Finite Group Databasebeta

← #11662461

Subgroups — #11662461

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
1C22×C3×#137C211yesmaximal, characteristic, complemented
2#87268C211yesmaximal, characteristic
3C2×#5816C211yesmaximal, characteristic, complemented
4C2×#5745C211yesmaximal, characteristic, complemented
5#84072C211yesmaximal, characteristic, complemented
6C2×#5225C211yesmaximal, characteristic, complemented
7#83986C218yesmaximal, complemented
8#83985C218yesmaximal, complemented
9#83887C214yesmaximal, complemented
10#83863C214yesmaximal, complemented
11#83752C211yesmaximal, characteristic, complemented
12#60455—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C2×C3×#137—28no—
17C2×C3×#137C2214yes—
18C2×C3×#137C2214yescomplemented
19C22×C3×Q16C2212yescomplemented
20C22×C3×SD16C2212yescomplemented
21C22×C3×M16C2211yescharacteristic
22#5833C2214yes—
23#5832—22no—
24#5830—21no—
25#5828C2214yes—
26#5825—21no—
27#5816—22no—
28#5816—22no—
29#5816C2214yes—
30#5816C2212yes—
31#5816C2214yescomplemented
32C2×#947—21no—
33C2×#946C2214yescomplemented
34C2×#944—21no—
35#5805C2212yes—
36#5804—22no—
37#5791—22no—
38C2×#929—22no—
39C2×#926C2211yescharacteristic
40C2×#926C2211yescharacteristic
41C22×#208C2211yescharacteristic
42#5745—28no—
43C2×#907C2214yescomplemented
44C2×D3×M16C2214yescomplemented
45C2×#901C2214yes—
46#5233C2214yescomplemented
47#5232C2214yes—
48#5225C2214yescomplemented
49#5225C2214yescomplemented
50#5225C2212yescomplemented
51C2×#835C2214yescomplemented
52#5215C2212yescomplemented
53#5189C22116yescomplemented
54#5188C22116yescomplemented
55#5174C22116yescomplemented
56#5171C22116yescomplemented
57#5121C2214yescomplemented
58#5121C2214yescomplemented
59#5106C2214yescomplemented
60C22×#137D311yescharacteristic, complemented
61C2×#568—31no—
62#3449—31no—
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—21no—
72C2×C3×#41—21no—
73C2×C3×#41—24no—
74C2×C3×#41—24no—
75C2×C3×#41C2312yes—
76C2×C3×#41C2312yes—
77C22×C3×Q8—21no—
78C22×C3×Q8—21no—
79C22×C3×Q8—21no—
80C22×C3×Q8—24no—
81C22×C3×Q8—24no—
82C22×C3×Q8C2312yes—
83C22×C3×Q8C2311yescharacteristic
84C22×C3×Q8C2312yes—
85C22×C3×Q8C2312yes—
86C22×C3×D4—21no—
87C22×C3×D4C2311yescharacteristic
88C23×C3×C4—21no—
89C23×C3×C4—21no—
90C23×C3×C4C2311yescharacteristic
91C22×#67D412yes—
92C2×#235—24no—
93C2×#235—24no—
94C2×#235D412yes—
95C2×#235D412yes—
96C2×#235C2312yes—
97C2×#235C2312yes—
98C22×D12D411yescharacteristic
99C22×C4×D3C2312yes—
100C22×Dic6D411yescharacteristic
101C3×#137—216no—
102C3×#137—216no—
103C2×C3×Q16—28no—
104C2×C3×Q16C2318yes—
105C2×C3×SD16—28no—
106C2×C3×SD16C2318yes—
107C2×C3×M16D418yescomplemented
108C2×C3×M16C2312yes—
109C2×C3×M16C2312yes—
110C22×C3×C8C2312yes—
111#953—24no—
112C2×#217—21no—
113C2×#217—21no—
114#947—24no—
115#947—24no—
116#947—22no—
117#946—28no—
118#946C23116yes—
119#946C2318yes—
120#944—24no—
