Finite Group Databasebeta

← #29721483

Subgroups — #29721483

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455635C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3C4×#14519803C211yesmaximal, characteristic
4#21454487C211yesmaximal, characteristic, complemented
5#21447268C212yesmaximal, complemented
6#21447111C212yesmaximal
7#21446807C211yesmaximal, characteristic, complemented
8#21446598C212yesmaximal, complemented
9#21446592C211yesmaximal, characteristic, complemented
10#21446483C212yesmaximal, complemented
11C61×#1800C211yesmaximal, characteristic, complemented
12C2×#14519803C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14C4×#11758222C2211yescharacteristic
15#16840058—21no—
16#16840053—21no—
17#16840052—21no—
18#16840052—21no—
19#16840052—22no—
20#16840048—21no—
21#16840043—21no—
22#16839872—21no—
23#16839862—21no—
24C4×#11758200—21no—
25C4×#11758199—21no—
26#16839822—22no—
27#16839818—21no—
28#16839814—21no—
29C2×#14519712C2212yes—
30#16839804C2212yes—
31#16839802C2211yescharacteristic
32C2×#14519708C2212yes—
33C2×C4⋊#92418C2211yescharacteristic
34#16839767C2211yescharacteristic
35#16839764C2212yes—
36#16839761C2211yescharacteristic
37C2×#14519698C2212yes—
38#16839730C2212yes—
39#16839720C2212yes—
40C4×#11758194C2211yescharacteristic
41#16839530C2212yes—
42#16839530C2212yes—
43#16839526C2212yes—
44C2×C4×D4×C61C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C2×#100×C61C2212yes—
48C61×#343C2212yes—
49C2×#11758222C2311yescharacteristic
50C22×Dic122—21no—
51#14519803—22no—
52#14519803—22no—
53C2×#11758200—21no—
54C2×#11758199—21no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C22×#92418C2312yes—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62C4×Dic122D411yescharacteristic
63C4×Dic122D411yescharacteristic
64C4×D244D411yescharacteristic
65C4×D244D411yescharacteristic
66C4×C22⋊D61—21no—
67C4×C22⋊D61—21no—
68#14519736C2311yescharacteristic
69#14519734—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C42×D61—21no—
73#14519712—24no—
74#14519711—22no—
75#14519710—21no—
76#14519709—22no—
77#14519708—21no—
78#14519708—22no—
79#14519708—22no—
80C4⋊#92418D411yescharacteristic
81C4⋊#92418D411yescharacteristic
82C4⋊#92418D412yes—
83#14519702—22no—
84#14519700—21no—
85#14519699—22no—
86#14519698—21no—
87#14519698—21no—
88#14519698—22no—
89#14519698—22no—
90#14519697—21no—
91#14519689—22no—
92#14519689—22no—
93#14519688—21no—
94#14519687—22no—
95C4×#92418—21no—
96C8×Dic61—22no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C2×M16×C61C2312yes—
100C2×M16×C61C2312yes—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—21no—
109C4⋊C8×C61—21no—
110C4⋊C8×C61—21no—
111C4⋊C8×C61—21no—
112#100×C61—24no—
113C24×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61#4112yes—
120C22×C4×C61—21no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61—82no—
124C22×C4×C61#4111yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—21no—
128C22×Dic61—22no—
129#11758222—21no—
130#11758222—21no—
131#11758222—21no—
132#11758222—21no—
133C2×D244C2×D411yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×Dic122—41no—
138C2×Dic122—41no—
139C2×Dic122—82no—
140C2×Dic122C2×D411yescharacteristic
141M16×C61—22no—
142M16×C61—22no—
143M16×C61—22no—
144M16×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C22⋊Dic61—21no—
156#11758200—24no—
157#11758199—24no—
158#11758198—21no—
159#11758198—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—41no—
163C4⋊Dic61—41no—
164C4⋊Dic61C2×D411yescharacteristic
165C4⋊Dic61C2×D411yescharacteristic
166#11758196—21no—
