Finite Group Databasebeta

← #10703020

Subgroups — #10703020

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×C3×#417C211yesmaximal, characteristic
2C2×#5234C211yesmaximal, characteristic
3C2×#5232C211yesmaximal, characteristic
4C2×#4558C211yesmaximal, characteristic, complemented
5C2×#4554C211yesmaximal, characteristic, complemented
6C2×#4530C211yesmaximal, characteristic, complemented
7C2×#4529C211yesmaximal, characteristic, complemented
8#11192—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C22×#208C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16C3×#417—22no—
17C3×#417C414yescomplemented
18C2×C3×#103—21no—
19C2×C3×#102—21no—
20#5234—22no—
21#5234—22no—
22#5232—22no—
23#5232—22no—
24C2×#835—21no—
25C2×#832—21no—
26C2×#830—22no—
27#4558—22no—
28#4554—22no—
29#4530—22no—
30#4529—22no—
31C2×#417—31no—
32C2×#417D311yescharacteristic
33C2×#408—31no—
34C2×#343—31no—
35C2×#343—31no—
36C2×#341—31no—
37C2×#340—31no—
38C2×C3×#41—21no—
39C2×C3×#41—21no—
40C2×C3×#41—21no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—21no—
46C2×C3×#41—21no—
47C2×C3×#41—21no—
48C2×C3×#41C2×C412yes—
49C2×C3×#41D412yes—
50C22×C3×Q8D411yescharacteristic
51C22×C3×D4—21no—
52C22×C3×D4D411yescharacteristic
53C23×C3×C4—21no—
54C23×C3×C4—21no—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16C2×C412yes—
63C22×C3×C8—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2×C412yes—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3D412yes—
76C2×#208—21no—
77C2×#208—21no—
78C2×#208—21no—
79C2×#208—21no—
80C2×#208—21no—
81C2×#208—21no—
82C2×#208—21no—
83C2×#208—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208D412yes—
87C2×#208D412yes—
88C22×C3⋊C8—21no—
89C22×C3⋊C8—21no—
90C22×C3⋊C8—21no—
91C22×C3⋊C8—21no—
92C22×C3⋊C8—21no—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95C22×C3⋊C8—21no—
96C22×C3⋊C8D412yes—
97C3×#103—24no—
98C3×#102—24no—
99#835—24no—
100#832—24no—
101#830—28no—
102C22×#41C2×D311yescharacteristic
103C22×M16—31no—
104C22×M16—31no—
105C22×M16C2×D311yescharacteristic
106C23×C8—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4C2×D311yescharacteristic
109#417—22no—
110#417—62no—
111#417—62no—
112#417Dic314yescomplemented
113C2×#103—21no—
114C2×#103—61no—
115C2×#102—21no—
116C2×#102—61no—
117#408—62no—
118#408—62no—
119C2×C22⋊C8—62no—
120#343—62no—
121#343—62no—
122#341—62no—
123#340—62no—
124C24×C3—21no—
125C24×C3#4111yescharacteristic
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—22no—
129C3×#41—22no—
130C3×#41—22no—
131C3×#41—22no—
132C3×#41—22no—
133C3×#41—22no—
134C3×#41—41no—
135C3×#41—41no—
136C3×#41—41no—
137C3×#41—41no—
138C3×#41—41no—
139C3×#41—41no—
140C3×#41—41no—
141C3×#41—41no—
142C2×C3×Q8—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8C22⋊C412yes—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—41no—
156C2×C3×D4—41no—
157C2×C3×D4—41no—
158C2×C3×D4—41no—
159C2×C3×D4C22⋊C412yes—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—41no—
165C22×C3×C4—41no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—81no—
171C22×C3×C4—81no—
172C22×C3×C4C22×C411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4#4111yescharacteristic
182C22×C3×C4#4111yescharacteristic
183C22×C3×C4#4111yescharacteristic
184C22×C3×C4#4111yescharacteristic
185C22×C3×C4#4111yescharacteristic
186C22×Dic3—22no—
187C22×Dic3—41no—
188C22×Dic3—41no—
189C22×Dic3—81no—
190C22×Dic3—81no—
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—41no—
194C3×M16—41no—
195C2×C3×C8—22no—
196C2×C3×C8—41no—
197C2×C3×C8—41no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—41no—
205C4⋊Dic3—41no—
206#208—22no—
207#208—22no—
208#208—22no—
209#208—22no—
210#208—41no—
211#208—41no—
212#208—41no—
213#208—41no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—41no—