121#944—24no—
122#944—22no—
123#940—22no—
124#939—28no—
125#938C2318yes—
126#937—24no—
127#936—22no—
128#935—24no—
129#930—24no—
130#929—28no—
131#929—28no—
132#929—24no—
133C4×#67—28no—
134C2×(C2×D3)⋊C4—24no—
135#926—21no—
136#926—21no—
137#926—21no—
138#926—21no—
139#926C2312yes—
140#926C2312yes—
141#926C2312yes—
142#926C2312yes—
143C2×C4⋊Dic3C2312yes—
144C2×C4⋊Dic3C2312yes—
145#924—24no—
146C2×Dic3⋊C4—24no—
147C2×C4×Dic3—21no—
148C2×C4×Dic3—21no—
149C2×#208—24no—
150C2×#208C2312yes—
151C2×#208C2312yes—
152C22×C3⋊C8C2312yes—
153#907—216no—
154D3×M16—216no—
155#901—216no—
156C2×ΓL(1,𝔽25)C2318yes—
157C2×D3×C8C2318yes—
158#835C23116yes—
159#835C2318yes—
160#832C2318yes—
161#818C2318yes—
162#817C2318yes—
163C22×#41—31no—
164C22×#41C2×D311yescharacteristic
165C23×Q8C2×D311yescharacteristic
166C2×#137—28no—
167C2×#137C2×D314yes—
168C2×#137C2×D314yescomplemented
169C22×Q16C2×D312yescomplemented
170C22×SD16C2×D312yescomplemented
171#568—68no—
172C2×#131—34no—
173C2×#131—34no—
174C22×M16—31no—
175C22×M16—34no—
176C22×M16C2×D311yescharacteristic
177#537—34no—
178#536—34no—
179#536—34no—
180#536—32no—
181#536—62no—
182#536—62no—
183#536—61no—
184#535—32no—
185#535—61no—
186#534—62no—
187#532—62no—
188C2×C4⋊Q8—61no—
189C2×#124—61no—
190#525—34no—
191#525—62no—
192C2×#123—62no—
193C2×C22⋊Q8—34no—
194C2×#117—31no—
195C2×#117—31no—
196#470—316no—
197#468—316no—
198#467—316no—
199#465—316no—
200#428—34no—
201#428—34no—
202#427—34no—
203#419—34no—
204#419—34no—
205#419—32no—
206#418—32no—
207#417—34no—
208#416—34no—
209C2×#103—34no—
210C24×C3—21no—
211C24×C3C2×D411yescharacteristic
212C23×D3C2×D412yes—
213C3×#41—24no—
214C3×#41—24no—
215C3×#41—24no—
216C3×#41—24no—
217C3×#41—44no—
218C3×#41—44no—
219C2×C3×Q8—22no—
220C2×C3×Q8—22no—
221C2×C3×Q8—21no—
222C2×C3×Q8—22no—
223C2×C3×Q8—24no—
224C2×C3×Q8—22no—
225C2×C3×Q8—24no—
226C2×C3×Q8—24no—
227C2×C3×Q8—24no—
228C2×C3×Q8—24no—
229C2×C3×Q8—24no—
230C2×C3×Q8—22no—
231C2×C3×Q8—44no—
232C2×C3×Q8—44no—
233C2×C3×Q8—44no—
234C2×C3×Q8C2414yes—
235C2×C3×Q8C2412yes—
236C2×C3×D4—22no—
237C2×C3×D4—22no—
238C2×C3×D4—22no—
239C2×C3×D4—21no—
240C2×C3×D4—22no—
241C2×C3×D4—44no—
242C2×C3×D4C2412yes—
243C22×C3×C4—22no—
244C22×C3×C4—21no—
245C22×C3×C4—22no—
246C22×C3×C4—22no—
247C22×C3×C4—22no—
248C22×C3×C4—22no—
249C22×C3×C4—44no—
250C22×C3×C4—44no—
251C22×C3×C4C2×D411yescharacteristic
252C22×C3×C4C2×D411yescharacteristic
253C22×C3×C4C2×D411yescharacteristic
254C22×C3×C4C2×D414yes—
255C22×C3×C4#4114yes—
256C22×C3×C4C2411yescharacteristic
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2411yescharacteristic
259C2×#67—28no—
260C2×#67—28no—
261C2×#67C2×D414yes—
262C2×#67C2×D414yes—
263C22×Dic3—22no—
264C22×Dic3—22no—
265C22×Dic3C2×D412yes—
266#235—28no—
267#235—28no—
268#235—24no—
269#235—24no—
270#235—24no—
271#235—24no—
272C2×D12—24no—
273C2×D12C2×D412yes—
274C2×D12C2×D412yes—
