Finite Group Databasebeta

← #10703022

Subgroups — #10703022

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5133C211yesmaximal, characteristic
2C2×#5133C211yesmaximal, characteristic
3C2×#5110C211yesmaximal, characteristic
4C2×C3×#341C211yesmaximal, characteristic, complemented
5C2×#4554C211yesmaximal, characteristic, complemented
6C2×#4554C211yesmaximal, characteristic, complemented
7C2×#4551C211yesmaximal, characteristic, complemented
8#11192—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5133—22no—
17#5133—22no—
18#5133—22no—
19#5133C414yescomplemented
20C2×#821—21no—
21C2×#821—21no—
22#5110—22no—
23#5110—22no—
24C2×(C2×D3)⋊C8—22no—
25C2×#816—21no—
26C2×#816—21no—
27C3×#341—22no—
28#4554—22no—
29#4554—22no—
30#4551—22no—
31C2×#417—31no—
32C2×#417—31no—
33C2×#408—31no—
34C2×#343—31no—
35C2×#343—31no—
36C2×#341D311yescharacteristic, complemented
37C2×#340—31no—
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C23×Dic3—21no—
41C23×Dic3—21no—
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235C2×C412yes—
53C2×#235D412yes—
54C22×D12D411yescharacteristic
55C22×C4×D3—21no—
56C22×Dic6D411yescharacteristic
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16C2×C412yes—
68C2×C3×M16D412yes—
69C22×C3×C8—21no—
70C22×C3×C8—21no—
71C22×C3×C8—21no—
72C22×C3×C8—21no—
73C22×C3×C8—21no—
74C22×C3×C8—21no—
75C22×C3×C8—21no—
76C22×C3×C8—21no—
77C22×C3×C8D412yes—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2×C412yes—
89C2×C4⋊Dic3D412yes—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208D412yes—
96C22×C3⋊C8—21no—
97#821—24no—
98#821—24no—
99(C2×D3)⋊C8—28no—
100#816—24no—
101#816—24no—
102C22×#41—31no—
103C22×M16—31no—
104C22×M16C2×D311yescharacteristic
105C22×M16C2×D311yescharacteristic
106C23×C8C2×D311yescharacteristic
107C22×C4⋊C4—31no—
108C22×C4⋊C4—31no—
109#417—34no—
110#417—62no—
111#417—62no—
112#417—62no—
113C2×#103—61no—
114C2×#103—61no—
115C2×#102—61no—
116C2×#102—61no—
117#408—62no—
118#408—62no—
119C2×C22⋊C8—62no—
120#343—62no—
121#343—62no—
122#341—22no—
123#340—62no—
124C24×C3#4111yescharacteristic
125C23×D3—21no—
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C2×#67—22no—
141C2×#67—22no—
142C2×#67—41no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—41no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—81no—
154C22×Dic3—81no—
155C22×Dic3—81no—
156C22×Dic3—81no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—22no—
163#235—22no—
164#235—22no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170#235—41no—
171#235—41no—
172#235—41no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12C22⋊C412yes—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—41no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6C22⋊C412yes—
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—22no—
195C3×M16—41no—
196C3×M16—41no—
197C3×M16—41no—
198C3×M16—41no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—41no—
205C2×C3×C8—41no—
206C2×C3×C8—41no—
207C2×C3×C8—41no—
208C2×C3×C8—41no—
209C2×C3×C8—41no—
210C2×C3×C8—41no—
211C2×C3×C8—41no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—41no—
217C4⋊Dic3—41no—
218C4⋊Dic3—41no—
219C4⋊Dic3—41no—
220#208—22no—
221#208—22no—
222#208—41no—
223#208—41no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—41no—
226C2×C3⋊C8—41no—
227C2×#41—32no—
228C2×#41—32no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C2×#41—61no—
236C2×#41—61no—
237C2×#41—61no—
238C2×#41—61no—
239C22×Q8—31no—
240C22×D4—31no—
241C22×D4—61no—
242C23×C4—61no—
243C23×C4—61no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—32no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16C4×D312yes—
263C2×M16#6712yes—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—61no—
273C22×C8D1212yes—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—32no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286#103—64no—
