Finite Group Databasebeta

← #10801477

Subgroups — #10801477

Order 768.

1124 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#442C211yesmaximal, characteristic
2C2×#5153C211yesmaximal, characteristic
3C4×#936C211yesmaximal, characteristic, complemented
4C4×D4×Dic3C211yesmaximal, characteristic, complemented
5C2×#5018C211yesmaximal, characteristic, complemented
6#72037C211yesmaximal, characteristic
7#71609C211yesmaximal, characteristic, complemented
8#70570C211yesmaximal, characteristic, complemented
9#70564C211yesmaximal, characteristic, complemented
10#70173C211yesmaximal, characteristic, complemented
11#70171C211yesmaximal, characteristic, complemented
12#70162C211yesmaximal, characteristic, complemented
13#70154C211yesmaximal, characteristic, complemented
14#69813C211yesmaximal, characteristic, complemented
15#69809C211yesmaximal, characteristic, complemented
16#21507—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C22×#214C2211yescharacteristic
24C22×C4×Dic3—21no—
25C2×C4×Dic6C2211yescharacteristic
26C3×#442—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31#5221C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C4×#217—21no—
36C4×#217—21no—
37C4×#217—21no—
38#5153C418yescomplemented
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43#5018—24no—
44#4977C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47#4973C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4945—21no—
51#4945—21no—
52Dic3×C22⋊C4—21no—
53Dic3×C22⋊C4—21no—
54C4×C4⋊Dic3C2211yescharacteristic
55C42×Dic3C2211yescharacteristic
56C2×#803C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#463—31no—
64C2×#442D311yescharacteristic
65C2×#440—31no—
66C4×#124—31no—
67C42×D4—31no—
68#1983—31no—
69#1949—31no—
70#1927—31no—
71#1925—31no—
72#1879—31no—
73#1877—31no—
74#1804—31no—
75#1767—31no—
76#1763—31no—
77#1762—31no—
78C22×C3×D4C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×C3×C4—21no—
82C23×Dic3—21no—
83C23×Dic3—21no—
84C22×Dic6C2311yescharacteristic
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C2×C3×D8C2×C414yes—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4D414yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105#936—22no—
106#936C2×C414yescomplemented
107D4×Dic3—22no—
108D4×Dic3—22no—
109D4×Dic3—22no—
110D4×Dic3C2×C414yescomplemented
111C2×#214—22no—
112C2×#214—22no—
113C2×#214C2×C414yescomplemented
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—44no—
123C2×C4×Dic3D411yescharacteristic
124C2×C4×Dic3D411yescharacteristic
125C2×C4×Dic3D411yescharacteristic
126C2×C4×Dic3D411yescharacteristic
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131C4×Dic6—22no—
132C4×Dic6—22no—
133C3×#106—22no—
134C3×#102—22no—
135C3×#102—22no—
136C3×#97—22no—
137#832—22no—
138#832—22no—
139#832C2×C414yescomplemented
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#816—22no—
148#816C2×C414yescomplemented
149Dic3⋊C8—22no—
150Dic3⋊C8C2×C414yescomplemented
151#810—22no—
152#807—22no—
153#803—22no—
154C22×SD16—31no—
155C22×D8C2×D311yescharacteristic
156C2×#124—31no—
157C2×C4×Q8—31no—
158C2×C4×D4—31no—
159C2×C4×D4—31no—
160C2×C4×D4C2×D311yescharacteristic
161C2×C4×D4C2×D311yescharacteristic
162C22×C42—61no—
163#463—38no—
164#442—24no—
165#440—64no—
166C2×#107—31no—
167C2×#106C2×D311yescharacteristic
168C2×C4⋊C8—31no—
169C2×C4⋊C8—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#102—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×#102C2×D311yescharacteristic
177C2×#102C2×D311yescharacteristic
178C2×#97—31no—
179C2×#97C2×D311yescharacteristic
180#391—31no—
181#390—31no—
182#388—61no—
183#388—61no—
184#382—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C22⋊C4—61no—
189C4×C22⋊C4—61no—
190C4×C22⋊C4—61no—
191C4×C22⋊C4—61no—
192C4×C22⋊C4—61no—
