Finite Group Databasebeta

← #10858725

Subgroups — #10858725

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
1#82564C211yesmaximal, characteristic
2#82561C211yesmaximal, characteristic, complemented
3#79242C211yesmaximal, characteristic, complemented
4#78873C211yesmaximal, characteristic, complemented
5C3×#1859C211yesmaximal, characteristic, complemented
6#70668C211yesmaximal, characteristic, complemented
7#70376C211yesmaximal, characteristic, complemented
8#70366C211yesmaximal, characteristic
9#70303C211yesmaximal, characteristic
10#69587C211yesmaximal, characteristic, complemented
11#69573C211yesmaximal, characteristic, complemented
12#68996C211yesmaximal, characteristic, complemented
13#68981C211yesmaximal, characteristic, complemented
14#68758C211yesmaximal, characteristic, complemented
15#68743C211yesmaximal, characteristic, complemented
16#26407—31nomaximal, Sylow
17C3×#520C2211yescharacteristic
18#5827C2211yescharacteristic
19C2×#949C2211yescharacteristic
20C2×#948C2211yescharacteristic
21#5521C2211yescharacteristic
22#5500C2211yescharacteristic
23C3×#428C2211yescharacteristic
24C2×C3×#104C2211yescharacteristic
25C3×#417C2211yescharacteristic
26C3×#384C2211yescharacteristic
27#5233C2211yescharacteristic
28#5131C2211yescharacteristic
29C2×#823C2211yescharacteristic
30C2×#822C2211yescharacteristic
31#5121C2211yescharacteristic
32#5119C2211yescharacteristic
33#5119C2211yescharacteristic
34#5026—21no—
35C4×#215—21no—
36#5014—21no—
37C4×#213—21no—
38#4998—21no—
39#4998C2211yescharacteristic
40#4997C2211yescharacteristic
41#4995—21no—
42#4995C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45#4964—21no—
46#4964C2211yescharacteristic
47#4964C2211yescharacteristic
48C2×#808—21no—
49C2×#808C2211yescharacteristic
50C2×#808C2211yescharacteristic
51#4939C2211yescharacteristic
52#4891—21no—
53#4890—21no—
54#4871—21no—
55#4870—21no—
56#4796—21no—
57#4791—21no—
58#4766—21no—
59#4761—21no—
60C3×#343C2211yescharacteristic
61C3×#339C2211yescharacteristic
62#4558C2211yescharacteristic
63#4554C2211yescharacteristic
64#4530C2211yescharacteristic
65#4524C2211yescharacteristic
66#4523C2211yescharacteristic
67#4523C2211yescharacteristic
68#3084—31no—
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#1859D311yescharacteristic, complemented
82#1856—31no—
83C2×C3×#41C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C3×C4×Q8—21no—
88C3×C4×D4—21no—
89C3×C4×D4—41no—
90C3×#117—21no—
91C3×#117C2311yescharacteristic
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—41no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#949—22no—
98#949Q814yescomplemented
99#948—22no—
100#948—22no—
101Q8×Dic3—21no—
102C2×#215—21no—
103D4×Dic3—21no—
104D4×Dic3—41no—
105C2×#213—21no—
106C2×(C2×D3)⋊C4—41no—
107#926—21no—
108#926C2311yescharacteristic
109#926C2311yescharacteristic
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112#924—41no—
113C2×#208—21no—
114C2×#208C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C22×C3⋊C8—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#893—21no—
120#893—21no—
121#890—21no—
122#890—21no—
123#888—21no—
124#888—21no—
125D3×C22⋊C4—41no—
126C4×D12—21no—
127C4×D12—21no—
128C3×#107—21no—
129C3×#106—21no—
130C3×#104—22no—
131C3×#103—21no—
132C3×#102—21no—
133#835D411yescharacteristic
134#835D411yescharacteristic
135#832D411yescharacteristic
136#832D411yescharacteristic
137#823—22no—
138#822—22no—
139#821—21no—
140#821—21no—
141#818—21no—
142#817—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8—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#808—21no—
158#808D411yescharacteristic
159#808D411yescharacteristic
160#807—21no—
161#807—21no—
162#807—21no—
163#807D411yescharacteristic
164#807D411yescharacteristic
165#803—21no—
166C4×C3⋊C8—21no—
167C2×AGL(1,ℤ/8)—31no—
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520C2×D311yescharacteristic
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×#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—31no—
197C2×#106—61no—
198#424—31no—
199#424—31no—
200C2×#104C2×D311yescharacteristic
201#418—31no—
202#417C2×D311yescharacteristic
203#416—31no—
204C2×#102—31no—
205C2×#102—31no—
206C2×#100—31no—
207#405—31no—
208#390—31no—
209#384C2×D311yescharacteristic
210#343—31no—
211#343—31no—
212#343C2×D311yescharacteristic
213#341—31no—
214#339—31no—
215#339—31no—
216#339C2×D311yescharacteristic
217#336—31no—
218C23×D3—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41C2×Q812yes—
222C3×#41#4112yes—
223C2×C3×Q8C2×D411yescharacteristic
224C2×C3×D4—21no—
225C2×C3×D4C2×D411yescharacteristic
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—41no—
231C22×C3×C4C2411yescharacteristic
232C22×Dic3—21no—
233C22×Dic3—41no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12C2×Q812yes—
240C2×C4×D3—41no—
241C2×C4×D3—41no—
242C2×Dic6—41no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—82no—
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C22⋊C4—21no—
258C3×C22⋊C4—41no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—41no—
263#217—21no—
264#217—21no—
265#217—41no—
266#215—24no—
267#213—24no—
268(C2×D3)⋊C4—41no—
269(C2×D3)⋊C4—41no—
270(C2×D3)⋊C4—82no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—41no—
274C4⋊Dic3—41no—
275C4⋊Dic3C2×D411yescharacteristic
276C4⋊Dic3C2×D411yescharacteristic
