Finite Group Databasebeta

← #10801802

Subgroups — #10801802

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#465C211yesmaximal, characteristic
2C2×#5164C211yesmaximal, characteristic, complemented
3C2×#5101C211yesmaximal, characteristic, complemented
4#82744C211yesmaximal, characteristic, complemented
5#82633C211yesmaximal, characteristic
6#72050C211yesmaximal, characteristic, complemented
7#71971C211yesmaximal, characteristic, complemented
8#71697C211yesmaximal, characteristic, complemented
9#71682C211yesmaximal, characteristic
10#70562C211yesmaximal, characteristic, complemented
11#70553C211yesmaximal, characteristic, complemented
12#70253C211yesmaximal, characteristic, complemented
13Dic3×#103C211yesmaximal, characteristic, complemented
14#70185C211yesmaximal, characteristic, complemented
15Dic3×#102C211yesmaximal, characteristic, complemented
16#21531—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C3×#465C418yescomplemented
26C2×C3×#106C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5221—21no—
33#5221C2211yescharacteristic
34#5220—21no—
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37#5212C2211yescharacteristic
38#5212C2211yescharacteristic
39#5164—24no—
40C2×#818C2211yescharacteristic
41C2×#818C2211yescharacteristic
42#5101—24no—
43C2×#815C2211yescharacteristic
44C2×#815C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46C2×C8×Dic3C2211yescharacteristic
47#4984—21no—
48#4984C2211yescharacteristic
49#4984C2211yescharacteristic
50#4980—21no—
51Dic3×C4⋊C4C2211yescharacteristic
52Dic3×C4⋊C4C2211yescharacteristic
53#4951—21no—
54#4951—21no—
55#4951—21no—
56#4950—21no—
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#465D311yescharacteristic
64C2×#440—31no—
65C2×#405—31no—
66#2324—31no—
67#2323—31no—
68#1973—31no—
69#1944—31no—
70#1934—31no—
71#1933—31no—
72#1895—31no—
73#1894—31no—
74#1808—31no—
75#1776—31no—
76C4×#103—31no—
77C4×#102—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16C2×C414yes—
86C22×C3×C8—22no—
87C22×C3×C8—22no—
88C22×C3×C8C2×C414yescomplemented
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8C2×C414yescomplemented
93C3×#121—22no—
94C3×#121C2×C414yes—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217D411yescharacteristic
106C2×#217D411yescharacteristic
107Q8×Dic3—22no—
108Q8×Dic3—22no—
109D4×Dic3—22no—
110D4×Dic3—22no—
111#926—22no—
112#926—22no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3D411yescharacteristic
117C2×C4⋊Dic3D411yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3C2311yescharacteristic
126C2×C4×Dic3C2311yescharacteristic
127C2×#208—22no—
128C2×#208—22no—
129C2×#208—22no—
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C3×#106C2×C414yescomplemented
133C3×#103C2×C414yescomplemented
134C3×#102C2×C414yes—
135#835—22no—
136#835—22no—
137#832—22no—
138#832—22no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#818—22no—
148#818—22no—
149#815—22no—
150#815D414yescomplemented
151C8×Dic3—22no—
152C8×Dic3—22no—
153C22×SD16C2×D311yescharacteristic
154C22×M16—31no—
155C23×C8C2×D311yescharacteristic
156C2×C22⋊Q8C2×D311yescharacteristic
157C2×#121C2×D311yescharacteristic
158C2×C4×Q8—31no—
159C2×C4×D4—31no—
160C2×#117—31no—
161#465Dic318yescomplemented
162#440—64no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#106C2×D311yescharacteristic
166C2×#103—31no—
167C2×#103—31no—
168C2×#103C2×D311yescharacteristic
169C2×#102—31no—
170C2×#102—31no—
171C2×#102C2×D311yescharacteristic
172#405—64no—
173C2×#97—31no—
174C2×#97—31no—
175C2×C4×C8—31no—
176C2×C4×C8—31no—
177#391—31no—
178#390—31no—
179#390—61no—
180#389—31no—
181#389—31no—
182#389—61no—
183#388—61no—
184#388—61no—
185#388—61no—
186#387—61no—
187#386—61no—
188#385—61no—
