Finite Group Databasebeta

← #10818125

Subgroups — #10818125

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2913C211yesmaximal, characteristic, complemented
2C2×#5141C211yesmaximal, characteristic
3#82571C211yesmaximal, characteristic, complemented
4#82564C211yesmaximal, characteristic
5#79168C211yesmaximal, characteristic, complemented
6#72009C211yesmaximal, characteristic, complemented
7#71667C211yesmaximal, characteristic
8#71666C211yesmaximal, characteristic, complemented
9#70677C211yesmaximal, characteristic, complemented
10#70674C211yesmaximal, characteristic, complemented
11#70673C211yesmaximal, characteristic, complemented
12#70672C211yesmaximal, characteristic, complemented
13#70660C211yesmaximal, characteristic, complemented
14#70629C211yesmaximal, characteristic, complemented
15#70625C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C3×#520C2211yescharacteristic
20C2×#951C2211yescharacteristic
21C2×#949C2211yescharacteristic
22C22×#204C2211yescharacteristic
23#5521C2211yescharacteristic
24#5516C2211yescharacteristic
25C3×#428—21no—
26C3×#428—21no—
27C3×#428C2211yescharacteristic
28C3×#428C2211yescharacteristic
29C3×#427—21no—
30C3×#427C2211yescharacteristic
31C2×C3×#107—21no—
32C2×C3×#107C2211yescharacteristic
33C2×#837C2211yescharacteristic
34#5233C2211yescharacteristic
35#5141C418yescomplemented
36C2×#824C2211yescharacteristic
37C2×#823C2211yescharacteristic
38C2×#819C2211yescharacteristic
39#5101C2211yescharacteristic
40C2×#815C2211yescharacteristic
41#5030—21no—
42C4×#216—21no—
43#5026—21no—
44C4×#215—21no—
45#5018—21no—
46C4×#214—21no—
47#5014—21no—
48C4×#213—21no—
49#5004C2211yescharacteristic
50#5001C2211yescharacteristic
51#4997C2211yescharacteristic
52#4997C2211yescharacteristic
53#4988C2211yescharacteristic
54#4973C2211yescharacteristic
55C2×#810C2211yescharacteristic
56C2×#809C2211yescharacteristic
57#4964C2211yescharacteristic
58#4964C2211yescharacteristic
59#4889—21no—
60#4889—21no—
61#4852—21no—
62#4852—21no—
63#4558C2211yescharacteristic
64#4555C2211yescharacteristic
65#4532C2211yescharacteristic
66#4532C2211yescharacteristic
67#4530C2211yescharacteristic
68#4530C2211yescharacteristic
69C2×#471—31no—
70#2949—31no—
71#2947—31no—
72#2946—31no—
73#2913D311yescharacteristic, complemented
74#1973—31no—
75#1952—31no—
76#1931—31no—
77#1930—31no—
78#1883—31no—
79#1882—31no—
80#1857—31no—
81#1856—31no—
82#1773—31no—
83#1765—31no—
84C2×C3×#41C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C3×#131—22no—
88C3×#131C2×C414yescomplemented
89C2×C3×M16—21no—
90C2×C3×M16C2311yescharacteristic
91C22×C3×C8—21no—
92C22×C3×C8C2311yescharacteristic
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—41no—
98C3×C4×D4—41no—
99C3×#117—21no—
100C3×#117—21no—
101C3×#117C2311yescharacteristic
102C3×#117C2311yescharacteristic
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107#951—22no—
108#951C2×C414yescomplemented
109#949—22no—
110#949C2×C414yes—
111C2×#216—21no—
112C2×#215—21no—
113C2×#214—21no—
114C2×#213—21no—
115C2×(C2×D3)⋊C4—41no—
116#926C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C2×#208C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×#204—21no—
123C2×#204—21no—
124C2×#204—22no—
125C2×#204C2×C414yescomplemented
126#893—21no—
127#893—21no—
128#887—21no—
129#887—21no—
130D3×C22⋊C4—41no—
131#877—41no—
132C4×D12—21no—
133C4×D12—21no—
134C4×Dic6—21no—
135C4×Dic6—21no—
136C3×#107—21no—
137C3×#107—21no—
