Finite Group Databasebeta

← #10877414

Subgroups — #10877414

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
1C3×#2439C211yesmaximal, characteristic, complemented
2#73515C211yesmaximal, characteristic
3#71792C211yesmaximal, characteristic, complemented
4#71787C211yesmaximal, characteristic, complemented
5#71586C211yesmaximal, characteristic
6#71580C211yesmaximal, characteristic, complemented
7#69615C211yesmaximal, characteristic, complemented
8#69615C211yesmaximal, characteristic, complemented
9#69613C211yesmaximal, characteristic, complemented
10#69612C211yesmaximal, characteristic, complemented
11#69557C211yesmaximal, characteristic, complemented
12#69547C211yesmaximal, characteristic, complemented
13#69515C211yesmaximal, characteristic
14#69503C211yesmaximal, characteristic, complemented
15C22⋊C4×C3⋊C8C211yesmaximal, characteristic, complemented
16#27696—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C3×#396—21no—
23C3×#393—21no—
24C3×#391—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#387—21no—
28C3×#382C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C3×C4×C22⋊C4C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#809C2211yescharacteristic
33C2×#809C2211yescharacteristic
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36#4961—21no—
37#4961—21no—
38#4939C2211yescharacteristic
39#4939C2211yescharacteristic
40#4936C2211yescharacteristic
41#4935C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43C2×C4×C3⋊C8C2211yescharacteristic
44#4670—21no—
45#4670—21no—
46#4529C2211yescharacteristic
47#4529C2211yescharacteristic
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2500—31no—
61#2439D311yescharacteristic, complemented
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#1875—31no—
71#1835—31no—
72#1825—31no—
73C8×C22⋊C4—31no—
74#1735—31no—
75C22×C3×Q8—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C22×D12C2311yescharacteristic
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8—41no—
81C3×#117—22no—
82C3×#117—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98#930—22no—
99#930—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105#926—22no—
106#926—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4×Dic3—21no—
113C22×C3⋊C8—22no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—41no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3⋊#95—21no—
128C3⋊#95—21no—
129#830—22no—
130#809—22no—
131#809—22no—
132#809—41no—
133#809—41no—
134C4×C3⋊C8—22no—
135C4×C3⋊C8—41no—
136C23×C8—31no—
137C2×C4⋊Q8C2×D311yescharacteristic
138C2×#121—31no—
139C2×#117—31no—
140C2×#117C2×D311yescharacteristic
141C2×#102—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8—31no—
148#396—21no—
149#394—61no—
150#394—61no—
151#393—21no—
152#391—21no—
153#390—31no—
154#390—31no—
155#389C2×D311yescharacteristic
156#389C2×D311yescharacteristic
157#388—61no—
158#388—61no—
159#387—21no—
160#384—31no—
161#382—31no—
162#382C2×D311yescharacteristic
163#380C2×D311yescharacteristic
164C4×C22⋊C4C2×D311yescharacteristic
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#340—31no—
176#340—31no—
177#336—31no—
178#336—31no—
179C24×C3C2×D411yescharacteristic
180C23×D3—21no—
181C2×C3×Q8—42no—
182C2×C3×Q8—42no—
183C2×C3×Q8—81no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4#4112yes—
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C2×#67—42no—
201C2×#67—81no—
202C2×#67—81no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—81no—
222C3×C4⋊C4#4112yes—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—81no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—42no—
232#217—42no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—81no—
245C4⋊Dic3—81no—
246C4×Dic3—42no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
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—21no—
261C22×D4—31no—
262C22×D4—61no—
263C23×C4C22×D311yescharacteristic
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8—62no—
269C22×C8—62no—
270C22×C8—121no—
271C4⋊Q8—22no—
272C4⋊Q8—41no—
273#121—62no—
274#121—62no—
275#117—22no—
276#117—41no—
277#117—62no—
278#117—62no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4#6711yescharacteristic
300C2×C22⋊C4#6711yescharacteristic
301C2×C42—21no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#102—62no—
306#102—62no—
307#102—121no—
308#102—121no—
309C22⋊C8—62no—
310C4×C8—62no—
311C4×C8—121no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—61no—
318#95—61no—
319#95#6711yescharacteristic
320#95#6711yescharacteristic
