Finite Group Databasebeta

← #10824893

Subgroups — #10824893

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
1C4×#942C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic, complemented
3C4×#908C211yesmaximal, characteristic
4#78519C211yesmaximal, characteristic, complemented
5C3×C4×#101C211yesmaximal, characteristic, complemented
6#70126C211yesmaximal, characteristic, complemented
7#70118C211yesmaximal, characteristic, complemented
8#70117C211yesmaximal, characteristic
9#70109C211yesmaximal, characteristic, complemented
10C4×#822C211yesmaximal, characteristic, complemented
11C4×#823C211yesmaximal, characteristic, complemented
12#69011C211yesmaximal, characteristic, complemented
13#68996C211yesmaximal, characteristic, complemented
14#68986C211yesmaximal, characteristic
15#68981C211yesmaximal, characteristic, complemented
16#23718—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#942C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#908C2211yescharacteristic
21C4×#235C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C3×#101C2211yescharacteristic
24C2×C3×#101C2211yescharacteristic
25C3×C4×M16C2211yescharacteristic
26C3×C4×M16C2211yescharacteristic
27#5225C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5133C2211yescharacteristic
30#5131C2211yescharacteristic
31C2×#823C2211yescharacteristic
32C2×#823C2211yescharacteristic
33#5129C2211yescharacteristic
34#5129C2211yescharacteristic
35C2×#822C2211yescharacteristic
36C2×#822C2211yescharacteristic
37#5121C2211yescharacteristic
38#5119C2211yescharacteristic
39#5119C2211yescharacteristic
40#5030—21no—
41C4×#216—21no—
42#5026—21no—
43C4×#215—21no—
44C4×#215—22no—
45C4×#215—22no—
46#4966C2211yescharacteristic
47C2×#809C2211yescharacteristic
48#4964C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C4×#208C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4711—21no—
53#4710—21no—
54#4709—22no—
55C4×D24—21no—
56C4×#204—21no—
57#4685—22no—
58C3×#343C2211yescharacteristic
59C3×#343C2211yescharacteristic
60#4558C2211yescharacteristic
61#4558C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64C4×#137—31no—
65C4×AGL(1,ℤ/8)—31no—
66C2×C4×SD16—31no—
67#2922—31no—
68#2090—31no—
69#2079—31no—
70#2064—31no—
71#2063—31no—
72#2052—31no—
73#2051—31no—
74#2046—31no—
75#2019—31no—
76C4×#108—31no—
77C4×#101D311yescharacteristic, complemented
78C4×#100—31no—
79C22×C3×Q8C2311yescharacteristic
80C2×#235C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C3×C4×Q8—21no—
88C3×C4×Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#216—21no—
96#942—22no—
97#942—22no—
98C2×#215—22no—
99C2×#215—21no—
100C2×#215—22no—
101C2×#215—22no—
102C4×#67—21no—
103C4×#67—21no—
104C2×(C2×D3)⋊C4—21no—
105#926C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#908—22no—
112#908—22no—
113C2×D24—21no—
114C2×#204—21no—
115C4×D12—21no—
116C4×D12—21no—
117C4×D12—21no—
118C4×D12—21no—
119C4×D12D411yescharacteristic
120C4×D12D411yescharacteristic
121#872—21no—
122C42×D3—21no—
123C4×Dic6D411yescharacteristic
124C4×Dic6D411yescharacteristic
125C3×#101—22no—
126C3×#101—22no—
127C3×#97—21no—
128C3×#97—21no—
129C3×C4×C8—21no—
130C3×C4×C8—21no—
131#835—21no—
132#835—21no—
133#835—22no—
134#823—22no—
135#823—22no—
136#822—22no—
137#822—22no—
138#821—21no—
139#821—21no—
140#821—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#818—21no—
144#817—21no—
145#816—21no—
146Dic3⋊C8—22no—
147Dic3⋊C8—22no—
148#810—21no—
149#809—21no—
150#809—22no—
151#808—21no—
152#808—22no—
153#807—21no—
154#803—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8D412yes—
157C4×C3⋊C8D412yes—
158C2×#137—31no—
159C2×AGL(1,ℤ/8)—31no—
160C22×SD16—31no—
161C4×#41—31no—
162C2×C4×Q8C2×D311yescharacteristic
163C2×C4×D4—31no—
164#442—61no—
165#441—61no—
166#440—61no—
167#440—61no—
168C4×Q16—61no—
169C4×SD16—61no—
170C4×SD16—62no—
171C4×SD16—62no—
172C4×SD16—61no—
173C4×D8—61no—
174#433—62no—
175#431—62no—
176C2×#108—31no—
177C2×#108—31no—
178#428—31no—
179#428—31no—
180C2×#106—31no—
181#424—31no—
182#424—31no—
183#419—31no—
184#418—31no—
185#417—31no—
186#417—31no—
187C2×#103—31no—
188C2×#102—31no—
189C2×#101C2×D311yescharacteristic
190C2×#101C2×D311yescharacteristic
191C2×#100—31no—
192C2×#100—31no—
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16C2×D311yescharacteristic
197C4×M16C2×D311yescharacteristic
198C2×C4×C8—31no—
199#343—31no—
200#343—31no—
201#343C2×D311yescharacteristic
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×C4#4111yescharacteristic
215C22×C3×C4#4111yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C2×#67—21no—
218C22×Dic3—21no—
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—21no—
243C3×M16—21no—
244C3×M16#4112yes—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—41no—
252C3×C4⋊C4—41no—
253C3×C42—41no—
254C3×C42—41no—
255C3×C42C2×D411yescharacteristic
256C3×C42C2×D411yescharacteristic
257C3×C42C2×D411yescharacteristic
258C3×C42C2×D411yescharacteristic
259#217—21no—
260#216—24no—
261#215—28no—
262#215—24no—
263(C2×D3)⋊C4—21no—
264(C2×D3)⋊C4—21no—
265(C2×D3)⋊C4—41no—
266(C2×D3)⋊C4—41no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3C2×D411yescharacteristic
270C4⋊Dic3C2×D411yescharacteristic
