Finite Group Databasebeta

← #10824894

Subgroups — #10824894

Order 768.

784 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79199C211yesmaximal, characteristic
2#79070C211yesmaximal, characteristic, complemented
3#78946C211yesmaximal, characteristic, complemented
4C2×#4788C211yesmaximal, characteristic, complemented
5C3×#2064C211yesmaximal, characteristic, complemented
6#70714C211yesmaximal, characteristic
7Dic3×#108C211yesmaximal, characteristic, complemented
8#70671C211yesmaximal, characteristic, complemented
9#70542C211yesmaximal, characteristic, complemented
10#70339C211yesmaximal, characteristic, complemented
11#70264C211yesmaximal, characteristic, complemented
12#70117C211yesmaximal, characteristic
13Dic3×#101C211yesmaximal, characteristic, complemented
14#69011C211yesmaximal, characteristic, complemented
15#68972C211yesmaximal, characteristic, complemented
16#23718—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C2×#942C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#908C2211yescharacteristic
21#5522C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C3×#424C2211yescharacteristic
26C3×#419C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#101C2211yescharacteristic
29C2×#836C2211yescharacteristic
30#5133C2211yescharacteristic
31C2×#823C2211yescharacteristic
32C2×#823C2211yescharacteristic
33#5129C2211yescharacteristic
34Dic3×M16C2211yescharacteristic
35Dic3×M16C2211yescharacteristic
36Dic3×M16C2211yescharacteristic
37C2×#821C2211yescharacteristic
38#5110C2211yescharacteristic
39C2×#819C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41#5001C2211yescharacteristic
42#4999C2211yescharacteristic
43#4995C2211yescharacteristic
44#4966C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4889—21no—
47#4872—21no—
48#4869—21no—
49#4852—21no—
50#4840—22no—
51#4833—22no—
52#4817—21no—
53#4790—21no—
54#4789—21no—
55#4788—21no—
56#4788—22no—
57#4788—22no—
58C3×#343C2211yescharacteristic
59#4558C2211yescharacteristic
60#4558C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4530C2211yescharacteristic
64C4×#137—31no—
65C4×AGL(1,ℤ/8)—31no—
66C2×C4×SD16—31no—
67#2922—31no—
68#2090—31no—
69#2079—31no—
70#2064D311yescharacteristic, complemented
71#2063—31no—
72#2052—31no—
73#2051—31no—
74#2046—31no—
75#2019—31no—
76C4×#108—31no—
77C4×#101—31no—
78C4×#100—31no—
79C22×C3×Q8C2311yescharacteristic
80C2×#235C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#117C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89Q8×Dic3—21no—
90Q8×Dic3—21no—
91Q8×Dic3—21no—
92Q8×Dic3—21no—
93C2×#216—21no—
94#942—22no—
95#942—22no—
96C2×#215—22no—
97C2×#215—21no—
98C2×#215—22no—
99C2×#215—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#908—22no—
110#908—22no—
111C2×D24—21no—
112C2×#204—21no—
113#893—21no—
114#893—21no—
115#893—21no—
116#893—21no—
117#893D411yescharacteristic
118#893D411yescharacteristic
119#887D411yescharacteristic
120#887D411yescharacteristic
121#67⋊C4—21no—
122#67⋊C4—21no—
123#872—21no—
124C42×D3—21no—
125C3×#108—22no—
126C3×#107—21no—
127C3×#106—21no—
128C3×C4⋊C8—22no—
129C3×#103—22no—
130C3×#103—21no—
131C3×#103—21no—
132C3×#101—22no—
133#836—22no—
134#823—22no—
135#823—22no—
136#821—21no—
137#821—22no—
138(C2×D3)⋊C8—22no—
139(C2×D3)⋊C8—22no—
140#819—22no—
141#816—21no—
142#815—21no—
143#815—21no—
144#815—21no—
145C8×Dic3—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148C8×Dic3D412yes—
149C8×Dic3D412yes—
150#810—21no—
151#809—21no—
152#809—21no—
153#809—21no—
154#808—21no—
155#808—22no—
156#807—21no—
157C4⋊(C3⋊C8)—22no—
158C2×#137—31no—
159C2×AGL(1,ℤ/8)—31no—
160C22×SD16—31no—
161C4×#41—31no—
162C2×C4×Q8—31no—
163C2×C4×D4—31no—
164#442—61no—
165#441—61no—
166#440—61no—
167#440—61no—
168C4×Q16—61no—
169C4×SD16—62no—
170C4×SD16—62no—
171C4×SD16—61no—
172C4×SD16—61no—
173C4×D8—61no—
174#433—62no—
175#431—62no—
176C2×#108—31no—
177C2×#108C2×D311yescharacteristic
178#428—31no—
179#428C2×D311yescharacteristic
180C2×#106—31no—
181#424—31no—
182#424C2×D311yescharacteristic
