Finite Group Databasebeta

← #11662467

Subgroups — #11662467

Order 768.

2678 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3449C211yesmaximal, characteristic
2C2×#5825C211yesmaximal, characteristic, complemented
3C22×#951C211yesmaximal, characteristic, complemented
4C2×#5819C211yesmaximal, characteristic, complemented
5#84072C211yesmaximal, characteristic, complemented
6#83423C218yesmaximal, complemented
7#83399C214yesmaximal, complemented
8#83377C214yesmaximal, complemented
9#82925C218yesmaximal, complemented
10C2×#5004C211yesmaximal, characteristic, complemented
11#82207C211yesmaximal, characteristic, complemented
12#60455—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C3×#537C2214yes—
17C3×#536—21no—
18C3×#536C2212yes—
19C3×#535—21no—
20C3×#535C2212yes—
21C3×#534—22no—
22C3×#532—22no—
23C3×#525—22no—
24C3×#525C2214yes—
25C2×C3×#117C2211yescharacteristic
26#5825—22no—
27#5825—22no—
28#5825C2214yes—
29#5825C2214yescomplemented
30C2×#951—28no—
31C2×#951C2214yes—
32C2×#951C2214yescomplemented
33#5819—28no—
34C2×#948C2214yes—
35C2×#948C2214yescomplemented
36C2×#944—21no—
37C22×#216C2212yescomplemented
38C2×#936—21no—
39C2×#935—22no—
40C22×#214C2212yescomplemented
41C2×#926C2211yescharacteristic
42C2×#924C2214yescomplemented
43C22×#208C2211yescharacteristic
44C22×#208C2211yescharacteristic
45C22×#208C2214yescomplemented
46#5233C2214yescomplemented
47#5232C2214yes—
48#5225C2212yescomplemented
49#5215C2212yescomplemented
50#5052C22116yescomplemented
51#5051C22116yescomplemented
52#5042C22116yescomplemented
53#5041C22116yescomplemented
54#5004C2214yescomplemented
55#5004C2214yescomplemented
56#4998C2214yescomplemented
57#4995C2214yescomplemented
58C2×#810C2214yescomplemented
59#4964C2214yescomplemented
60C22×#137—31no—
61C2×#568—31no—
62#3449D311yescharacteristic
63C2×#536—31no—
64#3063—38no—
65#3059—34no—
66#3057—34no—
67#3056—38no—
68#2911—31no—
69#2896—31no—
70C2×#419—31no—
71C2×C3×#41—24no—
72C2×C3×#41—21no—
73C2×C3×#41—21no—
74C2×C3×#41—24no—
75C2×C3×#41—24no—
76C2×C3×#41—24no—
77C2×C3×#41D412yes—
78C2×C3×#41D412yes—
79C2×C3×#41C2312yes—
80C2×C3×#41C2312yes—
81C2×C3×#41C2312yes—
82C2×C3×#41C2312yes—
83C22×C3×Q8D411yescharacteristic
84C22×C3×Q8C2311yescharacteristic
85C22×C3×D4—21no—
86C22×C3×D4D412yes—
87C22×C3×D4D411yescharacteristic
88C22×C3×D4C2311yescharacteristic
89C23×C3×C4—21no—
90C23×C3×C4C2312yes—
91C23×C3×C4C2311yescharacteristic
92C23×Dic3—21no—
93C22×Dic6—24no—
94C22×Dic6—24no—
95C22×Dic6—21no—
96C22×Dic6—21no—
97C22×Dic6—21no—
98C22×Dic6C2312yes—
99C22×Dic6C2312yes—
100C22×Dic6C2312yes—
101C3×C4⋊Q8—22no—
102C3×#127—22no—
103C3×#124—22no—
104C3×#124—22no—
105C3×#123—24no—
106C3×#123—24no—
107C3×C22⋊Q8—24no—
108C3×C22⋊Q8C2318yes—
109C3×#121—24no—
110C3×#121—28no—
111C3×#121C2318yes—
112C3×#120—24no—
113C3×#120—24no—
114C3×C4×D4—28no—
115C3×#117—21no—
116C3×#117—21no—
117C3×#117C2312yes—
118C3×#117C2312yes—
