Finite Group Databasebeta

← #10824479

Subgroups — #10824479

Order 768.

956 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#442C211yesmaximal, characteristic, complemented
2#82329C211yesmaximal, characteristic, complemented
3#82325C211yesmaximal, characteristic, complemented
4C2×C4×#213C211yesmaximal, characteristic, complemented
5#72044C211yesmaximal, characteristic
6#72040C211yesmaximal, characteristic, complemented
7C4×#832C211yesmaximal, characteristic, complemented
8#70585C211yesmaximal, characteristic, complemented
9#70571C211yesmaximal, characteristic, complemented
10#70212C211yesmaximal, characteristic, complemented
11#70194C211yesmaximal, characteristic, complemented
12#70177C211yesmaximal, characteristic, complemented
13#70166C211yesmaximal, characteristic, complemented
14#69832C211yesmaximal, characteristic, complemented
15#69812C211yesmaximal, characteristic, complemented
16#23688—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#940C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#442—24no—
24C2×C3×#106C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5221C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5213—21no—
35#5213—21no—
36#5212—21no—
37#5211—21no—
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C4×#213—24no—
43#4988C2211yescharacteristic
44#4984C2211yescharacteristic
45#4984C2211yescharacteristic
46#4977C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4956—21no—
50#4956—21no—
51#4951—21no—
52#4951—21no—
53C4×C4⋊Dic3C2211yescharacteristic
54#4932C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442D311yescharacteristic, complemented
63C2×C4×D8—31no—
64#2328—31no—
65#2319—31no—
66#2085—31no—
67#2073—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2054—31no—
72#2041—31no—
73#2011—31no—
74#1768—31no—
75#1762—31no—
76C4×#102—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4—21no—
81C22×#67—21no—
82C22×#67—21no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4D414yescomplemented
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#940—22no—
106#940—22no—
107C2×#213—22no—
108C2×#213—22no—
109C2×#213—22no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C4×D12—22no—
124C4×D12—22no—
125C3×#106—22no—
126C3×#102—22no—
127C3×#102—22no—
128C3×#97—22no—
129#832—22no—
130#832—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#821—22no—
142#821—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145#809—22no—
146#807—22no—
147C4×C3⋊C8—22no—
148C22×D8—31no—
149C22×D8C2×D311yescharacteristic
150C2×#127—31no—
151C2×C4×D4—31no—
152C2×C4×D4C2×D311yescharacteristic
153C2×C4×D4C2×D311yescharacteristic
154#442—24no—
155C4×D8—64no—
156C2×#107—31no—
157C2×#106C2×D311yescharacteristic
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#102—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97C2×D311yescharacteristic
169C2×C4×C8—31no—
170#390—31no—
171#390—31no—
172#390—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#386—61no—
179#386—61no—
180#386—61no—
181#386—61no—
182#382—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4C2×D412yes—
207C2×C3×D4#4112yes—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—21no—
211C22×C3×C4—42no—
212C22×C3×C4—81no—
213C22×C3×C4—81no—
214C22×C3×C4—81no—
215C22×C3×C4—81no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C2×#67—24no—
220C2×#67—24no—
221C2×#67—81no—
222C2×#67—81no—
223C2×#67—81no—
224C2×#67—81no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12#4112yes—
238C2×C4×D3—42no—
239C2×C4×D3—81no—
240C2×C4×D3—81no—
241C3×D8—28no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8#4112yes—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2×D412yes—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—42no—
254C3×C42C2×D412yes—
255C3×C42#4112yes—
256#217—81no—
257#217—81no—
258#217—81no—
259#217—81no—
260#217—81no—
261#217—81no—
262#213—28no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—81no—
265(C2×D3)⋊C4—81no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—81no—
273C4⋊Dic3—81no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—81no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8#4112yes—
285C22×D4—31no—
286C22×D4—61no—
287C22×D4—61no—
288C22×D4C22×D311yescharacteristic
289C22×D4C22×D311yescharacteristic
290C23×C4—21no—
291C23×C4—21no—
292C23×C4—61no—
293C2×D8—22no—
294C2×D8—22no—
295C2×D8—22no—
296C2×D8—62no—
297C2×D8—62no—
298C2×D8—62no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303#127—62no—
304#127—62no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—62no—
309C4×D4—62no—
310C4×D4#6714yescomplemented
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#106—22no—
335C4⋊C8—62no—
336C4⋊C8—62no—
337#102—22no—
338#102—22no—
339#102—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#97—22no—
346C4×C8—62no—
347#95—31no—
348#95—31no—