183#419C2×D311yescharacteristic
184#418—31no—
185#417—31no—
186#417—31no—
187C2×#103C2×D311yescharacteristic
188C2×#102—31no—
189C2×#101—31no—
190C2×#101C2×D311yescharacteristic
191C2×#100—31no—
192C2×#100—31no—
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16—31no—
197C4×M16—31no—
198C2×C4×C8—31no—
199#343—31no—
200#343—31no—
201#343—31no—
202#343C2×D311yescharacteristic
203#341—31no—
204#341—31no—
205C23×D3—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8#4112yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2411yescharacteristic
215C2×#67—21no—
216C22×Dic3—21no—
217C22×Dic3#4111yescharacteristic
218C22×Dic3#4111yescharacteristic
219#235—21no—
220#235—21no—
221#235—21no—
222#235—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12C2×D411yescharacteristic
229C2×D12#4112yes—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—41no—
234C2×C4×D3—41no—
235C2×C4×D3—82no—
236C2×Dic6C2×D411yescharacteristic
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—21no—
240C3×M16—21no—
241C3×M16—21no—
242C3×M16#4112yes—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C2×D412yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2×D411yescharacteristic
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C22⋊C4—21no—
254C3×C42—21no—
255#216—24no—
256#215—28no—
257#215—24no—
258(C2×D3)⋊C4—21no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—41no—
261(C2×D3)⋊C4—41no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—41no—
265C4⋊Dic3—41no—
266Dic3⋊C4—21no—
267Dic3⋊C4—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—41no—
271C4×Dic3—41no—
272C4×Dic3C2×D411yescharacteristic
273C4×Dic3C2×D411yescharacteristic
274C4×Dic3C2×D411yescharacteristic
275C4×Dic3C2×D411yescharacteristic
276#208—21no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208—21no—
281#208#4112yes—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8C2×D412yes—
288D24—24no—
289#204—24no—
290C2×#41—31no—
291C22×Q8C22×D311yescharacteristic
292C22×D4—31no—
293C23×C4—61no—
294#137—62no—
295#137—62no—
296AGL(1,ℤ/8)—62no—
297AGL(1,ℤ/8)—62no—
298C2×Q16—61no—
299C2×SD16—61no—
300C2×SD16—62no—
301C2×SD16—62no—
302C2×SD16—61no—
303C2×SD16—62no—
304C2×D8—61no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16C22×D311yescharacteristic
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C22×C8—31no—
312C22×C8C22×D311yescharacteristic
313C4×Q8—31no—
314C4×Q8—31no—
315C4×Q8—61no—
316C4×Q8—61no—
317C4×Q8—61no—
318C4×Q8—61no—
319C4×D4—31no—
320C4×D4—31no—
321C4×D4—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327#117—61no—
328#117C22×D311yescharacteristic
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C22⋊C4—61no—
333C2×C42—31no—
334C2×C42—61no—
335C2×C42—61no—
336#108—22no—
337#108—62no—
338#107—21no—
339#107—61no—
340#106—21no—
341#106—62no—
342#106—61no—
343C4⋊C8—22no—
344C4⋊C8—62no—
345#103—22no—
346#103—21no—
347#103—21no—
348#103—61no—
349#103—61no—
350#102—61no—
351#102—61no—
352#102—62no—
353#102—61no—
354#102—61no—
355#101—22no—
356#101—62no—
357#100—62no—
358#100—62no—
359C22⋊C8—62no—
360C22⋊C8—62no—
361#97—61no—
362#97—61no—
363#97—61no—
364C4×C8—32no—
365C4×C8—32no—
366C4×C8—61no—
367C4×C8—61no—
368C4×C8—61no—
369C23×C3C2×#4111yescharacteristic
370C22×D3—21no—
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—81no—
375C22×D3—81no—
376C3×Q8—22no—
377C3×Q8—22no—
378C3×Q8—41no—
379C3×Q8—41no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—81no—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393#67—41no—
394#67—41no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—81no—
400C2×Dic3#12111yescharacteristic
401C2×Dic3#12111yescharacteristic
402C2×Dic3#12111yescharacteristic
403C2×Dic3#12111yescharacteristic
404D12—22no—
405D12—21no—
406D12—22no—
407D12—21no—
408D12—41no—
409D12—41no—
410C4×D3—42no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—42no—
414C4×D3—82no—
415C4×D3—82no—
416Dic6—21no—
417Dic6—21no—
418C3×C8—22no—
419C3×C8—22no—
420C3×C8—41no—
