Finite Group Databasebeta

← #10780319

Subgroups — #10780319

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×#2969C211yesmaximal, characteristic, complemented
2#83150C211yesmaximal, characteristic
3#83123C211yesmaximal, characteristic, complemented
4#78528C211yesmaximal, characteristic, complemented
5#70967C211yesmaximal, characteristic, complemented
6#70960C211yesmaximal, characteristic, complemented
7#70947C211yesmaximal, characteristic, complemented
8#70940C211yesmaximal, characteristic, complemented
9#70876C211yesmaximal, characteristic, complemented
10#70869C211yesmaximal, characteristic, complemented
11#70855C211yesmaximal, characteristic, complemented
12#70848C211yesmaximal, characteristic, complemented
13#70779C211yesmaximal, characteristic, complemented
14#70775C211yesmaximal, characteristic, complemented
15#69851C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17C3×#529C2211yescharacteristic
18#5477C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×C8×Q8C2211yescharacteristic
22C3×#436—21no—
23C3×#436C2211yescharacteristic
24C3×#435—21no—
25C3×C8×D4C2211yescharacteristic
26C3×#433—21no—
27C3×#431C2211yescharacteristic
28C3×#405—21no—
29C3×C4×M16C2211yescharacteristic
30#5133—21no—
31#5133—21no—
32#5066—21no—
33#5064—21no—
34#5057—21no—
35#5055—21no—
36#5049—21no—
37#5043—21no—
38#5039—21no—
39#5033—21no—
40#5027C2211yescharacteristic
41#5021C2211yescharacteristic
42#5021C2211yescharacteristic
43#5020C2211yescharacteristic
44#5015C2211yescharacteristic
45#5008C2211yescharacteristic
46#5008C2211yescharacteristic
47#5007C2211yescharacteristic
48#4926C2211yescharacteristic
49#4924C2211yescharacteristic
50#4714—21no—
51#4714—21no—
52#4694C2211yescharacteristic
53#4694C2211yescharacteristic
54#4681C2211yescharacteristic
55#4681C2211yescharacteristic
56#4485C2211yescharacteristic
57#4485C2211yescharacteristic
58#4482C2211yescharacteristic
59#4482C2211yescharacteristic
60#4481C2211yescharacteristic
61#4481C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3251—31no—
69#3238—31no—
70#3141—31no—
71#2969D311yescharacteristic, complemented
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—
83C2×#235—21no—
84C22×Dic6—21no—
85C2×C3×M16—21no—
86C2×C3×M16D411yescharacteristic
87C2×C3×M16D411yescharacteristic
88C22×C3×C8—21no—
89C3×#126—21no—
90C3×#125D411yescharacteristic
91C3×#124D411yescharacteristic
92C3×#123—21no—
93C3×C22⋊Q8D411yescharacteristic
94C3×#121D411yescharacteristic
95C3×C4×Q8C2311yescharacteristic
96C3×C4×D4—21no—
97C3×C4×D4C2311yescharacteristic
98C3×#117—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#215—21no—
101C2×#213—21no—
102#930—21no—
103#930—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×#208—21no—
107#875—21no—
108#874C2311yescharacteristic
109C4×D12—21no—
110C4×D12—21no—
111#869—21no—
112#869—21no—
113#869—21no—
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C3×C4⋊C8—21no—
118C3×C4⋊C8—21no—
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C22⋊C8—21no—
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×#97—21no—
125C3×#97D411yescharacteristic
126C3×#97D411yescharacteristic
127C3×C4×C8—21no—
128C3×C4×C8—21no—
129C3×C4×C8C2311yescharacteristic
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#835—21no—
133#835—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#830—21no—
138#821—21no—
139#821—21no—
140#821—21no—
141#821—21no—
142#816—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146#809—21no—
147#809—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160#532—31no—
161C2×C4⋊Q8—31no—
162#529C2×D311yescharacteristic
163#501—61no—
164#489—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#483—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477—31no—
176#476—31no—
177#476—31no—
178#475—31no—
179#474—31no—
180#469—61no—
181#468—61no—
182#463—61no—
183#460—31no—
184#459—31no—
185#446C2×D311yescharacteristic
186C8×Q8C2×D311yescharacteristic
187#436—21no—
188#436C2×D311yescharacteristic
189#435—21no—
190C8×D4C2×D311yescharacteristic
191#433—21no—
192#432—31no—
193#431C2×D311yescharacteristic
194#423—31no—
195#417—61no—
196#417—61no—
197#405—21no—
198C4×M16C2×D311yescharacteristic
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×Q8C2×D411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4111yescharacteristic
218C2×#67—41no—
219C22×Dic3—41no—
220#235—42no—
221#235—42no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×C4×D3—41no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—41no—
230C2×Dic6—82no—
231C3×M16—41no—
232C3×M16—41no—
233C3×M16—42no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—42no—
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C42—21no—
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42#4111yescharacteristic
256C3×C42C2411yescharacteristic
257#215—81no—
258#213—81no—
259(C2×D3)⋊C4—41no—
260(C2×D3)⋊C4—41no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—41no—
268C4⋊Dic3—41no—