275C2×C4×D3—24no—
276C2×C4×D3—24no—
277C2×C4×D3C2×D414yes—
278C2×C4×D3C2414yes—
279C2×Dic6—24no—
280C2×Dic6C2×D412yes—
281C2×Dic6C2×D412yes—
282C3×Q16—216no—
283C3×SD16—216no—
284C3×M16C2×D418yes—
285C3×M16C2×D418yes—
286C2×C3×C8C2×D418yes—
287C2×C3×C8C2414yes—
288#217—22no—
289#217—22no—
290#217—22no—
291#217—22no—
292#217—28no—
293(C2×D3)⋊C4—216no—
294(C2×D3)⋊C4—28no—
295C4⋊Dic3—22no—
296C4⋊Dic3—22no—
297C4⋊Dic3C2414yes—
298C4⋊Dic3C2414yes—
299Dic3⋊C4—216no—
300Dic3⋊C4—28no—
301C4×Dic3—22no—
302C4×Dic3—22no—
303C4×Dic3—22no—
304C4×Dic3—22no—
305#208—24no—
306#208—24no—
307C2×C3⋊C8—24no—
308C2×C3⋊C8C2414yes—
309ΓL(1,𝔽25)—216no—
310D3×C8—216no—
311C2×#41—21no—
312C2×#41—21no—
313C2×#41—24no—
314C2×#41—24no—
315C2×#41—32no—
316C2×#41—32no—
317C2×#41—32no—
318C2×#41—32no—
319C2×#41—64no—
320C2×#41—64no—
321C2×#41C22×D312yes—
322C2×#41C22×D312yes—
323C22×Q8—21no—
324C22×Q8—21no—
325C22×Q8—21no—
326C22×Q8—24no—
327C22×Q8—24no—
328C22×Q8—31no—
329C22×Q8C22×D312yes—
330C22×Q8C22×D312yes—
331C22×Q8C22×D311yescharacteristic
332C22×Q8C22×D312yes—
333C22×D4—21no—
334C22×D4—32no—
335C22×D4—31no—
336C22×D4C22×D311yescharacteristic
337C23×C4—21no—
338C23×C4—21no—
339C23×C4—32no—
340C23×C4C22×D311yescharacteristic
341#137—216no—
342#137—216no—
343C2×Q16—28no—
344C2×Q16C22×D318yes—
345C2×SD16—28no—
346C2×SD16C22×D318yes—
347#131—616no—
348#131—616no—
349C2×M16—32no—
350C2×M16—32no—
351C2×M16—38no—
352C2×M16—64no—
353C2×M16—616no—
354C2×M16#6718yescomplemented
355C2×M16C22×D312yes—
356C2×M16C22×D312yes—
357C22×C8—32no—
358C22×C8—38no—
359C22×C8C22×D312yes—
360C4⋊Q8—64no—
361C4⋊Q8—64no—
362C4⋊Q8—62no—
363#127—62no—
364#124—62no—
365#124—64no—
366#124—64no—
367#124—62no—
368#123—68no—
369#123—68no—
370#123—64no—
371#123—64no—
372C22⋊Q8—316no—
373C22⋊Q8—38no—
374C22⋊Q8—64no—
375C22⋊Q8—68no—
376#121—38no—
377#121—64no—
378#121—68no—
379#120—64no—
380#120—64no—
381C4×D4—68no—
382#117—32no—
383#117—32no—
384#117—32no—
385#117—32no—
386#117—61no—
387#117—61no—
388#117—61no—
389#117—61no—
390C2×C4⋊C4—32no—
391C2×C4⋊C4—32no—
392C2×C4⋊C4—64no—
393C2×C22⋊C4—64no—
394C2×C22⋊C4—61no—
395C2×C22⋊C4—61no—
396C2×C42—61no—
397C2×C42—61no—
398#107—38no—
399#106—38no—
400#103—316no—
401#103—38no—
402#102—38no—
403C23×C3—22no—
404C23×C3—42no—
405C23×C3—42no—
406C23×C3—82no—
407C23×C3#12114yes—
408C23×C3C22×D411yescharacteristic
409C23×C3C22×D411yescharacteristic
410C23×C3C22×D411yescharacteristic
411C22×D3—24no—
412C22×D3—44no—
413C22×D3C22×D414yes—
414C3×Q8—24no—
415C3×Q8—22no—
416C3×Q8—24no—
417C3×Q8—22no—
418C3×Q8—44no—
419C3×Q8—44no—
420C3×Q8—44no—
421C3×Q8—42no—
422C3×D4—22no—
423C3×D4—22no—
424C3×D4—44no—
425C3×D4—44no—
426C3×D4—42no—
427C2×C3×C4—22no—
428C2×C3×C4—24no—
429C2×C3×C4—42no—
430C2×C3×C4—44no—
431C2×C3×C4—44no—
432C2×C3×C4—44no—
433C2×C3×C4—44no—
434C2×C3×C4—44no—
435C2×C3×C4#12114yes—
436C2×C3×C4#12114yes—
437C2×C3×C4#12114yes—
