Finite Group Databasebeta

← #10858720

Subgroups — #10858720

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×#3081C211yesmaximal, characteristic, complemented
2#83476C211yesmaximal, characteristic, complemented
3#82537C211yesmaximal, characteristic
4#79178C211yesmaximal, characteristic, complemented
5#71900C211yesmaximal, characteristic, complemented
6#71647C211yesmaximal, characteristic, complemented
7#70680C211yesmaximal, characteristic, complemented
8#70668C211yesmaximal, characteristic, complemented
9#70661C211yesmaximal, characteristic, complemented
10#70545C211yesmaximal, characteristic, complemented
11#70533C211yesmaximal, characteristic, complemented
12#70328C211yesmaximal, characteristic
13#70317C211yesmaximal, characteristic, complemented
14#70218C211yesmaximal, characteristic
15#70184C211yesmaximal, characteristic, complemented
16#26407—31nomaximal, Sylow
17C3×#544C2211yescharacteristic
18C3×#518C2211yescharacteristic
19C2×#932C2211yescharacteristic
20C2×#901C2211yescharacteristic
21#5523C2211yescharacteristic
22#5523C2211yescharacteristic
23C3×#478—21no—
24C3×#478—21no—
25C3×#476—21no—
26C3×#474—21no—
27C3×#428C2211yescharacteristic
28C3×#427C2211yescharacteristic
29C3×#424C2211yescharacteristic
30C3×#418C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32#5221C2211yescharacteristic
33C2×#833C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#825C2211yescharacteristic
36C2×#822C2211yescharacteristic
37C2×#817C2211yescharacteristic
38#5094—21no—
39#5084—21no—
40#5066—21no—
41#5063—21no—
42#5018—21no—
43C4×#214—21no—
44#5014—21no—
45C4×#213—21no—
46#5002C2211yescharacteristic
47#4999C2211yescharacteristic
48#4997C2211yescharacteristic
49#4995C2211yescharacteristic
50#4995C2211yescharacteristic
51#4995C2211yescharacteristic
52#4995C2211yescharacteristic
53#4975C2211yescharacteristic
54#4966C2211yescharacteristic
55C2×#808C2211yescharacteristic
56#4881—21no—
57#4881—21no—
58#4860—21no—
59D3×#106—21no—
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4552C2211yescharacteristic
63#4550C2211yescharacteristic
64#4532C2211yescharacteristic
65#4532C2211yescharacteristic
66#4532C2211yescharacteristic
67#4530C2211yescharacteristic
68#3084—31no—
69#3081D311yescharacteristic, complemented
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—
83C22×C3×D4C2311yescharacteristic
84C2×#235C2311yescharacteristic
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×D4—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×#117C2311yescharacteristic
97C3×#117C2311yescharacteristic
98C3×#117C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—41no—
101C2×#217—41no—
102C2×#214—21no—
103#932—22no—
104#932Q814yescomplemented
105C2×#213—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—41no—
109C2×C4×Dic3C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#901—22no—
115#901—22no—
116C2×ΓL(1,𝔽25)—21no—
117C2×D3×C8—21no—
118#892—21no—
119#892—21no—
120D3×C4⋊C4—21no—
121D3×C4⋊C4—21no—
122#67⋊C4—41no—
123#67⋊C4—41no—
124C4×D12—21no—
125C4×D12—21no—
126C4×Dic6—21no—
127C4×Dic6—21no—
128C3×#107—21no—
129C3×#107D411yescharacteristic
130C3×#107D411yescharacteristic
131C3×#106—21no—
132C3×#106D411yescharacteristic
133C3×#106D411yescharacteristic
134C3×C4⋊C8—21no—
135C3×C4⋊C8—21no—
136C3×#102—21no—
137C3×#102—21no—
138C3×#102—21no—
139C3×#102—21no—
140#833—22no—
141#832—21no—
142#832—21no—
143#825—22no—
144#822—22no—
145#817—21no—
146#817—21no—
147#810—21no—
148#810D411yescharacteristic
149#810D411yescharacteristic
150#809—21no—
151#809D411yescharacteristic
152#809D411yescharacteristic
153#808—21no—
154#808—21no—
155#808—21no—
156#808—21no—
157#808—21no—
158#808—21no—
159#807—21no—
160#807—21no—
161#807—21no—
162#807—21no—
