Finite Group Databasebeta

← #10778214

Subgroups — #10778214

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
1#81777C211yesmaximal, characteristic, complemented
2#80382C211yesmaximal, characteristic, complemented
3#80361C211yesmaximal, characteristic, complemented
4#79905C211yesmaximal, characteristic, complemented
5C3×#1668C211yesmaximal, characteristic, complemented
6#71283C211yesmaximal, characteristic, complemented
7#71023C211yesmaximal, characteristic, complemented
8#71022C211yesmaximal, characteristic, complemented
9#70943C211yesmaximal, characteristic, complemented
10#68583C211yesmaximal, characteristic, complemented
11#68568C211yesmaximal, characteristic, complemented
12#68401C211yesmaximal, characteristic, complemented
13#68400C211yesmaximal, characteristic, complemented
14#68386C211yesmaximal, characteristic, complemented
15#68385C211yesmaximal, characteristic, complemented
16#19184—31nomaximal, Sylow
17C2×#916—21no—
18#5716C2211yescharacteristic
19#5716C2211yescharacteristic
20C4×#239C2211yescharacteristic
21C3×#493C2211yescharacteristic
22C3×#460C2211yescharacteristic
23C3×#460C2211yescharacteristic
24C3×C8×Q8C2211yescharacteristic
25#5172—21no—
26#5166—21no—
27Dic3×SD16—21no—
28#5159—21no—
29#5159—21no—
30#5152—21no—
31#5152—21no—
32#5072C2211yescharacteristic
33#5072C2211yescharacteristic
34#5070C2211yescharacteristic
35#5070C2211yescharacteristic
36#5031C2211yescharacteristic
37Q8×C3⋊C8C2211yescharacteristic
38#5021C2211yescharacteristic
39#4842C2211yescharacteristic
40#4841C2211yescharacteristic
41#4838—21no—
42#4838C2211yescharacteristic
43#4835—21no—
44#4835C2211yescharacteristic
45#4833C2211yescharacteristic
46D3×C4⋊C8C2211yescharacteristic
47#4787—21no—
48#4779—21no—
49#4772—21no—
50#4756—21no—
51#4688—21no—
52#4688C2211yescharacteristic
53C4×ΓL(1,𝔽25)—21no—
54C4×ΓL(1,𝔽25)C2211yescharacteristic
55#4680C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#331C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4544C2211yescharacteristic
60#4544C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4457C2211yescharacteristic
68#3184—31no—
69#3106—31no—
70#3035—31no—
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#1668D311yescharacteristic, complemented
80#1565—31no—
81#1553—31no—
82#1551—31no—
83C2×#239D411yescharacteristic
84C2×#237—21no—
85C2×#237—21no—
86C2×C3×SD16—21no—
87C2×C3×SD16—21no—
88C2×C3×D8—21no—
89C2×C3×D8—21no—
90C3×#127C2311yescharacteristic
91C3×#127C2311yescharacteristic
92C3×C4×Q8C2311yescharacteristic
93Q8×Dic3D411yescharacteristic
94C2×#216—21no—
95#938—21no—
96#938—21no—
97#938—21no—
98#938—21no—
99#936—21no—
100#936—21no—
101D4×Dic3—21no—
102D4×Dic3—21no—
103D4×Dic3—21no—
104D4×Dic3—21no—
105C2×#213—21no—
106C2×#213—21no—
107#916—44no—
108C2×#204—21no—
109C2×ΓL(1,𝔽25)—21no—
110C2×ΓL(1,𝔽25)—21no—
111C2×ΓL(1,𝔽25)D411yescharacteristic
112C2×ΓL(1,𝔽25)D411yescharacteristic
113C2×D3×C8—21no—
114#893—21no—
115#893D411yescharacteristic
116#892—21no—
117#892D411yescharacteristic
118C4×D12—21no—
119C4×D12C2311yescharacteristic
120C42×D3—21no—
121C42×D3C2311yescharacteristic
122#869C2311yescharacteristic
123#869C2311yescharacteristic
124C3×C4⋊C8—21no—
125C3×C4⋊C8C2311yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×C4⋊C8C2311yescharacteristic
128C3×#102—21no—
129C3×#102—21no—
130C3×C4×C8—21no—
131C3×C4×C8C2311yescharacteristic
132#835—21no—
133#832—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#821—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144#817—21no—
145#817—21no—
146#817—21no—
147#816—21no—
148#815—21no—
149#815D411yescharacteristic
150#815D411yescharacteristic
151Dic3⋊C8—21no—
152C8×Dic3—21no—
153#810—21no—
154#807—21no—
155C4⋊(C3⋊C8)—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8—21no—
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—61no—
