Finite Group Databasebeta

← #10818127

Subgroups — #10818127

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
1C2×#4898C211yesmaximal, characteristic
2#79179C211yesmaximal, characteristic, complemented
3#79168C211yesmaximal, characteristic, complemented
4#78571C211yesmaximal, characteristic, complemented
5#78560C211yesmaximal, characteristic
6C3×#1773C211yesmaximal, characteristic, complemented
7#70701C211yesmaximal, characteristic, complemented
8#69892C211yesmaximal, characteristic, complemented
9#69891C211yesmaximal, characteristic, complemented
10#69839C211yesmaximal, characteristic, complemented
11#69826C211yesmaximal, characteristic, complemented
12#68776C211yesmaximal, characteristic
13#68775C211yesmaximal, characteristic, complemented
14#68764C211yesmaximal, characteristic, complemented
15#68758C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17C2×#909C2211yescharacteristic
18C2×#908C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#901C2211yescharacteristic
21#5523C2211yescharacteristic
22#5523C2211yescharacteristic
23#5521C2211yescharacteristic
24#5516C2211yescharacteristic
25C2×C3×#108C2211yescharacteristic
26C3×#427C2211yescharacteristic
27C3×#405C2211yescharacteristic
28C3×#405C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30C2×#824C2211yescharacteristic
31C2×#823C2211yescharacteristic
32#5121C2211yescharacteristic
33#5121C2211yescharacteristic
34#5113C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#816C2211yescharacteristic
38#5101C2211yescharacteristic
39#5004C2211yescharacteristic
40#5001C2211yescharacteristic
41#4995C2211yescharacteristic
42#4995C2211yescharacteristic
43#4939C2211yescharacteristic
44#4929C2211yescharacteristic
45C2×#805C2211yescharacteristic
46#4898C418yescomplemented
47#4889—21no—
48#4889—21no—
49#4880—21no—
50D3×#107—21no—
51#4861—21no—
52#4861—21no—
53#4852—21no—
54#4852—21no—
55#4716—21no—
56#4711—21no—
57#4710—21no—
58#4710—21no—
59C4×Dic12—21no—
60C4×D24—21no—
61C4×#204—21no—
62C4×#204—21no—
63C3×#339C2211yescharacteristic
64C3×#339C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4552C2211yescharacteristic
68#4523C2211yescharacteristic
69C2×#471—31no—
70#2949—31no—
71#2947—31no—
72#2946—31no—
73#2913—31no—
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#1773D311yescharacteristic, complemented
83#1765—31no—
84C2×#235C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#117C2311yescharacteristic
92C3×#117C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×(C2×D3)⋊C4—41no—
95#926C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×#208C2311yescharacteristic
99C2×#208C2311yescharacteristic
100#909—22no—
101#909C2×C414yescomplemented
102#908—22no—
103#908C2×C414yes—
104C2×Dic12—21no—
105C2×D24—21no—
106C2×#204—21no—
107C2×#204—21no—
108C2×#204—21no—
109C2×#204—21no—
110C2×#204—22no—
111C2×#204C2×C414yescomplemented
112#901—22no—
113#901C2×C414yescomplemented
114C2×ΓL(1,𝔽25)—21no—
115C2×D3×C8—21no—
116#893—21no—
117#893—21no—
118#892—21no—
119#892—21no—
120D3×C4⋊C4—21no—
121D3×C4⋊C4—21no—
122#887—21no—
123#887—21no—
124#67⋊C4—41no—
125#67⋊C4—41no—
126D3×C22⋊C4—41no—
127#877—41no—
128C4×D12—21no—
129C4×D12—21no—
130C4×D12—21no—
131C4×D12—21no—
132C4×Dic6—21no—
133C4×Dic6—21no—
134C4×Dic6—21no—