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C22⋊C4—21no—
121C2×C3×C42—21no—
122C2×#217—24no—
123C2×#217—21no—
124#951—216no—
125#951—216no—
126#948—216no—
127#948—216no—
128#944—24no—
129#944—24no—
130C2×#216—28no—
131C2×#216C2318yes—
132#936—24no—
133#936—24no—
134#935—28no—
135#935—28no—
136C2×#214—28no—
137C2×#214C2318yes—
138#926—21no—
139#926—21no—
140#926C2312yes—
141#926C2312yes—
142C2×C4⋊Dic3C2312yes—
143#924—28no—
144#924C23116yes—
145C2×Dic3⋊C4—24no—
146C2×C4×Dic3—21no—
147C2×#208—24no—
148C2×#208—216no—
149C2×#208D418yescomplemented
150C2×#208C2312yes—
151C2×#208C2312yes—
152C2×#208C2318yes—
153C2×#208C2312yes—
154C2×#208C2312yes—
155C22×C3⋊C8C2312yes—
156C22×C3⋊C8C2318yes—
157C22×C3⋊C8C2312yes—
158#835C2318yes—
159#832C2318yes—
160#810C23116yes—
161#808C2318yes—
162#807C2318yes—
163C22×#41C2×D311yescharacteristic
164C22×#41C2×D311yescharacteristic
165C23×Q8—31no—
166C2×#137—34no—
167C2×#137—34no—
168C2×#137—68no—
169C22×Q16—32no—
170C22×SD16—32no—
171#568—68no—
172C2×#131—34no—
173C2×#131—34no—
174C22×M16—31no—
175C22×M16—31no—
176C22×M16—34no—
177#537C2×D314yes—
178#536—21no—
179#536—34no—
180#536—34no—
181#536—62no—
182#536—62no—
183#536C2×D312yes—
184#535—21no—
185#535C2×D312yes—
186#534—22no—
187#532—22no—
188C2×C4⋊Q8—61no—
189C2×#124—61no—
190#525—22no—
191#525C2×D314yes—
192C2×#123—62no—
193C2×C22⋊Q8—34no—
194C2×#117—31no—
195C2×#117C2×D311yescharacteristic
196#470—316no—
197#468—316no—
198#467—316no—
199#465—316no—
200#428—34no—
201#428—34no—
202#427—34no—
203#419—32no—
204#419—34no—
205#419—34no—
206#418—32no—
207#417—34no—
208#416—34no—
209C2×#103—34no—
210C24×C3—21no—
211C24×C3C2×D412yes—
212C24×C3C2×D411yescharacteristic
213C3×#41—28no—
214C3×#41—28no—
215C3×#41—24no—
216C3×#41—24no—
217C3×#41—24no—
218C3×#41—24no—
219C3×#41—24no—
220C3×#41—24no—
221C3×#41—24no—
222C3×#41—24no—
223C3×#41—44no—
224C3×#41—44no—
225C2×C3×Q8—24no—
226C2×C3×Q8—21no—
227C2×C3×Q8—22no—
228C2×C3×Q8—22no—
229C2×C3×Q8C2×D412yes—
230C2×C3×Q8C2×D412yes—
231C2×C3×Q8C2412yes—
232C2×C3×D4—28no—
233C2×C3×D4—22no—
234C2×C3×D4—22no—
235C2×C3×D4—28no—
236C2×C3×D4—24no—
237C2×C3×D4—22no—
238C2×C3×D4—21no—
239C2×C3×D4—22no—
240C2×C3×D4—44no—
241C2×C3×D4C2×D414yes—
242C2×C3×D4C2×D414yes—
243C2×C3×D4C2×D412yes—
244C2×C3×D4C2×D412yes—
245C2×C3×D4C2412yes—
246C22×C3×C4—24no—
247C22×C3×C4—24no—
248C22×C3×C4—21no—
249C22×C3×C4—22no—
250C22×C3×C4—22no—
251C22×C3×C4—22no—
252C22×C3×C4—44no—
253C22×C3×C4C2×D414yes—
254C22×C3×C4C2×D412yes—
255C22×C3×C4C2×D414yes—
256C22×C3×C4C2×D411yescharacteristic
257C22×C3×C4C2×D411yescharacteristic
258C22×C3×C4C2×D411yescharacteristic
259C22×C3×C4#4114yes—
260C22×C3×C4C2414yes—
261C22×C3×C4C2411yescharacteristic
262C22×C3×C4C2411yescharacteristic
263C22×C3×C4C2411yescharacteristic
264C22×Dic3—22no—
265C22×Dic3—22no—
266C22×Dic3—22no—
267C22×Dic3—22no—
268C22×Dic3—44no—
269C2×Dic6—24no—
270C2×Dic6—24no—
271C2×Dic6—24no—