162#532—31no—
163#532—31no—
164C4×#41—31no—
165#498—31no—
166#493C2×D311yescharacteristic
167#486—31no—
168#486—31no—
169#482—61no—
170#477—31no—
171#476—31no—
172#469—61no—
173#469—61no—
174#464—61no—
175#463—61no—
176#462—31no—
177#461—31no—
178#460C2×D311yescharacteristic
179#460C2×D311yescharacteristic
180#459—31no—
181#459—31no—
182#449—61no—
183C8×Q8—31no—
184C8×Q8C2×D311yescharacteristic
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⋊C8—31no—
197#416—61no—
198C4×M16—31no—
199C4×M16—61no—
200#335—31no—
201#335C2×D311yescharacteristic
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#331C2×D311yescharacteristic
211#331C2×D311yescharacteristic
212C2×C3×Q8C2×D411yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×#67—41no—
220C2×#67—41no—
221C22×Dic3—41no—
222C22×Dic3—41no—
223#239—22no—
224#239—41no—
225#237—42no—
226#237—42no—
227#237—82no—
228C2×D12—21no—
229C2×D12C2×D411yescharacteristic
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3C2×D411yescharacteristic
234C2×C4×D3C2×D411yescharacteristic
235C2×C4×D3#4111yescharacteristic
236C2×Dic6—21no—
237C2×Dic6—21no—
238C3×SD16—42no—
239C3×SD16—42no—
240C3×D8—42no—
241C3×D8—42no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8C2×D411yescharacteristic
246C2×C3×C8C2×D411yescharacteristic
247C3×C4⋊C4—21no—
248C3×C4⋊C4C2×D411yescharacteristic
249C3×C42—21no—
250C3×C42C2411yescharacteristic
251#217—41no—
252#217—41no—
253#217—41no—
254#217—41no—
255#216—42no—
256#213—42no—
257#213—42no—
258(C2×D3)⋊C4—21no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3C2×D411yescharacteristic
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3C2×D411yescharacteristic
270C4×Dic3C2×D411yescharacteristic
271C4×Dic3#4111yescharacteristic
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277#204—42no—
278ΓL(1,𝔽25)—22no—
279ΓL(1,𝔽25)—41no—
280ΓL(1,𝔽25)—42no—
281ΓL(1,𝔽25)—42no—
282D3×C8—42no—
283C2×#41—31no—
284C2×#41—61no—
285C2×#41—61no—
286#135—124no—
287C2×Q16—61no—
288C2×SD16—21no—
289C2×SD16—21no—
290C2×SD16—61no—
291C2×D8—21no—
292C2×D8—21no—
293C2×D8—61no—
294C2×D8—61no—
295C2×M16—31no—
296C2×M16—31no—
297C2×M16—61no—
298C2×M16—61no—
299C22×C8—61no—
300C4⋊Q8—31no—
301C4⋊Q8—31no—
302#127C22×D311yescharacteristic
303#127C22×D311yescharacteristic
304#124—61no—
305#124—61no—
306#121—61no—
307#121—61no—
308#121—61no—
309#121—61no—
310C4×Q8—31no—
311C4×Q8C22×D311yescharacteristic
312C4×D4—31no—
313C4×D4—31no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317C4×D4—61no—
318C4×D4—61no—
319C4×D4—61no—
320#117—31no—
321#117—61no—
322C2×C42—31no—
323C2×C42—61no—
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—61no—
333C4⋊C8—61no—
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336C4⋊C8C22×D311yescharacteristic
337#103—61no—
338#103—61no—
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—21no—
357C4×C8—31no—
358C4×C8—61no—
359C4×C8—61no—
360C4×C8C22×D311yescharacteristic
361C23×C3—41no—
362C23×C3—41no—
363C22×D3—21no—
364C22×D3#12111yescharacteristic
365C3×Q8—21no—
366C3×Q8AGL(1,ℤ/8)12yes—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—81no—
376C3×D4—81no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382#67—82no—
383#67—82no—
384C2×Dic3—21no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—82no—
388C2×Dic3—82no—
389C2×Dic3#12111yescharacteristic
390C2×Dic3#12111yescharacteristic
391C2×Dic3#12111yescharacteristic
392D12—21no—
393D12—21no—
394D12—42no—
395C4×D3—21no—
396C4×D3—21no—
397C4×D3—21no—
398C4×D3—21no—
399C4×D3—41no—
400C4×D3—42no—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—81no—
404Dic6—41no—
405Dic6—41no—
406Dic6—81no—
407C3×C8—21no—
408C3×C8—41no—
409C3×C8—41no—
410C3×C8—41no—
411C3×C8#13512yescomplemented
412C3⋊C8—21no—
413C3⋊C8—21no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417#41—62no—