421C3×C8—41no—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C24—61no—
427#41—61no—
428#41—61no—
429#41—61no—
430#41—61no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—31no—
437C2×Q8#23912yes—
438C2×D4—31no—
439C2×D4—32no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—121no—
455C22×C4—121no—
456C22×C4—242no—
457C22×C4C23×D311yescharacteristic
458Q16—64no—
459SD16—64no—
460SD16—68no—
461SD16—64no—
462D8—64no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—32no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16—61no—
474M16#23712yes—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—32no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8D3×D412yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4D3×D411yescharacteristic
496C4⋊C4D3×D411yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—61no—
499C22⋊C4—61no—
500C22⋊C4—121no—
501C22⋊C4—121no—
502C42—21no—
503C42—31no—
504C42—31no—
505C42—31no—
506C42—31no—
507C42—61no—
508C42—61no—
509C42—121no—
510C42—121no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3C2×#12111yescharacteristic
514C22×C3#52911yescharacteristic
515C22×C3#52911yescharacteristic
516C2×D3—41no—
517C2×D3—42no—
518C2×D3—41no—
519C2×D3—41no—
520C2×D3—82no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—82no—
524C3×C4—42no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—81no—
528C3×C4#52511yescharacteristic
529C3×C4#52511yescharacteristic
530C3×C4#52511yescharacteristic
531C3×C4#53211yescharacteristic
532Dic3—21no—
533Dic3—21no—
534Dic3—21no—
535Dic3—41no—
536Dic3—81no—
537Dic3#47112yes—
538C23—61no—
539C23—62no—
540C23—121no—
541C23—121no—
542C23—241no—
543C23—241no—
544C23D3×#4111yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—61no—
550Q8—61no—
551D4—61no—
552D4—62no—
553D4—62no—
554D4—61no—
555D4—121no—
556D4—121no—
557D4—121no—
558D4—121no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—31no—
563C2×C4—31no—
564C2×C4—31no—
565C2×C4—31no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—81no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—242no—
579C2×C4—242no—
580C2×C4—241no—
581C2×C4C2×D3×D411yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×#23711yescharacteristic
584C2×C4C2×#23911yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C8—22no—
588C8—22no—
589C8—41no—
590C8—41no—
591C8—62no—
592C8—62no—
593C8—121no—
594C8—121no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3#239211yescharacteristic
598C2×C3C2×#47111yescharacteristic
599C2×C3#306711yescharacteristic
600D3—81no—
601D3—82no—
602C22—21no—
603C22—21no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—122no—
608C22—242no—
609C22—242no—
610C22—241no—
611C22—241no—
612C22D3×#12111yescharacteristic
613C22#562411yescharacteristic
614C22#570611yescharacteristic
615C4—32no—
616C4—41no—
617C4—41no—
618C4—42no—
619C4—61no—
620C4—61no—
621C4—61no—
622C4—81no—
623C4—121no—
624C4—241no—
625C4#559611yescharacteristic
626C4#560611yescharacteristic
627C4#562711yescharacteristic
628C4#562711yescharacteristic
629C3#2371811yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—241no—
633C2—242no—
634C2#7728111yescharacteristic
635C2D3×#47111yescharacteristic
636C2#8094311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2064C2
socleC22×C3C2×#121
prime radicalC2×#908C22
hypercenter#2064D3
hypocenterC3#23718
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2064

Complements

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

NH ≅ G/Ntype
2. #79070C2semidirect
3. #78946C2semidirect
4. C2×#4788C2semidirect
5. C3×#2064C2semidirect
7. Dic3×#108C2semidirect
8. #70671C2semidirect
9. #70542C2semidirect
10. #70339C2semidirect
11. #70264C2semidirect
13. Dic3×#101C2semidirect
14. #69011C2semidirect
15. #68972C2semidirect
70. #2064D3semidirect
629. C3#23718semidirect

Maximal subgroups

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

subgroupindex
#791992
#790702
#789462
C2×#47882
C3×#20642
#707142
Dic3×#1082
#706712
#705422
#703392
#702642
#701172
Dic3×#1012
#690112
#689722
#237183