193C43—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#336—31no—
199#336—31no—
200C24×C3—21no—
201C24×C3—21no—
202C2×C3×D4—21no—
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×D4C22×C412yes—
211C2×C3×D4C22×C412yes—
212C2×C3×D4C2×D412yes—
213C2×C3×D4#4112yes—
214C22×C3×C4—21no—
215C22×C3×C4—24no—
216C22×C3×C4—21no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4—42no—
220C22×C3×C4—42no—
221C22×C3×C4—42no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—24no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—24no—
230C22×Dic3—21no—
231C22×Dic3—42no—
232C22×Dic3—42no—
233C22×Dic3—42no—
234C22×Dic3—42no—
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6C22×C412yes—
245C3×D8—28no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8C22×C412yes—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4C2×D412yes—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—42no—
258C3×C42C2×D412yes—
259C3×C42#4112yes—
260#217—24no—
261#217—24no—
262#217—24no—
263#217—24no—
264#217—42no—
265#217—42no—
266#214—28no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3C22×C412yes—
272C4⋊Dic3#4112yes—
273Dic3⋊C4—42no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—22no—
277C4×Dic3—41no—
278C4×Dic3—41no—
279C4×Dic3—41no—
280C4×Dic3—42no—
281C4×Dic3—42no—
282C4×Dic3—42no—
283C4×Dic3—42no—
284C4×Dic3C22×C412yes—
285C4×Dic3#4112yes—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8C22×C412yes—
292C22×Q8—31no—
293C22×D4C22×D311yescharacteristic
294C22×D4C22×D311yescharacteristic
295C23×C4—21no—
296C23×C4—21no—
297C23×C4—61no—
298C23×C4—61no—
299C2×SD16—34no—
300C2×SD16—62no—
301C2×SD16—62no—
302C2×D8—22no—
303C2×D8—22no—
304C2×D8C4×D314yes—
305C22×C8—31no—
306C22×C8—31no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309#124—34no—
310#124—62no—
311C4×Q8—62no—
312C4×Q8—62no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—34no—
317C4×D4—62no—
318C4×D4—62no—
319C4×D4—62no—
320C4×D4#6714yes—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—61no—
342C2×C42—61no—
343C2×C42—61no—
344C2×C42—61no—
345C2×C42—124no—
346C2×C42C22×D311yescharacteristic
347#107—62no—
348#106—22no—
349C4⋊C8—34no—
350C4⋊C8—62no—
351#103—34no—
352#103—62no—
353#103—62no—
354#102—22no—
355#102—22no—
356#102—34no—
357#102—62no—
358#102—62no—
359#97—22no—
360#97—62no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366#95—61no—
367#95—61no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3C22×D411yescharacteristic
383C3×D4—22no—
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4C4×D414yes—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—48no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—84no—
403C2×C3×C4#12112yes—
404C2×C3×C4#12112yes—
405C2×C3×C4C23×C411yescharacteristic
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—48no—
424C2×Dic3—48no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3C4×D412yes—
435C2×Dic3C4×D412yes—
436C2×Dic3AGL(1,ℤ/8)12yes—
437Dic6—22no—
438Dic6—22no—
439Dic6—41no—
440Dic6—41no—
441C3×C8—22no—
442C3×C8—41no—
443C3⋊C8—22no—
444C3⋊C8—41no—
445C24—21no—
446C24—21no—
447C2×Q8—32no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×C4×D312yes—
462C2×D4C2×C4×D312yes—
463C2×D4#23512yes—
464C2×D4C2×#6712yes—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—64no—
478C22×C4—64no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—122no—
484C22×C4—122no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4D3×D411yescharacteristic
488C22×C4D3×D411yescharacteristic
489C22×C4C23×D311yescharacteristic
490SD16—68no—
491D8—28no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—32no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8C2×C4×D312yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—32no—
508C4⋊C4—32no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—122no—
