Finite Group Databasebeta

← #10778217

Subgroups — #10778217

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3035C211yesmaximal, characteristic, complemented
2#83295C211yesmaximal, characteristic, complemented
3#82529C211yesmaximal, characteristic, complemented
4#82519C211yesmaximal, characteristic, complemented
5#71808C211yesmaximal, characteristic, complemented
6#71055C211yesmaximal, characteristic, complemented
7#71052C211yesmaximal, characteristic, complemented
8#71043C211yesmaximal, characteristic, complemented
9#71040C211yesmaximal, characteristic, complemented
10#71022C211yesmaximal, characteristic, complemented
11#71021C211yesmaximal, characteristic, complemented
12#71010C211yesmaximal, characteristic, complemented
13#71009C211yesmaximal, characteristic, complemented
14#70433C211yesmaximal, characteristic, complemented
15#70420C211yesmaximal, characteristic, complemented
16#19184—31nomaximal, Sylow
17C2×C3×#135—21no—
18C3×#532C2211yescharacteristic
19C3×C4×#41C2211yescharacteristic
20#5477C2211yescharacteristic
21C3×#464—21no—
22C3×#463—21no—
23C3×#462C2211yescharacteristic
24C3×#461C2211yescharacteristic
25C3×#460C2211yescharacteristic
26C3×#459C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C3×#416—21no—
29#5238—21no—
30#5088C2211yescharacteristic
31#5080C2211yescharacteristic
32#5073C2211yescharacteristic
33#5070C2211yescharacteristic
34#5039—21no—
35#5039—21no—
36#5027C2211yescharacteristic
37C4×#215—21no—
38Q8×C3⋊C8C2211yescharacteristic
39Q8×C3⋊C8C2211yescharacteristic
40#5021C2211yescharacteristic
41#5015C2211yescharacteristic
42#5014—21no—
43#5014—21no—
44C4×#213—21no—
45#5012—21no—
46#5012—21no—
47#5012C2211yescharacteristic
48#5012C2211yescharacteristic
49#5008C2211yescharacteristic
50#5003—21no—
51#4926—21no—
52#4926C2211yescharacteristic
53#4925C2211yescharacteristic
54C4×#208—21no—
55C4×#208C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4544C2211yescharacteristic
59#4544C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3184—31no—
69#3106—31no—
70#3035D311yescharacteristic, complemented
71#2971—31no—
72#1701—31no—
73#1697—31no—
74#1691—31no—
75#1687—31no—
76#1682—31no—
77#1678—31no—
78#1672—31no—
79#1668—31no—
80#1565—31no—
81#1553—31no—
82#1551—31no—
83C2×C3×#41—21no—
84C2×C3×#41D411yescharacteristic
85C2×#235—21no—
86C3×#135—44no—
87C2×C3×Q16—21no—
88C2×C3×SD16—21no—
89C2×C3×SD16—21no—
90C2×C3×D8—21no—
91C22×C3×C8—21no—
92C3×C4⋊Q8C2311yescharacteristic
93C3×#127C2311yescharacteristic
94C3×#124—21no—
95C3×#121—21no—
96C3×#121—21no—
97C3×C4×Q8D411yescharacteristic
98C3×C4×Q8C2311yescharacteristic
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4D411yescharacteristic
104C3×C4×D4C2311yescharacteristic
105C3×#117—21no—
106C3×#117D411yescharacteristic
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×#215—21no—
110C2×#213—21no—
111C2×#213—21no—
112C2×#213—21no—
113#930—21no—
114#930—21no—
115C2×#208—21no—
116C2×#208—21no—
117C2×#208D411yescharacteristic
118C2×#208D411yescharacteristic
119#875—21no—
120#874C2311yescharacteristic
121C4×D12—21no—
122C4×D12—21no—
123#869C2311yescharacteristic
124C3×C4⋊C8—21no—
125C3×C4⋊C8C2311yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×#103—21no—
128C3×#103—21no—
129C3×#102—21no—
130C3×#102—21no—
131#835—21no—
132#832—21no—
133#832—21no—
134#832—21no—
135#830—21no—
136#830—21no—
137#830—21no—
138#830—21no—
139#809—21no—
140#809—21no—
141#809—21no—
142#809—21no—
143#808—21no—
144#808—21no—
145#808—21no—
146#807—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⋊C8—21no—
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—21no—
162#532—31no—
163#532C2×D311yescharacteristic
164C4×#41C2×D311yescharacteristic
165#498—31no—
166#493—31no—
167#486—31no—
168#486—31no—
169#482—61no—
170#477—31no—
171#476—31no—
172#469—61no—
173#469—61no—
174#464—21no—
175#463—21no—
