
97#121×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C42×C61C2311yescharacteristic
103#103×C61—22no—
104#103×C61—22no—
105#102×C61—22no—
106#102×C61—22no—
107#97×C61—21no—
108C4×C8×C61—21no—
109C23×D61—21no—
110#41×C61—22no—
111#41×C61—22no—
112#41×C61—22no—
113#41×C61—22no—
114C2×Q8×C61—41no—
115C2×Q8×C61—41no—
116C2×Q8×C61—82no—
117C2×Q8×C61C2×D412yes—
118C2×D4×C61—22no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61C2×D412yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61#4111yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244#4112yes—
134C2×C4×D61—41no—
135C2×C4×D61—41no—
136C2×C4×D61—82no—
137M16×C61—21no—
138M16×C61—21no—
139M16×C61—21no—
140M16×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61—41no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155#11758201—44no—
156#11758199—44no—
157#11758198—41no—
158#11758198—41no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161#11758194—21no—
162#11758194—21no—
163#11758194—22no—
164#11758194—22no—
165#11758194—21no—
166#11758194—44no—
167#11758194#4112yes—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418—44no—
174C2×#92418C2×D412yes—
175C2×#92418C2×D412yes—
176C23×C61—22no—
177C23×C61C2×#4111yescharacteristic
178C22×D61—22no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—81no—
182C22×D61—81no—
183Q8×C61—22no—
184Q8×C61—22no—
185Q8×C61—82no—
186Q8×C61—82no—
187D4×C61—22no—
188D4×C61—22no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—81no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—82no—
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61H(5,𝔽2)11yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C2×Dic61—41no—
211C2×Dic61—81no—
212D244—22no—
213D244—22no—
214D244—41no—
215D244—41no—
216C4×D61—82no—
217C4×D61—82no—
218C8×C61—41no—
219C8×C61—41no—
220#92418—22no—
221#92418—22no—
222#92418—22no—
223#92418—42no—
224#92418—41no—
225#25358—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61C2×#12111yescharacteristic
232C22×C61#52911yescharacteristic
233C22×C61#54211yescharacteristic
234C2×D61—42no—
235C2×D61—82no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—82no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—42no—
243C4×C61—82no—
244C4×C61#47212yes—
245C4×C61#54611yescharacteristic
246C4×C61#54611yescharacteristic
247C4×C61#54711yescharacteristic
248C4×C61#54811yescharacteristic
249Dic61—81no—
250#3141D6111yescharacteristic, complemented
251C2×#474—611no—
252#3036—611no—
253#2935—611no—
254#2078—612no—
255#2051—612no—
256#1997—611no—
257#1950—611no—
258#1907—612no—
259#1816—612no—
260C4×#100—611no—
261C2×C61—21no—
262C2×C61—21no—
263C2×C61—82no—
264C2×C61#242711yescharacteristic
265C2×C61C2×#47211yescharacteristic
266C2×C61#306911yescharacteristic
267D61—82no—
268C2×AGL(1,ℤ/8)—612no—
269C2×#131—612no—
270#532C2×D6111yescharacteristic
271C2×C4⋊Q8C2×D6111yescharacteristic
272C2×C4×D4—611no—
273#489—22no—
274#487—21no—
275#486—21no—
276#474—1221no—
277#474—1222no—
278#474—1221no—
279#463—1222no—
280#461—1221no—
281#459—1221no—
282#436—1221no—
283#435—1222no—
284C8×D4—1221no—
285C2×#108—612no—
286C2×#107—612no—
287#423—611no—
288C2×C4⋊C8—611no—
289#417—612no—
290#417C2×D6112yes—
291#417C2×D6112yes—
292C2×#102—612no—
293C2×#100—611no—
294C2×#100—611no—
295#408—612no—
296C4×M16—611no—
297C4×M16C2×D6111yescharacteristic
298#343—611no—
299#343—611no—
300#343—612no—
301#343—612no—
302#341—612no—
303C61#2535811yescharacteristic, complemented, Sylow
304C2×#41C22×D6112yes—
305C22×Q8—21no—
306C22×D4—611no—
307C23×C4—1221no—
308AGL(1,ℤ/8)—1224no—
309AGL(1,ℤ/8)—2442no—
310C2×SD16—1222no—
311C2×D8—1222no—
312#131—1224no—
313#131—2442no—
314C2×M16—611no—
315C2×M16—612no—
316C2×M16—611no—
317C2×M16—1222no—
318C2×M16C22×D6111yescharacteristic
319C2×M16C22×D6111yescharacteristic
320C22×C8—612no—
321C22×C8—1222no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8C22⋊D6111yescharacteristic
