Finite Group Databasebeta

← #10845242

Subgroups — #10845242

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#463C211yesmaximal, characteristic
2#82955C211yesmaximal, characteristic, complemented
3C2×#5055C211yesmaximal, characteristic, complemented
4#82379C211yesmaximal, characteristic, complemented
5#72051C211yesmaximal, characteristic, complemented
6#72048C211yesmaximal, characteristic, complemented
7#72040C211yesmaximal, characteristic
8#72037C211yesmaximal, characteristic
9#71609C211yesmaximal, characteristic, complemented
10#70469C211yesmaximal, characteristic, complemented
11#70459C211yesmaximal, characteristic, complemented
12#70233C211yesmaximal, characteristic, complemented
13#70221C211yesmaximal, characteristic, complemented
14#70165C211yesmaximal, characteristic, complemented
15#70153C211yesmaximal, characteristic, complemented
16#25334—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×D8C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×#924—21no—
24C2×#924—21no—
25C2×#869C2211yescharacteristic
26C3×#463—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5221C2211yescharacteristic
34#5221C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42#5055—24no—
43#4975C2211yescharacteristic
44#4975C2211yescharacteristic
45#4951—21no—
46#4951—21no—
47#4950—21no—
48#4949—21no—
49#4949—21no—
50Dic3×C22⋊C4—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#489—31no—
63C2×#463D311yescharacteristic
64#2542—31no—
65#2367—31no—
66#2072—31no—
67#2071—31no—
68#2055—31no—
69#2054—31no—
70#1987—31no—
71#1940—31no—
72#1880—31no—
73#1879—31no—
74#1803—31no—
75#1802—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×Dic3—21no—
82C22×Dic6—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—41no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C2×C3×D8—41no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#124—22no—
92C3×#124D414yes—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107Q8×Dic3—22no—
108Q8×Dic3—41no—
109D4×Dic3—22no—
110D4×Dic3—41no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119#924—44no—
120#924—44no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131#869—22no—
132#869—22no—
133C3×C4⋊C8—22no—
134C3×#103—22no—
135C3×#102—22no—
136#835—22no—
137#835—22no—
138#832—22no—
139#832—22no—
140#832—22no—
141#832—22no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95D411yescharacteristic
146C3⋊#95D411yescharacteristic
147Dic3⋊C8—41no—
148Dic3⋊C8—41no—
149#803—22no—
150C22×SD16C2×D311yescharacteristic
151C22×D8C2×D311yescharacteristic
152C2×C4⋊Q8—31no—
153C2×#124C2×D311yescharacteristic
154C2×C22⋊Q8—61no—
155C2×C22⋊Q8—61no—
156C2×C4×Q8—31no—
157C2×C4×D4—31no—
158C2×C4×D4C2×D311yescharacteristic
159#489—64no—
160#463—24no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102C2×D311yescharacteristic
172C2×#97—31no—
173#399—61no—
174#399—61no—
175#391—31no—
176#390—31no—
177#390—31no—
178#390—31no—
179#388—61no—
180#388—61no—
181#385—61no—
182#384—31no—
183#384—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C22⋊C4—61no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#336—31no—
194#336—31no—
195C24×C3—21no—
196C24×C3—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C2×D412yes—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4C2×D412yes—
214C2×C3×D4#4112yes—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4—42no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—42no—
228C22×Dic3—42no—
229C22×Dic3—81no—
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C2×Dic6—42no—
235C2×Dic6—42no—
236C2×Dic6—42no—
237C3×SD16—44no—
238C3×D8—44no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—42no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252#217—42no—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—42no—
258#217—81no—
259C4⋊Dic3—21no—
260C4⋊Dic3—22no—
261C4⋊Dic3—22no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—42no—
269C4⋊Dic3—42no—
270C4⋊Dic3#4112yes—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—42no—
280C4×Dic3—81no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C22×Q8—61no—
288C22×Q8C22×D311yescharacteristic
289C22×D4C22×D311yescharacteristic
290C22×D4C22×D311yescharacteristic
291C23×C4—21no—
292C23×C4—61no—
293C2×SD16—22no—
294C2×SD16—22no—
295C2×SD16—41no—
296C2×D8—22no—
297C2×D8—22no—
298C2×D8—41no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4⋊Q8—62no—
304C4⋊Q8—62no—
305#124—22no—
306#124#6714yes—
307C22⋊Q8—124no—
308C22⋊Q8—124no—
309C4×Q8—62no—
310C4×Q8—121no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—62no—