135C4×Dic6—21no—
136C3×#108C2×C414yescomplemented
137C3×#107D411yescharacteristic
138C3×#107D411yescharacteristic
139C3×#106D411yescharacteristic
140C3×#106D411yescharacteristic
141C3×#97—21no—
142C3×#97—21no—
143C3×#97—22no—
144C3×C4×C8—21no—
145C3×C4×C8—21no—
146C3⋊#95—41no—
147#824C2×C414yescomplemented
148#823C2×C414yescomplemented
149#821—21no—
150#821—21no—
151#821D411yescharacteristic
152#821D411yescharacteristic
153#818—21no—
154#818—21no—
155#818—21no—
156#818—21no—
157#817—21no—
158#817—21no—
159#816—21no—
160#816—21no—
161#816D411yescharacteristic
162#816D411yescharacteristic
163#815—21no—
164C8×Dic3—21no—
165#810—21no—
166#810—21no—
167#809—21no—
168#809—21no—
169#808—21no—
170#808—21no—
171#807—21no—
172#807—21no—
173#805—22no—
174C2×#137—31no—
175C2×AGL(1,ℤ/8)—31no—
176C22×SD16—31no—
177C2×#131—31no—
178#520—31no—
179#520—31no—
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×#108C2×D311yescharacteristic
196#428—31no—
197#428—31no—
198#428—31no—
199#428—31no—
200#428—61no—
201#428—61no—
202#427—61no—
203#427C2×D311yescharacteristic
204C2×#107—31no—
205C2×#107—61no—
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#405C2×D311yescharacteristic
216#405C2×D311yescharacteristic
217C2×#97C2×D311yescharacteristic
218#391—31no—
219#390—31no—
220#343—31no—
221#343—31no—
222#341—31no—
223#339—31no—
224#339C2×D311yescharacteristic
225#339C2×D311yescharacteristic
226C23×D3—21no—
227C22×C3×C4—21no—
228C22×C3×C4C2411yescharacteristic
229C2×#67—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233#235—21no—
234#235—21no—
235#235C22×C412yes—
236#235#4112yes—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12C22×C412yes—
243C2×D12C2×D411yescharacteristic
244C2×C4×D3—21no—
245C2×C4×D3—41no—
246C2×C4×D3—41no—
247C2×C4×D3—41no—
248C2×C4×D3—41no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6C22×C412yes—
255C2×Dic6C2×D411yescharacteristic
256C3×M16—21no—
257C3×M16—21no—
258C3×M16C22×C412yes—
259C3×M16C22×C412yes—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8C22×C412yes—
265C2×C3×C8C2×D411yescharacteristic
266C2×C3×C8C2×D411yescharacteristic
267C2×C3×C8C2×D411yescharacteristic
268C2×C3×C8C2×D411yescharacteristic
269C2×C3×C8#4112yes—
270C3×C4⋊C4C2×D411yescharacteristic
271C3×C4⋊C4C2×D411yescharacteristic
272C3×C4⋊C4C2×D411yescharacteristic
273C3×C4⋊C4C2×D411yescharacteristic
274C3×C22⋊C4—21no—
275C3×C22⋊C4—21no—
276C3×C42—21no—
277C3×C42—21no—
278C3×C42—21no—
279C3×C42—21no—
280#217—21no—
281(C2×D3)⋊C4—41no—
282(C2×D3)⋊C4—41no—
283(C2×D3)⋊C4—41no—
284(C2×D3)⋊C4—41no—
285(C2×D3)⋊C4—82no—
286C4⋊Dic3—21no—
287C4⋊Dic3—21no—
288C4⋊Dic3—21no—
289C4⋊Dic3—21no—
290C4⋊Dic3C2×D411yescharacteristic
291C4⋊Dic3C2×D411yescharacteristic
292Dic3⋊C4—41no—
293Dic3⋊C4—41no—
294Dic3⋊C4—41no—
295Dic3⋊C4—41no—
296C4×Dic3—21no—
297C4×Dic3—41no—
298C4×Dic3—41no—
299C4×Dic3—41no—
300C4×Dic3—41no—
301#208—21no—
302#208—21no—
303#208—21no—
304#208C22×C412yes—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307Dic12—24no—
308D24—24no—
309#204—24no—
310#204—24no—
311#204—24no—
312#204—24no—
313ΓL(1,𝔽25)—24no—
314D3×C8—24no—
315C2×#41—31no—
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—61no—
327C2×SD16—61no—
328C2×SD16—62no—
329C2×D8—61no—
330#131—34no—
331#131—62no—