272C2×Dic6—24no—
273C2×Dic6—24no—
274C2×Dic6—24no—
275C2×Dic6—22no—
276C2×Dic6—22no—
277C2×Dic6—22no—
278C2×Dic6—44no—
279C2×Dic6—44no—
280C2×Dic6—44no—
281C2×Dic6C2414yes—
282C3×C4⋊C4—22no—
283C3×C4⋊C4—28no—
284C3×C4⋊C4C2414yes—
285C3×C22⋊C4—22no—
286C3×C22⋊C4—22no—
287C3×C22⋊C4—28no—
288C3×C22⋊C4—28no—
289C3×C42—22no—
290C3×C42—22no—
291#217—216no—
292#217—22no—
293#217—22no—
294#216—216no—
295#214—216no—
296C4⋊Dic3—22no—
297C4⋊Dic3C2414yes—
298Dic3⋊C4—216no—
299C4×Dic3—22no—
300C4×Dic3—22no—
301#208—24no—
302#208—24no—
303#208—216no—
304#208C2×D418yes—
305#208C2×D418yes—
306C2×C3⋊C8—24no—
307C2×C3⋊C8—216no—
308C2×C3⋊C8C2×D418yes—
309C2×C3⋊C8C2414yes—
310C2×C3⋊C8C2414yes—
311C2×#41—24no—
312C2×#41—24no—
313C2×#41—21no—
314C2×#41—21no—
315C2×#41—24no—
316C2×#41—24no—
317C2×#41#6712yes—
318C2×#41#6712yes—
319C2×#41C22×D312yes—
320C2×#41C22×D312yes—
321C2×#41C22×D312yes—
322C2×#41C22×D312yes—
323C22×Q8—32no—
324C22×Q8—32no—
325C22×Q8—32no—
326C22×Q8—61no—
327C22×Q8—61no—
328C22×Q8—61no—
329C22×Q8—64no—
330C22×Q8—64no—
331C22×Q8#6711yescharacteristic
332C22×Q8C22×D311yescharacteristic
333C22×D4—21no—
334C22×D4#6712yes—
335C22×D4#6711yescharacteristic
336C22×D4C22×D311yescharacteristic
337C23×C4—21no—
338C23×C4—61no—
339C23×C4C22×D312yes—
340C23×C4C22×D311yescharacteristic
341#137—616no—
342#137—616no—
343C2×Q16—38no—
344C2×Q16—68no—
345C2×SD16—38no—
346C2×SD16—68no—
347#131—616no—
348#131—616no—
349C2×M16—38no—
350C2×M16—32no—
351C2×M16—32no—
352C2×M16—32no—
353C2×M16—32no—
354C2×M16—38no—
355C2×M16—616no—
356C2×M16—64no—
357C22×C8—32no—
358C22×C8—32no—
359C22×C8—38no—
360C4⋊Q8—22no—
361C4⋊Q8—64no—
362C4⋊Q8—64no—
363#127—22no—
364#124—22no—
365#124—22no—
366#124—64no—
367#124—64no—
368#123—24no—
369#123—24no—
370#123—68no—
371#123—68no—
372C22⋊Q8—24no—
373C22⋊Q8—316no—
374C22⋊Q8—68no—
375C22⋊Q8C22×D318yes—
376#121—24no—
377#121—28no—
378#121C22×D318yes—
379#120—24no—
380#120—24no—
381C4×D4—28no—
382#117—21no—
383#117—21no—
384#117—32no—
385#117—32no—
386#117—61no—
387#117—61no—
388#117C22×D312yes—
389#117C22×D312yes—
390C2×C4⋊C4—32no—
391C2×C4⋊C4—64no—
392C2×C4⋊C4C22×D312yes—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—61no—
395C2×C22⋊C4—64no—
396C2×C42—21no—
397C2×C42—61no—
398#107—38no—
399#106—38no—
400#103—38no—
401#103—316no—
402#102—38no—
403C23×C3—24no—
404C23×C3—22no—
405C23×C3—42no—
406C23×C3—44no—
407C23×C3—42no—
408C23×C3—82no—
409C23×C3#12114yes—
410C23×C3C22×D414yes—
411C23×C3C22×D411yescharacteristic
412C23×C3C22×D411yescharacteristic
413C23×C3C22×D411yescharacteristic
414C3×Q8—24no—
415C3×Q8—24no—
416C3×Q8—22no—
417C3×Q8—22no—
418C3×Q8—42no—
419C3×D4—28no—
420C3×D4—28no—
421C3×D4—24no—
422C3×D4—24no—
423C3×D4—22no—
424C3×D4—22no—
425C3×D4—44no—
426C3×D4—44no—
427C3×D4—42no—
428C3×D4—88no—
429C2×C3×C4—24no—
430C2×C3×C4—28no—
431C2×C3×C4—24no—
432C2×C3×C4—24no—
433C2×C3×C4—24no—
434C2×C3×C4—22no—
435C2×C3×C4—42no—
436C2×C3×C4—44no—