269#208—81no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×#41—61no—
275C22×Q8—61no—
276C2×SD16—61no—
277C2×D8—61no—
278C2×M16—21no—
279C2×M16—61no—
280C2×M16#6711yescharacteristic
281C2×M16#6711yescharacteristic
282C22×C8—21no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8—31no—
286C4⋊Q8—61no—
287C4⋊Q8—61no—
288C4⋊Q8—61no—
289#127—31no—
290#126—21no—
291#125D1211yescharacteristic
292#124—61no—
293#124D1211yescharacteristic
294#123—21no—
295C22⋊Q8D1211yescharacteristic
296#121—61no—
297#121—61no—
298#121D1211yescharacteristic
299C4×Q8C22×D311yescharacteristic
300C4×D4—21no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4C22×D311yescharacteristic
304#117—21no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C42C22×D311yescharacteristic
308#107—61no—
309#107—61no—
310#107—61no—
311#106—61no—
312#106—61no—
313#106—61no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—61no—
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—61no—
324#103—61no—
325#103—61no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#102—61no—
331#102—61no—
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—21no—
341C22⋊C8—21no—
342C22⋊C8—61no—
343#97—21no—
344#97—61no—
345#97#6711yescharacteristic
346#97#6711yescharacteristic
347C4×C8—21no—
348C4×C8—21no—
349C4×C8C22×D311yescharacteristic
350C4×C8C22×D311yescharacteristic
351C23×C3—21no—
352C23×C3#12111yescharacteristic
353C22×D3—41no—
354C3×Q8—21no—
355C3×Q8—21no—
356C3×D4—21no—
357C3×D4—21no—
358C3×D4—81no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4#12111yescharacteristic
370C2×C3×C4#12111yescharacteristic
371C2×C3×C4#12111yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375#67—82no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—82no—
380D12—41no—
381D12—41no—
382D12—41no—
383D12—41no—
384D12—81no—
385C4×D3—82no—
386Dic6—41no—
387Dic6—41no—
388Dic6—82no—
389Dic6—81no—
390Dic6—81no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—41no—
396C3×C8—41no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—122no—
402#41—122no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—121no—
407C2×Q8—242no—
408C2×Q8C2×D1211yescharacteristic
409C2×D4—21no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—121no—
414C2×D4C2×D1211yescharacteristic
415C22×C4—21no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4C2×D1211yescharacteristic
419C22×C4#23511yescharacteristic
420C22×C4C2×#6711yescharacteristic
421SD16—241no—
422D8—241no—
423M16—41no—
424M16—41no—
425M16—42no—
426M16—241no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—42no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8C2×#6711yescharacteristic
435C2×C8C2×#6711yescharacteristic
436C2×C8C2×#6711yescharacteristic
437C2×C8C2×#6711yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—121no—
449C4⋊C4—121no—
450C4⋊C4C2×D1211yescharacteristic
451C4⋊C4C2×D1211yescharacteristic
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C42—21no—
460C42C2×D1211yescharacteristic
461C42#23511yescharacteristic
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#57511yescharacteristic
482C3×C4#57511yescharacteristic
483C3×C4#57711yescharacteristic
484C3×C4#57811yescharacteristic
485Dic3—81no—
486Dic3—81no—
487Dic3—81no—
488C23—21no—
489C23—121no—
490C23#93011yescharacteristic
491Q8—21no—
492Q8—21no—
493Q8—121no—
494Q8—121no—
495Q8—241no—
496Q8—241no—
497Q8—242no—
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—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—242no—
521C2×C4—242no—
522C2×C4#93011yescharacteristic
523C2×C4#93011yescharacteristic
524C2×C4#93011yescharacteristic
525C2×C4C22×D1211yescharacteristic
526C2×C4C2×#23511yescharacteristic
527C2×C4C22×#6711yescharacteristic
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
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×#93011yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—241no—
562C4—241no—
563C4—241no—
564C4#575111yescharacteristic
565C4#575311yescharacteristic
566C4#582211yescharacteristic
567C4#582211yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—41no—
570C2—81no—
571C2—481no—
572C2#8252711yescharacteristic
573C2#8380811yescharacteristic
574C2#8381411yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2969C2
socleC22×C3C2×#121
prime radical#83150C2
hypercenter#2969D3
hypocenterC3#19382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2969

Complements

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

NH ≅ G/Ntype
1. C3×#2969C2semidirect
3. #83123C2semidirect
4. #78528C2semidirect
5. #70967C2semidirect
6. #70960C2semidirect
7. #70947C2semidirect
8. #70940C2semidirect
9. #70876C2semidirect
10. #70869C2semidirect
11. #70855C2semidirect
12. #70848C2semidirect
13. #70779C2semidirect
14. #70775C2semidirect
15. #69851C2semidirect
71. #2969D3semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
C3×#29692
#831502
#831232
#785282
#709672
#709602
#709472
#709402
#708762
#708692
#708552
#708482
#707792
#707752
#698512
#193823