138C3×#107—21no—
139C3×#107—21no—
140C3×#107D411yescharacteristic
141C3×#107D411yescharacteristic
142C3×#106—21no—
143C3×#106—21no—
144C3×#106D411yescharacteristic
145C3×#106D411yescharacteristic
146#837—22no—
147#835D411yescharacteristic
148#835D411yescharacteristic
149#832D411yescharacteristic
150#832D411yescharacteristic
151C3⋊#95—41no—
152#824C2×C414yescomplemented
153#823C2×C414yescomplemented
154#819C2×C414yescomplemented
155#815—21no—
156#815—22no—
157C8×Dic3—21no—
158#810—21no—
159#810—21no—
160#810—21no—
161#810—21no—
162#809—21no—
163#809—21no—
164#809—21no—
165#809—21no—
166#808—21no—
167#808—21no—
168#807—21no—
169#807—21no—
170#803—21no—
171#803—21no—
172C4×C3⋊C8—21no—
173C4×C3⋊C8—21no—
174C2×#137—31no—
175C2×AGL(1,ℤ/8)—31no—
176C22×SD16—31no—
177C2×#131C2×D311yescharacteristic
178#520C2×D311yescharacteristic
179#520C2×D311yescharacteristic
180#519—31no—
181#518—31no—
182#471—38no—
183#442—61no—
184#441—61no—
185#440—61no—
186#440—61no—
187#440—61no—
188#440—61no—
189C4×Q16—61no—
190C4×SD16—61no—
191C4×SD16—61no—
192C4×SD16—61no—
193C4×SD16—61no—
194C4×D8—61no—
195C2×#108—31no—
196#428—21no—
197#428—21no—
198#428—31no—
199#428—31no—
200#428C2×D311yescharacteristic
201#428C2×D311yescharacteristic
202#427—21no—
203#427C2×D311yescharacteristic
204C2×#107—21no—
205C2×#107C2×D311yescharacteristic
206C2×#104—31no—
207#419—31no—
208#418—31no—
209#416—31no—
210C2×#103—31no—
211C2×#102—31no—
212C2×#101—31no—
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405—31no—
217C2×#97—31no—
218#391—31no—
219#390—31no—
220#343—31no—
221#343—31no—
222#341—31no—
223#339—31no—
224#339—31no—
225#339—31no—
226C23×D3—21no—
227C3×#41—21no—
228C3×#41—21no—
229C3×#41C22×C412yes—
230C3×#41#4112yes—
231C2×C3×Q8C2×D411yescharacteristic
232C2×C3×D4—21no—
233C2×C3×D4C2×D411yescharacteristic
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—41no—
238C22×C3×C4—41no—
239C22×C3×C4C2411yescharacteristic
240C22×Dic3—21no—
241C22×Dic3—21no—
242C2×D12—21no—
243C2×D12—21no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12—21no—
247C2×D12C22×C412yes—
248C2×C4×D3—41no—
249C2×C4×D3—41no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6C22×C412yes—
256C3×M16—21no—
257C3×M16—24no—
258C3×M16C22×C412yes—
259C2×C3×C8—21no—
260C2×C3×C8—24no—
261C2×C3×C8C22×C412yes—
262C2×C3×C8C2×D411yescharacteristic
263C2×C3×C8C2×D411yescharacteristic
264C2×C3×C8#4112yes—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—41no—
270C3×C4⋊C4—41no—
271C3×C4⋊C4C2×D411yescharacteristic
272C3×C4⋊C4C2×D411yescharacteristic
273C3×C4⋊C4C2×D411yescharacteristic
274C3×C4⋊C4C2×D411yescharacteristic
275C3×C22⋊C4—21no—
276C3×C22⋊C4—21no—
277C3×C22⋊C4—41no—
278C3×C22⋊C4—41no—
279C3×C42—21no—
280C3×C42—21no—
281C3×C42—41no—
282C3×C42—41no—
283#217—21no—
284#216—24no—
285#215—24no—
286#214—24no—
287#213—24no—
288(C2×D3)⋊C4—41no—
289(C2×D3)⋊C4—41no—
290(C2×D3)⋊C4—82no—
291C4⋊Dic3C2×D411yescharacteristic
292C4⋊Dic3C2×D411yescharacteristic
293Dic3⋊C4—41no—
294Dic3⋊C4—41no—
295C4×Dic3—21no—
296C4×Dic3—21no—
297C4×Dic3—21no—
298C4×Dic3—41no—
299C4×Dic3—41no—
300#208—21no—
301#208—21no—
302#208—21no—
303#208—21no—
304#208C22×C412yes—
305#208C22×C412yes—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—21no—
308C2×C3⋊C8—21no—
309C2×C3⋊C8—21no—
310C2×C3⋊C8—21no—
311C2×C3⋊C8C2×D411yescharacteristic
312C2×C3⋊C8C2×D411yescharacteristic
313#204—24no—
314#204—24no—
315C2×#41C22×D311yescharacteristic
