Finite Group Databasebeta

← #72228271

Subgroups — #72228271

Order 1536.

1176 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10948647C212yesmaximal
2#10813709C211yesmaximal, characteristic, complemented
3#10813662C211yesmaximal, characteristic, complemented
4#10713271C212yesmaximal, complemented
5#10711879C212yesmaximal, complemented
6#10711872C212yesmaximal, complemented
7#10711613C211yesmaximal, characteristic, complemented
8#10711298C211yesmaximal, characteristic, complemented
9#10711288C211yesmaximal, characteristic, complemented
10#10710561C211yesmaximal, characteristic, complemented
11C3×#33463C211yesmaximal, characteristic
12#497868—31nomaximal, Sylow
13C3×#3414C2211yescharacteristic
14C2×C3×#443C2211yescharacteristic
15C2×C3×#443C2211yescharacteristic
16#83910C2212yes—
17#83899C2212yes—
18#83789—21no—
19#83784—21no—
20C3×#2179C414yescomplemented
21C3×#2178C414yescomplemented
22C3×#2161C2211yescharacteristic
23C3×#2143C2211yescharacteristic
24C3×#2143C2211yescharacteristic
25C3×C4×M32C2211yescharacteristic
26#72081C2212yes—
27C2×#4566C2212yes—
28#72015—21no—
29#72013—21no—
30#71521—24no—
31#71515—24no—
32C2×#4513C2212yes—
33C2×#4513C2212yes—
34#71425C2212yes—
35#70707C2212yes—
36#70700C2212yes—
37#70375—22no—
38#70363—22no—
39#69792C2211yescharacteristic
40#69779C2211yescharacteristic
41#69764C2211yescharacteristic
42C2×#4463C2211yescharacteristic
43#67813C2212yes—
44#67731C2212yes—
45#67730C2212yes—
46#33469—32no—
47#33463D311yescharacteristic
48#22694—31no—
49#22689—31no—
50#13175—32no—
51#12949—32no—
52#12948—32no—
53#12903—31no—
54#12825—31no—
55#12823—31no—
56#12667—31no—
57C2×C3×#135C2311yescharacteristic
58C2×C3×#135C2311yescharacteristic
59C2×C3×#131—21no—
60C2×C3×#131—21no—
61#5827—22no—
62#5819—22no—
63C3×#510D412yes—
64C3×#510Q812yes—
65C3×#509D412yes—
66C3×#509Q812yes—
67C3×#508C2×C414yes—
68C2×C3×M32C2311yescharacteristic
69C2×C3×M32C2311yescharacteristic
70C3×#443—21no—
71C3×#443—21no—
72C3×#443—21no—
73C3×#443—21no—
74C3×#443C2×C412yes—
75C3×#443C2×C412yes—
76C3×#443C2×C412yescomplemented
77C3×#443C2×C412yescomplemented
78C2×C3×#108C2311yescharacteristic
79C2×C3×#108C2311yescharacteristic
80C2×C3×#104—21no—
81C2×C3×#104—21no—
82C2×C3×C4×C8C2311yescharacteristic
83#5236—22no—
84#5232—22no—
85#5197—24no—
86#5196—42no—
87#5184—22no—
88Dic3×SD16—42no—
89#5152—22no—
90#5125—22no—
91#5125—42no—
92#5121—22no—
93#5121—42no—
94C2×#819—21no—
95C2×#819—42no—
96C2×#819D411yescharacteristic
97C2×#819C2312yes—
98#5106—41no—
99#5106D411yescharacteristic
100C2×#817—21no—
101C2×#817—41no—
102C2×#817C2312yes—
103#5101C2312yes—
104C2×#812C2312yes—
105#4935—21no—
106#4924—21no—
107C3×#368C2×C414yescomplemented
108C3×#359C2×C414yes—
