Finite Group Databasebeta

← #10877413

Subgroups — #10877413

Order 768.

752 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76453C211yesmaximal, characteristic
2#76223C211yesmaximal, characteristic, complemented
3C3×#1934C211yesmaximal, characteristic, complemented
4#70202C211yesmaximal, characteristic
5#70201C211yesmaximal, characteristic
6#70186C211yesmaximal, characteristic, complemented
7#70166C211yesmaximal, characteristic, complemented
8C8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
9#69777C211yesmaximal, characteristic, complemented
10#69638C211yesmaximal, characteristic, complemented
11#69612C211yesmaximal, characteristic, complemented
12#68873C211yesmaximal, characteristic, complemented
13#68866C211yesmaximal, characteristic, complemented
14#68848C211yesmaximal, characteristic, complemented
15#68812C211yesmaximal, characteristic, complemented
16#27696—31nomaximal, Sylow
17C2×#938C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#389C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4994—21no—
33#4987C2211yescharacteristic
34#4980—21no—
35#4977C2211yescharacteristic
36#4977C2211yescharacteristic
37#4972C2211yescharacteristic
38#4961—21no—
39#4961—21no—
40#4951—21no—
41#4951—21no—
42C4×(C2×D3)⋊C4C2211yescharacteristic
43#4935C2211yescharacteristic
44#4673—21no—
45#4646—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2500—31no—
61#2439—31no—
62#2101—31no—
63#2100—31no—
64#2073—31no—
65#2023—31no—
66#1990—31no—
67#1934—31no—
68#1934D311yescharacteristic, complemented
69#1893—31no—
70#1875—31no—
71#1835—31no—
72#1825—31no—
73C8×C22⋊C4—31no—
74#1735—31no—
75C22×C3×D4C2311yescharacteristic
76C22×#67—21no—
77C22×C4×D3C2311yescharacteristic
78C22×Dic6—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91#938—22no—
92#938—22no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×D3×C8—22no—
112C2×D3×C8—22no—
113C2×D3×C8—41no—
114#892—22no—
115#892—22no—
116#892—22no—
117#892—41no—
118#869—22no—
119#869—41no—
120C3×#102—22no—
121C3×#102—41no—
122C3×C4×C8—41no—
123C3×#95—21no—
124#832—22no—
125#832—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134(C2×D3)⋊C8—22no—
135C8×Dic3—22no—
136C23×C8—31no—
137C2×C4⋊Q8—31no—
138C2×#121—31no—
139C2×#117—31no—
140C2×#117—31no—
141C2×#102—31no—
142C2×#102—31no—
143C2×#102C2×D311yescharacteristic
144C2×#102C2×D311yescharacteristic
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8C2×D311yescharacteristic
148#396—61no—
149#394—61no—
150#394—61no—
151#393—61no—
152#391—61no—
153#390—31no—
154#390C2×D311yescharacteristic
155#389—31no—
156#389C2×D311yescharacteristic
157#388—61no—
158#388—61no—
159#387—61no—
160#384—31no—
161#382—31no—
162#382—31no—
163#380—31no—
164C4×C22⋊C4—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
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—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×#67—42no—
194C2×#67—81no—
195C2×#67—81no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3#4112yes—
208C2×C4×D3#4112yes—
209C2×Dic6—42no—
210C2×Dic6—42no—
211C2×Dic6—81no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8#4112yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C42—21no—
228C3×C42—21no—
229#217—42no—
230#217—81no—
231#217—81no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—22no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—81no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3—81no—
246C4⋊Dic3—81no—
247C4⋊Dic3#4112yes—
248Dic3⋊C4—81no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—42no—
252C4×Dic3—42no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259D3×C8—44no—
260C22×Q8—61no—
261C22×D4—61no—
262C22×D4C22×D311yescharacteristic
263C23×C4—31no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—62no—
267C22×C8—62no—
268C22×C8—121no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C4⋊Q8—62no—
272C4⋊Q8—121no—
273#121—62no—
274#121—62no—
275#117—62no—
276#117—62no—
277#117—62no—
278#117—121no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—31no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#102—22no—
306#102—41no—
307#102—62no—
308#102—121no—
309C22⋊C8—62no—
310C4×C8—41no—
311C4×C8—62no—
312#95—21no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3C22×D411yescharacteristic
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3#12412yes—