437C2×C3×C4—44no—
438C2×C3×C4—82no—
439C2×C3×C4#12114yes—
440C2×C3×C4#12114yes—
441C2×C3×C4#12114yes—
442C2×C3×C4C22×D412yes—
443C2×C3×C4C22×D412yes—
444C2×C3×C4C22×D412yes—
445C2×C3×C4C22×D414yes—
446C2×C3×C4C22×D411yescharacteristic
447C2×C3×C4C22×D411yescharacteristic
448C2×C3×C4C22×D411yescharacteristic
449C2×C3×C4C2×#4112yes—
450C2×C3×C4C2×#4112yes—
451C2×C3×C4C2×#4112yes—
452C2×C3×C4C2511yescharacteristic
453C2×Dic3—24no—
454C2×Dic3—24no—
455C2×Dic3—44no—
456C2×Dic3—44no—
457C2×Dic3—44no—
458C2×Dic3—44no—
459Dic6—24no—
460Dic6—24no—
461Dic6—44no—
462Dic6—44no—
463Dic6—44no—
464C3⋊C8—24no—
465C3⋊C8C22×D418yes—
466C24—21no—
467C24C2×#6712yes—
468C24C2×#6711yescharacteristic
469#41—28no—
470#41—28no—
471#41—24no—
472#41—24no—
473#41—24no—
474#41—24no—
475#41—24no—
476#41—24no—
477#41—24no—
478#41—24no—
479#41—44no—
480#41—44no—
481C2×Q8—24no—
482C2×Q8—21no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—34no—
486C2×Q8—62no—
487C2×Q8—62no—
488C2×Q8—62no—
489C2×Q8—64no—
490C2×Q8—64no—
491C2×Q8—64no—
492C2×Q8—64no—
493C2×Q8—64no—
494C2×Q8—64no—
495C2×Q8—124no—
496C2×Q8—124no—
497C2×Q8—124no—
498C2×Q8C2×#6712yes—
499C2×Q8C2×#6712yes—
500C2×Q8C23×D312yes—
501C2×D4—28no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—28no—
505C2×D4—24no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—44no—
510C2×D4C2×#6714yes—
511C2×D4C2×#6714yes—
512C2×D4C2×#6712yes—
513C2×D4C2×#6712yes—
514C2×D4C23×D312yes—
515C22×C4—24no—
516C22×C4—24no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—44no—
522C22×C4—62no—
523C22×C4—62no—
524C22×C4—62no—
525C22×C4—62no—
526C22×C4—124no—
527C22×C4D3×D414yes—
528C22×C4#23714yes—
529C22×C4C2×#6712yes—
530C22×C4C2×#6714yes—
531C22×C4C2×#6711yescharacteristic
532C22×C4C2×#6711yescharacteristic
533C22×C4C2×#6711yescharacteristic
534C22×C4C23×D314yes—
535C22×C4C23×D311yescharacteristic
536C22×C4C23×D311yescharacteristic
537C22×C4C23×D311yescharacteristic
538Q16—616no—
539SD16—616no—
540M16—38no—
541M16—38no—
542M16—616no—
543M16—64no—
544M16—64no—
545C2×C8—38no—
546C2×C8—34no—
547C2×C8—34no—
548C2×C8—616no—
549C2×C8—64no—
550C4⋊C4—22no—
551C4⋊C4—28no—
552C4⋊C4—34no—
553C4⋊C4—62no—
554C4⋊C4—616no—
555C4⋊C4C23×D314yes—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—28no—
559C22⋊C4—28no—
560C22⋊C4—62no—
561C22⋊C4—62no—
562C22⋊C4—616no—
563C42—22no—
564C42—22no—
565C42—62no—
566C42—62no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—22no—
570C22×C3—28no—
571C22×C3—24no—
572C22×C3—44no—
573C22×C3—44no—
574C22×C3—42no—
575C22×C3—84no—
576C22×C3—84no—
577C22×C3C2×#12112yes—
578C22×C3C2×#12112yes—
579C22×C3C2×#12112yes—
580C22×C3C23×D411yescharacteristic
581C3×C4—24no—
582C3×C4—44no—
583C3×C4—42no—
584C3×C4C2×#12112yes—
585C3×C4C2×#12112yes—
586C3×C4C2×#12112yes—
587C3×C4C23×D411yescharacteristic
588C3×C4C22×#4111yescharacteristic
589Dic3—44no—
590Dic3—44no—
591C23—24no—
592C23—22no—
593C23—42no—