514C4⋊C4C2×#6712yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C42—32no—
524C42—32no—
525C42—62no—
526C42—61no—
527C42—61no—
528C42—121no—
529C42—121no—
530C42—121no—
531C42—122no—
532C42—122no—
533C42—122no—
534C42—122no—
535C42#23712yes—
536C42C2×#6712yes—
537C22×C3—24no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—42no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3—81no—
551C22×C3C2×C4×D411yescharacteristic
552C22×C3C2×#12111yescharacteristic
553C22×C3#52511yescharacteristic
554C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
555C22×C3#57511yescharacteristic
556C22×C3#57511yescharacteristic
557C22×C3#57511yescharacteristic
558C3×C4—22no—
559C3×C4—22no—
560C3×C4—42no—
561C3×C4—81no—
562C3×C4C2×C4×D411yescharacteristic
563C3×C4C4×#4111yescharacteristic
564C3×C4#52911yescharacteristic
565C3×C4#53211yescharacteristic
566Dic3—22no—
567Dic3—22no—
568Dic3—22no—
569Dic3—41no—
570Dic3—42no—
571Dic3—42no—
572Dic3—42no—
573Dic3—81no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23C2×D3×D411yescharacteristic
589Q8—62no—
590Q8—62no—
591Q8—121no—
592Q8—121no—
593D4—22no—
594D4—22no—
595D4—22no—
596D4—22no—
597D4—22no—
598D4—41no—
599D4—41no—
600D4C4×#6714yes—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—32no—
607C2×C4—32no—
608C2×C4—32no—
609C2×C4—48no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—61no—
626C2×C4—62no—
627C2×C4—84no—
628C2×C4—128no—
629C2×C4—128no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4#93912yes—
640C2×C4#93912yes—
641C2×C4C22×C4×D311yescharacteristic
642C2×C4C2×#23511yescharacteristic
643C2×C4C2×#23711yescharacteristic
644C2×C4D3×#4111yescharacteristic
645C2×C4D3×#4111yescharacteristic
646C2×C4C22×#6711yescharacteristic
647C8—22no—
648C8—41no—
649C8—62no—
650C8—121no—
651C2×C3—44no—
652C2×C3—82no—
653C2×C3C4×#12111yescharacteristic
654C2×C3#294311yescharacteristic
655C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
656C2×C3#303611yescharacteristic
657C2×C3#304011yescharacteristic
658C2×C3#311111yescharacteristic
659C2×C3#311411yescharacteristic
660C22—24no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22C4×D3×D411yescharacteristic
675C22#554711yescharacteristic
676C22D3×AGL(1,ℤ/8)11yescharacteristic
677C22#577311yescharacteristic
678C22#577311yescharacteristic
679C22#577311yescharacteristic
680C22C2×#93911yescharacteristic
681C4—22no—
682C4—22no—
683C4—42no—
684C4—62no—
685C4—62no—
686C4—62no—
687C4—81no—
688C4—122no—
689C4—122no—
690C4—121no—
691C4—122no—
692C4—241no—
693C4C4×#23711yescharacteristic
694C4#553811yescharacteristic
695C4C2×C4×#6711yescharacteristic
696C4#583211yescharacteristic
697C3#2150711yescharacteristic, complemented, Sylow
698C2—44no—
699C2—82no—
700C2#7986411yescharacteristic
701C2#7986511yescharacteristic
702C2#7987011yescharacteristic
703C2#7987111yescharacteristic
704C2C4×#93911yescharacteristic
705C2#8393111yescharacteristic
706C2#8393311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
cosocleC22×C3×C4C24
hypercenterC2×#442D3
hypocenterC3#21507
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
3. C4×#936C2semidirect
4. C4×D4×Dic3C2semidirect
5. C2×#5018C2semidirect
7. #71609C2semidirect
8. #70570C2semidirect
9. #70564C2semidirect
10. #70173C2semidirect
11. #70171C2semidirect
12. #70162C2semidirect
13. #70154C2semidirect
14. #69813C2semidirect
15. #69809C2semidirect
38. #5153C4semidirect
106. #936C2×C4semidirect
110. D4×Dic3C2×C4semidirect
113. C2×#214C2×C4semidirect
139. #832C2×C4semidirect
148. #816C2×C4semidirect
150. Dic3⋊C8C2×C4semidirect
697. C3#21507semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4422
C2×#51532
C4×#9362
C4×D4×Dic32
C2×#50182
#720372
#716092
#705702
#705642
#701732
#701712
#701622
#701542
#698132
#698092
#215073