Finite Group Databasebeta

← #10823579

Subgroups — #10823579

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5603C211yesmaximal, characteristic
2C2×#4873C211yesmaximal, characteristic, complemented
3C2×#4757C211yesmaximal, characteristic, complemented
4#73871C211yesmaximal, characteristic, complemented
5C3×#2033C211yesmaximal, characteristic
6#71685C211yesmaximal, characteristic
7#70617C211yesmaximal, characteristic, complemented
8#70216C211yesmaximal, characteristic, complemented
9#70215C211yesmaximal, characteristic, complemented
10Dic3×#102C211yesmaximal, characteristic, complemented
11#69964C211yesmaximal, characteristic, complemented
12#69907C211yesmaximal, characteristic, complemented
13#69527C211yesmaximal, characteristic, complemented
14#68910C211yesmaximal, characteristic, complemented
15#68783C211yesmaximal, characteristic, complemented
16#23585—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×#936—21no—
19C2×#936C2211yescharacteristic
20C2×#935—21no—
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C22×Dic12C2211yescharacteristic
27#5603—24no—
28#5603—24no—
29C2×#888C2211yescharacteristic
30C2×#887C2211yescharacteristic
31C2×#887C2211yescharacteristic
32C2×#879—21no—
33C2×#878—21no—
34C2×C3×#107C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×#102C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#380C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#830C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46#4973C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#807C2211yescharacteristic
52#4945—21no—
53#4944—21no—
54#4873—24no—
55#4757—24no—
56#4655—21no—
57#4647—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#542—31no—
65C2×#440—31no—
66C2×C4×Q16—31no—
67#2323—31no—
68#2057—31no—
69#2042—31no—
70#2036—31no—
71#2033D311yescharacteristic
72#2024—31no—
73#2015—31no—
74#2001—31no—
75#2000—31no—
76#1763—31no—
77C4×#102—31no—
78#1750—31no—
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C22×Dic6—21no—
84C22×Dic6C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#121—22no—
89C3×#121—22no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217D411yescharacteristic
101C2×#217D411yescharacteristic
102#936—28no—
103#936—22no—
104#936—22no—
105#935—28no—
106D4×Dic3—22no—
107D4×Dic3D414yes—
108C2×#214—22no—
109C2×#214—22no—
110C2×#214—22no—
111#926—22no—
112#926—22no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117#924—22no—
118#924—22no—
119#924—22no—
120#924—22no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×C4×Dic3—21no—
128C2×C4×Dic3—21no—
129C2×C4×Dic3—21no—
130C2×C4×Dic3C2311yescharacteristic
131C2×C4×Dic3C2311yescharacteristic
132C22×C3⋊C8C2311yescharacteristic
133C22×C3⋊C8C2311yescharacteristic
134C2×Dic12—22no—
135C2×Dic12—22no—
136C2×Dic12—22no—
137#888—22no—
138#888—22no—
139#887—22no—
140#887—22no—
141#887—22no—
142#887D414yes—
143#879—28no—
144#878—28no—
145C3×#107—22no—
146C3×#102—22no—
147C3×#102—22no—
148C3×C22⋊C8—22no—
149C3×#95—21no—
150#832—22no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155#830—22no—
156#816—22no—
157#816—22no—
158#815—22no—
159C8×Dic3—22no—
160#810—22no—
161#810—22no—
162#810—22no—
163#807—22no—
164C22×Q16—31no—
165C22×SD16—31no—