594C23—44no—
595C23—42no—
596C23—82no—
597C23#93914yes—
598C23C22×#6714yes—
599C23C22×#6711yescharacteristic
600C23C22×#6711yescharacteristic
601C23C22×#6711yescharacteristic
602Q8—24no—
603Q8—24no—
604Q8—22no—
605Q8—22no—
606Q8—42no—
607Q8—64no—
608Q8—64no—
609Q8—124no—
610Q8—124no—
611Q8—124no—
612D4—28no—
613D4—28no—
614D4—22no—
615D4—22no—
616D4—24no—
617D4—24no—
618D4—44no—
619D4—44no—
620D4—42no—
621D4—88no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—64no—
632C2×C4—64no—
633C2×C4—82no—
634C2×C4—124no—
635C2×C4—124no—
636C2×C4—124no—
637C2×C4—124no—
638C2×C4#93914yes—
639C2×C4#93914yes—
640C2×C4#93914yes—
641C2×C4C2×D3×D412yes—
642C2×C4C2×D3×D412yes—
643C2×C4C2×D3×D412yes—
644C2×C4C2×#23712yes—
645C2×C4C2×#23712yes—
646C2×C4C2×#23712yes—
647C2×C4C22×#6714yes—
648C2×C4C22×#6711yescharacteristic
649C2×C4C22×#6711yescharacteristic
650C2×C4C22×#6711yescharacteristic
651C2×C4C24×D311yescharacteristic
652C8—38no—
653C8—64no—
654C2×C3—22no—
655C2×C3—22no—
656C2×C3—22no—
657C2×C3—44no—
658C2×C3—82no—
659C2×C3C22×#12111yescharacteristic
660C2×C3#358612yes—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—28no—
665C22—24no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—84no—
670C22—84no—
671C22C2×#93912yes—
672C22C2×#93912yes—
673C22C2×#93912yes—
674C22C23×#6711yescharacteristic
675C4—24no—
676C4—44no—
677C4—42no—
678C4—124no—
679C4—124no—
680C4C2×#93912yes—
681C4C2×#93912yes—
682C4C2×#93912yes—
683C4C22×D3×D411yescharacteristic
684C4C22×#23711yescharacteristic
685C3#6045511yescharacteristic, complemented, Sylow
686C2—22no—
687C2—22no—
688C2—22no—
689C2—44no—
690C2—82no—
691C2C22×#93911yescharacteristic
692C2#8729912yes—
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC3×#3449C2
socleC22×C3C23×D4
hypercenter#3449D3
hypocenterC3#60455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×#41 ◁ #3449

Complements

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

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C22×#951C2semidirect
4. C2×#5819C2semidirect
5. #84072C2semidirect
6. #83423C2semidirect
7. #83399C2semidirect
8. #83377C2semidirect
9. #82925C2semidirect
10. C2×#5004C2semidirect
11. #82207C2semidirect
29. #5825C22semidirect
32. C2×#951C22semidirect
35. C2×#948C22semidirect
37. C22×#216C22semidirect
40. C22×#214C22semidirect
42. C2×#924C22semidirect
45. C22×#208C22semidirect
46. #5233C22semidirect
48. #5225C22semidirect
49. #5215C22semidirect
50. #5052C22semidirect
51. #5051C22semidirect
52. #5042C22semidirect
53. #5041C22semidirect
54. #5004C22semidirect
55. #5004C22semidirect
56. #4998C22semidirect
57. #4995C22semidirect
58. C2×#810C22semidirect
59. #4964C22semidirect
149. C2×#208D4semidirect
685. C3#60455semidirect

Maximal subgroups

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

subgroupindex
C3×#34492
C2×#58252
C22×#9512
C2×#58192
#840722
#834232
#833992
#833772
#829252
C2×#50042
#822072
#604553