Finite Group Databasebeta

← #29721471

Subgroups — #29721471

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
1C2×#16840039C211yesmaximal, characteristic, complemented
2C4×#14519802C211yesmaximal, characteristic
3#21454485C211yesmaximal, characteristic, complemented
4#21447262C212yesmaximal, complemented
5#21447109C212yesmaximal
6#21446809C211yesmaximal, characteristic, complemented
7#21446601C212yesmaximal, complemented
8#21446592C211yesmaximal, characteristic, complemented
9#21446480C212yesmaximal, complemented
10#21445808C211yesmaximal, characteristic, complemented
11C61×#3079C211yesmaximal, characteristic, complemented
12C2×#14519802C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840039—21no—
15#16840039—21no—
16#16840039—22no—
17#16839872—21no—
18#16839860—21no—
19C4×#11758201—21no—
20C4×#11758199—21no—
21#16839823—22no—
22#16839819—21no—
23#16839815—21no—
24C2×#14519712C2212yes—
25#16839805C2211yescharacteristic
26C2×#14519710C2212yes—
27#16839786C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839766C2211yescharacteristic
30#16839763C2211yescharacteristic
31C2×#14519697C2212yes—
32C2×#14519693C2212yes—
33#16839729C2212yes—
34#16839718C2212yes—
35C4×#11758194C2211yescharacteristic
36#16839531C2212yes—
37#16839531C2212yes—
38#16839530C2212yes—
39#16839526C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C4×#41×C61C2211yescharacteristic
42C61×#477—21no—
43C61×#476—21no—
44C61×#475—21no—
45C61×#474—21no—
46C61×#428C2212yes—
47C61×#423C2211yescharacteristic
48C61×#417C2212yes—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51#14519802—22no—
52#14519802—22no—
53C2×#11758201—21no—
54C2×#11758199—21no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2312yes—
57C2×#11758194C2311yescharacteristic
58C22×#92418C2312yes—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244—21no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758198—21no—
65#14519712—24no—
66#14519710—21no—
67#14519710—22no—
68#14519710—22no—
69#14519708—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418D411yescharacteristic
73C4⋊#92418D411yescharacteristic
74C4⋊#92418D412yes—
75#14519701—21no—
76#14519698—21no—
77#14519697—21no—
78#14519697—21no—
79#14519697—22no—
80#14519697—22no—
81#14519693—24no—
82C22⋊#92418—22no—
83C22⋊#92418—22no—
84#14519688—21no—
85#14519688—22no—
86C4×#92418—21no—
87C4×#92418—22no—
88C2×#41×C61C2311yescharacteristic
89C22×Q8×C61—21no—
90C2×M16×C61C2312yes—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61—21no—
95C4×Q8×C61D411yescharacteristic
96C4×Q8×C61D411yescharacteristic
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61D411yescharacteristic
100C4×D4×C61D411yescharacteristic
101#117×C61—21no—
102#117×C61C2311yescharacteristic
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#107×C61—22no—
108#106×C61—22no—
109C4⋊C8×C61—21no—
110C4⋊C8×C61—21no—
111#103×C61—22no—
112#102×C61—22no—
113C23×D61—21no—
114#41×C61—21no—
115#41×C61—21no—
116#41×C61—21no—
117#41×C61—21no—
118C2×Q8×C61—41no—
119C2×Q8×C61—41no—
120C2×Q8×C61—82no—
121C2×Q8×C61C2×D411yescharacteristic
122C2×D4×C61—21no—
123C2×D4×C61C2×D411yescharacteristic
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61#4111yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244#4112yes—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61—82no—
141M16×C61—22no—
142M16×C61—22no—
143C2×C8×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61C2×D411yescharacteristic
151C4⋊C4×C61C2×D411yescharacteristic
152C22⋊C4×C61—21no—
