Finite Group Databasebeta

← #29721481

Subgroups — #29721481

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#21455634C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3#21454485C211yesmaximal, characteristic, complemented
4#21447267C212yesmaximal, complemented
5#21447108C212yesmaximal
6#21446808C211yesmaximal, characteristic, complemented
7#21446596C212yesmaximal, complemented
8#21446594C211yesmaximal, characteristic, complemented
9#21446484C212yesmaximal, complemented
10#21445811C211yesmaximal, characteristic, complemented
11C4×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840057—21no—
14#16840052—21no—
15#16840052—21no—
16#16840052—22no—
17#16840052—21no—
18#16840047—21no—
19#16840042—21no—
20#16839823—22no—
21#16839819—21no—
22#16839815—21no—
23C2×#14519712C2212yes—
24#16839805C2212yes—
25C2×#14519708C2212yes—
26#16839786C2211yescharacteristic
27#16839786C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839763C2212yes—
30C2×#14519698C2212yes—
31C2×#14519694C2212yes—
32#16839729C2212yes—
33#16839718C2212yes—
34#16839531C2212yes—
35#16839530C2212yes—
36#16839530C2212yes—
37#16839526C2212yes—
38C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
39C4×#41×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#442—21no—
42C61×#440—21no—
43C4×SD16×C61—21no—
44C4×D8×C61—21no—
45C61×#428C2211yescharacteristic
46C61×#418C2211yescharacteristic
47C61×#417C2211yescharacteristic
48C4×M16×C61C2211yescharacteristic
49C22×Dic122—21no—
50C2×#11758194C2312yes—
51C2×#11758194C2312yes—
52C22×#92418C2312yes—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2312yes—
59#14519712—24no—
60#14519710—22no—
61#14519708—22no—
62#14519708—22no—
63#14519708—22no—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68C4⋊#92418D411yescharacteristic
69C4⋊#92418D411yescharacteristic
70C4⋊#92418D412yes—
71#14519701—22no—
72#14519698—22no—
73#14519698—22no—
74#14519698—22no—
75#14519694—24no—
76C22⋊#92418—22no—
77C22⋊#92418—22no—
78#14519688—22no—
79C4×#92418—22no—
80C2×#41×C61C2311yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—21no—
83AGL(1,ℤ/8)×C61—22no—
84AGL(1,ℤ/8)×C61—22no—
85C2×SD16×C61—21no—
86C2×D8×C61—21no—
87C2×M16×C61C2311yescharacteristic
88C2×M16×C61C2311yescharacteristic
89C4×Q8×C61D411yescharacteristic
90C4×Q8×C61D411yescharacteristic
91C4×D4×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D411yescharacteristic
98C4×D4×C61D411yescharacteristic
99#117×C61—21no—
100#117×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105#107×C61—21no—
106#106×C61—21no—
107#103×C61—21no—
108#102×C61—21no—
109#102×C61—21no—
110#102×C61—21no—
111#97×C61—21no—
112C4×C8×C61—21no—
113C24×C61—21no—
114#41×C61—21no—
115#41×C61—21no—
116#41×C61—21no—
117#41×C61—21no—
118C2×Q8×C61C2×D411yescharacteristic
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61C2×D411yescharacteristic
126C2×D4×C61#4112yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—41no—
132C22×C4×C61—41no—
133C22×C4×C61—82no—
134C22×C4×C61#4111yescharacteristic
135C22×C4×C61#4111yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—22no—
138C2×Dic122—41no—
139C2×Dic122—41no—
140C2×Dic122—82no—
141SD16×C61—24no—
142D8×C61—24no—
143M16×C61—21no—
144M16×C61—21no—
145M16×C61—21no—
146M16×C61#4112yes—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D411yescharacteristic
154C4⋊C4×C61C2×D411yescharacteristic
155C22⋊C4×C61—21no—
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C22⋊C4×C61—41no—