316C22×Q8—31no—
317C22×D4—31no—
318#137—34no—
319#137—62no—
320AGL(1,ℤ/8)—34no—
321AGL(1,ℤ/8)—62no—
322C2×Q16—61no—
323C2×SD16—34no—
324C2×SD16—61no—
325C2×SD16—61no—
326C2×SD16—62no—
327C2×SD16—61no—
328C2×SD16—61no—
329C2×D8—61no—
330#131—22no—
331#131C4×D314yescomplemented
332C2×M16—21no—
333C2×M16—31no—
334C2×M16—31no—
335C2×M16—31no—
336C2×M16C22×D311yescharacteristic
337C22×C8—21no—
338C22×C8—31no—
339C22×C8C22×D311yescharacteristic
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8—61no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—41no—
349C4×D4—41no—
350C4×D4—61no—
351C4×D4—61no—
352C4×D4—61no—
353C4×D4—61no—
354#117—21no—
355#117—21no—
356#117—31no—
357#117—121no—
358#117C22×D311yescharacteristic
359#117C22×D311yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4C22×D311yescharacteristic
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—121no—
365C2×C22⋊C4—121no—
366C2×C42—31no—
367#108—34no—
368#107—21no—
369#107—21no—
370#107—21no—
371#107—21no—
372#107—61no—
373#107—61no—
374#107#6711yescharacteristic
375#107#6711yescharacteristic
376#106—21no—
377#106—21no—
378#106—61no—
379#106—61no—
380#106#6711yescharacteristic
381#106#6711yescharacteristic
382#104—62no—
383#103—31no—
384#103—31no—
385#103—61no—
386#103—61no—
387#103—61no—
388#103—61no—
389#102—31no—
390#102—31no—
391#102—61no—
392#102—61no—
393#102—61no—
394#102—61no—
395#101—34no—
396#100—34no—
397#97—61no—
398#97—61no—
399#97—62no—
400#97—61no—
401C4×C8—61no—
402C4×C8—61no—
403C4×C8—61no—
404#95—121no—
405C23×C3—21no—
406C23×C3H(5,𝔽2)11yescharacteristic
407C22×D3—22no—
408C22×D3—41no—
409C22×D3—41no—
410C22×D3—81no—
411C22×D3—81no—
412C3×Q8—21no—
413C3×Q8C4×D412yes—
414C3×D4—22no—
415C3×D4—22no—
416C3×D4—21no—
417C3×D4C4×D412yes—
418C2×C3×C4—21no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—21no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—81no—
428C2×C3×C4—81no—
429C2×C3×C4—161no—
430C2×C3×C4—161no—
431C2×C3×C4C23×C411yescharacteristic
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C2×#4111yescharacteristic
435C2×C3×C4C2×#4111yescharacteristic
436C2×C3×C42−1+411yescharacteristic
437C2×Dic3—22no—
438C2×Dic3—21no—
439C2×Dic3—21no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445D12—22no—
446D12—22no—
447D12—41no—
448D12—41no—
449C4×D3—161no—
450C4×D3—161no—
451Dic6—22no—
452Dic6—22no—
453Dic6—41no—
454Dic6—41no—
455C3×C8—22no—
456C3×C8C4×D412yes—
457C3×C8C4×D412yes—
458C3⋊C8—22no—
459C3⋊C8—22no—
460C3⋊C8—21no—
461C3⋊C8—21no—
462C3⋊C8—41no—
463C24—61no—
464#41—21no—
465#41—21no—
466#41C2×C4×D312yes—
467#41#23912yes—
468C2×Q8—32no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8D3×D411yescharacteristic
475C2×D4—21no—
476C2×D4—32no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—61no—
482C2×D4D3×D411yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4C23×D311yescharacteristic
493Q16—64no—
494SD16—64no—
495SD16—64no—
496SD16—64no—
497SD16—64no—
498D8—64no—
499M16—21no—
500M16—24no—
501M16—32no—
502M16—32no—
503M16—61no—
504M16—61no—
505M16—61no—
506M16—61no—
507M16C2×C4×D312yes—
508C2×C8—21no—
509C2×C8—24no—
510C2×C8—31no—
511C2×C8—31no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8C2×C4×D312yes—
518C2×C8#23512yes—
519C2×C8C2×#6711yescharacteristic
520C2×C8C2×#6711yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—31no—
526C4⋊C4—31no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—121no—
