Finite Group Databasebeta

← #10858694

Subgroups — #10858694

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79210C211yesmaximal, characteristic, complemented
2#79092C211yesmaximal, characteristic, complemented
3#78918C211yesmaximal, characteristic, complemented
4#78726C211yesmaximal, characteristic
5C3×#2063C211yesmaximal, characteristic, complemented
6#70722C211yesmaximal, characteristic, complemented
7#70700C211yesmaximal, characteristic, complemented
8#70667C211yesmaximal, characteristic, complemented
9#70534C211yesmaximal, characteristic, complemented
10#70328C211yesmaximal, characteristic
11#70182C211yesmaximal, characteristic, complemented
12#70088C211yesmaximal, characteristic
13#70068C211yesmaximal, characteristic, complemented
14#69003C211yesmaximal, characteristic, complemented
15#68980C211yesmaximal, characteristic, complemented
16#26405—31nomaximal, Sylow
17#5798C2211yescharacteristic
18C2×#932C2211yescharacteristic
19C2×#907C2211yescharacteristic
20#5523C2211yescharacteristic
21#5519C2211yescharacteristic
22#5504C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C3×#424C2211yescharacteristic
26C3×#418C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#100C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#825C2211yescharacteristic
32#5133C2211yescharacteristic
33#5121C2211yescharacteristic
34#5121C2211yescharacteristic
35#5121C2211yescharacteristic
36#5119C2211yescharacteristic
37#5113C2211yescharacteristic
38C2×#817C2211yescharacteristic
39#5101C2211yescharacteristic
40#4998C2211yescharacteristic
41#4977C2211yescharacteristic
42#4964C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4881—21no—
45D3×#107—21no—
46#4861—21no—
47#4860—21no—
48#4851—21no—
49#4847—21no—
50#4842—21no—
51#4829—21no—
52#4785—21no—
53#4766—21no—
54#4763—21no—
55#4761—21no—
56#4759—21no—
57#4758—21no—
58#4757—21no—
59#4756—21no—
60C3×#343C2211yescharacteristic
61#4558C2211yescharacteristic
62#4555C2211yescharacteristic
63#4555C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4550C2211yescharacteristic
67#4532C2211yescharacteristic
68#3084—31no—
69#3081—31no—
70#2946—31no—
71#2913—31no—
72#2092—31no—
73#2089—31no—
74#2063D311yescharacteristic, complemented
75#2061—31no—
76#1951—31no—
77#1922—31no—
78#1907—31no—
79#1883—31no—
80#1866—31no—
81#1860—31no—
82#1856—31no—
83C22×C3×D4C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#117C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×#217—41no—
92C2×#217—41no—
93D4×Dic3—21no—
94D4×Dic3—21no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97C2×#214—21no—
98#932—22no—
99#932Q814yescomplemented
100C2×#213—21no—
101C4×#67—41no—
102#926C2311yescharacteristic
103#926C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#907—22no—
110#907—22no—
111C2×ΓL(1,𝔽25)—21no—
112C2×D3×C8—21no—
113#893—21no—
114#892—21no—
115#892—21no—
116D3×C4⋊C4—21no—
117D3×C4⋊C4—21no—
118#890—21no—
119#888—21no—
120#887—21no—
121#67⋊C4—41no—
122#878—41no—
123C4×D12—21no—
124#870—21no—
125#869—21no—
126C4×Dic6—21no—
127C3×#108—22no—
128C3×#107—21no—
129C3×#106—21no—
130C3×C4⋊C8—21no—
131C3×C4⋊C8—21no—
132C3×#102—21no—
133C3×#102—21no—
134C3×#102—21no—
135C3×#102—21no—
136C3×#100—22no—
137#832—21no—
138#832—21no—
139C3⋊#95—41no—
140#825—22no—
141#821—21no—
142#821D411yescharacteristic
143#821D411yescharacteristic
144#818—21no—
145#818—21no—
146#818—21no—
147#817—21no—
148#817—21no—
149#817—21no—
150#817—21no—
151#817—21no—
152#816—21no—
153#816D411yescharacteristic
154#816D411yescharacteristic
155#815—21no—
156Dic3⋊C8—21no—
157Dic3⋊C8—21no—
158C8×Dic3—21no—
159#808—21no—
160#808D411yescharacteristic
161#808D411yescharacteristic
162#807—21no—
163#807—21no—
164#807—21no—
165#807D411yescharacteristic
166#807D411yescharacteristic
167C2×AGL(1,ℤ/8)—31no—
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520—31no—
172#518—31no—
173#479—61no—
174#478—61no—
175#478—61no—
176#478—61no—
177#476—61no—
178#476—61no—
179#475—61no—
180#474—61no—
181#442—61no—
182#440—61no—
183C4×SD16—61no—
184C4×D8—61no—
185C2×#108C2×D311yescharacteristic