314C4×D4—121no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—61no—
341C2×C42—61no—
342C2×C42C22×D311yescharacteristic
343C4⋊C8—22no—
344C4⋊C8—121no—
345C4⋊C8—121no—
346#103—22no—
347#103—62no—
348#103—62no—
349#102—22no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354#97—62no—
355#95—31no—
356#95—31no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3C22×D411yescharacteristic
375C3×Q8—22no—
376C3×Q8—22no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—41no—
386C3×D4—41no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—84no—
398C2×C3×C4#12112yes—
399C2×C3×C4#12112yes—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4H(5,𝔽2)11yescharacteristic
405C2×C3×C42−1+411yescharacteristic
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—84no—
429C2×Dic3AGL(1,ℤ/8)12yes—
430C2×Dic3AGL(1,ℤ/8)12yes—
431Dic6—84no—
432C3×C8—41no—
433C3×C8—41no—
434C3⋊C8—41no—
435C3⋊C8—41no—
436C24—21no—
437C24—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—122no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×Q8C2×#6712yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4#23712yes—
458C2×D4C2×#6712yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—44no—
479D8—44no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—32no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4—122no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C42—61no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—122no—
525C42—241no—
526C42#23712yes—
527C42C2×#6712yes—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3C2×#12111yescharacteristic
543C22×C3#54611yescharacteristic
544C22×C3#54811yescharacteristic
545C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
546C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
547C22×C3#57711yescharacteristic
548C22×C3#57811yescharacteristic
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—42no—
552C3×C4—81no—
553C3×C4#52911yescharacteristic
554C3×C4#54211yescharacteristic
555C3×C4#54611yescharacteristic
556C3×C4#54711yescharacteristic
557Dic3—22no—
558Dic3—22no—
559Dic3—41no—
560Dic3—42no—
561Dic3—41no—
562Dic3—81no—
563Dic3—81no—
564Dic3—81no—
565C23—22no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C2×D3×D411yescharacteristic
580Q8—22no—
581Q8—22no—
582Q8—41no—
583Q8—41no—
584Q8—244no—
585D4—22no—
586D4—22no—
587D4—22no—
588D4—22no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—32no—
599C2×C4—32no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—84no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—244no—
629C2×C4#93912yes—
630C2×C4#93912yes—
631C2×C4C2×#23711yescharacteristic
632C2×C4C2×#23711yescharacteristic
633C2×C4#100411yescharacteristic
634C2×C4D3×#4111yescharacteristic
635C2×C4#101011yescharacteristic
636C2×C4C22×#6711yescharacteristic
637C8—41no—
638C8—41no—
639C8—121no—
640C8—121no—
641C2×C3—82no—
642C2×C3—82no—
643C2×C3#242711yescharacteristic
644C2×C3#303811yescharacteristic
645C2×C3#304611yescharacteristic
646C2×C3#330611yescharacteristic
647C2×C3#330811yescharacteristic
648C2×C3#336711yescharacteristic
649C2×C3#337311yescharacteristic
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22#559811yescharacteristic
665C22#561211yescharacteristic
666C22#577011yescharacteristic
667C22#577111yescharacteristic
668C22D3×AGL(1,ℤ/8)11yescharacteristic
669C22D3×AGL(1,ℤ/8)11yescharacteristic
670C22C2×#93911yescharacteristic
671C4—22no—
672C4—22no—
673C4—42no—
674C4—62no—
675C4—62no—
676C4—81no—
677C4—121no—
678C4—122no—
679C4—121no—
680C4—241no—
681C4—241no—
682C4—241no—
683C4#560011yescharacteristic
684C4#560311yescharacteristic
685C4#580611yescharacteristic
686C4#582411yescharacteristic
687C3#2533411yescharacteristic, complemented, Sylow
688C2—82no—
689C2—82no—
690C2#8051811yescharacteristic
691C2#8052111yescharacteristic
692C2#8056111yescharacteristic
693C2#8056211yescharacteristic
694C2#8264711yescharacteristic
695C2#8390211yescharacteristic
696C2#8393211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#463C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×#463D3
hypocenterC3#25334
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
2. #82955C2semidirect
3. C2×#5055C2semidirect
4. #82379C2semidirect
5. #72051C2semidirect
6. #72048C2semidirect
9. #71609C2semidirect
10. #70469C2semidirect
11. #70459C2semidirect
12. #70233C2semidirect
13. #70221C2semidirect
14. #70165C2semidirect
15. #70153C2semidirect
687. C3#25334semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4632
#829552
C2×#50552
#823792
#720512
#720482
#720402
#720372
#716092
#704692
#704592
#702332
#702212
#701652
#701532
#253343