Finite Group Databasebeta

← #10858719

Subgroups — #10858719

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3084C211yesmaximal, characteristic, complemented
2#83474C211yesmaximal, characteristic, complemented
3#82561C211yesmaximal, characteristic, complemented
4#78820C211yesmaximal, characteristic
5#71900C211yesmaximal, characteristic, complemented
6#71667C211yesmaximal, characteristic
7#70674C211yesmaximal, characteristic, complemented
8#70671C211yesmaximal, characteristic, complemented
9#70587C211yesmaximal, characteristic, complemented
10#70572C211yesmaximal, characteristic, complemented
11#70542C211yesmaximal, characteristic, complemented
12#70534C211yesmaximal, characteristic, complemented
13#70303C211yesmaximal, characteristic
14#70297C211yesmaximal, characteristic, complemented
15#70292C211yesmaximal, characteristic, complemented
16#26407—31nomaximal, Sylow
17C3×#544C2211yescharacteristic
18C3×#520C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#949C2211yescharacteristic
21C2×#948C2211yescharacteristic
22#5521C2211yescharacteristic
23C3×#479—21no—
24C3×#478—21no—
25C3×#476—21no—
26C3×#475—21no—
27C3×#428C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C3×#424C2211yescharacteristic
30C3×#417C2211yescharacteristic
31C3×#416C2211yescharacteristic
32C2×#837C2211yescharacteristic
33C2×#823C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#819C2211yescharacteristic
36#5101C2211yescharacteristic
37#5087—21no—
38#5084—21no—
39#5066—21no—
40#5066—21no—
41#5001C2211yescharacteristic
42#4999C2211yescharacteristic
43#4998—21no—
44#4998C2211yescharacteristic
45#4995—21no—
46#4995C2211yescharacteristic
47#4995C2211yescharacteristic
48#4988C2211yescharacteristic
49#4975C2211yescharacteristic
50C2×#809C2211yescharacteristic
51#4964—21no—
52#4964C2211yescharacteristic
53#4964C2211yescharacteristic
54C2×#808—21no—
55C2×#808C2211yescharacteristic
56#4817—21no—
57#4788—21no—
58#4785—21no—
59#4756—21no—
60#4558C2211yescharacteristic
61#4555C2211yescharacteristic
62#4554C2211yescharacteristic
63#4550C2211yescharacteristic
64#4532C2211yescharacteristic
65#4532C2211yescharacteristic
66#4530C2211yescharacteristic
67#4530C2211yescharacteristic
68#3084D311yescharacteristic, complemented
69#3081—31no—
70#2946—31no—
71#2912—31no—
72#2094—31no—
73#2090—31no—
74#2063—31no—
75#2060—31no—
76#1951—31no—
77#1922—31no—
78#1907—31no—
79#1883—31no—
80#1866—31no—
81#1859—31no—
82#1856—31no—
83C2×C3×#41C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—21no—
89C3×#125—21no—
90C3×#125—21no—
91C3×C22⋊Q8—41no—
92C3×C4×Q8—21no—
93C3×C4×D4—21no—
94C3×C4×D4—41no—
95C3×#117—21no—
96C3×#117C2311yescharacteristic
97C3×#117C2311yescharacteristic
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101#949—22no—
102#949Q814yescomplemented
103#948—22no—
104#948—22no—
105Q8×Dic3—21no—
106C2×#215—21no—
107D4×Dic3—21no—
108D4×Dic3—41no—
109C2×#213—21no—
110C2×(C2×D3)⋊C4—41no—
111#926—21no—
112#926C2311yescharacteristic
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—41no—
116C2×C4×Dic3C2311yescharacteristic
117C2×#208—21no—
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C2×#208C2311yescharacteristic
121C22×C3⋊C8—21no—
122C22×C3⋊C8C2311yescharacteristic
123#893—21no—
124#893—21no—
125D3×C22⋊C4—41no—
126C4×D12—21no—
127C4×D12—21no—
128C3×#107—21no—
129C3×#106—21no—
130C3×#106—21no—
131C3×#106—21no—
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×#103—21no—
135C3×#103D411yescharacteristic
136C3×#103D411yescharacteristic
137C3×#102—21no—
138C3×#102D411yescharacteristic
139C3×#102D411yescharacteristic
140#837—22no—
141#823—22no—
142#821—21no—
143#821—21no—
144#819—22no—
145#815—21no—
146C8×Dic3—21no—
147#809—21no—
148#809—21no—
149#809—21no—
150#809—21no—
151#808—21no—
152#808—21no—
153#808—21no—
154#808—21no—
155#808—21no—
156#808—21no—
157#808D411yescharacteristic
158#808D411yescharacteristic
159#807—21no—
160#807—21no—
161#807—21no—
162#807—21no—
163#807D411yescharacteristic
164#807D411yescharacteristic
165C4⋊(C3⋊C8)—21no—
166C4⋊(C3⋊C8)—21no—
167C2×AGL(1,ℤ/8)—31no—
168C2×#131—31no—
169#544C2×D311yescharacteristic
170#520—31no—