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428—61no—
191#428—61no—
192#428C2×D311yescharacteristic
193#427—31no—
194#427—61no—
195C2×#107—31no—
196C2×#107—61no—
197C2×#106—31no—
198#424—31no—
199#424C2×D311yescharacteristic
200C2×#104—31no—
201#418C2×D311yescharacteristic
202#417—31no—
203#416—31no—
204C2×#102—31no—
205C2×#102C2×D311yescharacteristic
206C2×#100C2×D311yescharacteristic
207#405—31no—
208#390—31no—
209#382—31no—
210#343—31no—
211#343—31no—
212#343C2×D311yescharacteristic
213#341—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#336—31no—
218C24×C3—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4C2×Q812yes—
225C22×C3×C4—21no—
226C22×C3×C4C2411yescharacteristic
227C2×#67—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—41no—
232C22×Dic3—41no—
233#235—21no—
234#235—21no—
235#235C2×Q812yes—
236#235#4112yes—
237C2×D12C2×D411yescharacteristic
238C2×C4×D3—21no—
239C2×C4×D3—41no—
240C2×C4×D3—41no—
241C2×Dic6—41no—
242C2×Dic6C2×D411yescharacteristic
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16#4112yes—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8C2×D411yescharacteristic
255C2×C3×C8C2×D411yescharacteristic
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C22⋊C4—21no—
261C3×C42—21no—
262#217—21no—
263#217—21no—
264#217—41no—
265#217—41no—
266#217—82no—
267#214—24no—
268#213—24no—
269(C2×D3)⋊C4—41no—
270(C2×D3)⋊C4—41no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—41no—
276C4⋊Dic3C2×D411yescharacteristic
277C4⋊Dic3C2×D411yescharacteristic
278C4⋊Dic3C2×D411yescharacteristic
279C4⋊Dic3C2×D411yescharacteristic
280Dic3⋊C4—41no—
281Dic3⋊C4—41no—
282Dic3⋊C4—41no—
283Dic3⋊C4—41no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—41no—
289C4×Dic3—41no—
290#208—21no—
291#208C2×Q812yes—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8C2×D411yescharacteristic
296C2×C3⋊C8C2×D411yescharacteristic
297ΓL(1,𝔽25)—24no—
298D3×C8—24no—
299C2×#41—31no—
300C22×D4C22×D311yescharacteristic
301AGL(1,ℤ/8)—34no—
302AGL(1,ℤ/8)—62no—
303C2×SD16—61no—
304C2×D8—61no—
305#131—62no—
306#131—62no—
307C2×M16—31no—
308C2×M16—61no—
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C22×C8—31no—
313C22×C8—61no—
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—61no—
316C4⋊Q8—61no—
317#125—61no—
318#125—61no—
319C22⋊Q8—121no—
320C4×Q8—61no—
321C4×Q8—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—121no—
329C4×D4—121no—
330#117—31no—
331#117—31no—
332#117—61no—
333#117—61no—
334#117C22×D311yescharacteristic
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—121no—
341C2×C22⋊C4—121no—
342C2×C42—31no—
343#108—22no—
344#107—21no—
345#107—31no—
346#107—31no—
347#107—61no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#106—21no—
354#106—31no—
355#106—31no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366#104—62no—
367#103—31no—
368#103—31no—
369#103—61no—
370#102—21no—
371#102—21no—
372#102—21no—
373#102—21no—
374#102—31no—
375#102—31no—
376#102—61no—
377#102—61no—
378#102—61no—
379#100—22no—
380#97—61no—
381C4×C8—61no—
382#95—121no—
383C23×C3—22no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—81no—
387C23×C3—81no—
388C23×C3H(5,𝔽2)11yescharacteristic
389C22×D3—21no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—41no—
393C3×D4—41no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—81no—
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×Q811yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404#67—22no—
405#67—22no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—81no—
417C2×Dic3—81no—
418C2×Dic3—161no—
419C2×Dic3—161no—
420D12—21no—
421D12C22⋊Q812yes—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—161no—
425C4×D3—161no—
426Dic6—21no—
427Dic6—161no—
428Dic6C22⋊Q812yes—
429C3×C8—22no—
430C3×C8—21no—
431C3×C8—21no—
432C3×C8—41no—
433C3×C8—41no—
434C3⋊C8—22no—
435C3⋊C8—21no—
436C3⋊C8—21no—
437C24—21no—
438#41—32no—
439#41—32no—
440#41—61no—
441#41—61no—
442C2×Q8—31no—
443C2×Q8—121no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—31no—
450C2×D4—61no—
451C2×D4D3×Q812yes—