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3#12412yes—
328C23×C3C22×D411yescharacteristic
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C3×Q8—84no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
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—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4#12412yes—
366C2×C3×C4#13512yes—
367C2×C3×C4#13512yes—
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C42−1+411yescharacteristic
374#67—84no—
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—
384D12—41no—
385D12—41no—
386D12—41no—
387D12—41no—
388C3⋊C8—22no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24—61no—
393C24C2×#6711yescharacteristic
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—81no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—122no—
404C2×D4—241no—
405C2×D4—241no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4#23912yes—
418C22×C4#23912yes—
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C23×D311yescharacteristic
425C2×C8—32no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
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—21no—
444C4⋊C4—21no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—81no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4#23712yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4—81no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—241no—
466C22⋊C4—241no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—122no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3C2×#12411yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3#54711yescharacteristic
483C22×C3C2×#13511yescharacteristic
484C22×C3C2×#13511yescharacteristic
485C22×C3#57611yescharacteristic
486C22×C3#57711yescharacteristic
487C2×D3—81no—
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—42no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#52911yescharacteristic
504C3×C4#53911yescharacteristic
505C3×C4#55111yescharacteristic
506C3×C4#55211yescharacteristic
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23#94712yes—
524C23C22×#6711yescharacteristic
525Q8—84no—
526D4—121no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—244no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
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—41no—
557C2×C4—41no—
558C2×C4—41no—
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#94712yes—
569C2×C4#95012yes—
570C2×C4#95012yes—
571C2×C4C2×#23711yescharacteristic
572C2×C4#100411yescharacteristic
573C2×C4C2×#23911yescharacteristic
574C2×C4C2×#23911yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4#101011yescharacteristic
577C8—62no—
578C8—121no—
579C8—121no—
580C8—121no—
581C2×C3—42no—
582C2×C3#248811yescharacteristic
583C2×C3#313111yescharacteristic
584C2×C3#313611yescharacteristic
585C2×C3#329711yescharacteristic
586C2×C3#330111yescharacteristic
587C2×C3#334011yescharacteristic
588C2×C3#334111yescharacteristic
589D3—161no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—241no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22#580611yescharacteristic
605C22C2×#94711yescharacteristic
606C22C2×#95011yescharacteristic
607C22C2×#95011yescharacteristic
608C22#582411yescharacteristic
609C22#583511yescharacteristic
610C22#583711yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—241no—
621C4—241no—
622C4—241no—
623C4#562411yescharacteristic
624C4#563711yescharacteristic
625C4#564211yescharacteristic
626C4#564511yescharacteristic
627C3#2769611yescharacteristic, complemented, Sylow
628C2—42no—
629C2—481no—
630C2#8277111yescharacteristic
631C2#8333411yescharacteristic
632C2#8347011yescharacteristic
633C2#8349511yescharacteristic
634C2#8353511yescharacteristic
635C2#8364711yescharacteristic
636C2#8365811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2439C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2439D3
hypocenterC3#27696
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
1. C3×#2439C2semidirect
3. #71792C2semidirect
4. #71787C2semidirect
6. #71580C2semidirect
7. #69615C2semidirect
8. #69615C2semidirect
9. #69613C2semidirect
10. #69612C2semidirect
11. #69557C2semidirect
12. #69547C2semidirect
14. #69503C2semidirect
15. C22⋊C4×C3⋊C8C2semidirect
61. #2439D3semidirect
627. C3#27696semidirect

Maximal subgroups

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

subgroupindex
C3×#24392
#735152
#717922
#717872
#715862
#715802
#696152
#696132
#696122
#695572
#695472
#695152
#695032
C22⋊C4×C3⋊C82
#276963