438C2×C3×C4C22×D411yescharacteristic
439C2×C3×C4C22×D411yescharacteristic
440C2×C3×C4C22×D411yescharacteristic
441C2×C3×C4C22×D412yes—
442C2×C3×C4C22×D412yes—
443C2×C3×C4C22×D412yes—
444C2×C3×C4C2×#4112yes—
445C2×C3×C4C2×#4112yes—
446C2×C3×C4C2×#4112yes—
447C2×C3×C4C2511yescharacteristic
448#67—28no—
449#67—28no—
450#67—88no—
451C2×Dic3—24no—
452C2×Dic3—24no—
453C2×Dic3—24no—
454C2×Dic3—44no—
455C2×Dic3—82no—
456C2×Dic3C22×D414yes—
457D12—24no—
458D12—24no—
459C4×D3—28no—
460C4×D3—24no—
461C4×D3—24no—
462Dic6—24no—
463Dic6—24no—
464C3×C8C22×D418yes—
465C3⋊C8—24no—
466C24—21no—
467C24—32no—
468C24D3×D411yescharacteristic
469#41—24no—
470#41—24no—
471#41—24no—
472#41—24no—
473#41—44no—
474#41—44no—
475#41—68no—
476#41—68no—
477#41—64no—
478#41—64no—
479#41—64no—
480#41—64no—
481C2×Q8—22no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—21no—
485C2×Q8—24no—
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—22no—
489C2×Q8—24no—
490C2×Q8—22no—
491C2×Q8—24no—
492C2×Q8—24no—
493C2×Q8—32no—
494C2×Q8—32no—
495C2×Q8—44no—
496C2×Q8—44no—
497C2×Q8—44no—
498C2×Q8—64no—
499C2×Q8C23×D314yes—
500C2×Q8C23×D312yes—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—21no—
504C2×D4—22no—
505C2×D4—22no—
506C2×D4—34no—
507C2×D4—34no—
508C2×D4—32no—
509C2×D4—32no—
510C2×D4—44no—
511C2×D4—68no—
512C2×D4—68no—
513C2×D4—64no—
514C2×D4C23×D312yes—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—32no—
522C22×C4—34no—
523C22×C4—34no—
524C22×C4—44no—
525C22×C4—44no—
526C22×C4—64no—
527C22×C4—64no—
528C22×C4—62no—
529C22×C4—62no—
530C22×C4D3×D411yescharacteristic
531C22×C4D3×D411yescharacteristic
532C22×C4D3×D411yescharacteristic
533C22×C4#23714yes—
534C22×C4C2×#6714yes—
535C22×C4C23×D311yescharacteristic
536C22×C4C23×D311yescharacteristic
537C22×C4C23×D311yescharacteristic
538Q16—216no—
539SD16—216no—
540M16—64no—
541M16—64no—
542M16—616no—
543M16C2×#6718yes—
544M16C2×#6718yes—
545C2×C8—34no—
546C2×C8—64no—
547C2×C8—616no—
548C2×C8C2×#6718yes—
549C2×C8C23×D314yes—
550C4⋊C4—34no—
551C4⋊C4—34no—
552C4⋊C4—616no—
553C4⋊C4—68no—
554C4⋊C4—62no—
555C4⋊C4—62no—
556C22⋊C4—616no—
557C22⋊C4—68no—
558C22⋊C4—68no—
559C22⋊C4—62no—
560C22⋊C4—62no—
561C22⋊C4—62no—
562C22⋊C4—62no—
563C42—62no—
564C42—62no—
565C42—62no—
566C42—62no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—22no—
570C22×C3—44no—
571C22×C3—44no—
572C22×C3—42no—
573C22×C3—84no—
574C22×C3C2×#12112yes—
575C22×C3C2×#12112yes—
576C22×C3C2×#12112yes—
577C22×C3C23×D411yescharacteristic
578C2×D3—28no—
579C2×D3—24no—
580C2×D3—84no—
581C3×C4—42no—
582C3×C4—44no—
583C3×C4C2×#12112yes—
584C3×C4C2×#12112yes—
585C3×C4C2×#12112yes—
586C3×C4C23×D411yescharacteristic
587C3×C4C22×#4111yescharacteristic
588Dic3—24no—
589Dic3—44no—
590Dic3—44no—
591C23—22no—
592C23—34no—
593C23—42no—
594C23—42no—
595C23—64no—
596C23—82no—
597C23—124no—
598C23#93814yes—
599C23C2×D3×D411yescharacteristic