452C22×C4—21no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4C23×D311yescharacteristic
462SD16—64no—
463D8—64no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—32no—
470M16—64no—
471M16—61no—
472M16#23712yes—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—31no—
479C2×C8—31no—
480C2×C8—64no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8D3×D411yescharacteristic
485C2×C8D3×D411yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—31no—
489C4⋊C4—31no—
490C4⋊C4—31no—
491C4⋊C4—31no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—121no—
497C4⋊C4—121no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4D3×D411yescharacteristic
502C4⋊C4D3×D411yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—61no—
505C22⋊C4—61no—
506C22⋊C4—121no—
507C22⋊C4—121no—
508C22⋊C4—121no—
509C22⋊C4—121no—
510C22⋊C4—242no—
511C42—21no—
512C42—61no—
513C42—61no—
514C42—61no—
515C42—61no—
516C42—121no—
517C42—121no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—42no—
521C22×C3—82no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—161no—
525C22×C3#53911yescharacteristic
526C22×C3D4211yescharacteristic
527C22×C3#55411yescharacteristic
528C2×D3—22no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—41no—
532C3×C4—41no—
533C3×C4—81no—
534C3×C4C2×C22⋊Q811yescharacteristic
535C3×C4#52511yescharacteristic
536C3×C4#54711yescharacteristic
537C3×C4D4×Q811yescharacteristic
538Dic3—22no—
539Dic3—41no—
540Dic3—41no—
541Dic3—41no—
542Dic3—41no—
543Dic3—81no—
544Dic3—81no—
545Dic3—81no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—61no—
550C23—81no—
551C23—81no—
552C23#100911yescharacteristic
553Q8—32no—
554Q8—61no—
555Q8—481no—
556D4—22no—
557D4—22no—
558D4—32no—
559D4—41no—
560D4—41no—
561D4—62no—
562D4—62no—
563D4—61no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—61no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—81no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—241no—
581C2×C4—241no—
582C2×C4—481no—
583C2×C4—481no—
584C2×C4—481no—
585C2×C4—481no—
586C2×C4C2×D3×D411yescharacteristic
587C2×C4C2×D3×D411yescharacteristic
588C2×C4C2×#23711yescharacteristic
589C2×C4C2×D3×Q811yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C2×C4#101011yescharacteristic
592C8—22no—
593C8—21no—
594C8—21no—
595C8—41no—
596C8—41no—
597C8—61no—
598C8—61no—
599C8—62no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—82no—
603C2×C3#244911yescharacteristic
604C2×C3#328811yescharacteristic
605C2×C3#329011yescharacteristic
606D3—42no—
607C22—21no—
608C22—21no—
609C22—42no—
610C22—62no—
611C22—82no—
612C22—81no—
613C22—81no—
614C22—121no—
615C22—121no—
616C22—161no—
617C22#560911yescharacteristic
618C22#563511yescharacteristic
619C22#570711yescharacteristic
620C4—41no—
621C4—41no—
622C4—62no—
623C4—81no—
624C4—121no—
625C4—121no—
626C4—121no—
627C4—121no—
628C4—241no—
629C4—241no—
630C4—241no—
631C4#560611yescharacteristic
632C4#561111yescharacteristic
633C4D3×C22⋊Q811yescharacteristic
634C4#563211yescharacteristic
635C3#2640511yescharacteristic, complemented, Sylow
636C2—21no—
637C2—21no—
638C2—82no—
639C2—122no—
640C2#7741911yescharacteristic
641C2#8093411yescharacteristic
642C2#8093911yescharacteristic
subgroupG/H
centerC22#5609
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2063C2
socleC22×C3D42
prime radicalC2×#932C22
hypercenter#2063D3
hypocenterC3#26405
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2063

Complements

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

NH ≅ G/Ntype
1. #79210C2semidirect
2. #79092C2semidirect
3. #78918C2semidirect
5. C3×#2063C2semidirect
6. #70722C2semidirect
7. #70700C2semidirect
8. #70667C2semidirect
9. #70534C2semidirect
11. #70182C2semidirect
13. #70068C2semidirect
14. #69003C2semidirect
15. #68980C2semidirect
74. #2063D3semidirect
99. #932Q8semidirect
635. C3#26405semidirect

Maximal subgroups

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

subgroupindex
#792102
#790922
#789182
#787262
C3×#20632
#707222
#707002
#706672
#705342
#703282
#701822
#700882
#700682
#690032
#689802
#264053