Finite Group Databasebeta

← #10824902

Subgroups — #10824902

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C4×#949C211yesmaximal, characteristic, complemented
3#82514C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#72095C211yesmaximal, characteristic
6#72093C211yesmaximal, characteristic, complemented
7C4×#833C211yesmaximal, characteristic, complemented
8#70676C211yesmaximal, characteristic, complemented
9#70671C211yesmaximal, characteristic, complemented
10#70319C211yesmaximal, characteristic, complemented
11#70317C211yesmaximal, characteristic, complemented
12#70307C211yesmaximal, characteristic, complemented
13#70302C211yesmaximal, characteristic, complemented
14#69873C211yesmaximal, characteristic, complemented
15C4×#823C211yesmaximal, characteristic, complemented
16#23719—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#949C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#442—21no—
24C3×#440—21no—
25C3×C4×SD16—21no—
26C3×C4×D8—21no—
27C3×#428C2211yescharacteristic
28C3×#418C2211yescharacteristic
29C3×#417C2211yescharacteristic
30C3×C4×M16C2211yescharacteristic
31#5234C2211yescharacteristic
32#5234C2211yescharacteristic
33#5232C2211yescharacteristic
34C2×#833C2211yescharacteristic
35C2×#833C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#823C2211yescharacteristic
38C2×#823C2211yescharacteristic
39C2×#822C2211yescharacteristic
40C2×#822C2211yescharacteristic
41#5026—21no—
42C4×#215—21no—
43#5014—21no—
44#5014—22no—
45#5014—22no—
46C4×#213—21no—
47#5001C2211yescharacteristic
48#5000—22no—
49#4999C2211yescharacteristic
50#4999C2211yescharacteristic
51#4995C2211yescharacteristic
52C2×#809C2211yescharacteristic
53C2×#808C2211yescharacteristic
54#4926—22no—
55C4×#208C2211yescharacteristic
56C4×#208C2211yescharacteristic
57C2×#803C2211yescharacteristic
58#4558C2211yescharacteristic
59#4558C2211yescharacteristic
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4530C2211yescharacteristic
63#4530C2211yescharacteristic
64C4×AGL(1,ℤ/8)—31no—
65C4×AGL(1,ℤ/8)D311yescharacteristic, complemented
66C2×#442—31no—
67#2923—31no—
68#2090—31no—
69#2079—31no—
70#2063—31no—
71#2063—31no—
72#2051—31no—
73#2051—31no—
74#2045—31no—
75#2018—31no—
76#1778—31no—
77C4×#100—31no—
78C4×#100—31no—
79C2×C3×#41C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C3×AGL(1,ℤ/8)—22no—
85C3×AGL(1,ℤ/8)—22no—
86C2×C3×SD16—21no—
87C2×C3×D8—21no—
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C3×C4×Q8D411yescharacteristic
91C3×C4×Q8D411yescharacteristic
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4D411yescharacteristic
99C3×C4×D4D411yescharacteristic
100C3×#117—21no—
101C3×#117C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C42—21no—
105C2×C3×C42C2311yescharacteristic
106#949—22no—
107#949—22no—
108C2×#215—21no—
109C2×#213—21no—
110C2×#213—22no—
111C2×#213—22no—
112C2×#213—22no—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×D12—21no—
122C4×D12—21no—
123C4×D12—21no—
124C4×D12—21no—
125C3×#107—21no—
126C3×#106—21no—
127C3×#103—21no—
128C3×#102—21no—
129C3×#102—21no—
130C3×#102—21no—
131C3×#97—21no—
132C3×C4×C8—21no—
133#835—21no—
134#833—22no—
135#833—22no—
136#832—21no—
137#832—22no—
138#830—22no—
139#830—22no—
140#823—22no—
141#823—22no—
142#822—22no—
143#822—22no—
144#809—21no—
145#809—21no—
146#809—22no—
147#808—21no—
148#808—22no—
149#807—21no—
150C4⋊(C3⋊C8)—22no—
151C4⋊(C3⋊C8)—22no—
152#803—21no—
153#803—21no—
154#803D412yes—
155#803D412yes—
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C2×AGL(1,ℤ/8)—31no—
159C2×AGL(1,ℤ/8)C2×D311yescharacteristic
160C22×D8—31no—
161C4×#41C2×D311yescharacteristic
162C2×C4×D4—31no—
163C2×C4×D4C2×D311yescharacteristic
164#442—21no—
165#442—62no—
166#442—62no—
167#442—61no—
168#440—21no—
169#440—61no—
170C4×SD16—21no—
171C4×SD16—61no—
172C4×D8—21no—
173C4×D8—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108—31no—
178#428—31no—
179#428C2×D311yescharacteristic
180C2×#106—31no—
181#424—31no—
182#424—31no—
183#418—31no—
184#418C2×D311yescharacteristic