166#542—64no—
167#542—64no—
168C2×C4⋊Q8—31no—
169C2×#126—61no—
170C2×#124—31no—
171C2×#124—61no—
172C2×#123—61no—
173C2×C22⋊Q8—31no—
174C2×C22⋊Q8—31no—
175C2×C22⋊Q8—61no—
176C2×#121C2×D311yescharacteristic
177C2×C4×Q8—31no—
178C2×C4×Q8—31no—
179C2×C4×D4—31no—
180C2×#117—31no—
181#440—64no—
182C4×Q16—64no—
183C2×#107—31no—
184C2×#107C2×D311yescharacteristic
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#102—31no—
191C2×#102C2×D311yescharacteristic
192C2×#102C2×D311yescharacteristic
193C2×C22⋊C8—31no—
194C2×C22⋊C8C2×D311yescharacteristic
195C2×#97—31no—
196C2×C4×C8—31no—
197#391—31no—
198#389—31no—
199#389—61no—
200#388—61no—
201#388—61no—
202#387—61no—
203#380C2×D311yescharacteristic
204C4×C4⋊C4—31no—
205#340—31no—
206#340—31no—
207#340—31no—
208#340C2×D311yescharacteristic
209#336—31no—
210#336—31no—
211C24×C3—21no—
212C24×C3C2×D411yescharacteristic
213C2×C3×D4—24no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—81no—
220C2×C3×D4—81no—
221C2×C3×D4C2×D412yes—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2411yescharacteristic
233C22×Dic3—21no—
234C22×Dic3—21no—
235C22×Dic3—21no—
236C22×Dic3—21no—
237C22×Dic3—24no—
238C22×Dic3—81no—
239C22×Dic3—81no—
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D411yescharacteristic
242C22×Dic3C2×D411yescharacteristic
243C22×Dic3C2×D411yescharacteristic
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—24no—
255C2×Dic6—81no—
256C2×Dic6—81no—
257C2×Dic6C2×D412yes—
258C2×Dic6#4112yes—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8#4112yes—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4C2×D412yes—
271C3×C22⋊C4—24no—
272C3×C22⋊C4—81no—
273#217—24no—
274#217—24no—
275#217—24no—
276#217—24no—
277#217—24no—
278#217—22no—
279#217—22no—
280#214—28no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—22no—
284C4⋊Dic3—22no—
285C4⋊Dic3C2×D412yes—
286C4⋊Dic3#4112yes—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—24no—
290Dic3⋊C4—24no—
291Dic3⋊C4—42no—
292Dic3⋊C4—81no—
293C4×Dic3—21no—
294C4×Dic3—21no—
295C4×Dic3—24no—
296C4×Dic3—42no—
297C4×Dic3—81no—
298C4×Dic3C2×D412yes—
299C4×Dic3C2×D412yes—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8#4112yes—
306Dic12—28no—
307C22×Q8—31no—
308C22×Q8—31no—
309C22×Q8—61no—
310C22×D4—21no—
311C22×D4C22×D311yescharacteristic
312C23×C4—61no—
313C23×C4C22×D311yescharacteristic
314C2×Q16—62no—
315C2×Q16—62no—
316C2×Q16—62no—
317C2×SD16—62no—
318C2×SD16—62no—
319C2×SD16—62no—
320C22×C8—31no—
321C22×C8—31no—
322C22×C8C22×D311yescharacteristic
323C22×C8C22×D311yescharacteristic
324C4⋊Q8—62no—
325C4⋊Q8—62no—
326#126—68no—
327#124—62no—
328#124—68no—
329#124—62no—
330#123—68no—
331C22⋊Q8—68no—
332C22⋊Q8—62no—
333C22⋊Q8—62no—
334C22⋊Q8—62no—
335C22⋊Q8—62no—
336#121—22no—
337#121—22no—
338C4×Q8—34no—
339C4×Q8—62no—
340C4×Q8—62no—
341C4×Q8—62no—
342C4×D4—34no—
343C4×D4—62no—
344#117—62no—
345#117—62no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4C22×D311yescharacteristic
357C2×C4⋊C4C22×D311yescharacteristic
358C2×C4⋊C4C22×D311yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—31no—
362C2×C22⋊C4—31no—
363C2×C22⋊C4—61no—
364C2×C22⋊C4—61no—
365C2×C22⋊C4—61no—
366C2×C22⋊C4—61no—
367C2×C22⋊C4—61no—
368C2×C42—31no—
369C2×C42—31no—