153C22⋊C4×C61—21no—
154C22⋊C4×C61—21no—
155C42×C61—21no—
156C42×C61—21no—
157C42×C61—21no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162#11758201—24no—
163#11758199—24no—
164#11758198—41no—
165#11758198—41no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168#11758194—21no—
169#11758194—22no—
170#11758194—21no—
171#11758194—21no—
172#11758194—22no—
173#11758194#4112yes—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—21no—
179C2×#92418C2×D412yes—
180C2×#92418C2×D412yes—
181C23×C61—21no—
182C23×C61C2×#4111yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—81no—
187C22×D61—81no—
188Q8×C61—21no—
189Q8×C61—21no—
190Q8×C61—82no—
191Q8×C61—82no—
192D4×C61—21no—
193D4×C61—21no—
194D4×C61—41no—
195D4×C61—41no—
196C2×C4×C61—22no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—81no—
206C2×C4×C61—82no—
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61#12111yescharacteristic
209C2×C4×C61#12111yescharacteristic
210C2×C4×C61#12111yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C612−1+411yescharacteristic
216C2×C4×C612−1+411yescharacteristic
217C2×Dic61—41no—
218C2×Dic61—81no—
219D244—22no—
220D244—22no—
221D244—41no—
222D244—41no—
223C4×D61—82no—
224C4×D61—82no—
225C8×C61—42no—
226#92418—22no—
227#92418—22no—
228#92418—22no—
229#92418—41no—
230#92418—42no—
231#24720—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61C2×#12111yescharacteristic
238C22×C61#55211yescharacteristic
239C22×C61#55611yescharacteristic
240C2×D61—42no—
241C2×D61—82no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—82no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—82no—
252C4×C61#47212yes—
253C4×C61#52511yescharacteristic
254C4×C61#53611yescharacteristic
255C4×C61#54711yescharacteristic
256C4×C61#54811yescharacteristic
257Dic61—81no—
258#3079D6111yescharacteristic, complemented
259C2×#474—611no—
260C4×AGL(1,ℤ/8)—611no—
261#2925—611no—
262#2078—612no—
263#2051—612no—
264#1997—611no—
265#1952—612no—
266#1950—611no—
267#1931—612no—
268#1800—611no—
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—81no—
272C2×C61#241511yescharacteristic
273C2×C61C2×#47211yescharacteristic
274C2×C61#306911yescharacteristic
275D61—82no—
276C2×AGL(1,ℤ/8)—611no—
277C2×C4⋊Q8C2×D6111yescharacteristic
278C4×#41C2×D6111yescharacteristic
279C2×C4×D4—611no—
280#477—21no—
281#476—21no—
282#475—21no—
283#474—21no—
284#474—1221no—
285#474—1221no—
286#474—1222no—
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—611no—
296#428C2×D6112yes—
297C2×#107—612no—
298#423—611no—
299#423C2×D6111yescharacteristic
300C2×C4⋊C8—611no—
301#418—611no—
302#417—611no—
303#417C2×D6112yes—
304C2×#102—612no—
305C2×#100—612no—
306#408—612no—
307#405—612no—
308C4×M16—611no—
309#343—612no—
310#343—612no—
311#343—612no—
312#341—612no—
313C61#2472011yescharacteristic, complemented, Sylow
314C2×#41C22×D6111yescharacteristic
315C22×Q8—21no—
316C22×D4—611no—
317C23×C4—1221no—
318AGL(1,ℤ/8)—1222no—
319AGL(1,ℤ/8)—1222no—
320C2×SD16—1221no—
321C2×D8—1221no—
322C2×M16—611no—
323C2×M16—611no—
324C2×M16—612no—
325C2×M16C22×D6112yes—
326C22×C8—612no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4×Q8C22⋊D6111yescharacteristic
332C4×Q8C22⋊D6111yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4—1221no—
338C4×D4—1221no—
339C4×D4C22⋊D6111yescharacteristic
340C4×D4C22⋊D6111yescharacteristic
341#117—21no—
342#117C22×D6111yescharacteristic
343C2×C4⋊C4—21no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4C22×D6112yes—
346C2×C22⋊C4—1221no—
347C2×C42—21no—