189#380—31no—
190#380—31no—
191C4×C4⋊C4—31no—
192C4×C4⋊C4—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3—21no—
200C24×C3C2×D411yescharacteristic
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8C22×C412yes—
207C2×C3×D4—24no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—81no—
214C2×C3×D4—81no—
215C2×C3×D4C22×C412yes—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C22×C412yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4#4112yes—
226C22×C3×C4C2411yescharacteristic
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—42no—
232C22×Dic3—81no—
233C22×Dic3—81no—
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C3×SD16—28no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—28no—
242C2×C3×C8C22×C412yes—
243C2×C3×C8C22×C412yes—
244C2×C3×C8C2×D412yes—
245C2×C3×C8#4112yes—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C22×C412yes—
250C3×C4⋊C4C22×C412yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253#217—22no—
254#217—22no—
255#217—42no—
256#217—81no—
257#217—81no—
258#217—81no—
259#217—81no—
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—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—42no—
271C4⋊Dic3—81no—
272C4⋊Dic3—81no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—42no—
276C4×Dic3—81no—
277C4×Dic3—81no—
278C4×Dic3C2×D412yes—
279C4×Dic3#4112yes—
280#208—28no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D412yes—
286C2×C3⋊C8#4112yes—
287C22×Q8C22×D311yescharacteristic
288C22×D4—21no—
289C22×D4C22×D311yescharacteristic
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16C2×Dic314yes—
295C2×M16—62no—
296C2×M16—62no—
297C2×M16—62no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8C2×Dic314yescomplemented
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C22⋊Q8—22no—
306C22⋊Q8C2×Dic314yescomplemented
307#121—22no—
308#121C2×Dic314yes—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×D4—62no—
312C4×D4—62no—
313#117—62no—
314#117—62no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42—61no—
341#107—62no—
342#107—62no—
343#106C2×Dic314yescomplemented
344#103—62no—
345#103—62no—
346#103C2×Dic314yescomplemented
347#102—62no—
348#102—62no—
349#102C2×Dic314yes—
350#97—34no—
351#97—62no—
352C4×C8—62no—
353C4×C8—62no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—22no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3C4×D412yes—
376C23×C3C22×D411yescharacteristic
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—41no—
380C3×Q8—41no—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—84no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C4×D412yes—
402C2×C3×C4C23×C411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
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—41no—
429C2×Dic3—41no—
430C2×Dic3—84no—
431C2×Dic3#12112yes—
432C2×Dic3#12112yes—
433C3×C8—22no—
434C3×C8C4×D414yes—
435C3⋊C8—22no—
436C3⋊C8—22no—
437C24—21no—
438C24D3×D411yescharacteristic
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C22×Dic312yes—
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—81no—
452C2×D4—81no—
453C2×D4C22×Dic312yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—122no—
470C22×C4—241no—
471C22×C4—241no—
472C22×C4D3×D411yescharacteristic
473C22×C4#23712yes—
474C22×C4C22×Dic312yes—
475C22×C4C23×D311yescharacteristic
476SD16—28no—
477M16—68no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—28no—
481C2×C8—32no—
482C2×C8—32no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8D3×D412yes—
488C2×C8#23712yes—
489C2×C8C22×Dic312yes—
490C2×C8C22×Dic312yes—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—62no—
495C4⋊C4—62no—
496C4⋊C4—61no—
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—241no—
506C4⋊C4—241no—