277C4⋊Dic3C2×D411yescharacteristic
278C4⋊Dic3C2×D411yescharacteristic
279Dic3⋊C4—41no—
280Dic3⋊C4—41no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—41no—
284#208—24no—
285#208—21no—
286#208—21no—
287#208C2×Q812yes—
288#208#4112yes—
289C2×C3⋊C8—24no—
290C2×C3⋊C8—21no—
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
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—61no—
311C2×M16C22×D311yescharacteristic
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8—31no—
315C22×C8—61no—
316C4⋊Q8—61no—
317C4⋊Q8—61no—
318#125—61no—
319#125—61no—
320C22⋊Q8—121no—
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—31no—
334#117—61no—
335#117C22×D311yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—41no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—121no—
343C2×C22⋊C4—121no—
344C2×C42C22×D311yescharacteristic
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—31no—
355#106—31no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106—61no—
363#106—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366C4⋊C8—61no—
367C4⋊C8—61no—
368#104—22no—
369#103—21no—
370#103—31no—
371#103—31no—
372#102—21no—
373#102—31no—
374#102—31no—
375#102—61no—
376#102—61no—
377#102—61no—
378#102—61no—
379#102—61no—
380#102—61no—
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×Q8C22⋊Q812yes—
393C3×D4—22no—
394C3×D4—22no—
395C3×D4—21no—
396C3×D4C22⋊Q812yes—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
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—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—41no—
421C2×Dic3—81no—
422C2×Dic3—161no—
423D12—22no—
424D12—22no—
425D12—41no—
426D12—41no—
427C4×D3—161no—
428C4×D3—161no—
429Dic6—161no—
430C3×C8—41no—
431C3×C8—41no—
432C3⋊C8—22no—
433C3⋊C8—21no—
434C3⋊C8—22no—
435C3⋊C8—21no—
436C3⋊C8—21no—
437C3⋊C8—21no—
438C24—61no—
439#41—21no—
440#41—21no—
441#41#23512yes—
442#41D3×Q812yes—
443C2×Q8—121no—
444C2×Q8D3×D411yescharacteristic
445C2×D4—21no—
446C2×D4—32no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4D3×D411yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—61no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4C23×D311yescharacteristic
463SD16—64no—
464D8—64no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—32no—
470M16—32no—
471M16—64no—
472M16—61no—
473M16—61no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—31no—
477C2×C8—31no—
478C2×C8—31no—
479C2×C8—31no—
480C2×C8—64no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—31no—
495C4⋊C4—41no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—82no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4—121no—
502C4⋊C4—121no—
503C4⋊C4D3×D411yescharacteristic
504C4⋊C4D3×D411yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—61no—
508C22⋊C4—61no—
509C22⋊C4—121no—
510C22⋊C4—121no—
511C22⋊C4—121no—
512C22⋊C4—242no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—41no—
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—42no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—81no—
538C3×C4—81no—
539C3×C4C2×C22⋊Q811yescharacteristic
540C3×C4#52511yescharacteristic
541C3×C4#54711yescharacteristic
542C3×C4D4×Q811yescharacteristic
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548C23—21no—
549C23—62no—
550C23—121no—
551C23—121no—
552C23—241no—
553C23—241no—
554C23#100411yescharacteristic
555Q8—21no—
556Q8—481no—
557Q8#89612yes—
558D4—22no—
559D4—22no—
560D4—21no—
561D4—62no—
562D4—62no—
563D4—121no—
564D4—121no—
565D4#89612yes—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—121no—
583C2×C4—161no—
584C2×C4—241no—
585C2×C4—481no—
586C2×C4—481no—
587C2×C4—481no—
588C2×C4C2×#23511yescharacteristic
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4C2×D3×Q811yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4#101011yescharacteristic
594C8—41no—
595C8—41no—
596C8—61no—
597C8—61no—
598C8—62no—
599C8—61no—
600C8—61no—
601C8—62no—
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#556411yescharacteristic
620C22#559011yescharacteristic
621C22#563611yescharacteristic
622C4—22no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—121no—
629C4—121no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4C2×#89611yescharacteristic
634C4#554711yescharacteristic
635C4#563211yescharacteristic
636C4#565511yescharacteristic
637C3#2640711yescharacteristic, complemented, Sylow
638C2—21no—
639C2—21no—
640C2—42no—
641C2—242no—
642C2#7442111yescharacteristic
643C2#8028011yescharacteristic
644C2#8030211yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1859C2
socleC22×C3D42
prime radicalC2×#949C22
hypercenter#1859D3
hypocenterC3#26407
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1859

Complements

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

NH ≅ G/Ntype
2. #82561C2semidirect
3. #79242C2semidirect
4. #78873C2semidirect
5. C3×#1859C2semidirect
6. #70668C2semidirect
7. #70376C2semidirect
10. #69587C2semidirect
11. #69573C2semidirect
12. #68996C2semidirect
13. #68981C2semidirect
14. #68758C2semidirect
15. #68743C2semidirect
81. #1859D3semidirect
98. #949Q8semidirect
637. C3#26407semidirect

Maximal subgroups

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

subgroupindex
#825642
#825612
#792422
#788732
C3×#18592
#706682
#703762
#703662
#703032
#695872
#695732
#689962
#689812
#687582
#687432
#264073