Finite Group Databasebeta

← #10868011

Subgroups — #10868011

Order 768.

742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82341C211yesmaximal, characteristic
2#76239C211yesmaximal, characteristic, complemented
3C3×#1864C211yesmaximal, characteristic, complemented
4C4×#832C211yesmaximal, characteristic, complemented
5#70580C211yesmaximal, characteristic, complemented
6#70557C211yesmaximal, characteristic, complemented
7#70206C211yesmaximal, characteristic
8#70194C211yesmaximal, characteristic, complemented
9#70175C211yesmaximal, characteristic
10#70165C211yesmaximal, characteristic, complemented
11#69619C211yesmaximal, characteristic, complemented
12#68813C211yesmaximal, characteristic, complemented
13#68795C211yesmaximal, characteristic, complemented
14#68657C211yesmaximal, characteristic, complemented
15#68624C211yesmaximal, characteristic, complemented
16#27041—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C3×#389C2211yescharacteristic
24#5221C2211yescharacteristic
25C2×#832C2211yescharacteristic
26C2×#832C2211yescharacteristic
27#5213—21no—
28C4×#217—21no—
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4994—21no—
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4985—21no—
37#4984C2211yescharacteristic
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4957—21no—
43#4944—21no—
44C4×(C2×D3)⋊C4C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4669—21no—
49#4654—21no—
50C3×#340C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2497—31no—
61#2363—31no—
62#2071—31no—
63#2067—31no—
64#2033—31no—
65#2033—31no—
66#1947—31no—
67#1942—31no—
68#1923—31no—
69#1921—31no—
70#1919—31no—
71#1864D311yescharacteristic, complemented
72#1854—31no—
73#1825—31no—
74C4×#102—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×#67—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×D4—22no—
82C3×C4×D4—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92#938—22no—
93#938—22no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#892—22no—
114#892—41no—
115C3×#106—22no—
116C3×#102—22no—
117C3×#102—41no—
118C3×#95—21no—
119#832—22no—
120#832—22no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129(C2×D3)⋊C8—22no—
130(C2×D3)⋊C8—41no—
131#818—22no—
132Dic3⋊C8—22no—
133#807—22no—
134#807—41no—
135C4×C3⋊C8—41no—
136C2×#121—31no—
137C2×C4×D4C2×D311yescharacteristic
138C2×#117—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#107—31no—
142C2×#106C2×D311yescharacteristic
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102C2×D311yescharacteristic
147C2×#102C2×D311yescharacteristic
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×C4×C8—31no—
151#400—61no—
152#396—61no—
153#390—31no—
154#390—61no—
155#389—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#388—61no—
159#388—61no—
160#387—61no—
161#386—61no—
162#384—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C22⋊C4—31no—
168C4×C22⋊C4—61no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C24×C3—21no—
180C23×D3C2×D411yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4#4112yes—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4—81no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C2×#67—24no—
197C2×#67—81no—
198C2×#67—81no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4#4112yes—
225C3×C22⋊C4—42no—
226C3×C42—21no—
227C3×C42—21no—
228#217—24no—
229#217—42no—
230#217—81no—
231(C2×D3)⋊C4—22no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—81no—
235(C2×D3)⋊C4—81no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3#4112yes—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C4×Dic3—81no—
252C4×Dic3—81no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8#4112yes—
259C22×D4—61no—
260C22×D4C22×D311yescharacteristic
261C23×C4—21no—
262C23×C4—31no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#121—62no—
268#121—62no—
269C4×D4—22no—
270C4×D4—41no—
271#117—62no—
272#117—121no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—31no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—31no—
295C2×C42—61no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—62no—
300#107—62no—
301#107—121no—
302#106—22no—
303C4⋊C8—62no—
304#102—22no—
305#102—41no—
306#102—62no—
307#102—62no—
308C22⋊C8—62no—
309C22⋊C8—121no—
310C4×C8—121no—
311#95—21no—
312#95—31no—