336C3×D4—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—41no—
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#13512yes—
354C2×C3×C4#13512yes—
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#12412yes—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—22no—
386C4×D3—22no—
387Dic6—84no—
388C3×C8—22no—
389C3×C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24—21no—
393C24—31no—
394C2×Q8—122no—
395C2×Q8—122no—
396C2×Q8—241no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—122no—
404C2×D4—241no—
405C2×D4—241no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—31no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4D3×D411yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4C23×D311yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—124no—
437C2×C8#23512yes—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—32no—
445C4⋊C4—42no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—122no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4—241no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—62no—
460C22⋊C4—62no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—241no—
465C22⋊C4—241no—
466C22⋊C4—241no—
467C42—21no—
468C42—21no—
469C42—61no—
470C42—61no—
471C42—122no—
472C42—122no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3C2×#12411yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3#54711yescharacteristic
483C22×C3C2×#13511yescharacteristic
484C22×C3C2×#13511yescharacteristic
485C22×C3#57611yescharacteristic
486C22×C3#57711yescharacteristic
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#52911yescharacteristic
500C3×C4#53911yescharacteristic
501C3×C4#55111yescharacteristic
502C3×C4#55211yescharacteristic
503Dic3—22no—
504Dic3—41no—
505Dic3—41no—
506Dic3—42no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—32no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23—61no—
523C23—61no—
524C23C2×D3×D411yescharacteristic
525Q8—244no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—244no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—32no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—62no—
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—121no—
569C2×C4#91212yes—
570C2×C4#91612yes—
571C2×C4C2×#23511yescharacteristic
572C2×C4#100411yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4#101011yescharacteristic
577C8—22no—
578C8—41no—
579C8—121no—
580C8—121no—
581C2×C3—161no—
582C2×C3#248811yescharacteristic
583C2×C3#313111yescharacteristic
584C2×C3#313611yescharacteristic
585C2×C3#329711yescharacteristic
586C2×C3#330111yescharacteristic
587C2×C3#334011yescharacteristic
588C2×C3#334111yescharacteristic
589D3—42no—
590C22—62no—
591C22—62no—
592C22—62no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—121no—
601C22—121no—
602C22—121no—
603C22—121no—
604C22#555111yescharacteristic
605C22#561011yescharacteristic
606C22D3×#12411yescharacteristic
607C22C2×#91211yescharacteristic
608C22C2×#91611yescharacteristic
609C22#576911yescharacteristic
610C22#577011yescharacteristic
611C4—22no—
612C4—22no—
613C4—62no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—121no—
618C4—122no—
619C4—121no—
620C4—241no—
621C4—241no—
622C4—241no—
623C4#552811yescharacteristic
624C4#555611yescharacteristic
625C4#566011yescharacteristic
626C4#570811yescharacteristic
627C3#2769611yescharacteristic, complemented, Sylow
628C2—122no—
629C2—161no—
630C2#7586711yescharacteristic
631C2#7961611yescharacteristic
632C2#7969011yescharacteristic
633C2#8068111yescharacteristic
634C2#8073611yescharacteristic
635C2#8156211yescharacteristic
636C2#8163811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1934C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1934D3
hypocenterC3#27696
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #76223C2semidirect
3. C3×#1934C2semidirect
6. #70186C2semidirect
7. #70166C2semidirect
8. C8×(C2×D3)⋊C4C2semidirect
9. #69777C2semidirect
10. #69638C2semidirect
11. #69612C2semidirect
12. #68873C2semidirect
13. #68866C2semidirect
14. #68848C2semidirect
15. #68812C2semidirect
68. #1934D3semidirect
627. C3#27696semidirect

Maximal subgroups

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

subgroupindex
#764532
#762232
C3×#19342
#702022
#702012
#701862
#701662
C8×(C2×D3)⋊C42
#697772
#696382
#696122
#688732
#688662
#688482
#688122
#276963