Finite Group Databasebeta

← #10780323

Subgroups — #10780323

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3238C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic, complemented
3#83150C211yesmaximal, characteristic
4#83129C211yesmaximal, characteristic, complemented
5#71774C211yesmaximal, characteristic, complemented
6#71155C211yesmaximal, characteristic, complemented
7#71154C211yesmaximal, characteristic, complemented
8#71151C211yesmaximal, characteristic, complemented
9#71150C211yesmaximal, characteristic, complemented
10#71139C211yesmaximal, characteristic, complemented
11#71138C211yesmaximal, characteristic, complemented
12#71135C211yesmaximal, characteristic, complemented
13#71134C211yesmaximal, characteristic, complemented
14#70817C211yesmaximal, characteristic, complemented
15#70795C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C3×#529C2211yescharacteristic
19#5477C2211yescharacteristic
20C3×#501—21no—
21C3×#489—21no—
22C3×#483—21no—
23C3×#480—21no—
24C3×#477C2211yescharacteristic
25C3×#477C2211yescharacteristic
26C3×#476C2211yescharacteristic
27C3×#476C2211yescharacteristic
28C3×#432C2211yescharacteristic
29#5093C2211yescharacteristic
30#5093C2211yescharacteristic
31#5089—21no—
32#5089—21no—
33#5088C2211yescharacteristic
34#5088C2211yescharacteristic
35#5066—21no—
36#5057—21no—
37#5049—21no—
38#5039—21no—
39#5027C2211yescharacteristic
40#5024C2211yescharacteristic
41Q8×C3⋊C8C2211yescharacteristic
42#5020C2211yescharacteristic
43#5015C2211yescharacteristic
44#5012—21no—
45#5012C2211yescharacteristic
46#5011—21no—
47D4×C3⋊C8C2211yescharacteristic
48#5007C2211yescharacteristic
49#5000—21no—
50#4997—21no—
51#4966—21no—
52#4966—21no—
53#4925C2211yescharacteristic
54#4924C2211yescharacteristic
55C4×#208C2211yescharacteristic
56#4546C2211yescharacteristic
57#4546C2211yescharacteristic
58#4546C2211yescharacteristic
59#4546C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3251—31no—
69#3238D311yescharacteristic, complemented
70#3141—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1698—31no—
75#1697—31no—
76#1684—31no—
77#1683—31no—
78#1678—31no—
79#1677—31no—
80#1569—31no—
81#1553—31no—
82#1548—31no—
83C22×C3×Q8—21no—
84C2×#235—21no—
85C3×C4⋊Q8—21no—
86C3×C4⋊Q8—21no—
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×#126—21no—
91C3×#125D411yescharacteristic
92C3×#124D411yescharacteristic
93C3×#123—21no—
94C3×C22⋊Q8D411yescharacteristic
95C3×#121D411yescharacteristic
96C3×C4×Q8C2311yescharacteristic
97C3×C4×D4—21no—
98C3×C4×D4C2311yescharacteristic
99C3×#117—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C42C2311yescharacteristic
103C2×#215—21no—
104C2×#213—21no—
105#930—21no—
106#930—21no—
107C2×#208—21no—
108C2×#208—21no—
109C2×#208D411yescharacteristic
110C2×#208D411yescharacteristic
111C22×C3⋊C8—21no—
112#875—21no—
113#874C2311yescharacteristic
114C4×D12—21no—
115C4×D12—21no—
116#869C2311yescharacteristic
117C3×#107—21no—
118C3×#107—21no—
119C3×#106—21no—
120C3×#106—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×#103—21no—
124C3×#103—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×C22⋊C8—21no—
128C3×#97—21no—
129#835—21no—
130#832—21no—
131#830—21no—
132#830—21no—
133#830—21no—
134#810—21no—
135#810—21no—
136#810—21no—
137#810—21no—
138#809—21no—
139#809—21no—
140#809—21no—
141#809—21no—
142#809—21no—
143#809—21no—
144#808—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)—21no—
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152C4⋊(C3⋊C8)C2311yescharacteristic
153#803—21no—
154#803D411yescharacteristic
155#803D411yescharacteristic
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8C2×D311yescharacteristic
162#529C2×D311yescharacteristic
163#501—21no—
164#489—21no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#483—21no—
172#480—21no—
173#478—61no—
174#477C2×D311yescharacteristic
175#477C2×D311yescharacteristic
176#476C2×D311yescharacteristic
177#476C2×D311yescharacteristic
178#475—31no—
179#474—31no—
180#469—61no—
181#468—61no—
182#463—61no—
183#460—31no—
184#459—31no—
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432C2×D311yescharacteristic
193#431—31no—
194#423—31no—
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—41no—
214C2×C3×Q8—82no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—41no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4#4111yescharacteristic
223C2×#67—41no—
224#235—42no—
225#235—42no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×C4×D3—41no—
230C2×Dic6—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C42—21no—
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42#4111yescharacteristic
254C3×C42C2411yescharacteristic
255#215—81no—
256#213—81no—
257(C2×D3)⋊C4—41no—
258(C2×D3)⋊C4—41no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261#208—41no—
262#208—41no—
263#208—42no—
264#208—81no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—42no—
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×#41—61no—
275C22×Q8—21no—
276C2×SD16—61no—
277C2×D8—61no—