332C2×M16—31no—
333C2×M16—31no—
334C2×M16—61no—
335C2×M16C22×D311yescharacteristic
336C2×M16C22×D311yescharacteristic
337C22×C8—61no—
338C22×C8C22×D311yescharacteristic
339C22×C8C22×D311yescharacteristic
340C4×Q8—61no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×D4—61no—
347C4×D4—61no—
348C4×D4—61no—
349C4×D4—61no—
350C4×D4—61no—
351C4×D4—61no—
352C4×D4—121no—
353C4×D4—121no—
354#117—31no—
355#117—61no—
356#117—61no—
357#117—121no—
358#117C22×D311yescharacteristic
359#117C22×D311yescharacteristic
360C2×C4⋊C4—31no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4—61no—
364C2×C22⋊C4—121no—
365C2×C22⋊C4—121no—
366C2×C42C22×D311yescharacteristic
367#108C4×D314yescomplemented
368#107—61no—
369#107—61no—
370#107—61no—
371#107—61no—
372#107—61no—
373#107—61no—
374#107D1211yescharacteristic
375#107D1211yescharacteristic
376#106—61no—
377#106—61no—
378#106—61no—
379#106—61no—
380#106D1211yescharacteristic
381#106D1211yescharacteristic
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—22no—
398#97—21no—
399#97—21no—
400#97—61no—
401C4×C8—21no—
402C4×C8—21no—
403C4×C8—61no—
404#95—121no—
405C23×C3H(5,𝔽2)11yescharacteristic
406C22×D3—21no—
407C22×D3—22no—
408C22×D3—41no—
409C22×D3—41no—
410C22×D3—81no—
411C22×D3—81no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4C23×C411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C42−1+411yescharacteristic
425#67—22no—
426#67—22no—
427C2×Dic3—21no—
428C2×Dic3—22no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—81no—
436C2×Dic3—81no—
437D12—22no—
438D12—22no—
439D12—21no—
440D12—41no—
441D12—41no—
442D12C4×D412yes—
443C4×D3—22no—
444C4×D3—22no—
445C4×D3—161no—
446C4×D3—161no—
447C4×D3—161no—
448C4×D3—161no—
449Dic6—21no—
450Dic6—22no—
451Dic6—22no—
452Dic6—41no—
453Dic6—41no—
454Dic6C4×D412yes—
455C3×C8—22no—
456C3×C8—22no—
457C3×C8—21no—
458C3×C8—21no—
459C3×C8C4×D412yes—
460C3×C8C4×D412yes—
461C3⋊C8—22no—
462C3⋊C8—41no—
463C24—61no—
464#41—32no—
465#41—32no—
466#41—61no—
467#41—61no—
468C2×Q8—32no—
469C2×Q8—31no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×D4—32no—
476C2×D4—31no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—61no—
482C2×D4—61no—
483C22×C4—21no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—121no—
489C22×C4—121no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4C23×D311yescharacteristic
493Q16—64no—
494SD16—64no—
495SD16—64no—
496SD16—64no—
497SD16—64no—
498D8—64no—
499M16—21no—
500M16—21no—
501M16—32no—
502M16—61no—
503M16—64no—
504M16—61no—
505M16—61no—
506M16C2×C4×D312yes—
507M16C2×C4×D312yes—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—61no—
513C2×C8—64no—
514C2×C8—61no—
515C2×C8C2×C4×D312yes—
516C2×C8C2×D1211yescharacteristic
517C2×C8C2×D1211yescharacteristic
518C2×C8#23512yes—
519C2×C8D3×D411yescharacteristic
520C2×C8D3×D411yescharacteristic
521C4⋊C4—31no—
522C4⋊C4—31no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—121no—
528C4⋊C4—121no—
529C4⋊C4—121no—
530C4⋊C4—121no—
531C4⋊C4C2×D1211yescharacteristic
532C4⋊C4C2×D1211yescharacteristic
533C4⋊C4C2×D1211yescharacteristic
534C4⋊C4C2×D1211yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—61no—
538C22⋊C4—121no—
539C22⋊C4—121no—
540C22⋊C4—121no—
541C22⋊C4—121no—
542C22⋊C4—242no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—61no—