313#95—31no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320C23×C3—22no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3C22×D411yescharacteristic
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—41no—
338C3×D4—41no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—84no—
354C2×C3×C4#13712yes—
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C42−1+411yescharacteristic
361#67—84no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3#13512yes—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3⋊C8—22no—
390C3⋊C8—41no—
391C24—21no—
392C24—31no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—64no—
399C2×D4—241no—
400C2×D4—241no—
401C2×D4#23512yes—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—81no—
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423C22×C4C23×D311yescharacteristic
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—32no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—32no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—81no—
450C4⋊C4—122no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C4⋊C4#23712yes—
455C22⋊C4—42no—
456C22⋊C4—62no—
457C22⋊C4—64no—
458C22⋊C4—62no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—241no—
462C22⋊C4—241no—
463C22⋊C4—241no—
464C42—21no—
465C42—21no—
466C42—61no—
467C42—61no—
468C42—122no—
469C42—241no—
470C42—241no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3#52511yescharacteristic
479C22×C3#54611yescharacteristic
480C22×C3#54711yescharacteristic
481C22×C3C2×#13511yescharacteristic
482C22×C3C2×#13711yescharacteristic
483C22×C3#57611yescharacteristic
484C22×C3#57611yescharacteristic
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C3×C4—22no—
493C3×C4—41no—
494C3×C4—42no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4#52911yescharacteristic
498C3×C4#53911yescharacteristic
499C3×C4#55311yescharacteristic
500C3×C4#55611yescharacteristic
501Dic3—22no—
502Dic3—41no—
503Dic3—42no—
504Dic3—41no—
505Dic3—81no—
506Dic3—81no—
507Dic3—81no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23C2×D3×D411yescharacteristic
523D4—22no—
524D4—22no—
525D4—41no—
526D4—41no—
527D4—244no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—32no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—84no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4#91512yes—
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4#100411yescharacteristic
572C2×C4#100711yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C8—41no—
576C8—41no—
577C8—62no—
578C8—121no—
579C2×C3—82no—
580C2×C3#246611yescharacteristic
581C2×C3#310711yescharacteristic
582C2×C3#312211yescharacteristic
583C2×C3#330511yescharacteristic
584C2×C3#331411yescharacteristic
585C2×C3#333911yescharacteristic
586C2×C3#335711yescharacteristic
587D3—81no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—121no—
596C22—121no—
597C22—121no—
598C22—121no—
599C22—121no—
600C22—121no—
601C22—121no—
602C22#554711yescharacteristic
603C22#559111yescharacteristic
604C22#563111yescharacteristic
605C22C2×#91511yescharacteristic
606C22D3×#13511yescharacteristic
607C22#577411yescharacteristic
608C22#577411yescharacteristic
609C4—22no—
610C4—42no—
611C4—41no—
612C4—62no—
613C4—81no—
614C4—81no—
615C4—122no—
616C4—121no—
617C4—121no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4#551511yescharacteristic
622C4#553811yescharacteristic
623C4#564111yescharacteristic
624C4#566111yescharacteristic
625C3#2704111yescharacteristic, complemented, Sylow
626C2—82no—
627C2—241no—
628C2#7443411yescharacteristic
629C2#7965511yescharacteristic
630C2#7987411yescharacteristic
631C2#8010511yescharacteristic
632C2#8016311yescharacteristic
633C2#8108611yescharacteristic
634C2#8119511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1864D3
hypocenterC3#27041
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #76239C2semidirect
3. C3×#1864C2semidirect
4. C4×#832C2semidirect
5. #70580C2semidirect
6. #70557C2semidirect
8. #70194C2semidirect
10. #70165C2semidirect
11. #69619C2semidirect
12. #68813C2semidirect
13. #68795C2semidirect
14. #68657C2semidirect
15. #68624C2semidirect
71. #1864D3semidirect
625. C3#27041semidirect

Maximal subgroups

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

subgroupindex
#823412
#762392
C3×#18642
C4×#8322
#705802
#705572
#702062
#701942
#701752
#701652
#696192
#688132
#687952
#686572
#686242
#270413