Finite Group Databasebeta

← #10824801

Subgroups — #10824801

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#442C211yesmaximal, characteristic, complemented
2#82396C211yesmaximal, characteristic, complemented
3#82396C211yesmaximal, characteristic, complemented
4C2×#4709C211yesmaximal, characteristic, complemented
5#72043C211yesmaximal, characteristic
6#72040C211yesmaximal, characteristic, complemented
7C4×#832C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic, complemented
9#70583C211yesmaximal, characteristic, complemented
10#70568C211yesmaximal, characteristic, complemented
11#70180C211yesmaximal, characteristic, complemented
12#70179C211yesmaximal, characteristic, complemented
13#70176C211yesmaximal, characteristic, complemented
14#70175C211yesmaximal, characteristic, complemented
15#69814C211yesmaximal, characteristic, complemented
16#23712—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#938C2211yescharacteristic
21C2×#938C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C3×#442—24no—
24C2×C3×#106C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5221C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5213—21no—
35#5213—21no—
36#5210—21no—
37#5210—21no—
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×(C2×D3)⋊C8C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43C2×Dic3⋊C8C2211yescharacteristic
44#4987C2211yescharacteristic
45#4987C2211yescharacteristic
46#4972C2211yescharacteristic
47#4972C2211yescharacteristic
48#4957—21no—
49#4957—21no—
50#4945—21no—
51#4945—21no—
52C4×C4⋊Dic3C2211yescharacteristic
53C4×C4⋊Dic3C2211yescharacteristic
54C2×C4×C3⋊C8C2211yescharacteristic
55#4709—24no—
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442D311yescharacteristic, complemented
63C2×#433—31no—
64#2324—31no—
65#2324—31no—
66#2088—31no—
67#2077—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2034—31no—
72#2033—31no—
73#1999—31no—
74#1776—31no—
75C4×#102—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×C4×D3C2311yescharacteristic
84C2×C3×D8—22no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104#938—22no—
105#938—22no—
106#938—22no—
107#938—22no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4D411yescharacteristic
111C2×(C2×D3)⋊C4D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#872—22no—
126#872—22no—
127C3×#106—22no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#97D414yescomplemented
131#832—22no—
132#832—22no—
133#832—22no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#818—22no—
144#817—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147C4×C3⋊C8—22no—
148C22×D8C2×D311yescharacteristic
149C2×#121—31no—
150C2×#121—31no—
151C2×C4×D4C2×D311yescharacteristic
152C2×C4×D4C2×D311yescharacteristic
153C2×#117—31no—
154#442—24no—
155#433—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102C2×D311yescharacteristic
166C2×#102C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×#97C2×D311yescharacteristic
170C2×C4×C8—31no—
171#390—31no—
172#390—31no—
173#390—61no—
174#390—61no—
175#389—31no—
176#389—31no—
177#388—61no—
178#388—61no—
179#386—61no—
180#386—61no—
181#385—61no—
182#385—61no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3—21no—
194C24×C3—21no—
195C23×D3C2×D411yescharacteristic
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×D4#4112yes—
207C2×C3×D4#4112yes—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—42no—
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×Dic3—21no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C3×D8—28no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253#217—24no—
254#217—24no—
255#217—81no—
256#217—81no—
257#217—81no—
258#217—81no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3#4112yes—
270C4⋊Dic3#4112yes—
271Dic3⋊C4—22no—
272Dic3⋊C4—22no—
273Dic3⋊C4—81no—
274Dic3⋊C4—81no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—81no—
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—21no—
285C22×D4—61no—
286C22×D4—61no—
287C22×D4C22×D311yescharacteristic
288C22×D4C22×D311yescharacteristic
289C23×C4—21no—
290C23×C4—21no—
291C23×C4—31no—
292C2×D8—22no—
293C2×D8—22no—
294C2×D8—22no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299#121—62no—
300#121—62no—
301#121—62no—
302#121—62no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307#117—62no—
308#117—62no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
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×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C42—31no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335#107—62no—
336#106—22no—
337#106—62no—
338C4⋊C8—62no—
339C4⋊C8—62no—
340#102—22no—