171#520C2×D311yescharacteristic
172#518—31no—
173#479—21no—
174#478—21no—
175#478—61no—
176#478—61no—
177#476—21no—
178#476—61no—
179#475—21no—
180#474—61no—
181#442—61no—
182#440—61no—
183C4×SD16—61no—
184C4×D8—61no—
185C2×#108—31no—
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×#106—31no—
196C2×#106—61no—
197C2×#106C2×D311yescharacteristic
198#424—31no—
199#424C2×D311yescharacteristic
200C2×#104—31no—
201#418—31no—
202#417C2×D311yescharacteristic
203#416C2×D311yescharacteristic
204C2×#102—31no—
205C2×#102—31no—
206C2×#100—31no—
207#405—31no—
208#390—31no—
209#384—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#341—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#336—31no—
218C23×D3—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41C2×Q812yes—
222C3×#41#4112yes—
223C2×C3×Q8—41no—
224C2×C3×Q8C2×D411yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4C2×D411yescharacteristic
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—41no—
231C22×C3×C4C2411yescharacteristic
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—41no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12C2×Q812yes—
241C2×C4×D3—41no—
242C2×C4×D3—41no—
243C3×M16—21no—
244C3×M16—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C2×D411yescharacteristic
249C2×C3×C8C2×D411yescharacteristic
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—41no—
256C3×C4⋊C4—41no—
257C3×C4⋊C4—41no—
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C22⋊C4—21no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—41no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—41no—
268#217—21no—
269#217—41no—
270#215—24no—
271#213—24no—
272(C2×D3)⋊C4—41no—
273(C2×D3)⋊C4—41no—
274(C2×D3)⋊C4—82no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—41no—
278C4⋊Dic3C2×D411yescharacteristic
279C4⋊Dic3C2×D411yescharacteristic
280Dic3⋊C4—82no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—41no—
285#208—24no—
286#208—21no—
287#208—21no—
288#208—21no—
289#208—21no—
290#208C2×Q812yes—
291#208#4112yes—
292C2×C3⋊C8—24no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8C2×D411yescharacteristic
299C2×C3⋊C8C2×D411yescharacteristic
300C2×#41C22×D311yescharacteristic
301C22×D4—31no—
302AGL(1,ℤ/8)—34no—
303AGL(1,ℤ/8)—62no—
304C2×SD16—61no—
305C2×D8—61no—
306#131—62no—
307#131—62no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16—61no—
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8—61no—
315C22×C8C22×D311yescharacteristic
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318#125—21no—
319#125—21no—
320C22⋊Q8—41no—
321C4×Q8—21no—
322C4×Q8—61no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—61no—
329C4×D4—61no—
330C4×D4—121no—
331#117—21no—
332#117—31no—
333#117—61no—
334#117C22×D311yescharacteristic
335#117C22×D311yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—121no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—121no—
343C2×C22⋊C4—121no—
344C2×C42—31no—
345#108—62no—
346#107—21no—
347#107—31no—
348#107—31no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#106—21no—
354#106—21no—
355#106—21no—
356#106—31no—
357#106—31no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106—61no—
363#106—61no—
364C4⋊C8—21no—
365C4⋊C8—21no—
366C4⋊C8—61no—
367C4⋊C8—61no—
368#104—62no—
369#103—21no—
370#103#6711yescharacteristic
371#103#6711yescharacteristic
372#102—21no—
373#102—61no—
374#102—61no—
375#102—61no—
376#102—61no—
377#102—61no—
378#102—61no—
379#102#6711yescharacteristic
380#102#6711yescharacteristic
381#100—62no—
382#97—61no—
383C4×C8—61no—
384C23×C3—21no—
385C23×C3H(5,𝔽2)11yescharacteristic
386C22×D3—22no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3—81no—
390C22×D3—81no—
391C3×Q8—21no—
392C3×Q8—161no—
393C3×Q8C22⋊Q812yes—
394C3×D4—22no—
395C3×D4—22no—
396C3×D4—21no—
397C3×D4C22⋊Q812yes—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—81no—
408C2×C3×C4—81no—
409C2×C3×C4—161no—
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C22×Q811yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C42−1+411yescharacteristic
416C2×Dic3—22no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—81no—
423C2×Dic3—161no—