159C22⋊C4×C61—41no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—41no—
170C4⋊Dic61—41no—
171#11758194—22no—
172#11758194—22no—
173#11758194—22no—
174#11758194—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418C2×D412yes—
180C2×#92418C2×D412yes—
181C23×C61—21no—
182C23×C61—22no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—81no—
186C23×C61—81no—
187C23×C61C2×#4111yescharacteristic
188Q8×C61—21no—
189Q8×C61—21no—
190D4×C61—21no—
191D4×C61—22no—
192D4×C61—21no—
193D4×C61—22no—
194D4×C61—41no—
195D4×C61—41no—
196D4×C61—41no—
197D4×C61—41no—
198C2×C4×C61—22no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—81no—
208C2×C4×C61—82no—
209C2×C4×C61—82no—
210C2×C4×C61—81no—
211C2×C4×C61#12111yescharacteristic
212C2×C4×C61#12111yescharacteristic
213C2×C4×C61#12111yescharacteristic
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C612−1+411yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×Dic61—42no—
222C2×Dic61—82no—
223Dic122—82no—
224Dic122—82no—
225C8×C61—22no—
226C8×C61—41no—
227#92418—22no—
228#92418—22no—
229#92418—42no—
230#92418—42no—
231#24721—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—41no—
235C22×C61—42no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61C2×#12111yescharacteristic
243C22×C61#55211yescharacteristic
244C22×C61#55611yescharacteristic
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—81no—
252C4×C61#47112yes—
253C4×C61#52511yescharacteristic
254C4×C61#53611yescharacteristic
255C4×C61#54711yescharacteristic
256C4×C61#54811yescharacteristic
257Dic61—82no—
258#3079—611no—
259C2×#476—611no—
260C4×AGL(1,ℤ/8)D6111yescharacteristic
261#2925—611no—
262#2079—612no—
263#2051—612no—
264#1997—611no—
265#1951—612no—
266#1950—611no—
267#1931—612no—
268#1800—611no—
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—81no—
272C2×C61—82no—
273C2×C61#241511yescharacteristic
274C2×C61C2×#47111yescharacteristic
275C2×C61#306811yescharacteristic
276C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
277C2×C4⋊Q8—611no—
278C4×#41C2×D6111yescharacteristic
279C2×C4×D4C2×D6111yescharacteristic
280#477—1221no—
281#476—1221no—
282#476—1221no—
283#476—1222no—
284#476—1221no—
285#475—1221no—
286#474—1221no—
287#442—21no—
288#440—21no—
289C4×SD16—21no—
290C4×D8—21no—
291#433—1222no—
292#432—1221no—
293#431—1221no—
294C2×#108—612no—
295#428—612no—
296#428C2×D6111yescharacteristic
297C2×#106—612no—
298#423—611no—
299#423—611no—
300C2×C4⋊C8—611no—
301#418C2×D6111yescharacteristic
302#417—612no—
303#417C2×D6111yescharacteristic
304C2×#102—612no—
305C2×#100—612no—
306#408—612no—
307#405—612no—
308C4×M16C2×D6111yescharacteristic
309#343—612no—
310#343—612no—
311#343—612no—
312#341—612no—
313C61#2472111yescharacteristic, complemented, Sylow
314C2×#41C22×D6111yescharacteristic
315C22×Q8—1221no—
316C22×D4C22×D6111yescharacteristic
317C23×C4—21no—
318AGL(1,ℤ/8)—22no—
319AGL(1,ℤ/8)—22no—
320C2×SD16—21no—
321C2×D8—21no—
322C2×M16—612no—
323C2×M16—612no—
324C2×M16C22×D6111yescharacteristic
325C2×M16C22×D6111yescharacteristic
326C22×C8—612no—
327C4⋊Q8—1221no—
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4×Q8C22⋊D6111yescharacteristic
332C4×Q8C22⋊D6111yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4C22⋊D6111yescharacteristic
340C4×D4C22⋊D6111yescharacteristic
341#117—21no—
342#117C22×D6111yescharacteristic
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—21no—
347C2×C42—21no—
348C2×C42C22×D6111yescharacteristic
349#108—1224no—
350#107—21no—
351#107—1222no—
352#106—21no—
353#106—1222no—