109C3×#359C2×C414yescomplemented
110C3×#346C2×C412yescomplemented
111C3×C4×C16C2×C412yescomplemented
112#4566—24no—
113#4565—22no—
114#4563—22no—
115#4559—22no—
116#4554—22no—
117#4523D412yes—
118#4513—24no—
119#4513—42no—
120#4512—24no—
121#4463—21no—
122#4463—21no—
123#3414C2×D311yescharacteristic
124#2957—32no—
125C2×#443C2×D311yescharacteristic
126C2×#443C2×D311yescharacteristic
127#2952—32no—
128#2916—61no—
129#2912—61no—
130AGL(1,ℤ/16)—64no—
131#2180—64no—
132#2179Dic314yescomplemented
133#2178Dic314yescomplemented
134#2161C2×D311yescharacteristic
135C2×#365—32no—
136C2×#365—32no—
137C2×#364—31no—
138#2148—32no—
139C2×#360—32no—
140#2143C2×D311yescharacteristic
141#2143C2×D311yescharacteristic
142#2110—32no—
143C4×M32C2×D311yescharacteristic
144#1953—32no—
145#1951—32no—
146#1861—62no—
147#1859—62no—
148#1848—31no—
149#1844—31no—
150#1815—61no—
151#1814—61no—
152#1772—31no—
153#1385—32no—
154#1383—32no—
155#1378—32no—
156C2×C3×#41—21no—
157C2×C3×#41—21no—
158C3×#135—21no—
159C3×#135—21no—
160C3×#135—21no—
161C3×#135—21no—
162C3×#135C22×C412yes—
163C3×#135C22×C412yes—
164C3×#135C2×D412yes—
165C3×#135C2×Q812yes—
166C2×C3×Q16C2×D411yescharacteristic
167C2×C3×Q16C2×Q811yescharacteristic
168C2×C3×SD16—21no—
169C2×C3×SD16—21no—
170C2×C3×D8C2×D411yescharacteristic
171C2×C3×D8C2×Q811yescharacteristic
172C3×#131—22no—
173C3×#131—22no—
174C3×#131—22no—
175C3×#131—22no—
176C3×#131—41no—
177C3×#131—41no—
178C3×#131—41no—
179C3×#131—41no—
180C2×C3×M16—21no—
181C2×C3×M16—21no—
182C2×C3×M16—41no—
183C2×C3×M16—41no—
184C22×C3×C8—41no—
185C22×C3×C8—41no—
186C22×C3×C8C2×D411yescharacteristic
187C22×C3×C8C2411yescharacteristic
188C2×C3×C42C2×D411yescharacteristic
189#948—44no—
190#948—82no—
191Q8×Dic3—42no—
192D4×Dic3—42no—
193D4×Dic3—82no—
194#926—42no—
195#926C2×D412yes—
196C2×C4⋊Dic3—22no—
197C2×C4⋊Dic3—42no—
198C2×C4⋊Dic3—81no—
199C2×#208—22no—
200C2×#208—44no—
201C2×#208C2×D412yes—
202C3×Q32C4⋊C414yescomplemented
203C3×SD32C4⋊C414yescomplemented
204C3×SD32C4⋊C414yescomplemented
205C3×D16C4⋊C414yescomplemented
206C3×M32—21no—
207C3×M32—21no—
208C3×M32C2×D412yes—
209C3×M32C2×Q812yes—
210C2×C3×C16C22×C412yes—
211C2×C3×C16C22×C412yes—
212C3×#108—21no—
213C3×#108—21no—
214C3×#108C22×C412yes—
215C3×#108C22×C412yes—
216C3×#104—22no—
217C3×#104—22no—
218C3×#104—22no—
219C3×#104—22no—
220C3×C4×C8C22×C411yescharacteristic
221C3×C4×C8C22×C411yescharacteristic
222C3×C4×C8#4111yescharacteristic
223C3×C4×C8#4111yescharacteristic
224#837—44no—
225#835—42no—
226#832—42no—
227#819—21no—
228#819—21no—
229#819—22no—
230#819—22no—
231#819—44no—
232#819—41no—
233#819—41no—
234#818—21no—
235#818—42no—
236#817—21no—
237#817—22no—
238#817—22no—
239#817—42no—