424D12—22no—
425D12—22no—
426D12—41no—
427D12—41no—
428C4×D3—161no—
429C4×D3—161no—
430C3×C8—21no—
431C3×C8—21no—
432C3×C8—41no—
433C3⋊C8—22no—
434C3⋊C8—21no—
435C3⋊C8—22no—
436C3⋊C8—21no—
437C3⋊C8—41no—
438C24—61no—
439#41—21no—
440#41—21no—
441#41D3×Q812yes—
442#41#23912yes—
443C2×Q8—41no—
444C2×Q8C2×#6711yescharacteristic
445C2×D4—21no—
446C2×D4—32no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4C2×#6711yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4C23×D311yescharacteristic
463SD16—64no—
464D8—64no—
465M16—21no—
466M16—21no—
467M16—32no—
468M16—32no—
469M16—64no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16—61no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—31no—
478C2×C8—31no—
479C2×C8—64no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8C2×#6711yescharacteristic
486C2×C8C2×#6711yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—121no—
500C4⋊C4—242no—
501C4⋊C4D3×D411yescharacteristic
502C4⋊C4D3×D411yescharacteristic
503C4⋊C4C2×#6711yescharacteristic
504C4⋊C4C2×#6711yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—41no—
508C22⋊C4—61no—
509C22⋊C4—121no—
510C22⋊C4—121no—
511C22⋊C4—121no—
512C22⋊C4—242no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—61no—
517C42—61no—
518C42—61no—
519C42—121no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—22no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3#53911yescharacteristic
526C22×C3D4211yescharacteristic
527C22×C3#55411yescharacteristic
528C2×D3—42no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—82no—
532C2×D3—161no—
533C3×C4—22no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4C2×C22⋊Q811yescharacteristic
541C3×C4#52511yescharacteristic
542C3×C4#54711yescharacteristic
543C3×C4D4×Q811yescharacteristic
544Dic3—42no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548C23—21no—
549C23—62no—
550C23—121no—
551C23—121no—
552C23—241no—
553C23—241no—
554C23#100411yescharacteristic
555Q8—21no—
556Q8—161no—
557Q8#94612yes—
558D4—22no—
559D4—22no—
560D4—21no—
561D4—62no—
562D4—62no—
563D4—121no—
564D4—121no—
565D4#94612yes—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—161no—
584C2×C4—241no—
585C2×C4—481no—
586C2×C4—481no—
587C2×C4—481no—
588C2×C4C2×D3×D411yescharacteristic
589C2×C4C2×D3×Q811yescharacteristic
590C2×C4C2×#23911yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4#101011yescharacteristic
593C2×C4C22×#6711yescharacteristic
594C8—21no—
595C8—21no—
596C8—41no—
597C8—62no—
598C8—61no—
599C8—61no—
600C8—62no—
601C8—121no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—42no—
605C2×C3#244911yescharacteristic
606C2×C3#328811yescharacteristic
607C2×C3#329111yescharacteristic
608D3—82no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—122no—
615C22—242no—
616C22—241no—
617C22—241no—
618C22—481no—
619C22#563611yescharacteristic
620C22#564511yescharacteristic
621C22D4×#6711yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—122no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4#562711yescharacteristic
634C4#563211yescharacteristic
635C4C2×#94611yescharacteristic
636C4#582411yescharacteristic
637C3#2640711yescharacteristic, complemented, Sylow
638C2—21no—
639C2—21no—
640C2—42no—
641C2—242no—
642C2#8276411yescharacteristic
643C2#8401411yescharacteristic
644C2#8401511yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3084C2
socleC22×C3D42
prime radicalC2×#949C22
hypercenter#3084D3
hypocenterC3#26407
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3084

Complements

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

NH ≅ G/Ntype
1. C3×#3084C2semidirect
2. #83474C2semidirect
3. #82561C2semidirect
5. #71900C2semidirect
7. #70674C2semidirect
8. #70671C2semidirect
9. #70587C2semidirect
10. #70572C2semidirect
11. #70542C2semidirect
12. #70534C2semidirect
14. #70297C2semidirect
15. #70292C2semidirect
68. #3084D3semidirect
102. #949Q8semidirect
637. C3#26407semidirect

Maximal subgroups

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

subgroupindex
C3×#30842
#834742
#825612
#788202
#719002
#716672
#706742
#706712
#705872
#705722
#705422
#705342
#703032
#702972
#702922
#264073