348C2×C42C22×D6111yescharacteristic
349#108—1224no—
350#107—22no—
351#107—1222no—
352#107—1222no—
353#107—1221no—
354#106—22no—
355#106—1221no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—611no—
359C4⋊C8—611no—
360C4⋊C8—612no—
361C4⋊C8—1221no—
362C4⋊C8—1221no—
363#103—22no—
364#103—1221no—
365#102—22no—
366#102—1222no—
367#102—1222no—
368#102—1221no—
369#102—1221no—
370#102—1221no—
371#100—1224no—
372C22⋊C8—1222no—
373C22⋊C8—1222no—
374#97—1222no—
375#97—1221no—
376C4×C8—1222no—
377C4×C8—1221no—
378C24—1221no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—82no—
386C2×Q8C2×C22⋊D6111yescharacteristic
387C2×D4—21no—
388C2×D4—612no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4C2×C22⋊D6111yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—1221no—
400C22×C4—4882no—
401C22×C4—2441no—
402C22×C4—2441no—
403C22×C4#1175822411yescharacteristic
404C22×C4#1175822411yescharacteristic
405C22×C4C23×D6111yescharacteristic
406SD16—1224no—
407D8—1224no—
408M16—22no—
409M16—22no—
410M16—612no—
411M16—1221no—
412M16—1222no—
413M16—1221no—
414M16—1221no—
415M16—1222no—
416C2×C8—22no—
417C2×C8—612no—
418C2×C8—612no—
419C2×C8—1221no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1221no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4C2×C22⋊D6111yescharacteristic
433C4⋊C4C2×C22⋊D6111yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42D4×D6111yescharacteristic
443C42D4×D6111yescharacteristic
444C42C2×C22⋊D6111yescharacteristic
445C42C2×C22⋊D6111yescharacteristic
446C23—21no—
447C23—1222no—
448C23—4881no—
449C23—4881no—
450C23—2441no—
451C23—2441no—
452C23C2×#1175822411yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—82no—
456Q8—82no—
457D4—21no—
458D4—21no—
459D4—41no—
460D4—41no—
461D4—1222no—
462D4—1222no—
463D4—2441no—
464D4—2441no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—4882no—
477C2×C4—4882no—
478C2×C4—4881no—
479C2×C4—2441no—
480C2×C4#1451974911yescharacteristic
481C2×C4#1451974911yescharacteristic
482C2×C4#1451974911yescharacteristic
483C2×C4#1451974911yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4#41×D6111yescharacteristic
487C2×C4#41×D6111yescharacteristic
488C2×C4#1451982511yescharacteristic
489C2×C4#1451982511yescharacteristic
490C8—42no—
491C8—1222no—
492C8—1222no—
493C8—1222no—
494C8—2441no—
495C8—2442no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—4882no—
502C22—4882no—
503C22—4881no—
504C22—4881no—
505C22—2442no—
506C22C2×#1451974911yescharacteristic
507C22#1684044611yescharacteristic
508C22#1684048311yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—4881no—
517C4#1684003012yes—
518C4#1684026911yescharacteristic
519C4#1684033611yescharacteristic
520C4#1684041411yescharacteristic
521C4#1684042211yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—4882no—
526C2#2144924911yescharacteristic
527C2C2×#1684003011yescharacteristic
528C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3079C2
socleC22×C61C2×#121
prime radicalC2×#14519802C22
hypercenter#3079D61
hypocenterC61#24720
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
1. C2×#16840039C2semidirect
3. #21454485C2semidirect
4. #21447262C2semidirect
6. #21446809C2semidirect
7. #21446601C2semidirect
8. #21446592C2semidirect
9. #21446480C2semidirect
10. #21445808C2semidirect
11. C61×#3079C2semidirect
258. #3079D61semidirect
313. C61#24720semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
C4×#145198022
#214544852
#214472622
#214471092
#214468092
#214466012
#214465922
#214464802
#214458082
C61×#30792
#2472061