176#462C2×D311yescharacteristic
177#461C2×D311yescharacteristic
178#460—31no—
179#460C2×D311yescharacteristic
180#459—31no—
181#459C2×D311yescharacteristic
182#449—61no—
183C8×Q8—31no—
184C8×Q8—31no—
185#442—61no—
186#442—61no—
187C4×SD16—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#436—61no—
192#436—61no—
193#432—31no—
194#432—61no—
195#431—31no—
196C2×C4⋊C8C2×D311yescharacteristic
197#416—21no—
198C4×M16—31no—
199C4×M16—61no—
200#335—31no—
201#335—31no—
202#334—31no—
203#334—31no—
204#333—31no—
205#333—31no—
206#332—31no—
207#332—31no—
208#331—31no—
209#331—31no—
210#331—31no—
211#331—31no—
212C3×#41—22no—
213C3×#41—41no—
214C3×#41—42no—
215C3×#41—42no—
216C2×C3×Q8—21no—
217C2×C3×Q8C2×D411yescharacteristic
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—41no—
223C2×C3×D4C2×D411yescharacteristic
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—41no—
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4#4111yescharacteristic
231C2×#67—41no—
232#235—82no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×C4×D3—41no—
237C2×Dic6—21no—
238C3×Q16—42no—
239C3×SD16—42no—
240C3×SD16—42no—
241C3×D8—42no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—42no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2×D411yescharacteristic
250C3×C4⋊C4C2×D411yescharacteristic
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4—21no—
254C3×C22⋊C4—41no—
255C3×C22⋊C4—41no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42C2×D411yescharacteristic
260C3×C42C2×D411yescharacteristic
261C3×C42#4111yescharacteristic
262C3×C42C2411yescharacteristic
263#215—42no—
264#213—42no—
265#213—42no—
266#213—42no—
267(C2×D3)⋊C4—41no—
268(C2×D3)⋊C4—41no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271#208—22no—
272#208—41no—
273#208—42no—
274#208—42no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C2×D411yescharacteristic
280C2×C3⋊C8C2×D411yescharacteristic
281C2×C3⋊C8C2×D411yescharacteristic
282C2×C3⋊C8C2×D411yescharacteristic
283C2×#41—21no—
284C2×#41—61no—
285C2×#41#6711yescharacteristic
286#135—44no—
287C2×Q16—21no—
288C2×SD16—21no—
289C2×SD16—21no—
290C2×SD16—61no—
291C2×D8—21no—
292C2×D8—61no—
293C2×D8—61no—
294C2×D8—61no—
295C2×M16—31no—
296C2×M16—31no—
297C2×M16—61no—
298C2×M16—61no—
299C22×C8—21no—
300C4⋊Q8—31no—
301C4⋊Q8C22×D311yescharacteristic
302#127—31no—
303#127C22×D311yescharacteristic
304#124—21no—
305#124—61no—
306#121—21no—
307#121—21no—
308#121—61no—
309#121—61no—
310C4×Q8#6711yescharacteristic
311C4×Q8C22×D311yescharacteristic
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—61no—
317C4×D4—61no—
318C4×D4#6711yescharacteristic
319C4×D4C22×D311yescharacteristic
320#117—21no—
321#117#6711yescharacteristic
322C2×C42—21no—
323C2×C42C22×D311yescharacteristic
324#107—61no—
325#106—61no—
326#106—61no—
327#106—61no—
328C4⋊C8—21no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—31no—
333C4⋊C8—61no—
334C4⋊C8—61no—
335C4⋊C8C22×D311yescharacteristic
336C4⋊C8C22×D311yescharacteristic
337#103—21no—
338#103—21no—
339#103—61no—
340#102—21no—
341#102—21no—
342#102—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348#102—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352C22⋊C8—61no—
353#97—31no—
354#97—31no—
355#97—61no—
356C4×C8—31no—
357C4×C8—31no—
358C4×C8—61no—
359C4×C8—61no—
360C4×C8—61no—
361C23×C3—21no—
362C23×C3—41no—
363C23×C3#12111yescharacteristic
364C22×D3—41no—
365C3×Q8—21no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8AGL(1,ℤ/8)12yes—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—42no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—81no—
377C3×D4—82no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—42no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—81no—
391C2×C3×C4—82no—
392C2×C3×C4#12111yescharacteristic
393C2×C3×C4#12111yescharacteristic
394C2×C3×C4#12111yescharacteristic