327#127C22⋊D6111yescharacteristic
328#124C22⋊D6112yes—
329#121—21no—
330#121—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336C4×D4—1221no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4C22×D6112yes—
340C2×C22⋊C4—1221no—
341C2×C42C22×D6111yescharacteristic
342#108—2442no—
343#107—1222no—
344#107—1222no—
345C4⋊C8—611no—
346C4⋊C8—612no—
347C4⋊C8—611no—
348C4⋊C8—1221no—
349C4⋊C8—1221no—
350#103—22no—
351#103—22no—
352#103—1222no—
353#102—22no—
354#102—22no—
355#102—1222no—
356#102—1222no—
357#102—1222no—
358#100—1222no—
359#100—1222no—
360C22⋊C8—1222no—
361C22⋊C8—1222no—
362#97—21no—
363#97—1221no—
364C4×C8—21no—
365C4×C8—1221no—
366C24—1221no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—22no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—82no—
374C2×Q8C2×C22⋊D6112yes—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—612no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—1221no—
384C2×D4C2×C22⋊D6112yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—1221no—
388C22×C4—4882no—
389C22×C4—2441no—
390C22×C4—2441no—
391C22×C4#1175822411yescharacteristic
392C22×C4#1175822411yescharacteristic
393C22×C4C23×D6111yescharacteristic
394SD16—2444no—
395D8—2444no—
396M16—21no—
397M16—21no—
398M16—21no—
399M16—21no—
400M16—612no—
401M16—1221no—
402M16—1221no—
403M16—1222no—
404M16—1222no—
405M16—1221no—
406M16—2444no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—612no—
410C2×C8—612no—
411C2×C8—1221no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—1221no—
415C2×C8—1222no—
416C2×C8—2444no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C42D4×D6111yescharacteristic
430C42D4×D6111yescharacteristic
431C42C2×C22⋊D6111yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C23—22no—
434C23—1222no—
435C23—4881no—
436C23—4881no—
437C23—2441no—
438C23—2441no—
439C23C2×#1175822411yescharacteristic
440Q8—22no—
441Q8—22no—
442Q8—82no—
443Q8—82no—
444D4—22no—
445D4—22no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—81no—
450D4—1222no—
451D4—1222no—
452D4—2441no—
453D4—2441no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—4882no—
462C2×C4—4882no—
463C2×C4—4881no—
464C2×C4—2441no—
465C2×C4#1451974911yescharacteristic
466C2×C4#1451974911yescharacteristic
467C2×C4#1451974911yescharacteristic
468C2×C4#1451974911yescharacteristic
469C2×C4C2×D4×D6111yescharacteristic
470C2×C4C22×C22⋊D6111yescharacteristic
471C2×C4#41×D6111yescharacteristic
472C2×C4#41×D6111yescharacteristic
473C2×C4#1451982311yescharacteristic
474C2×C4#1451982411yescharacteristic
475C8—41no—
476C8—41no—
477C8—1222no—
478C8—1222no—
479C8—1222no—
480C8—2442no—
481C8—2441no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—4882no—
488C22—4882no—
489C22—4881no—
490C22—4881no—
491C22—2442no—
492C22C2×#1451974911yescharacteristic
493C22#1684029111yescharacteristic
494C22#1684038111yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—82no—
500C4—4881no—
501C4#1684003012yes—
502C4#1684040011yescharacteristic
503C4#1684040511yescharacteristic
504C4#1684041111yescharacteristic
505C4#1684042011yescharacteristic
506C2—21no—
507C2—21no—
508C2—82no—
509C2—4882no—
510C2#2144939911yescharacteristic
511C2C2×#1684003011yescharacteristic
512C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3141C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#3141D61
hypocenterC61#25358
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
1. C2×#16840039C2semidirect
3. #21454563C2semidirect
4. #21447262C2semidirect
5. #21447109C2semidirect
6. #21446809C2semidirect
7. #21446594C2semidirect
8. #21446325C2semidirect
9. #21445881C2semidirect
10. C4×#14519693C2semidirect
11. C61×#3141C2semidirect
250. #3141D61semidirect
303. C61#25358semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
#214552792
#214545632
#214472622
#214471092
#214468092
#214465942
#214463252
#214458812
C4×#145196932
C61×#31412
#2535861