Finite Group Databasebeta

← #10818121

Subgroups — #10818121

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×#5141C211yesmaximal, characteristic
2#83785C211yesmaximal, characteristic, complemented
3#78821C211yesmaximal, characteristic, complemented
4#78820C211yesmaximal, characteristic
5#78572C211yesmaximal, characteristic, complemented
6C3×#1765C211yesmaximal, characteristic, complemented
7#70366C211yesmaximal, characteristic
8#70365C211yesmaximal, characteristic, complemented
9#69894C211yesmaximal, characteristic, complemented
10#69829C211yesmaximal, characteristic, complemented
11#69826C211yesmaximal, characteristic, complemented
12#68931C211yesmaximal, characteristic, complemented
13#68930C211yesmaximal, characteristic, complemented
14#68764C211yesmaximal, characteristic, complemented
15#68758C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18#5827C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#951C2211yescharacteristic
21C2×#949C2211yescharacteristic
22C22×#204C2211yescharacteristic
23#5521C2211yescharacteristic
24#5516C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#416C2211yescharacteristic
27C3×#405C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29#5141C418yescomplemented
30C2×#824C2211yescharacteristic
31#5131C2211yescharacteristic
32C2×#823C2211yescharacteristic
33#5121—21no—
34#5121—21no—
35#5121C2211yescharacteristic
36#5121C2211yescharacteristic
37#5120C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#819C2211yescharacteristic
40#5106—21no—
41#5106C2211yescharacteristic
42C2×#818—21no—
43C2×#818C2211yescharacteristic
44C2×#816C2211yescharacteristic
45#5101C2211yescharacteristic
46#5101C2211yescharacteristic
47#4995C2211yescharacteristic
48#4995C2211yescharacteristic
49#4939C2211yescharacteristic
50#4929C2211yescharacteristic
51#4817—21no—
52#4790—21no—
53#4789—21no—
54#4788—21no—
55#4785—21no—
56#4758—21no—
57#4757—21no—
58#4756—21no—
59#4710—21no—
60#4710—21no—
61C4×#204—21no—
62C4×#204—21no—
63C3×#341C2211yescharacteristic
64C3×#339C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4523C2211yescharacteristic
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#1773—31no—
83#1765D311yescharacteristic, complemented
84C2×C3×#41C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C3×#131—22no—
88C3×#131C2×C414yescomplemented
89C2×C3×M16—21no—
90C2×C3×M16C2311yescharacteristic
91C2×C3×M16C2311yescharacteristic
92C22×C3×C8—21no—
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2311yescharacteristic
95C3×#117C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#951—22no—
98#951C2×C414yescomplemented
99#949—22no—
100#949C2×C414yes—
101Q8×Dic3—21no—
102Q8×Dic3—21no—
103C2×#216—21no—
104C2×#215—21no—
105D4×Dic3—21no—
106D4×Dic3—21no—
107D4×Dic3—41no—
108D4×Dic3—41no—
109C2×#214—21no—
110C2×#213—21no—
111C2×(C2×D3)⋊C4—41no—
112#926—21no—
113#926—21no—
114#926C2311yescharacteristic
115#926C2311yescharacteristic
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×#208C2311yescharacteristic
121C2×#208C2311yescharacteristic
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×#104—22no—
137C3×#103D411yescharacteristic
138C3×#103D411yescharacteristic
139C3×#102D411yescharacteristic
140C3×#102D411yescharacteristic
141C3×#97—21no—
142C3×#97—22no—
143C3×C4×C8—21no—
144C3⋊#95—41no—
145#824C2×C414yescomplemented
146#823C2×C414yescomplemented
147#821—21no—
148#821—21no—
149#821—21no—
150#821—21no—
151#819C2×C414yescomplemented
152#818—21no—
153#818—21no—
154#818—21no—
155#818—21no—
156#818D411yescharacteristic
157#818D411yescharacteristic
158#817—21no—
159#817—21no—
160#817D411yescharacteristic
161#817D411yescharacteristic