185#417—31no—
186#417C2×D311yescharacteristic
187C2×#102—31no—
188C2×#102—31no—
189C2×#100—31no—
190C2×#100—31no—
191C2×#100—31no—
192C2×#100—31no—
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16—31no—
197C4×M16C2×D311yescharacteristic
198C2×#97—31no—
199#343—31no—
200#343—31no—
201#343—31no—
202#343—31no—
203#341—31no—
204#341—31no—
205C24×C3—21no—
206C23×D3—21no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—21no—
210C3×#41—21no—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4C2×D411yescharacteristic
219C2×C3×D4#4112yes—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—41no—
225C22×C3×C4—41no—
226C22×C3×C4—82no—
227C22×C3×C4#4111yescharacteristic
228C22×C3×C4#4111yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12#4112yes—
237C2×C4×D3—41no—
238C2×C4×D3—41no—
239C2×C4×D3—82no—
240C3×SD16—24no—
241C3×D8—24no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16#4112yes—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C4⋊C4C2×D411yescharacteristic
254C3×C22⋊C4—21no—
255C3×C22⋊C4—21no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—41no—
258C3×C22⋊C4—41no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42C2×D411yescharacteristic
263C3×C42C2×D411yescharacteristic
264C3×C42C2×D411yescharacteristic
265C3×C42C2×D411yescharacteristic
266#215—24no—
267#213—28no—
268#213—24no—
269(C2×D3)⋊C4—41no—
270(C2×D3)⋊C4—41no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273#208—21no—
274#208—21no—
275#208—21no—
276#208—21no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208#4112yes—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8C2×D412yes—
290C2×C3⋊C8C2×D412yes—
291C2×#41C22×D311yescharacteristic
292C22×D4—31no—
293C22×D4C22×D311yescharacteristic
294C23×C4—21no—
295C23×C4—61no—
296AGL(1,ℤ/8)—22no—
297AGL(1,ℤ/8)—22no—
298AGL(1,ℤ/8)—62no—
299AGL(1,ℤ/8)—62no—
300C2×SD16—21no—
301C2×SD16—61no—
302C2×D8—21no—
303C2×D8—62no—
304C2×D8—62no—
305C2×D8—62no—
306C2×D8—61no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16C22×D311yescharacteristic
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8—31no—
315C4×Q8#6711yescharacteristic
316C4×Q8#6711yescharacteristic
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327C4×D4#6711yescharacteristic
328C4×D4#6711yescharacteristic
329#117—21no—
330#117C22×D311yescharacteristic
331C2×C4⋊C4—31no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—61no—
335C2×C42—21no—
336C2×C42C22×D311yescharacteristic
337#108—62no—
338#108—62no—
339#107—21no—
340#107—61no—
341#106—21no—
342#106—62no—
343#106—61no—
344C4⋊C8—62no—
345C4⋊C8—62no—
346#103—21no—
347#103—61no—
348#102—21no—
349#102—21no—
350#102—21no—
351#102—62no—
352#102—62no—
353#102—61no—
354#102—61no—
355#102—61no—
356#100—62no—
357#100—62no—
358#100—62no—
359#100—62no—
360C22⋊C8—62no—
361C22⋊C8—62no—
362#97—21no—
363#97—32no—
364#97—32no—
365#97—61no—
366#97—61no—
367C4×C8—21no—
368C4×C8—61no—
369C4×C8—61no—
370C23×C3—21no—
371C23×C3—22no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—81no—
375C23×C3—81no—
376C23×C3C2×#4111yescharacteristic
377C22×D3—22no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—81no—
381C22×D3—81no—
382C3×Q8—21no—
383C3×Q8—21no—
384C3×D4—22no—
385C3×D4—21no—
386C3×D4—21no—
387C3×D4—22no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—81no—
401C2×C3×C4—81no—
402C2×C3×C4—82no—
403C2×C3×C4—82no—
404C2×C3×C4—81no—
405C2×C3×C4#12111yescharacteristic
406C2×C3×C4#12111yescharacteristic
407C2×C3×C4#12111yescharacteristic
408C2×C3×C4#12111yescharacteristic
409C2×C3×C4C22×D411yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×Dic3—41no—
416C2×Dic3—81no—
417D12—22no—
418D12—22no—
419D12—41no—
420D12—41no—
421C4×D3—82no—
422C4×D3—82no—
423C3×C8—22no—
424C3×C8—41no—
425C3⋊C8—22no—
426C3⋊C8—22no—
427C3⋊C8—22no—
428C3⋊C8—41no—
429C3⋊C8—41no—
430C3⋊C8—41no—
431C24—21no—
432C24—61no—
433#41—21no—
434#41—21no—
435#41—21no—
436#41—21no—
437C2×Q8C2×#6711yescharacteristic
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—32no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4#23512yes—