370C2×C42—61no—
371C2×C42—61no—
372C2×C42—61no—
373#107—22no—
374#107—62no—
375#103—62no—
376#103—62no—
377#103—62no—
378#103—62no—
379#103—62no—
380#102—22no—
381#102—22no—
382#102—62no—
383C22⋊C8—22no—
384C22⋊C8—62no—
385#97—62no—
386C4×C8—62no—
387#95—21no—
388#95—61no—
389#95—61no—
390#95—61no—
391#95—61no—
392C23×C3—21no—
393C23×C3—22no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3—21no—
397C23×C3—21no—
398C23×C3—21no—
399C23×C3—21no—
400C23×C3—41no—
401C23×C3—41no—
402C23×C3—41no—
403C23×C3—41no—
404C23×C3—41no—
405C23×C3—41no—
406C23×C3C22×D411yescharacteristic
407C3×D4—22no—
408C3×D4—22no—
409C3×D4—41no—
410C3×D4—41no—
411C3×D4—84no—
412C2×C3×C4—22no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C2×#4111yescharacteristic
431C2×C3×C4C2×#4111yescharacteristic
432C2×C3×C4C2×#4111yescharacteristic
433C2×C3×C4C2×#4111yescharacteristic
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—22no—
440C2×Dic3—21no—
441C2×Dic3—22no—
442C2×Dic3—22no—
443C2×Dic3—22no—
444C2×Dic3—21no—
445C2×Dic3—21no—
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451C2×Dic3—41no—
452C2×Dic3—41no—
453C2×Dic3—41no—
454C2×Dic3—41no—
455C2×Dic3—41no—
456C2×Dic3—84no—
457C2×Dic3#12012yes—
458C2×Dic3#12012yes—
459Dic6—22no—
460Dic6—22no—
461Dic6—22no—
462Dic6—22no—
463Dic6—41no—
464Dic6—41no—
465Dic6—41no—
466Dic6—41no—
467Dic6—84no—
468C3×C8—22no—
469C3×C8—41no—
470C3⋊C8—22no—
471C3⋊C8—41no—
472C24—21no—
473C24D3×D411yescharacteristic
474C2×Q8—32no—
475C2×Q8—32no—
476C2×Q8—64no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×Q8—61no—
486C2×Q8—61no—
487C2×Q8—241no—
488C2×Q8—241no—
489C2×D4—24no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—81no—
496C2×D4—81no—
497C2×D4D3×D412yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—31no—
508C22×C4—31no—
509C22×C4—31no—
510C22×C4—31no—
511C22×C4—64no—
512C22×C4—61no—
513C22×C4—61no—
514C22×C4—61no—
515C22×C4—61no—
516C22×C4—241no—
517C22×C4—241no—
518C22×C4D3×D411yescharacteristic
519C22×C4C23×D311yescharacteristic
520Q16—68no—
521SD16—68no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—32no—
528C2×C8—61no—
529C2×C8—61no—
530C2×C8—61no—
531C2×C8—61no—
532C2×C8—61no—
533C2×C8#23712yes—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—32no—
540C4⋊C4—32no—
541C4⋊C4—62no—
542C4⋊C4—62no—
543C4⋊C4—64no—
544C4⋊C4—64no—
545C4⋊C4—64no—
546C4⋊C4—64no—
547C4⋊C4—61no—
548C4⋊C4—61no—
549C4⋊C4—122no—
550C4⋊C4—241no—
551C4⋊C4D3×D412yes—
552C22⋊C4—24no—
553C22⋊C4—62no—
554C22⋊C4—62no—
555C22⋊C4—64no—
556C22⋊C4—64no—
557C22⋊C4—64no—
558C22⋊C4—64no—
559C22⋊C4—64no—
560C22⋊C4—81no—
561C42—32no—
562C42—32no—
563C42—64no—
564C42—61no—
565C42—61no—
566C42—122no—
567C42—241no—
568C22×C3—41no—
569C22×C3—41no—
570C22×C3—41no—
571C22×C3—42no—
572C22×C3—41no—
573C22×C3—41no—
574C22×C3—41no—
575C22×C3—42no—
576C22×C3—42no—
577C22×C3—41no—
578C22×C3—81no—
579C22×C3—81no—
580C22×C3—81no—
581C22×C3—81no—
582C22×C3C2×#12011yescharacteristic
583C22×C3#52511yescharacteristic
584C22×C3#53211yescharacteristic
585C22×C3#57611yescharacteristic
586C22×C3#57711yescharacteristic
587C22×C3#57811yescharacteristic
588C22×C3#57811yescharacteristic
589C3×C4—41no—
590C3×C4—42no—
591C3×C4—81no—
592C3×C4—81no—
593C3×C4#52511yescharacteristic