167#11758196—21no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—21no—
171#11758194—21no—
172#11758194—21no—
173#11758194—21no—
174#11758194#4112yes—
175C2×#92418—21no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—21no—
179C2×#92418C2×D412yes—
180C2×#92418C2×D412yes—
181C23×C61—22no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—81no—
185C23×C61—81no—
186C23×C61C2×#4111yescharacteristic
187C22×D61—21no—
188D4×C61—22no—
189D4×C61—22no—
190D4×C61—41no—
191D4×C61—41no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—82no—
196C2×C4×C61—82no—
197C2×C4×C61—81no—
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C612−1+411yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C22⋊D61—41no—
209C22⋊D61—41no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—42no—
214C2×Dic61—82no—
215D244—21no—
216D244—21no—
217C4×D61—41no—
218C4×D61—41no—
219C4×D61—42no—
220C4×D61—81no—
221Dic122—21no—
222Dic122—21no—
223Dic122—82no—
224Dic122—82no—
225C8×C61—42no—
226C8×C61—42no—
227#92418—22no—
228#92418—22no—
229#92418—22no—
230#92418—41no—
231#24721—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—42no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61C2×#12111yescharacteristic
240C22×C61#55211yescharacteristic
241C22×C61#55611yescharacteristic
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—81no—
249C4×C61#47112yes—
250C4×C61#52511yescharacteristic
251C4×C61#53611yescharacteristic
252C4×C61#54711yescharacteristic
253C4×C61#54811yescharacteristic
254Dic61—41no—
255Dic61—41no—
256Dic61—41no—
257Dic61—82no—
258#3079—611no—
259C2×#476—611no—
260C4×AGL(1,ℤ/8)—611no—
261#2925—611no—
262#2079—612no—
263#2051—612no—
264#1997—611no—
265#1951—612no—
266#1950—611no—
267#1931—612no—
268#1800D6111yescharacteristic, complemented
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—82no—
272C2×C61#241511yescharacteristic
273C2×C61C2×#47111yescharacteristic
274C2×C61#306811yescharacteristic
275D61—81no—
276C2×AGL(1,ℤ/8)—611no—
277C2×C4⋊Q8—611no—
278C4×#41—611no—
279C2×C4×D4C2×D6111yescharacteristic
280#477—1221no—
281#476—1221no—
282#476—1221no—
283#476—1221no—
284#476—1222no—
285#475—1221no—
286#474—1221no—
287#442—1221no—
288#440—1221no—
289C4×SD16—1221no—
290C4×D8—1221no—
291#433—1222no—
292#432—1221no—
293#431—1221no—
294C2×#108—612no—
295#428—612no—
296#428—611no—
297C2×#106—612no—
298#423C2×D6111yescharacteristic
299#423C2×D6111yescharacteristic
300C2×C4⋊C8—611no—
301#418—611no—
302#417—612no—
303#417—611no—
304C2×#102—612no—
305C2×#100C2×D6112yes—
306#408—612no—
307#405—612no—
308C4×M16—611no—
309#343—612no—
310#343—612no—
311#343C2×D6112yes—
312#341—612no—
313C61#2472111yescharacteristic, complemented, Sylow
314C2×#41—611no—
315C22×Q8—1221no—
316C22×D4C22×D6111yescharacteristic
317C23×C4—21no—
318AGL(1,ℤ/8)—1222no—
319AGL(1,ℤ/8)—1222no—
320C2×SD16—1221no—
321C2×D8—1221no—
322C2×M16—611no—
323C2×M16—611no—
324C2×M16C22×D6112yes—
325C2×M16C22×D6112yes—
326C22×C8—612no—
327C4⋊Q8—1221no—
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4×Q8—611no—
332C4×Q8—611no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—611no—
338C4×D4—611no—
339C4×D4—1221no—
340C4×D4—1221no—
341#117—611no—
342#117—1221no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—21no—
347C2×C42—1221no—
348C2×C42C22×D6111yescharacteristic
349#108—1224no—
350#107—1221no—
351#107—1222no—
352#106—1221no—
353#106—1222no—
354#106—1222no—
355#106—1222no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8—611no—
361C4⋊C8—611no—
362C4⋊C8—612no—
363#103—1221no—
364#103—1222no—
365#102—1221no—
366#102—1221no—
