Finite Group Databasebeta

← #10877406

Subgroups — #10877406

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#82663C211yesmaximal, characteristic
2#82248C211yesmaximal, characteristic, complemented
3C3×#1875C211yesmaximal, characteristic
4#71681C211yesmaximal, characteristic
5#71580C211yesmaximal, characteristic, complemented
6#70186C211yesmaximal, characteristic, complemented
7#70185C211yesmaximal, characteristic, complemented
8#70166C211yesmaximal, characteristic, complemented
9#70163C211yesmaximal, characteristic, complemented
10#69947C211yesmaximal, characteristic, complemented
11#69926C211yesmaximal, characteristic, complemented
12#68848C211yesmaximal, characteristic, complemented
13#68846C211yesmaximal, characteristic, complemented
14#68811C211yesmaximal, characteristic, complemented
15#68803C211yesmaximal, characteristic, complemented
16#27696—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25#5222—21no—
26#5222—21no—
27#5221C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5212C2211yescharacteristic
32C4×#217C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4984C2211yescharacteristic
36#4977C2211yescharacteristic
37#4977C2211yescharacteristic
38#4975C2211yescharacteristic
39#4972C2211yescharacteristic
40#4948—21no—
41#4947—21no—
42#4944—21no—
43#4944—21no—
44#4933—21no—
45#4929—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
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#1934—31no—
69#1893—31no—
70#1875D311yescharacteristic
71#1835—31no—
72#1825—31no—
73C8×C22⋊C4—31no—
74#1735—31no—
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C22×Dic6—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#121—22no—
82C3×#121—22no—
83C3×#117—22no—
84C3×#117—41no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217D411yescharacteristic
98C2×#217D411yescharacteristic
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8—22no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—41no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#888—22no—
119#888—41no—
120C3×#102—22no—
121C3×#102—41no—
122C3×C22⋊C8—22no—
123#832—22no—
124#832—41no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C8×Dic3—22no—
135C8×Dic3—41no—
136C23×C8—31no—
137C2×C4⋊Q8—31no—
138C2×#121C2×D311yescharacteristic
139C2×#117—31no—
140C2×#117C2×D311yescharacteristic
141C2×#102—31no—
142C2×#102—31no—
143C2×#102C2×D311yescharacteristic
144C2×#102C2×D311yescharacteristic
145C2×C22⋊C8C2×D311yescharacteristic
146C2×C4×C8—31no—
147C2×C4×C8—31no—
148#396—61no—
149#394—61no—
150#394—61no—
151#393—61no—
152#391—61no—
153#390—31no—
154#390—31no—
155#389—31no—
156#389—31no—
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—
180C24×C3C2×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—
187C2×C3×D4—42no—
188C2×C3×D4—81no—
189C2×C3×D4—81no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4#4112yes—
199C22×C3×C4#4112yes—
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
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—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4#4112yes—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C42—42no—
227#217—22no—
228#217—22no—
229#217—42no—
230#217—42no—
231#217—42no—
232#217—81no—
233#217—81no—
234#217—81no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—81no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
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—44no—
259C2×C3⋊C8#4112yes—
260C22×Q8—61no—
261C22×D4—21no—
262C22×D4C22×D311yescharacteristic
263C23×C4C22×D311yescharacteristic
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—22no—
274#121—22no—
275#117—22no—
276#117—41no—
277#117—62no—
278#117—62no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
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—21no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—61no—
305#102—22no—
306#102—41no—
307#102—62no—
308#102—121no—
309C22⋊C8—22no—
310C4×C8—62no—
311C4×C8—121no—
312#95—31no—
313#95—31no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3#12412yes—
335C23×C3C22×D411yescharacteristic
336C3×D4—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—84no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4#12412yes—
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C42−1+411yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
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—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3#13512yes—
386C2×Dic3#13512yes—
387Dic6—84no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—22no—
391C3⋊C8—41no—
392C24—21no—
393C24D3×D411yescharacteristic
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—42no—
404C2×D4—81no—
405C2×D4—81no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4#23512yes—
422C22×C4D3×D411yescharacteristic
423C22×C4#23712yes—
424C22×C4C23×D311yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—32no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—124no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4#23712yes—
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—42no—
468C42—61no—
469C42—61no—
470C42—61no—
471C42—61no—
472C42—122no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3C2×#12411yescharacteristic
488C22×C3#54611yescharacteristic
489C22×C3#54711yescharacteristic
490C22×C3C2×#13511yescharacteristic
491C22×C3C2×#13511yescharacteristic
492C22×C3#57611yescharacteristic
493C22×C3#57711yescharacteristic
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#52911yescharacteristic
499C3×C4#53911yescharacteristic
500C3×C4#55111yescharacteristic
501C3×C4#55211yescharacteristic
502Dic3—22no—
503Dic3—22no—
504Dic3—41no—
505Dic3—41no—
506Dic3—81no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23#88512yes—
524C23C2×D3×D411yescharacteristic
525Q8—244no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—84no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—32no—
539C2×C4—32no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
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—121no—
570C2×C4#88512yes—
571C2×C4C2×#23511yescharacteristic
572C2×C4C2×#23711yescharacteristic
573C2×C4C2×#23711yescharacteristic
574C2×C4#100411yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4#101011yescharacteristic
577C8—41no—
578C8—41no—
579C8—62no—
580C8—121no—
581C2×C3—42no—
582C2×C3—161no—
583C2×C3#248811yescharacteristic
584C2×C3#313111yescharacteristic
585C2×C3#313611yescharacteristic
586C2×C3#329711yescharacteristic
587C2×C3#330111yescharacteristic
588C2×C3#334011yescharacteristic
589C2×C3#334111yescharacteristic
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22C2×#88511yescharacteristic
605C22#559111yescharacteristic
606C22#565511yescharacteristic
607C22D3×#13511yescharacteristic
608C22D3×#13511yescharacteristic
609C22#577411yescharacteristic
610C22#577511yescharacteristic
611C4—22no—
612C4—42no—
613C4—62no—
614C4—62no—
615C4—81no—
616C4—81no—
617C4—121no—
618C4—121no—
619C4—241no—
620C4—241no—
621C4—241no—
622C4—241no—
623C4#554111yescharacteristic
624C4#555611yescharacteristic
625C4#560011yescharacteristic
626C4#560211yescharacteristic
627C3#2769611yescharacteristic, complemented, Sylow
628C2—42no—
629C2—161no—
630C2#7474011yescharacteristic
631C2#7882511yescharacteristic
632C2#7892011yescharacteristic
633C2#8010211yescharacteristic
634C2#8010311yescharacteristic
635C2#8136411yescharacteristic
636C2#8138911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1875C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#1875D3
hypocenterC3#27696
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. #82248C2semidirect
5. #71580C2semidirect
6. #70186C2semidirect
7. #70185C2semidirect
8. #70166C2semidirect
9. #70163C2semidirect
10. #69947C2semidirect
11. #69926C2semidirect
12. #68848C2semidirect
13. #68846C2semidirect
14. #68811C2semidirect
15. #68803C2semidirect
627. C3#27696semidirect

Maximal subgroups

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

subgroupindex
#826632
#822482
C3×#18752
#716812
#715802
#701862
#701852
#701662
#701632
#699472
#699262
#688482
#688462
#688112
#688032
#276963