162#816—21no—
163#816—21no—
164#816—21no—
165#816—21no—
166#815—21no—
167#815—21no—
168C8×Dic3—21no—
169C8×Dic3—21no—
170#808—21no—
171#808—21no—
172#807—21no—
173#807—21no—
174C2×#137—31no—
175C2×AGL(1,ℤ/8)—31no—
176C22×SD16—31no—
177C2×#131C2×D311yescharacteristic
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×#108—31no—
196#428—31no—
197#428—31no—
198#428—31no—
199#428—31no—
200#428—61no—
201#428—61no—
202#427—31no—
203#427—61no—
204C2×#107—31no—
205C2×#107—61no—
206C2×#104C2×D311yescharacteristic
207#419—31no—
208#418—31no—
209#416C2×D311yescharacteristic
210C2×#103—31no—
211C2×#102—31no—
212C2×#101—31no—
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405C2×D311yescharacteristic
217C2×#97C2×D311yescharacteristic
218#391—31no—
219#390—31no—
220#343—31no—
221#343—31no—
222#341C2×D311yescharacteristic
223#339—31no—
224#339—31no—
225#339C2×D311yescharacteristic
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×C4C2411yescharacteristic
237C22×Dic3—21no—
238C22×Dic3—21no—
239C22×Dic3—21no—
240C22×Dic3—41no—
241C22×Dic3—41no—
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×M16—21no—
259C3×M16—21no—
260C3×M16C22×C412yes—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—24no—
265C2×C3×C8—21no—
266C2×C3×C8C22×C412yes—
267C2×C3×C8C2×D411yescharacteristic
268C2×C3×C8C2×D411yescharacteristic
269C2×C3×C8C2×D411yescharacteristic
270C2×C3×C8C2×D411yescharacteristic
271C2×C3×C8#4112yes—
272C3×C4⋊C4C2×D411yescharacteristic
273C3×C4⋊C4C2×D411yescharacteristic
274C3×C22⋊C4—21no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42—21no—
278#217—21no—
279#217—21no—
280#217—41no—
281#217—41no—
282#216—24no—
283#215—24no—
284#214—24no—
285#213—24no—
286(C2×D3)⋊C4—41no—
287(C2×D3)⋊C4—41no—
288(C2×D3)⋊C4—82no—
289C4⋊Dic3—21no—
290C4⋊Dic3—21no—
291C4⋊Dic3—21no—
292C4⋊Dic3—21no—
293C4⋊Dic3—41no—
294C4⋊Dic3—41no—
295C4⋊Dic3C2×D411yescharacteristic
296C4⋊Dic3C2×D411yescharacteristic
297C4⋊Dic3C2×D411yescharacteristic
298C4⋊Dic3C2×D411yescharacteristic
299Dic3⋊C4—41no—
300Dic3⋊C4—41no—
301C4×Dic3—21no—
302C4×Dic3—21no—
303C4×Dic3—41no—
304C4×Dic3—41no—
305C4×Dic3—41no—
306C4×Dic3—41no—
307#208—21no—
308#208—21no—
309#208C22×C412yes—
310#208C22×C412yes—
311C2×C3⋊C8—21no—
312C2×C3⋊C8—21no—
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—61no—
327C2×SD16—61no—
328C2×SD16—62no—
329C2×D8—61no—
330#131—22no—
331#131C4×D314yescomplemented
332C2×M16—21no—
333C2×M16—31no—
334C2×M16—31no—
335C2×M16C22×D311yescharacteristic
336C2×M16C22×D311yescharacteristic
337C22×C8—21no—
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—31no—
356#117—61no—
357#117—61no—
358#117—121no—
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#108—34no—
368#107—31no—
369#107—31no—
370#107—61no—
371#107—61no—
372#107—61no—
373#107—61no—
374#107—61no—
375#107—61no—
376#106—31no—
377#106—31no—
378#106—61no—
379#106—61no—
380#106—61no—
381#106—61no—
382#104—22no—
383#103—61no—
384#103—61no—
385#103—61no—
386#103—61no—
387#103D1211yescharacteristic
388#103D1211yescharacteristic
389#102—61no—
390#102—61no—
391#102—61no—
392#102—61no—
393#102D1211yescharacteristic
394#102D1211yescharacteristic
395#101—34no—
396#100—34no—
397#97—21no—
398#97—22no—
399#97—61no—
400#97—61no—
401C4×C8—21no—
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—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4C23×C411yescharacteristic
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C2×#4111yescharacteristic
430C2×C3×C4C2×#4111yescharacteristic
431C2×C3×C42−1+411yescharacteristic
432C2×Dic3—22no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—81no—
442C2×Dic3—81no—