240#817—41no—
241#817—41no—
242#815—22no—
243C8×Dic3—22no—
244#812—22no—
245#812—22no—
246C2×C3⋊C16—22no—
247C4⋊(C3⋊C8)—81no—
248C2×#135C22×D311yescharacteristic
249C2×#135C22×D311yescharacteristic
250#568—62no—
251C2×#131—21no—
252C2×#131—21no—
253#520—62no—
254#510Dic612yes—
255#510D1212yes—
256#509Dic612yes—
257#509D1212yes—
258#508C2×Dic314yes—
259C2×M32—32no—
260C2×M32C22×D311yescharacteristic
261C2×M32C22×D311yescharacteristic
262#444—64no—
263#443—21no—
264#443—21no—
265#443—21no—
266#443—21no—
267#443—122no—
268#443C2×Dic312yes—
269#443C2×Dic312yes—
270#443C2×Dic312yescomplemented
271#443C2×Dic312yescomplemented
272#442—62no—
273#441—62no—
274C4×SD16—122no—
275#430—62no—
276#430—122no—
277C2×#108—31no—
278C2×#108—32no—
279C2×#108—61no—
280C2×#108—122no—
281C2×#108C22×D311yescharacteristic
282C2×#108C22×D311yescharacteristic
283#428—62no—
284#428—122no—
285#427—31no—
286#427—121no—
287C2×#106—32no—
288C2×#106—61no—
289C2×#106—121no—
290#423—61no—
291#421—62no—
292C2×#104—21no—
293C2×#104—21no—
294#417—62no—
295#405—32no—
296C2×C4×C8C22×D311yescharacteristic
297#384—61no—
298#368C2×Dic314yescomplemented
299#365—64no—
300#365—122no—
301#364—61no—
302#364—61no—
303#362—62no—
304#361—62no—
305#360—64no—
306#359C2×Dic314yes—
307#359C2×Dic314yescomplemented
308#347—64no—
309#346C2×Dic312yescomplemented
310C4×C16C2×Dic312yescomplemented
311#344—62no—
312#343—62no—
313#339—32no—
314C3×#41—22no—
315C3×#41—22no—
316C3×#41—22no—
317C3×#41—22no—
318C3×#41—41no—
319C3×#41—41no—
320C3×#41—41no—
321C3×#41—41no—
322C2×C3×Q8—21no—
323C2×C3×Q8—21no—
324C2×C3×D4—21no—
325C2×C3×D4—21no—
326C2×C3×D4—41no—
327C2×C3×D4—41no—
328C22×C3×C4—21no—
329C22×C3×C4—41no—
330C22×C3×C4—41no—
331C22×C3×C4C22×D411yescharacteristic
332C22×Dic3—42no—
333C22×Dic3—82no—
334C3×Q16—21no—
335C3×Q16—21no—
336C3×Q16C2×C4⋊C412yes—
337C3×Q16C2×C4⋊C412yes—
338C3×SD16—22no—
339C3×SD16—22no—
340C3×SD16—22no—
341C3×SD16—22no—
342C3×D8—21no—
343C3×D8—21no—
344C3×D8C2×C4⋊C412yes—
345C3×D8C2×C4⋊C412yes—
346C3×M16—22no—
347C3×M16—22no—
348C3×M16—41no—
349C3×M16—41no—
350C3×M16—42no—
351C3×M16—42no—
352C3×M16—42no—
353C3×M16—42no—
354C2×C3×C8—41no—
355C2×C3×C8—41no—
356C2×C3×C8—42no—
357C2×C3×C8—42no—
358C2×C3×C8—42no—
359C2×C3×C8—42no—
360C2×C3×C8C4×D411yescharacteristic
361C2×C3×C8C4×D411yescharacteristic
362C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
363C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
364C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
365C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
366C2×C3×C8C23×C411yescharacteristic
367C2×C3×C8C22×D411yescharacteristic
368C2×C3×C8C22×Q811yescharacteristic
369C2×C3×C8C2×#4111yescharacteristic