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398#67—82no—
399C2×Dic3—41no—
400D12—41no—
401D12—41no—
402D12—41no—
403D12—41no—
404D12—81no—
405C4×D3—82no—
406Dic6—81no—
407C3×C8—41no—
408C3×C8—41no—
409C3⋊C8—21no—
410C3⋊C8—21no—
411C3⋊C8—21no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8#13512yescomplemented
417#41—22no—
418#41—41no—
419#41—42no—
420#41—42no—
421#41—242no—
422C2×Q8—21no—
423C2×Q8—61no—
424C2×Q8C2×#6711yescharacteristic
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—41no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—121no—
434C2×D4C2×#6711yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—41no—
439C22×C4—121no—
440C22×C4D3×D411yescharacteristic
441C22×C4#23711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443Q16—42no—
444SD16—42no—
445SD16—42no—
446SD16—122no—
447D8—42no—
448D8—122no—
449D8—122no—
450D8—122no—
451M16—62no—
452M16—122no—
453M16—121no—
454M16—122no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—31no—
458C2×C8—31no—
459C2×C8—31no—
460C2×C8—31no—
461C2×C8—42no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4C2×#6711yescharacteristic
473C4⋊C4C2×#6711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—121no—
480C22⋊C4—121no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42D3×D411yescharacteristic
485C42#23711yescharacteristic
486C42C2×#6711yescharacteristic
487C42C23×D311yescharacteristic
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3C2×#12111yescharacteristic
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—22no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—81no—
509C3×C4C2×#13511yescharacteristic
510C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
511C3×C4#57511yescharacteristic
512C3×C4#57511yescharacteristic
513Dic3—81no—
514C23—21no—
515C23—41no—
516C23—121no—
517C23#93911yescharacteristic
518Q8—21no—
519Q8—41no—
520Q8—41no—
521Q8—241no—
522Q8#93212yes—
523D4—21no—
524D4—21no—
525D4—42no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—81no—
531D4—82no—
532D4—121no—
533D4—121no—
534D4—121no—
535D4—121no—
536D4—241no—
537D4—242no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—121no—
553C2×C4—242no—
554C2×C4#93911yescharacteristic
555C2×C4#93911yescharacteristic
556C2×C4#93911yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4C22×#6711yescharacteristic
560C8—32no—
561C8—41no—
562C8—41no—
563C8—61no—
564C8—61no—
565C8—61no—
566C8—121no—
567C8—121no—
568C8—121no—
569C8—121no—
570C2×C3—41no—
571C2×C3—81no—
572C2×C3—161no—
573C2×C3#303511yescharacteristic
574C2×C3#305511yescharacteristic
575C2×C3#306811yescharacteristic
576D3—161no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—241no—
587C22—241no—
588C22—241no—
589C22C2×#93911yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—241no—
599C4D3×#13511yescharacteristic
600C4#576811yescharacteristic
601C4C2×#93211yescharacteristic
602C4#582211yescharacteristic
603C3#1918411yescharacteristic, complemented, Sylow
604C2—41no—
605C2—81no—
606C2—161no—
607C2—481no—
608C2#8262111yescharacteristic
609C2#8390011yescharacteristic
610C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3035C2
socleC22×C3C2×#121
hypercenter#3035D3
hypocenterC3#19184
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. C3×#3035C2semidirect
2. #83295C2semidirect
3. #82529C2semidirect
4. #82519C2semidirect
5. #71808C2semidirect
6. #71055C2semidirect
7. #71052C2semidirect
8. #71043C2semidirect
9. #71040C2semidirect
10. #71022C2semidirect
11. #71021C2semidirect
12. #71010C2semidirect
13. #71009C2semidirect
14. #70433C2semidirect
15. #70420C2semidirect
70. #3035D3semidirect
416. C3⋊C8#135semidirect
603. C3#19184semidirect

Maximal subgroups

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

subgroupindex
C3×#30352
#832952
#825292
#825192
#718082
#710552
#710522
#710432
#710402
#710222
#710212
#710102
#710092
#704332
#704202
#191843