418#41—121no—
419#41—122no—
420#41—122no—
421#41—242no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8D3×D411yescharacteristic
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—31no—
432C2×D4—61no—
433C2×D4—121no—
434C2×D4—121no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—121no—
442C22×C4—121no—
443Q16—122no—
444SD16—42no—
445SD16—42no—
446SD16—122no—
447D8—42no—
448D8—42no—
449D8—122no—
450D8—122no—
451M16—62no—
452M16—121no—
453M16—122no—
454M16—122no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—31no—
459C2×C8—31no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—122no—
464C2×C8D3×D411yescharacteristic
465C2×C8D3×D411yescharacteristic
466C4⋊C4—21no—
467C4⋊C4—31no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4D3×D411yescharacteristic
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—61no—
477C22⋊C4—121no—
478C22⋊C4—121no—
479C22⋊C4—121no—
480C22⋊C4—121no—
481C42—21no—
482C42—31no—
483C42—31no—
484C42—31no—
485C42—61no—
486C42—61no—
487C42C23×D311yescharacteristic
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3C2×#12111yescharacteristic
495C2×D3—21no—
496C2×D3—21no—
497C2×D3—21no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—41no—
501C3×C4—22no—
502C3×C4—21no—
503C3×C4—41no—
504C3×C4C2×#13511yescharacteristic
505C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
506C3×C4#57511yescharacteristic
507C3×C4#57511yescharacteristic
508Dic3—21no—
509Dic3—21no—
510Dic3—21no—
511Dic3—41no—
512Dic3—81no—
513Dic3—81no—
514C23—31no—
515C23—41no—
516C23—41no—
517C23—61no—
518Q8—21no—
519Q8—121no—
520Q8—121no—
521Q8—241no—
522Q8#91112yes—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—61no—
532D4—61no—
533D4—81no—
534D4—81no—
535D4—122no—
536D4—242no—
537D4—242no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—31no—
541C2×C4—31no—
542C2×C4—31no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—121no—
549C2×C4—122no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—241no—
555C2×C4—242no—
556C2×C4—242no—
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—61no—
565C8—61no—
566C8—121no—
567C8—121no—
568C8—121no—
569C8#91612yescomplemented
570C2×C3—161no—
571C2×C3—161no—
572C2×C3#303511yescharacteristic
573C2×C3#305511yescharacteristic
574C2×C3#306811yescharacteristic
575D3—41no—
576D3—81no—
577C22—61no—
578C22—61no—
579C22—61no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—121no—
587C22—121no—
588C22—121no—
589C22D3×#12111yescharacteristic
590C4—21no—
591C4—22no—
592C4—41no—
593C4—61no—
594C4—61no—
595C4—61no—
596C4—121no—
597C4—241no—
598C4—241no—
599C4C2×#91111yescharacteristic
600C4C2×#91611yescharacteristic
601C4#576811yescharacteristic
602C4#577311yescharacteristic
603C3#1918411yescharacteristic, complemented, Sylow
604C2—121no—
605C2—161no—
606C2—161no—
607C2—241no—
608C2#8038211yescharacteristic
609C2#8071511yescharacteristic
610C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1668C2
socleC22×C3C2×#121
hypercenter#1668D3
hypocenterC3#19184
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1668

Complements

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

NH ≅ G/Ntype
1. #81777C2semidirect
2. #80382C2semidirect
3. #80361C2semidirect
4. #79905C2semidirect
5. C3×#1668C2semidirect
6. #71283C2semidirect
7. #71023C2semidirect
8. #71022C2semidirect
9. #70943C2semidirect
10. #68583C2semidirect
11. #68568C2semidirect
12. #68401C2semidirect
13. #68400C2semidirect
14. #68386C2semidirect
15. #68385C2semidirect
79. #1668D3semidirect
411. C3×C8#135semidirect
569. C8#916semidirect
603. C3#19184semidirect

Maximal subgroups

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

subgroupindex
#817772
#803822
#803612
#799052
C3×#16682
#712832
#710232
#710222
#709432
#685832
#685682
#684012
#684002
#683862
#683852
#191843