370C2×C3×C8H(5,𝔽2)11yescharacteristic
371C2×C3×C82−1+411yescharacteristic
372C3×C42—21no—
373C3×C42C4×D411yescharacteristic
374C3×C42C4×D411yescharacteristic
375#217—22no—
376#217—82no—
377C4⋊Dic3—22no—
378C4⋊Dic3—42no—
379C4⋊Dic3—42no—
380C4⋊Dic3—82no—
381C4⋊Dic3—162no—
382C4×Dic3—22no—
383C4×Dic3—82no—
384#208—22no—
385#208—22no—
386#208—42no—
387#208—42no—
388#208—84no—
389C2×C3⋊C8—22no—
390C2×C3⋊C8—42no—
391C2×C3⋊C8—84no—
392C3×C16—22no—
393C3×C16C2×C4⋊C414yes—
394C3⋊C16—42no—
395C2×#41—21no—
396C2×#41—21no—
397#135—21no—
398#135—21no—
399#135—21no—
400#135—21no—
401#135C2×Dic612yes—
402#135C2×D1212yes—
403#135C22×Dic312yes—
404#135C22×Dic312yes—
405C2×Q16C2×Dic611yescharacteristic
406C2×Q16C2×D1211yescharacteristic
407C2×SD16—21no—
408C2×SD16—21no—
409C2×D8C2×Dic611yescharacteristic
410C2×D8C2×D1211yescharacteristic
411#131—22no—
412#131—22no—
413#131—22no—
414#131—22no—
415#131—41no—
416#131—41no—
417#131—41no—
418#131—41no—
419#131—124no—
420#131—242no—
421C2×M16—21no—
422C2×M16—21no—
423C2×M16—32no—
424C2×M16—41no—
425C2×M16—41no—
426C2×M16—62no—
427C2×M16—124no—
428C22×C8—41no—
429C22×C8—41no—
430C22×C8D3×D411yescharacteristic
431C22×C8C23×D311yescharacteristic
432C4×Q8—122no—
433C4×D4—122no—
434C4×D4—242no—
435#117—32no—
436#117—122no—
437C2×C4⋊C4—62no—
438C2×C4⋊C4—122no—
439C2×C4⋊C4—241no—
440C2×C42D3×D411yescharacteristic
441Q32C4⋊Dic314yescomplemented
442SD32C4⋊Dic314yescomplemented
443SD32C4⋊Dic314yescomplemented
444D16C4⋊Dic314yescomplemented
445M32—21no—
446M32—21no—
447M32—62no—
448M32—62no—
449M32C2×Dic612yes—
450M32C2×D1212yes—
451C2×C16—62no—
452C2×C16C22×Dic312yes—
453C2×C16C22×Dic312yes—
454#108—21no—
455#108—21no—
456#108—61no—
457#108—61no—
458#108—62no—
459#108—62no—
460#108—124no—
461#108—121no—
462#108—121no—
463#108C22×Dic312yes—
464#108C22×Dic312yes—
465#107—61no—
466#107—122no—
467#106—61no—
468#106—62no—
469#106—62no—
470#106—122no—
471#106—121no—
472#106—121no—
473C4⋊C8—241no—
474#104—22no—
475#104—22no—
476#104—22no—
477#104—22no—
478#104—124no—
479#103—122no—
480#102—122no—
481#97—62no—
482C4×C8—62no—
483C4×C8#23711yescharacteristic
484C4×C8#23711yescharacteristic
485C4×C8C22×Dic311yescharacteristic
486C4×C8C22×Dic311yescharacteristic
487C23×C3—41no—
488C23×C3—41no—
489C23×C3#57711yescharacteristic
490C3×Q8—22no—
491C3×Q8—22no—
492C3×Q8—41no—
493C3×Q8—41no—
494C3×D4—22no—
495C3×D4—22no—
496C3×D4—42no—
497C3×D4—42no—
498C3×D4—42no—
499C3×D4—42no—
500C3×D4—41no—
501C3×D4—41no—
502C2×C3×C4—21no—
503C2×C3×C4—21no—
504C2×C3×C4—42no—
505C2×C3×C4—41no—
506C2×C3×C4—41no—
507C2×C3×C4—42no—
508C2×C3×C4—42no—
509C2×C3×C4—42no—
510C2×C3×C4—42no—
511C2×C3×C4C2×C4×D411yescharacteristic
512C2×C3×C4D4211yescharacteristic