600C23C2×D3×D411yescharacteristic
601C23C2×D3×D411yescharacteristic
602Q8—24no—
603Q8—22no—
604Q8—24no—
605Q8—22no—
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—42no—
610Q8—64no—
611Q8—64no—
612D4—22no—
613D4—22no—
614D4—44no—
615D4—44no—
616D4—42no—
617D4—68no—
618D4—68no—
619D4—64no—
620D4—64no—
621D4—248no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—34no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—64no—
632C2×C4—64no—
633C2×C4—64no—
634C2×C4—68no—
635C2×C4—64no—
636C2×C4—64no—
637C2×C4—124no—
638C2×C4—242no—
639C2×C4#93814yes—
640C2×C4#93814yes—
641C2×C4#93814yes—
642C2×C4C2×D3×D411yescharacteristic
643C2×C4C2×D3×D411yescharacteristic
644C2×C4C2×D3×D411yescharacteristic
645C2×C4C2×#23712yes—
646C2×C4C2×#23712yes—
647C2×C4C2×#23712yes—
648C2×C4C22×#6712yes—
649C2×C4C22×#6712yes—
650C2×C4C22×#6712yes—
651C2×C4C24×D311yescharacteristic
652C8—64no—
653C8C22×#6718yes—
654C2×C3—22no—
655C2×C3—22no—
656C2×C3—22no—
657C2×C3—82no—
658C2×C3C22×#12111yescharacteristic
659C2×C3#358612yes—
660D3—44no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—44no—
665C22—44no—
666C22—42no—
667C22—68no—
668C22—64no—
669C22—84no—
670C22—244no—
671C22C2×#93812yes—
672C22C2×#93812yes—
673C22C2×#93812yes—
674C22C22×D3×D411yescharacteristic
675C4—42no—
676C4—44no—
677C4—64no—
678C4—124no—
679C4—124no—
680C4C2×#93812yes—
681C4C2×#93812yes—
682C4C2×#93812yes—
683C4C22×#23711yescharacteristic
684C4C23×#6711yescharacteristic
685C3#6045511yescharacteristic, complemented, Sylow
686C2—22no—
687C2—22no—
688C2—22no—
689C2—82no—
690C2—124no—
691C2#8719712yes—
692C2C22×#93811yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×#137C2
socleC22×C3C23×D4
prime radical#87268C2
hypercenterC22×#137D3
hypocenterC3#60455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C22×C3×#137C2semidirect
3. C2×#5816C2semidirect
4. C2×#5745C2semidirect
5. #84072C2semidirect
6. C2×#5225C2semidirect
7. #83986C2semidirect
8. #83985C2semidirect
9. #83887C2semidirect
10. #83863C2semidirect
11. #83752C2semidirect
18. C2×C3×#137C22semidirect
19. C22×C3×Q16C22semidirect
20. C22×C3×SD16C22semidirect
31. #5816C22semidirect
33. C2×#946C22semidirect
43. C2×#907C22semidirect
44. C2×D3×M16C22semidirect
46. #5233C22semidirect
48. #5225C22semidirect
49. #5225C22semidirect
50. #5225C22semidirect
51. C2×#835C22semidirect
52. #5215C22semidirect
53. #5189C22semidirect
54. #5188C22semidirect
55. #5174C22semidirect
56. #5171C22semidirect
57. #5121C22semidirect
58. #5121C22semidirect
59. #5106C22semidirect
60. C22×#137D3semidirect
107. C2×C3×M16D4semidirect
168. C2×#137C2×D3semidirect
169. C22×Q16C2×D3semidirect
170. C22×SD16C2×D3semidirect
354. C2×M16#67semidirect
685. C3#60455semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1372
#872682
C2×#58162
C2×#57452
#840722
C2×#52252
#839862
#839852
#838872
#838632
#837522
#604553