443C2×Dic3—161no—
444C2×Dic3—161no—
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×C8—21no—
457C3×C8—21no—
458C3×C8—41no—
459C3×C8C4×D412yes—
460C3×C8C4×D412yes—
461C3⋊C8—22no—
462C3⋊C8—22no—
463C24—61no—
464#41—21no—
465#41—21no—
466#41C2×C4×D312yes—
467#41#23512yes—
468C2×Q8—32no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8C2×D1211yescharacteristic
475C2×D4—21no—
476C2×D4—32no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—61no—
482C2×D4C2×D1211yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
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—24no—
501M16—21no—
502M16—21no—
503M16—32no—
504M16—32no—
505M16—61no—
506M16—61no—
507M16C2×C4×D312yes—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—24no—
512C2×C8—21no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8C2×C4×D312yes—
516C2×C8C2×D1211yescharacteristic
517C2×C8C2×D1211yescharacteristic
518C2×C8D3×D411yescharacteristic
519C2×C8D3×D411yescharacteristic
520C2×C8#23712yes—
521C4⋊C4—31no—
522C4⋊C4—31no—
523C4⋊C4—31no—
524C4⋊C4—31no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—121no—
530C4⋊C4—121no—
531C4⋊C4—121no—
532C4⋊C4—121no—
533C4⋊C4C2×D1211yescharacteristic
534C4⋊C4C2×D1211yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—61no—
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—61no—
547C42—61no—
548C42—121no—
549C42—121no—
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—81no—
569C3×C4C2×C4×D411yescharacteristic
570C3×C4C2×C4×D411yescharacteristic
571C3×C4#54711yescharacteristic
572C3×C4#54811yescharacteristic
573Dic3—42no—
574Dic3—41no—
575Dic3—41no—
576Dic3—41no—
577Dic3—41no—
578Dic3—81no—
579Dic3—81no—
580C23—21no—
581C23—62no—
582C23—121no—
583C23—121no—
584C23—241no—
585C23—241no—
586C23#100911yescharacteristic
587Q8—21no—
588Q8—62no—
589Q8—62no—
590Q8—121no—
591Q8—121no—
592Q8C4×D1212yes—
593D4—22no—
594D4—22no—
595D4—21no—
596D4—62no—
597D4—62no—
598D4—121no—
599D4—121no—
600D4C4×D1212yes—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
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×C4C2×#23711yescharacteristic
629C2×C4#101011yescharacteristic
630C8—22no—
631C8—21no—
632C8—21no—
633C8—41no—
634C8—62no—
635C8—62no—
636C8#67⋊C412yes—
637C8#67⋊C412yes—
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#552311yescharacteristic
656C22D4×D1211yescharacteristic
657C22#556511yescharacteristic
658C4—22no—
659C4—41no—
660C4—41no—
661C4—81no—
662C4—121no—
663C4—121no—
664C4—121no—
665C4—121no—
666C4—122no—
667C4—241no—
668C4—241no—
669C4C2×C4×D1211yescharacteristic
670C4C2×#67⋊C411yescharacteristic
671C4#555211yescharacteristic
672C4#555511yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—42no—
677C2—242no—
678C2#7346711yescharacteristic
679C2#7990011yescharacteristic
680C2#7990111yescharacteristic
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1765C2
socleC22×C3D42
prime radicalC2×#949C22
cosocleC22×C3×C4C24
hypercenter#1765D3
hypocenterC3#23049
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #83785C2semidirect
3. #78821C2semidirect
5. #78572C2semidirect
6. C3×#1765C2semidirect
8. #70365C2semidirect
9. #69894C2semidirect
10. #69829C2semidirect
11. #69826C2semidirect
12. #68931C2semidirect
13. #68930C2semidirect
14. #68764C2semidirect
15. #68758C2semidirect
29. #5141C4semidirect
83. #1765D3semidirect
88. C3×#131C2×C4semidirect
98. #951C2×C4semidirect
125. C2×#204C2×C4semidirect
145. #824C2×C4semidirect
146. #823C2×C4semidirect
151. #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
C2×#51412
#837852
#788212
#788202
#785722
C3×#17652
#703662
#703652
#698942
#698292
#698262
#689312
#689302
#687642
#687582
#230493