Finite Group Databasebeta

← #10836639

Subgroups — #10836639

Order 768.

706 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
1C2×#4890C211yesmaximal, characteristic, complemented
2#78870C211yesmaximal, characteristic, complemented
3#78866C211yesmaximal, characteristic
4#78650C211yesmaximal, characteristic, complemented
5C3×#2078C211yesmaximal, characteristic, complemented
6#70625C211yesmaximal, characteristic, complemented
7#70307C211yesmaximal, characteristic
8#70301C211yesmaximal, characteristic, complemented
9#70003C211yesmaximal, characteristic
10#68994C211yesmaximal, characteristic, complemented
11#68985C211yesmaximal, characteristic, complemented
12#68983C211yesmaximal, characteristic, complemented
13#68974C211yesmaximal, characteristic, complemented
14#68931C211yesmaximal, characteristic, complemented
15#68920C211yesmaximal, characteristic, complemented
16#24720—31nomaximal, Sylow
17Dic3×#41C2211yescharacteristic
18C2×#949C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#888C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C3×#417C2211yescharacteristic
23C3×#417C2211yescharacteristic
24C3×#408C2211yescharacteristic
25#5234C2211yescharacteristic
26#5232C2211yescharacteristic
27#5131C2211yescharacteristic
28C2×#823C2211yescharacteristic
29C2×#823C2211yescharacteristic
30C2×#822C2211yescharacteristic
31#5122C2211yescharacteristic
32#5121C2211yescharacteristic
33#5118C2211yescharacteristic
34#5118C2211yescharacteristic
35Dic3×M16C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#819C2211yescharacteristic
38#5101C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×#809C2211yescharacteristic
41#4964C2211yescharacteristic
42#4964C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4890—21no—
45#4890—21no—
46#4890—21no—
47#4890—21no—
48#4817—21no—
49#4793—21no—
50#4791—21no—
51#4788—21no—
52#4785—21no—
53#4761—21no—
54#4759—21no—
55#4756—21no—
56#4720—21no—
57#4720—21no—
58#4719—21no—
59#4718—21no—
60C3×#343C2211yescharacteristic
61C3×#343C2211yescharacteristic
62C3×#341C2211yescharacteristic
63#4558C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4530C2211yescharacteristic
67#4530C2211yescharacteristic
68#3079—31no—
69C2×#474—31no—
70C4×AGL(1,ℤ/8)—31no—
71#2925—31no—
72#2078—31no—
73#2078D311yescharacteristic, complemented
74#2051—31no—
75#2051—31no—
76#1997—31no—
77#1952—31no—
78#1952—31no—
79#1950—31no—
80#1931—31no—
81#1931—31no—
82#1800—31no—
83C2×C3×#41C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×Dic6—21no—
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93#949—22no—
94#949—22no—
95Q8×Dic3D411yescharacteristic
96Q8×Dic3D411yescharacteristic
97C2×#215—21no—
98D4×Dic3—21no—
99D4×Dic3—21no—
100D4×Dic3D411yescharacteristic
101D4×Dic3D411yescharacteristic
102C2×#213—21no—
103C2×(C2×D3)⋊C4—21no—
104#926—21no—
105#926C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#893—21no—
115#893—21no—
116#893—21no—
117#893—21no—
118#888—21no—
119#888—21no—
120#888—21no—
121#888—21no—
122C3×#107—21no—
123C3×#107—21no—
124C3×#103—21no—
125C3×#103—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C22⋊C8—21no—
130#835—21no—
131#832—21no—
132#830—21no—
133#830—21no—
134#823—22no—
135#823—22no—
136#822—22no—
137#821—21no—
138#821—21no—
139#821—21no—
140#821—21no—
141#819—22no—
142#818—21no—
143#817—21no—
144#815—21no—
145#815—21no—
146#815—21no—
147Dic3⋊C8—21no—
148Dic3⋊C8—21no—
149Dic3⋊C8—21no—
150Dic3⋊C8—21no—
151Dic3⋊C8D411yescharacteristic
152Dic3⋊C8D411yescharacteristic
153Dic3⋊C8D411yescharacteristic
154Dic3⋊C8D411yescharacteristic
155C8×Dic3—21no—
156C8×Dic3—21no—
157C8×Dic3—21no—
158#809—21no—
159#809—21no—
160#808—21no—
161#808—21no—
162#807—21no—
163#807—21no—
164#807—21no—
165#807—21no—
166C2×AGL(1,ℤ/8)—31no—
167C2×C4⋊Q8—31no—
168C4×#41—31no—
169C2×C4×D4—31no—
170#477—61no—
171#476—61no—
172#475—61no—
173#474—61no—
174#474—61no—
175#474—61no—
176#474—61no—
177#474—61no—
178#442—61no—
179#440—61no—
180C4×SD16—61no—
181C4×D8—61no—
182#433—61no—
183#433—61no—
184#432—61no—
185#431—61no—
186C2×#108—31no—
187C2×#108—31no—