367#102—1221no—
368#102—1222no—
369#102—1222no—
370#102—1222no—
371#100—24no—
372C22⋊C8—1222no—
373C22⋊C8—1222no—
374#97—1222no—
375#97—1221no—
376C4×C8—1222no—
377C4×C8—1221no—
378C24—21no—
379#41—1221no—
380#41—1221no—
381#41—1221no—
382#41—1221no—
383C2×Q8—611no—
384C2×Q8—4882no—
385C2×Q8—2441no—
386C2×Q8—2441no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—611no—
393C2×D4—1221no—
394C2×D4#1175822412yes—
395C22×C4—21no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—82no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1222no—
403C22×C4#1175822211yescharacteristic
404C22×C4#1175822211yescharacteristic
405C22×C4C23×D6111yescharacteristic
406SD16—1224no—
407D8—1224no—
408M16—22no—
409M16—22no—
410M16—22no—
411M16—22no—
412M16—612no—
413M16—1221no—
414M16—1221no—
415M16—1221no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—612no—
419C2×C8—612no—
420C2×C8—1221no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1221no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—611no—
427C4⋊C4—611no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—2441no—
433C4⋊C4—2441no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—1221no—
437C22⋊C4—1221no—
438C22⋊C4—1221no—
439C42—1221no—
440C42—1221no—
441C42—1221no—
442C42D4×D6111yescharacteristic
443C42D4×D6111yescharacteristic
444C42D4×D6111yescharacteristic
445C42D4×D6111yescharacteristic
446C23—22no—
447C23—41no—
448C23—41no—
449C23—81no—
450C23—81no—
451C23—1221no—
452C23C2×#1175822211yescharacteristic
453Q8—1221no—
454Q8—1221no—
455Q8—4882no—
456Q8—4882no—
457D4—22no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—1221no—
462D4—1221no—
463D4—2441no—
464D4—2441no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—1221no—
474C2×C4—4881no—
475C2×C4—4882no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4#1451975011yescharacteristic
481C2×C4#1451975011yescharacteristic
482C2×C4#1451975011yescharacteristic
483C2×C4#1451975011yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4#41×D6111yescharacteristic
487C2×C4C2×#1175822411yescharacteristic
488C2×C4#1451982411yescharacteristic
489C2×C4#1451982511yescharacteristic
490C8—42no—
491C8—42no—
492C8—1222no—
493C8—1222no—
494C8—1222no—
495C8—2441no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—82no—
500C22—82no—
501C22—81no—
502C22—81no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22C2×#1451975011yescharacteristic
507C22#1684044711yescharacteristic
508C22#1684048011yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—81no—
513C4—4882no—
514C4—2441no—
515C4—2441no—
516C4—2441no—
517C4#1684002412yes—
518C4#1684027211yescharacteristic
519C4#1684033511yescharacteristic
520C4#1684041411yescharacteristic
521C4#1684041811yescharacteristic
522C2—21no—
523C2—21no—
524C2—82no—
525C2—4881no—
526C2#2144925011yescharacteristic
527C2C2×#1684002411yescharacteristic
528C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1800C2
socleC22×C61C2×#121
prime radicalC2×#14519803C22
hypercenter#1800D61
hypocenterC61#24721
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
1. #21455635C2semidirect
2. C2×#16840052C2semidirect
4. #21454487C2semidirect
5. #21447268C2semidirect
7. #21446807C2semidirect
8. #21446598C2semidirect
9. #21446592C2semidirect
10. #21446483C2semidirect
11. C61×#1800C2semidirect
268. #1800D61semidirect
313. C61#24721semidirect

Maximal subgroups

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

subgroupindex
#214556352
C2×#168400522
C4×#145198032
#214544872
#214472682
#214471112
#214468072
#214465982
#214465922
#214464832
C61×#18002
#2472161