287#103—64no—
288#102—64no—
289#102—64no—
290C22⋊C8—68no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3C4×D411yescharacteristic
296C23×C3#12111yescharacteristic
297C23×C3#12311yescharacteristic
298C23×C3#12311yescharacteristic
299C23×C3#12411yescharacteristic
300C23×C3#12611yescharacteristic
301C23×C3#12611yescharacteristic
302C22×D3—22no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C2×C3×C4C2×C22⋊C411yescharacteristic
310C2×C3×C4#11711yescharacteristic
311C2×C3×C4#11711yescharacteristic
312C2×C3×C4C4×D411yescharacteristic
313C2×C3×C4C4×D411yescharacteristic
314C2×C3×C4C4×D411yescharacteristic
315C2×C3×C4#12011yescharacteristic
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12111yescharacteristic
319C2×C3×C4#12111yescharacteristic
320C2×C3×C4#12111yescharacteristic
321C2×C3×C4#12111yescharacteristic
322C2×C3×C4#12111yescharacteristic
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4#12311yescharacteristic
325C2×C3×C4#12311yescharacteristic
326C2×C3×C4#12311yescharacteristic
327C2×C3×C4#12311yescharacteristic
328C2×C3×C4#12311yescharacteristic
329C2×C3×C4#12311yescharacteristic
330C2×C3×C4#12311yescharacteristic
331C2×C3×C4#12311yescharacteristic
332C2×C3×C4#12411yescharacteristic
333C2×C3×C4#12411yescharacteristic
334C2×C3×C4#12411yescharacteristic
335C2×C3×C4#12611yescharacteristic
336C2×C3×C4#12611yescharacteristic
337#67—42no—
338#67—42no—
339#67—42no—
340#67—42no—
341C2×Dic3—22no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—82no—
347C2×Dic3—82no—
348C2×Dic3—81no—
349C2×Dic3—81no—
350C2×Dic3—81no—
351C2×Dic3—81no—
352C2×Dic3—82no—
353C2×Dic3—82no—
354D12—22no—
355D12—22no—
356D12—41no—
357D12—41no—
358C4×D3—42no—
359C4×D3—42no—
360C4×D3—42no—
361C4×D3—42no—
362Dic6—22no—
363Dic6—22no—
364Dic6—41no—
365Dic6—41no—
366C3×C8—42no—
367C3×C8—42no—
368C3×C8—42no—
369C3⋊C8—42no—
370C24—61no—
371C24#23911yescharacteristic
372#41—62no—
373#41—62no—
374#41—62no—
375#41—62no—
376#41—62no—
377#41—62no—
378#41—62no—
379#41—62no—
380#41—121no—
381#41—121no—
382#41—121no—
383#41—121no—
384#41—121no—
385#41—121no—
386#41—121no—
387#41—121no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×D4—32no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—121no—
403C2×D4—121no—
404C2×D4—121no—
405C2×D4—121no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4—121no—
414C22×C4—121no—
415C22×C4—121no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4—121no—
419C22×C4—241no—
420C22×C4—241no—
421C22×C4—241no—
422C22×C4—241no—
423C22×C4C2×C4×D311yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4#23511yescharacteristic
426C22×C4#23511yescharacteristic
427C22×C4#23511yescharacteristic
428C22×C4#23511yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4#23711yescharacteristic
434C22×C4#23711yescharacteristic
435C22×C4#23911yescharacteristic
436C22×C4C2×#6711yescharacteristic
437M16—22no—
438M16—22no—
439M16—22no—
440M16—22no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—62no—
446M16—62no—
447M16—121no—
448M16—121no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—62no—
463C2×C8—121no—
464C2×C8—121no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—121no—
470C4⋊C4—121no—
471C4⋊C4—121no—
472C4⋊C4—121no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3#38611yescharacteristic
486C22×C3#38811yescharacteristic
487C22×C3#38811yescharacteristic
488C22×C3#39411yescharacteristic
489C22×C3#39711yescharacteristic
490C22×C3#39711yescharacteristic
491C22×C3#39711yescharacteristic
492C22×C3#47111yescharacteristic
493C22×C3#47111yescharacteristic
494C22×C3#47211yescharacteristic
495C22×C3#47311yescharacteristic
496C2×D3—42no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—81no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C3×C4#38111yescharacteristic
504C3×C4#38611yescharacteristic
