Finite Group Databasebeta

← #10844649

Subgroups — #10844649

Order 768.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#491C211yesmaximal, characteristic, complemented
2#83240C211yesmaximal, characteristic
3#83025C211yesmaximal, characteristic, complemented
4C2×#5066C211yesmaximal, characteristic
5#71778C211yesmaximal, characteristic, complemented
6#70262C211yesmaximal, characteristic, complemented
7#70261C211yesmaximal, characteristic, complemented
8#70245C211yesmaximal, characteristic
9#70245C211yesmaximal, characteristic
10#70242C211yesmaximal, characteristic, complemented
11#70241C211yesmaximal, characteristic, complemented
12#70230C211yesmaximal, characteristic
13#70230C211yesmaximal, characteristic
14#69836C211yesmaximal, characteristic, complemented
15#69812C211yesmaximal, characteristic
16#25289—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C3×#491—24no—
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#5066—24no—
39#4992—21no—
40#4992—21no—
41#4987—21no—
42#4987—21no—
43#4984—21no—
44#4984C2211yescharacteristic
45#4984C2211yescharacteristic
46#4980—21no—
47#4980—21no—
48Dic3×C4⋊C4—21no—
49Dic3×C4⋊C4C2211yescharacteristic
50Dic3×C4⋊C4C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#809C2211yescharacteristic
53C2×#808C2211yescharacteristic
54C2×#807C2211yescharacteristic
55C4×C4⋊Dic3C2211yescharacteristic
56C4×C4⋊Dic3C2211yescharacteristic
57C2×C4⋊(C3⋊C8)C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×#491D311yescharacteristic, complemented
63C2×#478—31no—
64#2547—31no—
65#2370—31no—
66#2068—31no—
67#2066—31no—
68#2042—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1927—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—22no—
85C3×#125—22no—
86C3×#125D414yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94#947—22no—
95#947—22no—
96#947—41no—
97#947—41no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C4×D12—22no—
118C4×D12—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#97—22no—
124#835—22no—
125#835—22no—
126#835—41no—
127#835—41no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#821—22no—
140#821—22no—
141Dic3⋊C8—41no—
142Dic3⋊C8—41no—
143#809—22no—
144#809—22no—
145#808—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C2×C4⋊Q8C2×D311yescharacteristic
149C2×#125C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×D4—31no—
153#491—24no—
154#478—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97C2×D311yescharacteristic
173#397—61no—
174#397—61no—
175#391—31no—
176#391—31no—
177#389—31no—
178#389—31no—
179#389—61no—
180#387—61no—
181#387—61no—
182#380—61no—
183#380—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—61no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
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—21no—
205C2×C3×Q8—21no—
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×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12#4112yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4C2×D412yes—
242C3×C4⋊C4C2×D412yes—
243C3×C42C2×D412yes—
244C3×C42#4112yes—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—41no—
256C4⋊Dic3—41no—
257C4⋊Dic3—42no—
258C4⋊Dic3—81no—
259C4⋊Dic3#4112yes—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—42no—
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
275C22×D4—31no—
276C23×C4—61no—
277C22×C8—31no—
278C22×C8—31no—
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283#125—22no—
284#125#6714yes—
285#124—62no—
286#124—62no—
287#124—121no—
288#124—121no—
289C4×D4—62no—
290C4×D4—62no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
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—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—62no—
316#106—62no—
317C4⋊C8—62no—
318C4⋊C8—121no—
319C4⋊C8—121no—
320#103—22no—
321#103—22no—
322#103—22no—
323#103—22no—
324#103—62no—
325#103—62no—
326#103—121no—
327#103—121no—
328#102—62no—
329#102—62no—
330#102—62no—
331#102—62no—
332#97—22no—
333#95—31no—
334#95—31no—
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—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—41no—
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×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C42−1+411yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405D12—22no—
406D12—22no—
407D12—41no—
408D12—41no—
409C4×D3—84no—
410C3×C8—41no—
411C3×C8—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C24—61no—
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—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×D4—32no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—121no—
475C4⋊C4—121no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C4⋊C4C2×#6712yes—
479C4⋊C4C2×#6712yes—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—61no—
486C42—61no—
487C42—61no—
488C42—61no—
489C42—122no—
490C42—241no—
491C42—241no—
492C42#23712yes—
493C42C2×#6712yes—
494C22×C3C2×#12111yescharacteristic
495C22×C3#54611yescharacteristic
496C22×C3#54811yescharacteristic
497C22×C3#57511yescharacteristic
498C22×C3#57511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#52911yescharacteristic
515C3×C4#54211yescharacteristic
516C3×C4#54811yescharacteristic
517C3×C4#55011yescharacteristic
518Dic3—42no—
519Dic3—41no—
520Dic3—41no—
521Dic3—41no—
522Dic3—41no—
523Dic3—81no—
524Dic3—81no—
525C23—62no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C2×D3×D411yescharacteristic
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541D4—62no—
542D4—62no—
543D4—121no—
544D4—121no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—244no—
583C2×C4#93912yes—
584C2×C4#93912yes—
585C2×C4C2×#23711yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4#100911yescharacteristic
589C2×C4#101011yescharacteristic
590C2×C4C22×#6711yescharacteristic
591C8—41no—
592C8—41no—
593C8—121no—
594C8—121no—
595C2×C3#242611yescharacteristic
596C2×C3#304011yescharacteristic
597C2×C3#304611yescharacteristic
598C2×C3#330711yescharacteristic
599C2×C3#331511yescharacteristic
600C2×C3#337611yescharacteristic
601C2×C3#337711yescharacteristic
602D3—82no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22#567611yescharacteristic
611C22#567711yescharacteristic
612C22#577711yescharacteristic
613C22#577711yescharacteristic
614C22#577811yescharacteristic
615C22#577911yescharacteristic
616C22C2×#93911yescharacteristic
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—122no—
624C4—121no—
625C4—121no—
626C4—121no—
627C4—121no—
628C4—241no—
629C4—241no—
630C4#568011yescharacteristic
631C4#568211yescharacteristic
632C4#583111yescharacteristic
633C4#583411yescharacteristic
634C3#2528911yescharacteristic, complemented, Sylow
635C2—242no—
636C2#8173411yescharacteristic
637C2#8173511yescharacteristic
638C2#8174811yescharacteristic
639C2#8175011yescharacteristic
640C2#8299511yescharacteristic
641C2#8396811yescharacteristic
642C2#8397111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#491C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#491D3
hypocenterC3#25289
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
1. C2×C3×#491C2semidirect
3. #83025C2semidirect
5. #71778C2semidirect
6. #70262C2semidirect
7. #70261C2semidirect
10. #70242C2semidirect
11. #70241C2semidirect
14. #69836C2semidirect
62. C2×#491D3semidirect
634. C3#25289semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4912
#832402
#830252
C2×#50662
#717782
#702622
#702612
#702452
#702422
#702412
#702302
#698362
#698122
#252893