188#428—31no—
189#428—31no—
190#428—31no—
191C2×#107—31no—
192C2×#107C2×D311yescharacteristic
193#423—31no—
194#423—31no—
195C2×C4⋊C8—31no—
196#418—31no—
197#417—31no—
198#417C2×D311yescharacteristic
199#417C2×D311yescharacteristic
200C2×#102—31no—
201C2×#102—31no—
202C2×#100—31no—
203C2×#100—31no—
204#408—31no—
205#408C2×D311yescharacteristic
206#405—31no—
207#405—31no—
208C4×M16—31no—
209#343—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#343C2×D311yescharacteristic
214#343C2×D311yescharacteristic
215#341—31no—
216#341C2×D311yescharacteristic
217C23×D3—21no—
218C3×#41—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41—21no—
222C2×C3×Q8C2×D411yescharacteristic
223C2×C3×D4—21no—
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4C2411yescharacteristic
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3#4111yescharacteristic
233C22×Dic3#4111yescharacteristic
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12#4112yes—
240C2×C4×D3—41no—
241C2×C4×D3—41no—
242C2×C4×D3—82no—
243C2×Dic6—41no—
244C2×Dic6—41no—
245C2×Dic6—82no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—21no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8C2×D411yescharacteristic
258C2×C3×C8C2×D411yescharacteristic
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263#217—21no—
264#217—21no—
265#217—21no—
266#215—24no—
267#213—24no—
268(C2×D3)⋊C4—41no—
269(C2×D3)⋊C4—41no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3C2×D411yescharacteristic
275C4⋊Dic3C2×D411yescharacteristic
276Dic3⋊C4—41no—
277Dic3⋊C4—41no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3C2×D411yescharacteristic
282C4×Dic3C2×D411yescharacteristic
283C4×Dic3C2×D411yescharacteristic
284C4×Dic3C2×D411yescharacteristic
285#208—21no—
286#208—21no—
287#208—21no—
288#208—21no—
289#208—21no—
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
298C2×#41C22×D311yescharacteristic
299C22×Q8—61no—
300C22×D4—31no—
301C23×C4—61no—
302AGL(1,ℤ/8)—62no—
303AGL(1,ℤ/8)—62no—
304C2×SD16—61no—
305C2×D8—61no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C22×C8—31no—
313C22×C8C22×D311yescharacteristic
314C4⋊Q8—61no—
315C4⋊Q8—61no—
316C4⋊Q8—61no—
317C4⋊Q8—61no—
318C4×Q8—31no—
319C4×Q8—31no—
320C4×D4—31no—
321C4×D4—31no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327C4×D4—61no—
328#117—31no—
329#117—61no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C4⋊C4C22×D311yescharacteristic
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—61no—
337#108—62no—
338#108—62no—
339#107—21no—
340#107—21no—
341#107—61no—
342#107—61no—
343#107—61no—
344#107—61no—
345#107—61no—
346#106—61no—
347#106—61no—
348#106—61no—
349C4⋊C8—31no—
350C4⋊C8—31no—
351C4⋊C8—31no—
352C4⋊C8—31no—
353C4⋊C8—61no—
354C4⋊C8—61no—
355C4⋊C8—61no—
356C4⋊C8—61no—
357#103—21no—
358#103—21no—
359#103—61no—
360#102—21no—
361#102—21no—
362#102—61no—
363#102—61no—
364#102—61no—
365#102—61no—
366#102—61no—
367#102—61no—
368#102—61no—
369#100—62no—
370#100—62no—
371C22⋊C8—21no—
372C22⋊C8—21no—
373C22⋊C8—61no—
374C22⋊C8—61no—
375#97—61no—
376#97—61no—
377#97—61no—
378C4×C8—61no—
379C4×C8—61no—
380C4×C8—61no—
381C23×C3—21no—
382C23×C3C2×#4111yescharacteristic
383C22×D3—22no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3—81no—
387C22×D3—81no—
388C3×Q8—21no—
389C3×Q8—21no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—41no—
393C3×D4—41no—
394C2×C3×C4—21no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—81no—
400C2×C3×C4—81no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C42−1+411yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—42no—
412C2×Dic3—41no—
413C2×Dic3—81no—
414C2×Dic3—81no—
415C2×Dic3#12111yescharacteristic
416C2×Dic3#12111yescharacteristic
417C2×Dic3#12111yescharacteristic
418C2×Dic3#12111yescharacteristic
419D12—22no—
420D12—22no—
421D12—41no—
422D12—41no—
423C4×D3—82no—
424C4×D3—82no—
425Dic6—82no—
426Dic6—82no—
427C3×C8—21no—
428C3×C8—21no—
429C3×C8—41no—
430C3×C8—41no—
431C3×C8—41no—
432C3⋊C8—22no—
433C3⋊C8—21no—
434C3⋊C8—21no—
435C3⋊C8—41no—
436C3⋊C8—41no—
437C24—61no—
438#41—21no—
439#41—21no—