513C2×C3×C4D4×Q811yescharacteristic
514C2×C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
515C2×C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
516C2×C3×C4#57711yescharacteristic
517C2×Dic3—42no—
518C2×Dic3—82no—
519C2×Dic3—162no—
520C2×Dic3—322no—
521C3×C8—42no—
522C3×C8—42no—
523C3×C8#44211yescharacteristic
524C3×C8#44211yescharacteristic
525C3×C8#44211yescharacteristic
526C3×C8#44211yescharacteristic
527C3×C8C22×C4⋊C411yescharacteristic
528C3×C8#52011yescharacteristic
529C3×C8#54711yescharacteristic
530C3×C8#55411yescharacteristic
531C3⋊C8—42no—
532C3⋊C8—82no—
533#41—22no—
534#41—22no—
535#41—22no—
536#41—22no—
537#41—41no—
538#41—41no—
539#41—41no—
540#41—41no—
541C2×Q8—21no—
542C2×Q8—21no—
543C2×D4—21no—
544C2×D4—21no—
545C2×D4—41no—
546C2×D4—41no—
547C22×C4—21no—
548C22×C4—41no—
549C22×C4—41no—
550C22×C4—122no—
551C22×C4—242no—
552C22×C4C2×D3×D411yescharacteristic
553Q16—21no—
554Q16—21no—
555Q16C2×C4⋊Dic312yes—
556Q16C2×C4⋊Dic312yes—
557SD16—22no—
558SD16—22no—
559SD16—22no—
560SD16—22no—
561D8—21no—
562D8—21no—
563D8C2×C4⋊Dic312yes—
564D8C2×C4⋊Dic312yes—
565M16—22no—
566M16—22no—
567M16—41no—
568M16—41no—
569M16—42no—
570M16—42no—
571M16—42no—
572M16—42no—
573M16—62no—
574M16—62no—
575M16—122no—
576M16—122no—
577M16—244no—
578C2×C8—41no—
579C2×C8—41no—
580C2×C8—42no—
581C2×C8—42no—
582C2×C8—42no—
583C2×C8—42no—
584C2×C8—62no—
585C2×C8—122no—
586C2×C8—244no—
587C2×C8#91111yescharacteristic
588C2×C8#91111yescharacteristic
589C2×C8#91111yescharacteristic
590C2×C8#91111yescharacteristic
591C2×C8D4×Dic311yescharacteristic
592C2×C8D4×Dic311yescharacteristic
593C2×C8C22×Dic611yescharacteristic
594C2×C8C22×D1211yescharacteristic
595C2×C8C2×#23711yescharacteristic
596C2×C8#100911yescharacteristic
597C2×C8#101011yescharacteristic
598C2×C8C23×Dic311yescharacteristic
599C4⋊C4—62no—
600C4⋊C4—122no—
601C4⋊C4—122no—
602C4⋊C4—242no—
603C4⋊C4—482no—
604C22⋊C4—62no—
605C22⋊C4—242no—
606C42—21no—
607C42—62no—
608C42—242no—
609C42D4×Dic311yescharacteristic
610C42D4×Dic311yescharacteristic
611C16—22no—
612C16—122no—
613C16C2×C4⋊Dic314yes—
614C22×C3—41no—
615C22×C3—42no—
616C22×C3—42no—
617C22×C3—82no—
618C22×C3—161no—
619C22×C3#295211yescharacteristic
620C22×C3#327211yescharacteristic
621C22×C3#338411yescharacteristic
622C3×C4—22no—
623C3×C4—41no—
624C3×C4—42no—
625C3×C4—42no—
626C3×C4D4×C4⋊C411yescharacteristic
627C3×C4C2×#44211yescharacteristic
628C3×C4#327211yescharacteristic
629C3×C4#338411yescharacteristic
630Dic3—82no—
631Dic3—162no—
632C23—41no—
633C23—41no—
634C23#577011yescharacteristic
635Q8—22no—
636Q8—22no—
637Q8—41no—
638Q8—41no—
639D4—22no—
640D4—22no—
641D4—42no—
642D4—42no—
643D4—42no—
644D4—42no—
645D4—41no—