530C4⋊C4—121no—
531C4⋊C4C2×#6711yescharacteristic
532C4⋊C4C2×#6711yescharacteristic
533C4⋊C4C2×#6711yescharacteristic
534C4⋊C4C2×#6711yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—61no—
540C22⋊C4—121no—
541C22⋊C4—121no—
542C22⋊C4—242no—
543C42—21no—
544C42—21no—
545C42—41no—
546C42—41no—
547C42—61no—
548C42—61no—
549C42—61no—
550C42—121no—
551C42—121no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—22no—
555C22×C3—41no—
556C22×C3—41no—
557C22×C3#52011yescharacteristic
558C22×C3D4211yescharacteristic
559C22×C3#55311yescharacteristic
560C2×D3—42no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—82no—
564C2×D3—161no—
565C3×C4—22no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—41no—
569C3×C4—41no—
570C3×C4—81no—
571C3×C4—81no—
572C3×C4C2×C4×D411yescharacteristic
573C3×C4C2×C4×D411yescharacteristic
574C3×C4#54711yescharacteristic
575C3×C4#54811yescharacteristic
576Dic3—42no—
577Dic3—41no—
578Dic3—41no—
579Dic3—81no—
580C23—21no—
581C23—62no—
582C23—121no—
583C23—121no—
584C23—241no—
585C23—241no—
586C23#100411yescharacteristic
587Q8—21no—
588Q8—62no—
589Q8—62no—
590Q8—121no—
591Q8—121no—
592Q8#89312yes—
593D4—22no—
594D4—22no—
595D4—21no—
596D4—62no—
597D4—62no—
598D4—121no—
599D4—121no—
600D4#89312yes—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—161no—
621C2×C4—161no—
622C2×C4—481no—
623C2×C4—481no—
624C2×C4C22×C4×D311yescharacteristic
625C2×C4C2×#23511yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4C2×#23911yescharacteristic
628C2×C4#100711yescharacteristic
629C2×C4C22×#6711yescharacteristic
630C8—22no—
631C8—62no—
632C8—62no—
633C8—61no—
634C8—61no—
635C8—121no—
636C8C4×#6712yes—
637C8C4×#6712yes—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—42no—
641C2×C3#236111yescharacteristic
642C2×C3#328811yescharacteristic
643C2×C3#328911yescharacteristic
644D3—82no—
645C22—21no—
646C22—21no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—122no—
651C22—242no—
652C22—241no—
653C22—241no—
654C22—481no—
655C22#550411yescharacteristic
656C22#551411yescharacteristic
657C22D4×#6711yescharacteristic
658C4—22no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—121no—
666C4—121no—
667C4—122no—
668C4—241no—
669C4C2×#89311yescharacteristic
670C4#550711yescharacteristic
671C4C2×C4×#6711yescharacteristic
672C4#581711yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—42no—
677C2—242no—
678C2#8216311yescharacteristic
679C2#8400911yescharacteristic
680C2#8401911yescharacteristic
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2913C2
socleC22×C3D42
prime radicalC2×#949C22
cosocleC22×C3×C4C24
hypercenter#2913D3
hypocenterC3#23049
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. C3×#2913C2semidirect
3. #82571C2semidirect
5. #79168C2semidirect
6. #72009C2semidirect
8. #71666C2semidirect
9. #70677C2semidirect
10. #70674C2semidirect
11. #70673C2semidirect
12. #70672C2semidirect
13. #70660C2semidirect
14. #70629C2semidirect
15. #70625C2semidirect
35. #5141C4semidirect
73. #2913D3semidirect
88. C3×#131C2×C4semidirect
108. #951C2×C4semidirect
125. C2×#204C2×C4semidirect
152. #824C2×C4semidirect
153. #823C2×C4semidirect
154. #819C2×C4semidirect
331. #131C4×D3semidirect
673. C3#23049semidirect

Maximal subgroups

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

subgroupindex
C3×#29132
C2×#51412
#825712
#825642
#791682
#720092
#716672
#716662
#706772
#706742
#706732
#706722
#706602
#706292
#706252
#230493