341#102—22no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C22⋊C8—62no—
347C22⋊C8—62no—
348#97#6714yescomplemented
349C4×C8—62no—
350#95—61no—
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—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
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—41no—
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—22no—
412C2×Dic3—22no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
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—
429C4×D3—41no—
430C4×D3—41no—
431C4×D3—41no—
432C4×D3—41no—
433C3×C8—22no—
434C3×C8—22no—
435C3⋊C8—41no—
436C3⋊C8—41no—
437C24—21no—
438C24—21no—
439C24—31no—
440C2×D4—21no—
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—64no—
451C2×D4—64no—
452C2×D4—241no—
453C2×D4—241no—
454C2×D4—241no—
455C2×D4—241no—
456C2×D4#23512yes—
457C2×D4#23512yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482D8—28no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8C2×#6712yes—
494C2×C8C2×#6712yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—32no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—241no—
510C4⋊C4—241no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—64no—
514C22⋊C4—64no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—241no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—61no—
524C42—61no—
525C42—241no—
526C42—241no—
527C42#23712yes—
528C42C2×#6712yes—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3C2×#12111yescharacteristic
544C22×C3#52511yescharacteristic
545C22×C3#52511yescharacteristic
546C22×C3#57611yescharacteristic
547C22×C3#57611yescharacteristic
548C22×C3#57711yescharacteristic
549C22×C3#57711yescharacteristic
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—81no—
560C3×C4—81no—
561C3×C4#52511yescharacteristic
562C3×C4#52911yescharacteristic
563C3×C4#52911yescharacteristic
564C3×C4#53211yescharacteristic
565Dic3—42no—
566Dic3—42no—
567Dic3—41no—
568Dic3—41no—
569Dic3—41no—
570Dic3—81no—
571Dic3—81no—
572C23—22no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23C2×D3×D411yescharacteristic
594D4—22no—
595D4—22no—
596D4—22no—
597D4—22no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—244no—
603D4—244no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—61no—
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—84no—
626C2×C4—84no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
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—121no—
640C2×C4—121no—
641C2×C4#93812yes—
642C2×C4#93812yes—
643C2×C4C2×#23511yescharacteristic
644C2×C4C2×#23511yescharacteristic
645C2×C4C2×#23711yescharacteristic
646C2×C4D3×#4111yescharacteristic
647C2×C4D3×#4111yescharacteristic
648C2×C4C22×#6711yescharacteristic
649C8—22no—
650C8—22no—
651C8—121no—
652C8—121no—
653C2×C3—82no—
654C2×C3—82no—
655C2×C3#239211yescharacteristic
656C2×C3#306211yescharacteristic
657C2×C3#306211yescharacteristic
658C2×C3#311811yescharacteristic
659C2×C3#311911yescharacteristic
660C2×C3#312211yescharacteristic
661C2×C3#312511yescharacteristic
662D3—81no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—121no—
678C22—121no—
679C22—121no—
680C22—121no—
681C22—121no—
682C22—121no—
683C22—121no—
684C22#554711yescharacteristic
685C22#554711yescharacteristic
686C22#577411yescharacteristic
687C22#577411yescharacteristic
688C22#577511yescharacteristic
689C22#577511yescharacteristic
690C22C2×#93811yescharacteristic
691C4—22no—
692C4—22no—
693C4—81no—
694C4—81no—
695C4—122no—
696C4—122no—
697C4—121no—
698C4—121no—
699C4—121no—
700C4—241no—
701C4—241no—
702C4#553811yescharacteristic
703C4#553811yescharacteristic
704C4#579111yescharacteristic
705C4#583211yescharacteristic
706C3#2371211yescharacteristic, complemented, Sylow
707C2—82no—
708C2—82no—
709C2—241no—
710C2#7987311yescharacteristic
711C2#7987411yescharacteristic
712C2#7987511yescharacteristic
713C2#7988611yescharacteristic
714C2#8240111yescharacteristic
715C2#8395111yescharacteristic
716C2#8395111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
prime radical#72043C2
hypercenterC2×#442D3
hypocenterC3#23712
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. #82396C2semidirect
3. #82396C2semidirect
4. C2×#4709C2semidirect
6. #72040C2semidirect
7. C4×#832C2semidirect
8. C4×#832C2semidirect
9. #70583C2semidirect
10. #70568C2semidirect
11. #70180C2semidirect
12. #70179C2semidirect
13. #70176C2semidirect
14. #70175C2semidirect
15. #69814C2semidirect
62. C2×#442D3semidirect
130. C3×#97D4semidirect
348. #97#67semidirect
706. C3#23712semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4422
#823962
C2×#47092
#720432
#720402
C4×#8322
#705832
#705682
#701802
#701792
#701762
#701752
#698142
#237123