505C3×C4#38811yescharacteristic
506C3×C4#38811yescharacteristic
507C3×C4#39411yescharacteristic
508C3×C4#39411yescharacteristic
509C3×C4#39411yescharacteristic
510C3×C4#39711yescharacteristic
511Dic3—42no—
512Dic3—82no—
513Dic3—82no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—62no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23#89311yescharacteristic
526C23#89411yescharacteristic
527C23#89411yescharacteristic
528C23#89511yescharacteristic
529C23#89811yescharacteristic
530C23#89811yescharacteristic
531C23#94711yescharacteristic
532Q8—62no—
533Q8—62no—
534Q8—121no—
535Q8—121no—
536D4—62no—
537D4—62no—
538D4—122no—
539D4—122no—
540D4—122no—
541D4—122no—
542D4—121no—
543D4—121no—
544C2×C4—62no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—242no—
554C2×C4—242no—
555C2×C4—241no—
556C2×C4—241no—
557C2×C4—241no—
558C2×C4—241no—
559C2×C4—242no—
560C2×C4—242no—
561C2×C4#87211yescharacteristic
562C2×C4C4×D1211yescharacteristic
563C2×C4#87511yescharacteristic
564C2×C4#87611yescharacteristic
565C2×C4D3×C22⋊C411yescharacteristic
566C2×C4#67⋊C411yescharacteristic
567C2×C4#88211yescharacteristic
568C2×C4#88311yescharacteristic
569C2×C4#88311yescharacteristic
570C2×C4#88311yescharacteristic
571C2×C4#88411yescharacteristic
572C2×C4#88411yescharacteristic
573C2×C4#88411yescharacteristic
574C2×C4#88511yescharacteristic
575C2×C4#88511yescharacteristic
576C2×C4#88611yescharacteristic
577C2×C4#89211yescharacteristic
578C2×C4#89411yescharacteristic
579C2×C4#89411yescharacteristic
580C2×C4#89411yescharacteristic
581C2×C4#89511yescharacteristic
582C2×C4#89811yescharacteristic
583C2×C4C4×#6711yescharacteristic
584C2×C4#92911yescharacteristic
585C2×C4#93011yescharacteristic
586C2×C4#93711yescharacteristic
587C2×C4#93811yescharacteristic
588C2×C4#93911yescharacteristic
589C8—42no—
590C8—42no—
591C8—42no—
592C8—122no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3#144011yescharacteristic
598C2×C3#190711yescharacteristic
599C2×C3#190811yescharacteristic
600C2×C3#205111yescharacteristic
601C2×C3#205211yescharacteristic
602C2×C3#207811yescharacteristic
603C2×C3#207911yescharacteristic
604D3—82no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22—241no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22#466511yescharacteristic
625C22#466911yescharacteristic
626C22#467411yescharacteristic
627C22#467511yescharacteristic
628C22#489611yescharacteristic
629C22#489711yescharacteristic
630C22#489811yescharacteristic
631C22#489811yescharacteristic
632C22#499011yescharacteristic
633C22#499111yescharacteristic
634C22#499211yescharacteristic
635C4—122no—
636C4—242no—
637C4—242no—
638C4#466411yescharacteristic
639C4#467011yescharacteristic
640C4#467211yescharacteristic
641C4#467511yescharacteristic
642C4#494011yescharacteristic
643C4#495611yescharacteristic
644C4#495911yescharacteristic
645C4#496111yescharacteristic
646C3#1119211yescharacteristic, complemented, Sylow
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—242no—
652C2#6892111yescharacteristic
653C2#6892211yescharacteristic
654C2#6894811yescharacteristic
655C2#6894911yescharacteristic
656C2#6942911yescharacteristic
657C2#7071411yescharacteristic
658C2#7071611yescharacteristic
subgroupG/H
centerC23#895
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#341C2
socleC23×C3#121
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#341D3
hypocenterC3#11192
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#341

Complements

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

NH ≅ G/Ntype
4. C2×C3×#341C2semidirect
5. C2×#4554C2semidirect
6. C2×#4554C2semidirect
7. C2×#4551C2semidirect
19. #5133C4semidirect
36. C2×#341D3semidirect
646. C3#11192semidirect

Maximal subgroups

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

subgroupindex
C2×#51332
C2×#51102
C2×C3×#3412
C2×#45542
C2×#45512
#111923