451C2×D4C2×#6711yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—61no—
459C22×C4—82no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4—242no—
463C22×C4#23711yescharacteristic
464C22×C4#23711yescharacteristic
465C22×C4C23×D311yescharacteristic
466SD16—24no—
467SD16—64no—
468D8—24no—
469D8—68no—
470D8—64no—
471M16—21no—
472M16—21no—
473M16—21no—
474M16—32no—
475M16—61no—
476M16—61no—
477M16—61no—
478M16—61no—
479M16—61no—
480M16—61no—
481M16—61no—
482M16#23512yes—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—32no—
486C2×C8—32no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4C2×#6711yescharacteristic
502C4⋊C4C2×#6711yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—121no—
509C22⋊C4—121no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42D3×D411yescharacteristic
514C42D3×D411yescharacteristic
515C42C2×#6711yescharacteristic
516C42C2×#6711yescharacteristic
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—41no—
520C22×C3—42no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3C2×#12111yescharacteristic
528C22×C3#52911yescharacteristic
529C22×C3#52911yescharacteristic
530C2×D3—42no—
531C2×D3—82no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—82no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—81no—
543C3×C4#52511yescharacteristic
544C3×C4#52511yescharacteristic
545C3×C4#52511yescharacteristic
546C3×C4#53211yescharacteristic
547Dic3—81no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—62no—
553C23—81no—
554C23—81no—
555C23—121no—
556C23—121no—
557C23—241no—
558C23—241no—
559C23C2×#23711yescharacteristic
560Q8—21no—
561Q8—21no—
562D4—21no—
563D4—22no—
564D4—22no—
565D4—21no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—62no—
571D4—62no—
572D4—121no—
573D4—121no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—121no—
588C2×C4—242no—
589C2×C4—242no—
590C2×C4—241no—
591C2×C4#93911yescharacteristic
592C2×C4#93911yescharacteristic
593C2×C4#93911yescharacteristic
594C2×C4#93911yescharacteristic
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×#23511yescharacteristic
597C2×C4C2×D3×D411yescharacteristic
598C2×C4D3×#4111yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4C22×#6711yescharacteristic
601C8—22no—
602C8—41no—
603C8—62no—
604C8—62no—
605C8—62no—
606C8—121no—
607C8—121no—
608C8—121no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—81no—
612C2×C3—82no—
613C2×C3#239211yescharacteristic
614C2×C3#306811yescharacteristic
615C2×C3#306811yescharacteristic
616D3—82no—
617C22—21no—
618C22—21no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—82no—
624C22—82no—
625C22—81no—
626C22—81no—
627C22—122no—
628C22—242no—
629C22—242no—
630C22—241no—
631C22—241no—
632C22#553811yescharacteristic
633C22#555711yescharacteristic
634C22C2×#93911yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—241no—
644C4#554711yescharacteristic
645C4#554811yescharacteristic
646C4#579111yescharacteristic
647C4#582811yescharacteristic
648C3#2371911yescharacteristic, complemented, Sylow
649C2—21no—
650C2—21no—
651C2—81no—
652C2—82no—
653C2—242no—
654C2#8234511yescharacteristic
655C2#8394311yescharacteristic
656C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×AGL(1,ℤ/8)C2
socleC22×C3C2×#121
prime radical#72095C2
hypercenterC4×AGL(1,ℤ/8)D3
hypocenterC3#23719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C3×C4×AGL(1,ℤ/8)C2semidirect
2. C4×#949C2semidirect
3. #82514C2semidirect
4. C2×#5014C2semidirect
6. #72093C2semidirect
7. C4×#833C2semidirect
8. #70676C2semidirect
9. #70671C2semidirect
10. #70319C2semidirect
11. #70317C2semidirect
12. #70307C2semidirect
13. #70302C2semidirect
14. #69873C2semidirect
15. C4×#823C2semidirect
65. C4×AGL(1,ℤ/8)D3semidirect
648. C3#23719semidirect

Maximal subgroups

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

subgroupindex
C3×C4×AGL(1,ℤ/8)2
C4×#9492
#825142
C2×#50142
#720952
#720932
C4×#8332
#706762
#706712
#703192
#703172
#703072
#703022
#698732
C4×#8232
#237193