278C2×M16—31no—
279C2×M16—31no—
280C2×M16—61no—
281C2×M16—61no—
282C22×C8—61no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—31no—
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
289#127—31no—
290#126—21no—
291#125#6711yescharacteristic
292#124—61no—
293#124#6711yescharacteristic
294#123—21no—
295C22⋊Q8#6711yescharacteristic
296#121—61no—
297#121—61no—
298#121#6711yescharacteristic
299C4×Q8C22×D311yescharacteristic
300C4×D4—21no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4C22×D311yescharacteristic
304#117—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C42C22×D311yescharacteristic
308#107—21no—
309#107—21no—
310#107—61no—
311#106—21no—
312#106—21no—
313#106—61no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8—61no—
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—21no—
324#103—21no—
325#103—61no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#102—21no—
331#102—21no—
332#102—61no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—21no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—21no—
344#97—31no—
345#97—31no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—61no—
350C4×C8—61no—
351C23×C3—21no—
352C23×C3#12111yescharacteristic
353C22×D3—41no—
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—81no—
357C3×Q8—81no—
358C3×Q8—82no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—81no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—82no—
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4#12111yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381#67—82no—
382C2×Dic3—41no—
383D12—41no—
384D12—41no—
385D12—41no—
386D12—41no—
387D12—81no—
388C4×D3—82no—
389Dic6—41no—
390Dic6—41no—
391C3×C8—41no—
392C3×C8—41no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—122no—
402#41—122no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—41no—
406C2×Q8—61no—
407C2×Q8—82no—
408C2×Q8C2×#6711yescharacteristic
409C2×D4—21no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—121no—
414C2×D4C2×#6711yescharacteristic
415C22×C4—21no—
416C22×C4—41no—
417C22×C4—121no—
418C22×C4D3×D411yescharacteristic
419C22×C4#23711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421SD16—241no—
422D8—241no—
423M16—121no—
424M16—121no—
425M16—122no—
426M16—241no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—31no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—122no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4C2×#6711yescharacteristic
451C4⋊C4C2×#6711yescharacteristic
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C42—21no—
460C42D3×D411yescharacteristic
461C42#23711yescharacteristic
462C42C2×#6711yescharacteristic
463C42C23×D311yescharacteristic
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3C2×#12111yescharacteristic
471C2×D3—81no—
472C2×D3—81no—
473C2×D3—81no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4#57511yescharacteristic
484C3×C4#57511yescharacteristic
485C3×C4#57711yescharacteristic
486C3×C4#57811yescharacteristic
487Dic3—81no—
488C23—21no—
489C23—121no—
490C23#93911yescharacteristic
491Q8—21no—
492Q8—21no—
493Q8—81no—
494Q8—81no—
495Q8—82no—
496Q8—121no—
497Q8—121no—
498D4—21no—
499D4—21no—
500D4—81no—
501D4—121no—
502D4—121no—
503D4—121no—
504D4—121no—
505D4—241no—
506D4—242no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—121no—
521C2×C4—242no—
522C2×C4#93911yescharacteristic
523C2×C4#93911yescharacteristic
524C2×C4#93911yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×#23711yescharacteristic
527C2×C4C22×#6711yescharacteristic
528C8—41no—
529C8—41no—
530C8—61no—
531C8—61no—
532C8—61no—
533C8—61no—
534C8—121no—
535C8—121no—
536C8—121no—
537C8—121no—
538C2×C3—41no—
539C2×C3—81no—
540C2×C3#304011yescharacteristic
541C2×C3#306311yescharacteristic
542C2×C3#306811yescharacteristic
543D3—161no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—241no—
551C22—241no—
552C22—241no—
553C22C2×#93911yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—241no—
564C4#577711yescharacteristic
565C4#577711yescharacteristic
566C4#583711yescharacteristic
567C4#583811yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—41no—
570C2—81no—
571C2—481no—
572C2#8340011yescharacteristic
573C2#8394311yescharacteristic
574C2#8396811yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3238C2
socleC22×C3C2×#121
prime radical#83150C2
hypercenter#3238D3
hypocenterC3#19382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3238

Complements

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

NH ≅ G/Ntype
1. C3×#3238C2semidirect
2. #83199C2semidirect
4. #83129C2semidirect
5. #71774C2semidirect
6. #71155C2semidirect
7. #71154C2semidirect
8. #71151C2semidirect
9. #71150C2semidirect
10. #71139C2semidirect
11. #71138C2semidirect
12. #71135C2semidirect
13. #71134C2semidirect
14. #70817C2semidirect
15. #70795C2semidirect
69. #3238D3semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
C3×#32382
#831992
#831502
#831292
#717742
#711552
#711542
#711512
#711502
#711392
#711382
#711352
#711342
#708172
#707952
#193823