594C3×C4#52511yescharacteristic
595C3×C4#53211yescharacteristic
596C3×C4#53211yescharacteristic
597Dic3—22no—
598Dic3—22no—
599Dic3—42no—
600Dic3—42no—
601Dic3—42no—
602Dic3—41no—
603Dic3—41no—
604Dic3—42no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—22no—
611C23—21no—
612C23—21no—
613C23—41no—
614C23—41no—
615C23—41no—
616C23—41no—
617C23—41no—
618C23—41no—
619C23C2×D3×D411yescharacteristic
620Q8—62no—
621Q8—62no—
622Q8—62no—
623Q8—62no—
624Q8—121no—
625Q8—121no—
626Q8—121no—
627Q8—121no—
628Q8—244no—
629D4—22no—
630D4—22no—
631D4—41no—
632D4—41no—
633D4—84no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—32no—
639C2×C4—32no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—61no—
653C2×C4—61no—
654C2×C4—61no—
655C2×C4—61no—
656C2×C4—62no—
657C2×C4—62no—
658C2×C4—61no—
659C2×C4—61no—
660C2×C4—61no—
661C2×C4—61no—
662C2×C4—61no—
663C2×C4—62no—
664C2×C4—61no—
665C2×C4—62no—
666C2×C4—121no—
667C2×C4—121no—
668C2×C4—121no—
669C2×C4—121no—
670C2×C4—121no—
671C2×C4—121no—
672C2×C4—121no—
673C2×C4—121no—
674C2×C4—244no—
675C2×C4C2×D3×D411yescharacteristic
676C2×C4C2×D3×D411yescharacteristic
677C2×C4C2×#23711yescharacteristic
678C2×C4D3×#4111yescharacteristic
679C2×C4D3×#4111yescharacteristic
680C2×C4D3×#4111yescharacteristic
681C8—22no—
682C8—41no—
683C8—62no—
684C8—121no—
685C2×C3—81no—
686C2×C3—82no—
687C2×C3#238811yescharacteristic
688C2×C3#301111yescharacteristic
689C2×C3#301211yescharacteristic
690C2×C3#312111yescharacteristic
691C2×C3#312411yescharacteristic
692C2×C3#317511yescharacteristic
693C2×C3#317811yescharacteristic
694C22—41no—
695C22—41no—
696C22—42no—
697C22—41no—
698C22—42no—
699C22—41no—
700C22—42no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22D3×#12011yescharacteristic
709C22#559611yescharacteristic
710C22#565611yescharacteristic
711C22#575711yescharacteristic
712C22#577111yescharacteristic
713C22#577511yescharacteristic
714C22#577911yescharacteristic
715C4—41no—
716C4—42no—
717C4—62no—
718C4—62no—
719C4—81no—
720C4—81no—
721C4—122no—
722C4—122no—
723C4—121no—
724C4—122no—
725C4—121no—
726C4—122no—
727C4#559611yescharacteristic
728C4#560611yescharacteristic
729C4#565611yescharacteristic
730C4#566811yescharacteristic
731C3#2358511yescharacteristic, complemented, Sylow
732C2—81no—
733C2—82no—
734C2#7688911yescharacteristic
735C2#8000511yescharacteristic
736C2#8023011yescharacteristic
737C2#8069811yescharacteristic
738C2#8078811yescharacteristic
739C2#8127311yescharacteristic
740C2#8135311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2033C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2033D3
hypocenterC3#23585
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
2. C2×#4873C2semidirect
3. C2×#4757C2semidirect
4. #73871C2semidirect
7. #70617C2semidirect
8. #70216C2semidirect
9. #70215C2semidirect
10. Dic3×#102C2semidirect
11. #69964C2semidirect
12. #69907C2semidirect
13. #69527C2semidirect
14. #68910C2semidirect
15. #68783C2semidirect
731. C3#23585semidirect

Maximal subgroups

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

subgroupindex
C2×#56032
C2×#48732
C2×#47572
#738712
C3×#20332
#716852
#706172
#702162
#702152
Dic3×#1022
#699642
#699072
#695272
#689102
#687832
#235853