440#41—21no—
441#41—21no—
442C2×Q8—121no—
443C2×Q8—121no—
444C2×Q8—242no—
445C2×Q8D3×D411yescharacteristic
446C2×D4—21no—
447C2×D4—32no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4D3×D411yescharacteristic
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—121no—
463C22×C4—121no—
464C22×C4—242no—
465C22×C4C23×D311yescharacteristic
466SD16—64no—
467D8—64no—
468M16—21no—
469M16—21no—
470M16—21no—
471M16—21no—
472M16—21no—
473M16—21no—
474M16—32no—
475M16—61no—
476M16—61no—
477M16—61no—
478M16—61no—
479M16—61no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—31no—
486C2×C8—31no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8D3×D411yescharacteristic
493C2×C8D3×D411yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—31no—
499C4⋊C4—31no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—121no—
505C4⋊C4—121no—
506C22⋊C4—61no—
507C22⋊C4—61no—
508C22⋊C4—61no—
509C22⋊C4—121no—
510C22⋊C4—121no—
511C42—31no—
512C42—31no—
513C42—31no—
514C42—31no—
515C42—61no—
516C42—61no—
517C42—61no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3C2×#12111yescharacteristic
524C22×C3#55211yescharacteristic
525C22×C3#55611yescharacteristic
526C2×D3—42no—
527C2×D3—82no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—82no—
531C3×C4—41no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4#52511yescharacteristic
535C3×C4#53611yescharacteristic
536C3×C4#54711yescharacteristic
537C3×C4#54811yescharacteristic
538Dic3—21no—
539Dic3—21no—
540Dic3—21no—
541Dic3—41no—
542Dic3—41no—
543Dic3—81no—
544Dic3#47212yes—
545C23—21no—
546C23—62no—
547C23—121no—
548C23—121no—
549C23—241no—
550C23—241no—
551C23D3×#4111yescharacteristic
552Q8—21no—
553Q8—21no—
554Q8—242no—
555Q8—242no—
556D4—21no—
557D4—21no—
558D4—41no—
559D4—41no—
560D4—62no—
561D4—62no—
562D4—121no—
563D4—121no—
564C2×C4—21no—
565C2×C4—31no—
566C2×C4—31no—
567C2×C4—31no—
568C2×C4—31no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—62no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—122no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—242no—
582C2×C4—242no—
583C2×C4—241no—
584C2×C4—241no—
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×D3×D411yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4#101011yescharacteristic
590C2×C4#101011yescharacteristic
591C8—21no—
592C8—21no—
593C8—41no—
594C8—41no—
595C8—41no—
596C8—61no—
597C8—61no—
598C8—62no—
599C8—121no—
600C8—121no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—81no—
604C2×C3#241511yescharacteristic
605C2×C3C2×#47211yescharacteristic
606C2×C3#306911yescharacteristic
607D3—82no—
608C22—21no—
609C22—21no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—122no—
614C22—242no—
615C22—242no—
616C22—241no—
617C22—241no—
618C22D3×#12111yescharacteristic
619C22#570211yescharacteristic
620C22#570811yescharacteristic
621C4—32no—
622C4—41no—
623C4—61no—
624C4—61no—
625C4—61no—
626C4—81no—
627C4—81no—
628C4—121no—
629C4—121no—
630C4—241no—
631C4#564811yescharacteristic
632C4#565511yescharacteristic
633C4#565611yescharacteristic
634C4#567711yescharacteristic
635C3#2472011yescharacteristic, complemented, Sylow
636C2—21no—
637C2—21no—
638C2—81no—
639C2—242no—
640C2#7746211yescharacteristic
641C2D3×#47211yescharacteristic
642C2#8094411yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2078C2
socleC22×C3C2×#121
prime radicalC2×#949C22
hypercenter#2078D3
hypocenterC3#24720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2078

Complements

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

NH ≅ G/Ntype
1. C2×#4890C2semidirect
2. #78870C2semidirect
4. #78650C2semidirect
5. C3×#2078C2semidirect
6. #70625C2semidirect
8. #70301C2semidirect
10. #68994C2semidirect
11. #68985C2semidirect
12. #68983C2semidirect
13. #68974C2semidirect
14. #68931C2semidirect
15. #68920C2semidirect
73. #2078D3semidirect
635. C3#24720semidirect

Maximal subgroups

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

subgroupindex
C2×#48902
#788702
#788662
#786502
C3×#20782
#706252
#703072
#703012
#700032
#689942
#689852
#689832
#689742
#689312
#689202
#247203