163C4⋊(C3⋊C8)—21no—
164C4⋊(C3⋊C8)—21no—
165#803—21no—
166C4×C3⋊C8—21no—
167C2×AGL(1,ℤ/8)—31no—
168C2×#131—31no—
169#544C2×D311yescharacteristic
170#520—31no—
171#520—31no—
172#518C2×D311yescharacteristic
173#479—61no—
174#478—21no—
175#478—21no—
176#478—61no—
177#476—21no—
178#476—61no—
179#475—61no—
180#474—21no—
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—61no—
194#427C2×D311yescharacteristic
195C2×#106—31no—
196C2×#106—31no—
197C2×#106—61no—
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×#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—
218C24×C3—21no—
219C2×C3×Q8—41no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4C2×Q812yes—
226C22×C3×C4—21no—
227C22×C3×C4—41no—
228C22×C3×C4—41no—
229C22×C3×C4C2411yescharacteristic
230C2×#67—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234#235—21no—
235#235—21no—
236#235C2×Q812yes—
237#235#4112yes—
238C2×D12C2×D411yescharacteristic
239C2×C4×D3—21no—
240C2×C4×D3—41no—
241C2×C4×D3—41no—
242C2×Dic6C2×D411yescharacteristic
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—41no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C22⋊C4—21no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—41no—
265C3×C22⋊C4—41no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269#217—82no—
270#214—24no—
271#213—24no—
272(C2×D3)⋊C4—41no—
273(C2×D3)⋊C4—41no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278Dic3⋊C4—41no—
279Dic3⋊C4—41no—
280Dic3⋊C4—82no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—41no—
284C4×Dic3—41no—
285#208—21no—
286#208—21no—
287#208—21no—
288#208—21no—
289#208C2×Q812yes—
290#208#4112yes—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8C2×D411yescharacteristic
297C2×C3⋊C8C2×D411yescharacteristic
298ΓL(1,𝔽25)—24no—
299D3×C8—24no—
300C2×#41—31no—
301C22×D4C22×D311yescharacteristic
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—61no—
322C4×Q8—61no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—61no—
328C4×D4—61no—
329C4×D4—121no—
330C4×D4—121no—
331#117—61no—
332#117—61no—
333#117C22×D311yescharacteristic
334#117C22×D311yescharacteristic
335#117C22×D311yescharacteristic
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—121no—
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—41no—
343C2×C22⋊C4—121no—
344C2×C42—31no—
345#108—62no—
346#107—21no—
347#107—61no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107#6711yescharacteristic
352#107#6711yescharacteristic
353#106—21no—
354#106—61no—
355#106—61no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106#6711yescharacteristic
363#106#6711yescharacteristic
364C4⋊C8—21no—
365C4⋊C8—21no—
366C4⋊C8—61no—
367C4⋊C8—61no—
368#104—62no—
369#103—31no—
370#103—31no—
371#103—61no—
372#102—21no—
373#102—21no—
374#102—21no—
375#102—21no—
376#102—31no—
377#102—31no—
378#102—61no—
379#102—61no—
380#102—61no—
381#100—62no—
382#97—61no—
383C4×C8—61no—
384C23×C3—22no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—81no—
388C23×C3—81no—
389C23×C3H(5,𝔽2)11yescharacteristic
390C22×D3—21no—
391C3×Q8—161no—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—41no—
395C3×D4—41no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—41no—
403C2×C3×C4—81no—
404C2×C3×C4—161no—
405C2×C3×C4—161no—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×Q811yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412#67—22no—
413#67—22no—
414C2×Dic3—21no—
415C2×Dic3—22no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—81no—
421C2×Dic3—81no—
422D12—21no—
423D12C22⋊Q812yes—
424C4×D3—22no—
425C4×D3—22no—
426C4×D3—161no—
427C4×D3—161no—
428Dic6—21no—
429Dic6C22⋊Q812yes—
430C3×C8—21no—
431C3×C8—21no—
432C3×C8—41no—
433C3⋊C8—22no—
434C3⋊C8—22no—
435C3⋊C8—21no—
436C3⋊C8—21no—