548C42—121no—
549C42—121no—
550C42—121no—
551C42—121no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3#52011yescharacteristic
555C22×C3D4211yescharacteristic
556C22×C3#55311yescharacteristic
557C2×D3—22no—
558C2×D3—41no—
559C2×D3—41no—
560C2×D3—42no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—82no—
564C2×D3—161no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—41no—
569C3×C4—81no—
570C3×C4C2×C4×D411yescharacteristic
571C3×C4C2×C4×D411yescharacteristic
572C3×C4#54711yescharacteristic
573C3×C4#54811yescharacteristic
574Dic3—22no—
575Dic3—42no—
576Dic3—41no—
577Dic3—41no—
578Dic3—81no—
579Dic3—81no—
580C23—61no—
581C23—62no—
582C23—121no—
583C23—121no—
584C23—241no—
585C23—241no—
586C23#100911yescharacteristic
587Q8—32no—
588Q8—61no—
589Q8—62no—
590Q8—62no—
591Q8—121no—
592Q8—121no—
593D4—32no—
594D4—62no—
595D4—62no—
596D4—62no—
597D4—61no—
598D4—62no—
599D4—121no—
600D4—121no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—61no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—241no—
619C2×C4—241no—
620C2×C4—481no—
621C2×C4—481no—
622C2×C4—481no—
623C2×C4—481no—
624C2×C4C22×C4×D311yescharacteristic
625C2×C4C22×D1211yescharacteristic
626C2×C4C2×#23511yescharacteristic
627C2×C4C2×D3×D411yescharacteristic
628C2×C4#100711yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C8—21no—
631C8—21no—
632C8—22no—
633C8—22no—
634C8—62no—
635C8—121no—
636C8C4×D1212yes—
637C8C4×D1212yes—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3#236111yescharacteristic
641C2×C3#328811yescharacteristic
642C2×C3#328911yescharacteristic
643D3—42no—
644D3—82no—
645C22—21no—
646C22—21no—
647C22—62no—
648C22—121no—
649C22—121no—
650C22—122no—
651C22—242no—
652C22—241no—
653C22—241no—
654C22—481no—
655C22#553011yescharacteristic
656C22D4×D1211yescharacteristic
657C22#557411yescharacteristic
658C4—41no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—62no—
663C4—81no—
664C4—121no—
665C4—121no—
666C4—122no—
667C4—241no—
668C4—241no—
669C4C2×C4×D1211yescharacteristic
670C4#550711yescharacteristic
671C4C4×D3×D411yescharacteristic
672C4#557611yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—122no—
677C2—242no—
678C2#7377411yescharacteristic
679C2#7990011yescharacteristic
680C2#7990111yescharacteristic
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1773C2
socleC22×C3D42
prime radicalC2×#908C22
cosocleC22×C3×C4C24
hypercenter#1773D3
hypocenterC3#23049
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. #79179C2semidirect
3. #79168C2semidirect
4. #78571C2semidirect
6. C3×#1773C2semidirect
7. #70701C2semidirect
8. #69892C2semidirect
9. #69891C2semidirect
10. #69839C2semidirect
11. #69826C2semidirect
13. #68775C2semidirect
14. #68764C2semidirect
15. #68758C2semidirect
46. #4898C4semidirect
82. #1773D3semidirect
101. #909C2×C4semidirect
111. C2×#204C2×C4semidirect
113. #901C2×C4semidirect
136. C3×#108C2×C4semidirect
147. #824C2×C4semidirect
148. #823C2×C4semidirect
367. #108C4×D3semidirect
673. C3#23049semidirect

Maximal subgroups

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

subgroupindex
C2×#48982
#791792
#791682
#785712
#785602
C3×#17732
#707012
#698922
#698912
#698392
#698262
#687762
#687752
#687642
#687582
#230493