220C2×C3⋊C8—41no—
221C2×C3⋊C8—41no—
222C2×C3⋊C8—41no—
223C2×C3⋊C8—41no—
224C2×C3⋊C8—41no—
225C2×C3⋊C8—41no—
226C2×C3⋊C8—41no—
227C2×#41—21no—
228C2×#41—21no—
229C2×#41—21no—
230C2×#41—21no—
231C2×#41—21no—
232C2×#41—21no—
233C2×#41—21no—
234C2×#41—21no—
235C2×#41—21no—
236C2×#41—21no—
237C2×#41C2×Dic312yes—
238C2×#41#6712yes—
239C22×Q8#6711yescharacteristic
240C22×D4—21no—
241C22×D4#6711yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—32no—
252C2×M16—32no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16—61no—
263C2×M16C2×Dic312yes—
264C22×C8—21no—
265C22×C8—32no—
266C22×C8—61no—
267C22×C8—61no—
268C22×C8—61no—
269C22×C8—61no—
270C22×C8—61no—
271C22×C8—61no—
272C22×C8—61no—
273C22×C8—61no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4C2×Dic312yes—
286#103—24no—
287#103—64no—
288#102—24no—
289#102—64no—
290C22⋊C8—68no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3C4×D411yescharacteristic
303C23×C3#12111yescharacteristic
304C23×C3#12311yescharacteristic
305C23×C3#12311yescharacteristic
306C23×C3#12411yescharacteristic
307C23×C3#12611yescharacteristic
308C23×C3#12611yescharacteristic
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—41no—
312C3×Q8—41no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—41no—
320C3×D4—41no—
321C2×C3×C4—22no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—82no—
330C2×C3×C4—81no—
331C2×C3×C4—81no—
332C2×C3×C4—82no—
333C2×C3×C4C2×C22⋊C411yescharacteristic
334C2×C3×C4#11711yescharacteristic
335C2×C3×C4#11711yescharacteristic
336C2×C3×C4C4×D411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4C4×D411yescharacteristic
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4#12111yescharacteristic
346C2×C3×C4#12111yescharacteristic
347C2×C3×C4#12111yescharacteristic
348C2×C3×C4#12311yescharacteristic
349C2×C3×C4#12311yescharacteristic
350C2×C3×C4#12311yescharacteristic
351C2×C3×C4#12311yescharacteristic
352C2×C3×C4#12311yescharacteristic
353C2×C3×C4#12311yescharacteristic
354C2×C3×C4#12311yescharacteristic
355C2×C3×C4#12311yescharacteristic
356C2×C3×C4#12411yescharacteristic
357C2×C3×C4#12411yescharacteristic
358C2×C3×C4#12411yescharacteristic
359C2×C3×C4#12611yescharacteristic
360C2×C3×C4#12611yescharacteristic
361C2×Dic3—42no—
362C2×Dic3—82no—
363C2×Dic3—81no—
364C2×Dic3—81no—
365C2×Dic3—82no—
366C3×C8—42no—
367C3⋊C8—42no—
368C3⋊C8—42no—
369C3⋊C8—42no—
370C24—21no—
371C24#23711yescharacteristic
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8#21712yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4#21712yes—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—62no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—121no—
420C22×C4—121no—
421C22×C4—241no—
422C22×C4—241no—
423C22×C4#23511yescharacteristic
424C22×C4#23511yescharacteristic
425C22×C4#23511yescharacteristic
426C22×C4#23511yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4#23711yescharacteristic
432C22×C4#23711yescharacteristic
433C22×C4#23711yescharacteristic
434C22×C4C22×Dic311yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437M16—22no—
438M16—22no—
439M16—41no—
440M16—41no—
441M16—62no—
442M16—62no—
443M16—62no—
444M16—62no—
445M16—121no—
446M16—121no—
447M16—121no—
448M16—121no—
449C2×C8—22no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8—121no—
458C2×C8—121no—
459C2×C8—121no—
460C2×C8—121no—
461C2×C8—121no—
462C2×C8—121no—
463C2×C8—121no—
464C2×C8—121no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—62no—
470C4⋊C4—62no—
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—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3#38611yescharacteristic
493C22×C3#38811yescharacteristic
494C22×C3#38811yescharacteristic
495C22×C3#39411yescharacteristic
496C22×C3#39711yescharacteristic
497C22×C3#39711yescharacteristic