354#106—1222no—
355#106—1222no—
356C4⋊C8—611no—
357C4⋊C8—611no—
358C4⋊C8—612no—
359C4⋊C8—1221no—
360C4⋊C8—1221no—
361C4⋊C8—1221no—
362C4⋊C8—1221no—
363#103—21no—
364#103—1222no—
365#102—21no—
366#102—21no—
367#102—21no—
368#102—1222no—
369#102—1222no—
370#102—1222no—
371#100—1224no—
372C22⋊C8—1222no—
373C22⋊C8—1222no—
374#97—21no—
375#97—1222no—
376C4×C8—21no—
377C4×C8—1222no—
378C24—21no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—4882no—
384C2×Q8—2441no—
385C2×Q8—2441no—
386C2×Q8C2×C22⋊D6111yescharacteristic
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4#1175822212yes—
394C2×D4C2×C22⋊D6111yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—82no—
402C22×C4—1222no—
403C22×C4#1175822411yescharacteristic
404C22×C4#1175822411yescharacteristic
405C22×C4C23×D6111yescharacteristic
406SD16—24no—
407D8—24no—
408M16—21no—
409M16—21no—
410M16—21no—
411M16—1222no—
412M16—1222no—
413M16—1222no—
414M16—1222no—
415M16#1175822212yes—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—612no—
419C2×C8—612no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1222no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4C2×C22⋊D6111yescharacteristic
433C4⋊C4C2×C22⋊D6111yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42D4×D6111yescharacteristic
443C42D4×D6111yescharacteristic
444C42C2×C22⋊D6111yescharacteristic
445C42C2×C22⋊D6111yescharacteristic
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—81no—
451C23—81no—
452C23C2×#1175822411yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—4882no—
456Q8—4882no—
457D4—21no—
458D4—22no—
459D4—21no—
460D4—22no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
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—41no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—4882no—
479C2×C4—2442no—
480C2×C4#1451974911yescharacteristic
481C2×C4#1451974911yescharacteristic
482C2×C4#1451974911yescharacteristic
483C2×C4#1451974911yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4C2×#1175822211yescharacteristic
487C2×C4C2×#1175822211yescharacteristic
488C2×C4#1451982411yescharacteristic
489C2×C4#1451982411yescharacteristic
490C8—22no—
491C8—41no—
492C8—1222no—
493C8—1222no—
494C8—2442no—
495C8—2442no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—42no—
500C22—41no—
501C22—41no—
502C22—82no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22C2×#1451974911yescharacteristic
507C22#1684045211yescharacteristic
508C22#1684048211yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—4882no—
517C4#1684002312yes—
518C4#1684027311yescharacteristic
519C4#1684033311yescharacteristic
520C4#1684041711yescharacteristic
521C4#1684042111yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—82no—
526C2#2144924811yescharacteristic
527C2C2×#1684002311yescharacteristic
528C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×AGL(1,ℤ/8)×C61C2
socleC22×C61C2×#121
prime radicalC2×AGL(1,ℤ/8)×C61C22
hypercenterC4×AGL(1,ℤ/8)D61
hypocenterC61#24721
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #21455634C2semidirect
2. C2×#16840052C2semidirect
3. #21454485C2semidirect
4. #21447267C2semidirect
6. #21446808C2semidirect
7. #21446596C2semidirect
8. #21446594C2semidirect
9. #21446484C2semidirect
10. #21445811C2semidirect
313. C61#24721semidirect

Maximal subgroups

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

subgroupindex
#214556342
C2×#168400522
#214544852
#214472672
#214471082
#214468082
#214465962
#214465942
#214464842
#214458112
C4×AGL(1,ℤ/8)×C612
#2472161