Finite Group Databasebeta

← #10824816

Subgroups — #10824816

Order 768.

820 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#441C211yesmaximal, characteristic, complemented
2#82488C211yesmaximal, characteristic
3#82488C211yesmaximal, characteristic
4C2×#4709C211yesmaximal, characteristic
5#72071C211yesmaximal, characteristic
6#72068C211yesmaximal, characteristic, complemented
7C4×#835C211yesmaximal, characteristic, complemented
8C4×#835C211yesmaximal, characteristic, complemented
9#70583C211yesmaximal, characteristic
10#70568C211yesmaximal, characteristic, complemented
11#70248C211yesmaximal, characteristic, complemented
12#70247C211yesmaximal, characteristic, complemented
13#70244C211yesmaximal, characteristic
14#70243C211yesmaximal, characteristic
15#69814C211yesmaximal, characteristic, complemented
16#23713—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×#946C2211yescharacteristic
21C2×#946C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C3×#441—24no—
24C2×C3×#106C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#4990—21no—
41#4990—21no—
42#4987C2211yescharacteristic
43#4987C2211yescharacteristic
44#4976—21no—
45#4976—21no—
46#4972C2211yescharacteristic
47#4972C2211yescharacteristic
48#4940—21no—
49#4940—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52#4933—21no—
53#4933—21no—
54C2×C4×C3⋊C8C2211yescharacteristic
55#4709—24no—
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#441D311yescharacteristic, complemented
63C2×#433—31no—
64#2323—31no—
65#2323—31no—
66#2088—31no—
67#2077—31no—
68#2068—31no—
69#2066—31no—
70#2056—31no—
71#2036—31no—
72#2035—31no—
73#2001—31no—
74#1776—31no—
75C4×#103—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C2×C3×Q16—22no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96#946—22no—
97#946—22no—
98#946—22no—
99#946—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#872—22no—
122#872—22no—
123C3×#106—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#97D414yes—
127#835—22no—
128#835—22no—
129#835—22no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
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×Q16C2×D311yescharacteristic
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×Q8C2×D311yescharacteristic
153C2×#117—31no—
154#441—24no—
155#433—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×#97C2×D311yescharacteristic
170C2×C4×C8—31no—
171#391—31no—
172#391—31no—
173#389—31no—
174#389—31no—
175#389—61no—
176#389—61no—
177#388—61no—
178#388—61no—
179#388—61no—
180#388—61no—
181#387—61no—
182#387—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—
193C23×D3C2×D411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8#4112yes—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C3×Q16—28no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8C2×D412yes—
232C2×C3×C8C2×D412yes—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C42—42no—
240C3×C42—42no—
241C3×C42C2×D412yes—
242C3×C42#4112yes—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—81no—
248(C2×D3)⋊C4—81no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3#4112yes—
256C4⋊Dic3#4112yes—
257Dic3⋊C4—24no—
258Dic3⋊C4—24no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261Dic3⋊C4—81no—
262Dic3⋊C4—81no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—81no—
266C4×Dic3—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C22×Q8C22×D311yescharacteristic
274C22×Q8C22×D311yescharacteristic
275C23×C4—31no—
276C2×Q16—22no—
277C2×Q16—22no—
278C2×Q16—22no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C22⋊Q8—62no—
284C22⋊Q8—62no—
285C22⋊Q8—62no—
286C22⋊Q8—62no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291#117—62no—
292#117—62no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—31no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#107—62no—
320#106—22no—
321#106—62no—
322C4⋊C8—62no—
323C4⋊C8—62no—
324#103—22no—
325#103—22no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330C22⋊C8—62no—
331C22⋊C8—62no—
332#97#6714yes—
333C4×C8—62no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3C22×D411yescharacteristic
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4#12112yes—
377C2×C3×C4#12112yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C4×D3—41no—
406C4×D3—41no—
407C4×D3—41no—
408C4×D3—41no—
409C3×C8—22no—
410C3×C8—22no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C24—31no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8#23512yes—
425C2×Q8#23512yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4C23×D311yescharacteristic
446Q16—28no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8C2×#6712yes—
458C2×C8C2×#6712yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—32no—
464C4⋊C4—32no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—241no—
478C4⋊C4—241no—
479C22⋊C4—64no—
480C22⋊C4—64no—
481C22⋊C4—62no—
482C22⋊C4—62no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—42no—
486C42—42no—
487C42—61no—
488C42—61no—
489C42—241no—
490C42—241no—
491C42#23712yes—
492C42C2×#6712yes—
493C22×C3C2×#12111yescharacteristic
494C22×C3#52511yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#57711yescharacteristic
497C22×C3#57711yescharacteristic
498C22×C3#57811yescharacteristic
499C22×C3#57811yescharacteristic
500C2×D3—41no—
501C2×D3—41no—
502C2×D3—41no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
506C2×D3—41no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52511yescharacteristic
514C3×C4#52911yescharacteristic
515C3×C4#52911yescharacteristic
516C3×C4#53211yescharacteristic
517Dic3—42no—
518Dic3—42no—
519Dic3—41no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23C2×D3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4#93812yes—
582C2×C4#93812yes—
583C2×C4C2×#23511yescharacteristic
584C2×C4C2×#23511yescharacteristic
585C2×C4C2×#23711yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4C22×#6711yescharacteristic
589C8—22no—
590C8—22no—
591C8—121no—
592C8—121no—
593C2×C3#239211yescharacteristic
594C2×C3#306311yescharacteristic
595C2×C3#306311yescharacteristic
596C2×C3#312011yescharacteristic
597C2×C3#312111yescharacteristic
598C2×C3#312511yescharacteristic
599C2×C3#312711yescharacteristic
600D3—81no—
601C22—121no—
602C22—121no—
603C22—121no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—121no—
608C22#554711yescharacteristic
609C22#554711yescharacteristic
610C22#577811yescharacteristic
611C22#577811yescharacteristic
612C22#577911yescharacteristic
613C22#577911yescharacteristic
614C22C2×#93811yescharacteristic
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—121no—
622C4—121no—
623C4—121no—
624C4—122no—
625C4—122no—
626C4—241no—
627C4—241no—
628C4#553811yescharacteristic
629C4#553811yescharacteristic
630C4#579111yescharacteristic
631C4#583211yescharacteristic
632C3#2371311yescharacteristic, complemented, Sylow
633C2—241no—
634C2#7983511yescharacteristic
635C2#7983611yescharacteristic
636C2#7984911yescharacteristic
637C2#7985011yescharacteristic
638C2#8240111yescharacteristic
639C2#8398611yescharacteristic
640C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#441C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#441D3
hypocenterC3#23713
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
1. C2×C3×#441C2semidirect
6. #72068C2semidirect
7. C4×#835C2semidirect
8. C4×#835C2semidirect
10. #70568C2semidirect
11. #70248C2semidirect
12. #70247C2semidirect
15. #69814C2semidirect
62. C2×#441D3semidirect
632. C3#23713semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4412
#824882
C2×#47092
#720712
#720682
C4×#8352
#705832
#705682
#702482
#702472
#702442
#702432
#698142
#237133