498C22×C3#39711yescharacteristic
499C22×C3#47111yescharacteristic
500C22×C3#47111yescharacteristic
501C22×C3#47211yescharacteristic
502C22×C3#47311yescharacteristic
503C3×C4—42no—
504C3×C4—82no—
505C3×C4#38111yescharacteristic
506C3×C4#38611yescharacteristic
507C3×C4#38811yescharacteristic
508C3×C4#38811yescharacteristic
509C3×C4#39411yescharacteristic
510C3×C4#39411yescharacteristic
511C3×C4#39411yescharacteristic
512C3×C4#39711yescharacteristic
513Dic3—82no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23#87911yescharacteristic
526C23#87911yescharacteristic
527C23#88311yescharacteristic
528C23#88511yescharacteristic
529C23D4×Dic311yescharacteristic
530C23#93511yescharacteristic
531C23#93911yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—41no—
543D4—41no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—122no—
557C2×C4—242no—
558C2×C4—241no—
559C2×C4—241no—
560C2×C4—242no—
561C2×C4#87911yescharacteristic
562C2×C4#87911yescharacteristic
563C2×C4#88311yescharacteristic
564C2×C4#88311yescharacteristic
565C2×C4#88311yescharacteristic
566C2×C4#88411yescharacteristic
567C2×C4#88411yescharacteristic
568C2×C4#88411yescharacteristic
569C2×C4#88411yescharacteristic
570C2×C4#88511yescharacteristic
571C2×C4#88511yescharacteristic
572C2×C4#88511yescharacteristic
573C2×C4#92611yescharacteristic
574C2×C4#92611yescharacteristic
575C2×C4#92911yescharacteristic
576C2×C4#92911yescharacteristic
577C2×C4#92911yescharacteristic
578C2×C4#92911yescharacteristic
579C2×C4D4×Dic311yescharacteristic
580C2×C4D4×Dic311yescharacteristic
581C2×C4D4×Dic311yescharacteristic
582C2×C4#93511yescharacteristic
583C2×C4#93711yescharacteristic
584C2×C4#93711yescharacteristic
585C2×C4#93911yescharacteristic
586C2×C4#93911yescharacteristic
587C2×C4#93911yescharacteristic
588C2×C4C2×#21711yescharacteristic
589C8—42no—
590C8—122no—
591C8—122no—
592C8—122no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—82no—
598C2×C3#144011yescharacteristic
599C2×C3#190711yescharacteristic
600C2×C3#190811yescharacteristic
601C2×C3#205111yescharacteristic
602C2×C3#205211yescharacteristic
603C2×C3#207811yescharacteristic
604C2×C3#207911yescharacteristic
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—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22#495111yescharacteristic
625C22#495111yescharacteristic
626C22#495211yescharacteristic
627C22#495211yescharacteristic
628C22#495211yescharacteristic
629C22#495911yescharacteristic
630C22#519911yescharacteristic
631C22#520011yescharacteristic
632C22#520311yescharacteristic
633C22#520411yescharacteristic
634C22#522011yescharacteristic
635C4—42no—
636C4—82no—
637C4—242no—
638C4#495111yescharacteristic
639C4#495111yescharacteristic
640C4#495211yescharacteristic
641C4#495911yescharacteristic
642C4#495911yescharacteristic
643C4#495911yescharacteristic
644C4#521111yescharacteristic
645C4#522011yescharacteristic
646C3#1119211yescharacteristic, complemented, Sylow
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—82no—
652C2#7030111yescharacteristic
653C2#7030211yescharacteristic
654C2#7030711yescharacteristic
655C2#7031011yescharacteristic
656C2#7154811yescharacteristic
657C2#7209111yescharacteristic
658C2#7210311yescharacteristic
subgroupG/H
centerC23#939
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#417C2
socleC23×C3#121
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenterC2×#417D3
hypocenterC3#11192
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
4. C2×#4558C2semidirect
5. C2×#4554C2semidirect
6. C2×#4530C2semidirect
7. C2×#4529C2semidirect
17. C3×#417C4semidirect
112. #417Dic3semidirect
646. C3#11192semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4172
C2×#52342
C2×#52322
C2×#45582
C2×#45542
C2×#45302
C2×#45292
#111923