271Dic3⋊C4—21no—
272Dic3⋊C4—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276#208—21no—
277#208—21no—
278#208—21no—
279#208#4112yes—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8C2×D412yes—
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×M16C22×D311yescharacteristic
308C2×M16C22×D311yescharacteristic
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C22×C8—31no—
312C22×C8—31no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—31no—
318C4×Q8—31no—
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—31no—
328#117—61no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C22⋊C4—61no—
333C2×C42—21no—
334C2×C42—61no—
335C2×C42C22×D311yescharacteristic
336#108—62no—
337#108—62no—
338#107—61no—
339#107—61no—
340#106—61no—
341#106—62no—
342#106—61no—
343C4⋊C8—62no—
344C4⋊C8—62no—
345#103—61no—
346#103—61no—
347#103—61no—
348#103—61no—
349#103—62no—
350#102—62no—
351#102—61no—
352#102—61no—
353#102—61no—
354#102—61no—
355#101—22no—
356#101—22no—
357#100—62no—
358#100—62no—
359C22⋊C8—62no—
360C22⋊C8—62no—
361#97—21no—
362#97—21no—
363#97—61no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8—32no—
367C4×C8—32no—
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—22no—
381C2×C3×C4—22no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—81no—
387C2×C3×C4#12111yescharacteristic
388C2×C3×C4#12111yescharacteristic
389C2×C3×C4#12111yescharacteristic
390C2×C3×C4#12111yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397#67—41no—
398#67—41no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—81no—
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—41no—
420C3×C8—41no—
421C3×C8—41no—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C3⋊C8—22no—
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#23512yes—
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—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—121no—
453C22×C4—121no—
454C22×C4—242no—
455C22×C4#23511yescharacteristic
456C22×C4#23511yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—64no—
459SD16—64no—
460SD16—68no—
461SD16—64no—
462D8—64no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—21no—
470M16—32no—
471M16—61no—
472M16—61no—
473M16—61no—
474M16#23512yes—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—32no—
480C2×C8—32no—
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—31no—
490C4⋊C4—31no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C22⋊C4—61no—
498C22⋊C4—61no—
499C22⋊C4—61no—
500C22⋊C4—121no—
501C22⋊C4—121no—
502C42—41no—
503C42—41no—
504C42—61no—
505C42—61no—
506C42—61no—
507C42D3×D411yescharacteristic
508C42D3×D411yescharacteristic
509C42D3×D411yescharacteristic
510C42D3×D411yescharacteristic
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—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—42no—
528C3×C4—81no—
529C3×C4#47112yes—
530C3×C4#52511yescharacteristic
531C3×C4#52511yescharacteristic
532C3×C4#52511yescharacteristic
533C3×C4#53211yescharacteristic
534Dic3—41no—
535Dic3—41no—
536Dic3—41no—
537Dic3—81no—
538C23—61no—
539C23—62no—
540C23—121no—
541C23—121no—
542C23—241no—
543C23—241no—
544C23C2×#23511yescharacteristic
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—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—81no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—242no—
575C2×C4—242no—
576C2×C4—241no—
577C2×C4#88411yescharacteristic
578C2×C4#88411yescharacteristic
579C2×C4#88411yescharacteristic
580C2×C4#88411yescharacteristic
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×#23511yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4C2×D3×D411yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C8—22no—
588C8—41no—
589C8—41no—
590C8—41no—
591C8—62no—
592C8—62no—
593C8—62no—
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—122no—
606C22—121no—
607C22—121no—
608C22—242no—
609C22—242no—
610C22—241no—
611C22—241no—
612C22#547911yescharacteristic
613C22C2×#88411yescharacteristic
614C22#552811yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—81no—
620C4—121no—
621C4—121no—
622C4—121no—
623C4—241no—
624C4#475412yes—
625C4#554711yescharacteristic
626C4#554711yescharacteristic
627C4#554711yescharacteristic
628C4#554811yescharacteristic
629C3#2371811yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—241no—
633C2—242no—
634C2#7358111yescharacteristic
635C2C2×#475411yescharacteristic
636C2#7871911yescharacteristic
subgroupG/H
centerC2×C4#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#101C2
socleC22×C3C2×#121
prime radicalC2×#908C22
hypercenterC4×#101D3
hypocenterC3#23718
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
1. C4×#942C2semidirect
2. C2×C4×#215C2semidirect
4. #78519C2semidirect
5. C3×C4×#101C2semidirect
6. #70126C2semidirect
7. #70118C2semidirect
9. #70109C2semidirect
10. C4×#822C2semidirect
11. C4×#823C2semidirect
12. #69011C2semidirect
13. #68996C2semidirect
15. #68981C2semidirect
77. C4×#101D3semidirect
629. C3#23718semidirect

Maximal subgroups

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

subgroupindex
C4×#9422
C2×C4×#2152
C4×#9082
#785192
C3×C4×#1012
#701262
#701182
#701172
#701092
C4×#8222
C4×#8232
#690112
#689962
#689862
#689812
#237183