437C3⋊C8—41no—
438C24—21no—
439#41—32no—
440#41—32no—
441#41—61no—
442#41—61no—
443C2×Q8—31no—
444C2×Q8—41no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—31no—
451C2×D4—61no—
452C2×D4D3×Q812yes—
453C22×C4—21no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
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—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—121no—
497C4⋊C4—121no—
498C4⋊C4—242no—
499C4⋊C4D3×D411yescharacteristic
500C4⋊C4D3×D411yescharacteristic
501C4⋊C4C2×#6711yescharacteristic
502C4⋊C4C2×#6711yescharacteristic
503C4⋊C4C2×#6711yescharacteristic
504C4⋊C4C2×#6711yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—121no—
511C22⋊C4—121no—
512C22⋊C4—242no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—61no—
517C42—61no—
518C42—121no—
519C42—121no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—42no—
523C22×C3—82no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—161no—
527C22×C3#53911yescharacteristic
528C22×C3D4211yescharacteristic
529C22×C3#55411yescharacteristic
530C2×D3—22no—
531C2×D3—41no—
532C2×D3—41no—
533C3×C4—41no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—81no—
540C3×C4C2×C22⋊Q811yescharacteristic
541C3×C4#52511yescharacteristic
542C3×C4#54711yescharacteristic
543C3×C4D4×Q811yescharacteristic
544Dic3—22no—
545Dic3—42no—
546Dic3—81no—
547Dic3—81no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—61no—
552C23—81no—
553C23—81no—
554C23#100411yescharacteristic
555Q8—32no—
556Q8—61no—
557Q8—161no—
558D4—22no—
559D4—22no—
560D4—32no—
561D4—41no—
562D4—41no—
563D4—62no—
564D4—62no—
565D4—61no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—81no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—161no—
583C2×C4—161no—
584C2×C4—241no—
585C2×C4—241no—
586C2×C4—481no—
587C2×C4—481no—
588C2×C4C2×D3×D411yescharacteristic
589C2×C4C2×D3×Q811yescharacteristic
590C2×C4#100711yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4C22×#6711yescharacteristic
594C8—21no—
595C8—21no—
596C8—41no—
597C8—62no—
598C8—62no—
599C8—61no—
600C8—61no—
601C8—121no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—82no—
605C2×C3#244911yescharacteristic
606C2×C3#328811yescharacteristic
607C2×C3#329111yescharacteristic
608D3—42no—
609C22—21no—
610C22—21no—
611C22—42no—
612C22—62no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—121no—
617C22—121no—
618C22—161no—
619C22#563611yescharacteristic
620C22#563811yescharacteristic
621C22D4×#6711yescharacteristic
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—62no—
629C4—81no—
630C4—122no—
631C4—241no—
632C4—241no—
633C4D3×C22⋊Q811yescharacteristic
634C4#563111yescharacteristic
635C4Q8×#6711yescharacteristic
636C4#582811yescharacteristic
637C3#2640711yescharacteristic, complemented, Sylow
638C2—21no—
639C2—21no—
640C2—82no—
641C2—122no—
642C2#8281711yescharacteristic
643C2#8400911yescharacteristic
644C2#8401411yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3081C2
socleC22×C3D42
prime radicalC2×#932C22
hypercenter#3081D3
hypocenterC3#26407
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3081

Complements

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

NH ≅ G/Ntype
1. C3×#3081C2semidirect
2. #83476C2semidirect
4. #79178C2semidirect
5. #71900C2semidirect
6. #71647C2semidirect
7. #70680C2semidirect
8. #70668C2semidirect
9. #70661C2semidirect
10. #70545C2semidirect
11. #70533C2semidirect
13. #70317C2semidirect
15. #70184C2semidirect
69. #3081D3semidirect
104. #932Q8semidirect
637. C3#26407semidirect

Maximal subgroups

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

subgroupindex
C3×#30812
#834762
#825372
#791782
#719002
#716472
#706802
#706682
#706612
#705452
#705332
#703282
#703172
#702182
#701842
#264073