507C4⋊C4C22×Dic312yes—
508C4⋊C4C22×Dic312yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—62no—
512C22⋊C4—62no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C42—32no—
519C42—32no—
520C42—61no—
521C42—61no—
522C42—122no—
523C42—241no—
524C42—241no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3C2×C4×D411yescharacteristic
540C22×C3C2×#12111yescharacteristic
541C22×C3#53211yescharacteristic
542C22×C3#57611yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57711yescharacteristic
545C22×C3#57811yescharacteristic
546C3×C4—22no—
547C3×C4—42no—
548C3×C4—42no—
549C3×C4—42no—
550C3×C4C2×C4×D411yescharacteristic
551C3×C4C4×#4111yescharacteristic
552C3×C4#52511yescharacteristic
553C3×C4#52911yescharacteristic
554Dic3—22no—
555Dic3—22no—
556Dic3—41no—
557Dic3—41no—
558Dic3—81no—
559Dic3—81no—
560Dic3—81no—
561Dic3—81no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23D4×Dic312yes—
576C23C2×D3×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581D4—22no—
582D4—22no—
583D4—41no—
584D4—41no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—32no—
596C2×C4—32no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—61no—
604C2×C4—61no—
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—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
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—244no—
626C2×C4D4×Dic312yes—
627C2×C4C2×D3×D411yescharacteristic
628C2×C4C2×#23711yescharacteristic
629C2×C4C2×#23711yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4C23×Dic311yescharacteristic
633C8—22no—
634C8—62no—
635C8—62no—
636C8D4×Dic314yes—
637C2×C3—42no—
638C2×C3—82no—
639C2×C3C4×#12111yescharacteristic
640C2×C3#294911yescharacteristic
641C2×C3#295111yescharacteristic
642C2×C3#306211yescharacteristic
643C2×C3#306311yescharacteristic
644C2×C3#317811yescharacteristic
645C2×C3#317911yescharacteristic
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22#559611yescharacteristic
661C22D3×#12111yescharacteristic
662C22#576211yescharacteristic
663C22#576211yescharacteristic
664C22#576911yescharacteristic
665C22#577111yescharacteristic
666C22C2×D4×Dic311yescharacteristic
667C4—22no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—62no—
672C4—62no—
673C4—121no—
674C4—121no—
675C4—241no—
676C4—241no—
677C4—241no—
678C4—241no—
679C4#560011yescharacteristic
680C4#560611yescharacteristic
681C4C2×D4×Dic311yescharacteristic
682C4Dic3×#4111yescharacteristic
683C3#2153111yescharacteristic, complemented, Sylow
684C2—42no—
685C2—82no—
686C2#8063611yescharacteristic
687C2#8063711yescharacteristic
688C2#8066111yescharacteristic
689C2#8066211yescharacteristic
690C2Dic3×#12111yescharacteristic
691C2#8384311yescharacteristic
692C2#8389811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#465C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
cosocleC22×C3×C4C24
hypercenterC2×#465D3
hypocenterC3#21531
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
2. C2×#5164C2semidirect
3. C2×#5101C2semidirect
4. #82744C2semidirect
6. #72050C2semidirect
7. #71971C2semidirect
8. #71697C2semidirect
10. #70562C2semidirect
11. #70553C2semidirect
12. #70253C2semidirect
13. Dic3×#103C2semidirect
14. #70185C2semidirect
15. Dic3×#102C2semidirect
25. C3×#465C4semidirect
88. C22×C3×C8C2×C4semidirect
92. C3×C22⋊Q8C2×C4semidirect
132. C3×#106C2×C4semidirect
133. C3×#103C2×C4semidirect
150. #815D4semidirect
161. #465Dic3semidirect
302. C22×C8C2×Dic3semidirect
306. C22⋊Q8C2×Dic3semidirect
343. #106C2×Dic3semidirect
346. #103C2×Dic3semidirect
683. C3#21531semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4652
C2×#51642
C2×#51012
#827442
#826332
#720502
#719712
#716972
#716822
#705622
#705532
#702532
Dic3×#1032
#701852
Dic3×#1022
#215313