646D4—41no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—122no—
657C2×C4—242no—
658C2×C4—482no—
659C2×C4—962no—
660C2×C4D4×Dic611yescharacteristic
661C2×C4D4×D1211yescharacteristic
662C2×C4C2×#91111yescharacteristic
663C2×C4C2×#91111yescharacteristic
664C2×C4#577011yescharacteristic
665C2×C4C2×D4×Dic311yescharacteristic
666C8—42no—
667C8—42no—
668C8—122no—
669C8—242no—
670C8#515211yescharacteristic
671C8#515211yescharacteristic
672C8#515211yescharacteristic
673C8#515211yescharacteristic
674C8#554311yescharacteristic
675C8#555511yescharacteristic
676C8C22×C4⋊Dic311yescharacteristic
677C8#582711yescharacteristic
678C2×C3—21no—
679C2×C3—21no—
680C2×C3—21no—
681C2×C3—82no—
682C2×C3—82no—
683C2×C3#2267011yescharacteristic
684C22—41no—
685C22—42no—
686C22—42no—
687C22—82no—
688C22—161no—
689C22#7957611yescharacteristic
690C22#7960311yescharacteristic
691C22#8389911yescharacteristic
692C4—22no—
693C4—41no—
694C4—42no—
695C4—42no—
696C4—242no—
697C4—482no—
698C4#7957611yescharacteristic
699C4#7960311yescharacteristic
700C4D4×C4⋊Dic311yescharacteristic
701C4C2×#515211yescharacteristic
702C3#49786811yescharacteristic, complemented, Sylow
703C2—21no—
704C2—21no—
705C2—21no—
706C2—82no—
707C2—82no—
708C2#1081346711yescharacteristic
subgroupG/H
centerC2#10813467
commutatorC2×C3×C8C23×C4
FrattiniC22×C8C23×D3
FittingC3×#33463C2
socleC2×C3#22670
prime radicalC3×#3414C22
cosocleC22×C3×C8C24
hypercenter#33463D3
hypocenterC3#497868
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ #33463

Complements

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

NH ≅ G/Ntype
2. #10813709C2semidirect
3. #10813662C2semidirect
4. #10713271C2semidirect
5. #10711879C2semidirect
6. #10711872C2semidirect
7. #10711613C2semidirect
8. #10711298C2semidirect
9. #10711288C2semidirect
10. #10710561C2semidirect
20. C3×#2179C4semidirect
21. C3×#2178C4semidirect
76. C3×#443C2×C4semidirect
77. C3×#443C2×C4semidirect
107. C3×#368C2×C4semidirect
109. C3×#359C2×C4semidirect
110. C3×#346C2×C4semidirect
111. C3×C4×C16C2×C4semidirect
132. #2179Dic3semidirect
133. #2178Dic3semidirect
202. C3×Q32C4⋊C4semidirect
203. C3×SD32C4⋊C4semidirect
204. C3×SD32C4⋊C4semidirect
205. C3×D16C4⋊C4semidirect
270. #443C2×Dic3semidirect
271. #443C2×Dic3semidirect
298. #368C2×Dic3semidirect
307. #359C2×Dic3semidirect
309. #346C2×Dic3semidirect
310. C4×C16C2×Dic3semidirect
441. Q32C4⋊Dic3semidirect
442. SD32C4⋊Dic3semidirect
443. SD32C4⋊Dic3semidirect
444. D16C4⋊Dic3semidirect
702. C3#497868semidirect

Maximal subgroups

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

subgroupindex
#109486472
#108137092
#108136622
#107132712
#107118792
#107118722
#107116132
#107112982
#107112882
#107105612
C3×#334632
#4978683