349#95—31no—
350#95—31no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
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×C3C22×D411yescharacteristic
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—41no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—84no—
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×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406#67—84no—
407#67—84no—
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—21no—
419C2×Dic3—21no—
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—
429D12—22no—
430D12—22no—
431D12—41no—
432D12—41no—
433C4×D3—84no—
434C3×C8—22no—
435C3×C8—41no—
436C3⋊C8—22no—
437C3⋊C8—41no—
438C24—21no—
439C24—21no—
440C24—61no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—32no—
452C2×D4—64no—
453C2×D4—64no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—241no—
460C2×D4—241no—
461C2×D4—241no—
462C2×D4—241no—
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—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—122no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—28no—
487D8—68no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—32no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8#23512yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—241no—
510C4⋊C4—241no—
511C4⋊C4C2×#6712yes—
512C22⋊C4—24no—
513C22⋊C4—42no—
514C22⋊C4—122no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—61no—
524C42—61no—
525C42—61no—
526C42—61no—
527C42—241no—
528C42#23712yes—
529C42C2×#6712yes—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3C2×#12111yescharacteristic
545C22×C3#52511yescharacteristic
546C22×C3#53211yescharacteristic
547C22×C3#57611yescharacteristic
548C22×C3#57611yescharacteristic
549C22×C3#57611yescharacteristic
550C22×C3#57711yescharacteristic
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—42no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C3×C4—22no—
559C3×C4—22no—
560C3×C4—42no—
561C3×C4—81no—
562C3×C4#52511yescharacteristic
563C3×C4#52511yescharacteristic
564C3×C4#52911yescharacteristic
565C3×C4#52911yescharacteristic
566Dic3—41no—
567Dic3—41no—
568Dic3—41no—
569Dic3—41no—
570Dic3—81no—
571Dic3—81no—
572Dic3—81no—
573C23—22no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—62no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23C2×D3×D411yescharacteristic
595D4—22no—
596D4—22no—
597D4—22no—
598D4—22no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—41no—
603D4—62no—
604D4—62no—
605D4—121no—
606D4—121no—
607D4—244no—
608D4—244no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—244no—
643C2×C4#93912yes—
644C2×C4#93912yes—
645C2×C4C2×#23511yescharacteristic
646C2×C4C2×#23511yescharacteristic
647C2×C4C2×#23711yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4C22×#6711yescharacteristic
651C8—22no—
652C8—41no—
653C8—62no—
654C8—121no—
655C2×C3—82no—
656C2×C3—82no—
657C2×C3#239211yescharacteristic
658C2×C3#304111yescharacteristic
659C2×C3#304511yescharacteristic
660C2×C3#311811yescharacteristic
661C2×C3#312211yescharacteristic
662C2×C3#317611yescharacteristic
663C2×C3#317911yescharacteristic
664D3—82no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—122no—
680C22—122no—
681C22—122no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22#554711yescharacteristic
687C22#554811yescharacteristic
688C22#577411yescharacteristic
689C22#577411yescharacteristic
690C22#577411yescharacteristic
691C22#577511yescharacteristic
692C22C2×#93911yescharacteristic
693C4—22no—
694C4—22no—
695C4—42no—
696C4—81no—
697C4—121no—
698C4—121no—
699C4—121no—
700C4—121no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4#553811yescharacteristic
705C4#555711yescharacteristic
706C4#579111yescharacteristic
707C4#582811yescharacteristic
708C3#2368811yescharacteristic, complemented, Sylow
709C2—82no—
710C2—82no—
711C2—242no—
712C2#7986011yescharacteristic
713C2#7986111yescharacteristic
714C2#7987411yescharacteristic
715C2#7988611yescharacteristic
716C2#8234511yescharacteristic
717C2#8393411yescharacteristic
718C2#8393611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
prime radical#72044C2
hypercenterC2×#442D3
hypocenterC3#23688
LCSG ▷ C22×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
1. C2×C3×#442C2semidirect
2. #82329C2semidirect
3. #82325C2semidirect
4. C2×C4×#213C2semidirect
6. #72040C2semidirect
7. C4×#832C2semidirect
8. #70585C2semidirect
9. #70571C2semidirect
10. #70212C2semidirect
11. #70194C2semidirect
12. #70177C2semidirect
13. #70166C2semidirect
14. #69832C2semidirect
15. #69812C2semidirect
62. C2×#442D3semidirect
93. C3×C4×D4D4semidirect
310. C4×D4#67semidirect
708. C3#23688semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4422
#823292
#823252
C2×C4×#2132
#720442
#720402
C4×#8322
#705852
#705712
#702122
#701942
#701772
#701662
#698322
#698122
#236883