Finite Group Databasebeta

← #10877429

Subgroups — #10877429

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2440C211yesmaximal, characteristic
2#73517C211yesmaximal, characteristic
3#71794C211yesmaximal, characteristic, complemented
4#71787C211yesmaximal, characteristic, complemented
5#71559C211yesmaximal, characteristic
6C2×#4529C211yesmaximal, characteristic
7#69614C211yesmaximal, characteristic, complemented
8#69614C211yesmaximal, characteristic, complemented
9#69612C211yesmaximal, characteristic, complemented
10#69612C211yesmaximal, characteristic, complemented
11#69589C211yesmaximal, characteristic, complemented
12#69574C211yesmaximal, characteristic, complemented
13#69514C211yesmaximal, characteristic
14#69502C211yesmaximal, characteristic
15C22⋊C4×C3⋊C8C211yesmaximal, characteristic
16#27697—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C3×#398—21no—
24C3×#395—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#385—21no—
28C3×#384C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C3×C4×C22⋊C4C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#810C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36#4950—21no—
37#4950—21no—
38#4935C2211yescharacteristic
39#4935C2211yescharacteristic
40#4929C2211yescharacteristic
41#4929C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43C2×C4×C3⋊C8C2211yescharacteristic
44#4661—21no—
45#4661—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#2502—31no—
61#2440D311yescharacteristic
62#2100—31no—
63#2100—31no—
64#2072—31no—
65#2024—31no—
66#1990—31no—
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1872—31no—
71#1834—31no—
72#1824—31no—
73C8×C22⋊C4—31no—
74C2×#340—31no—
75C22×C3×Q8—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C22×Dic6C2311yescharacteristic
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—41no—
82C3×C22⋊Q8—44no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—41no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111#924—22no—
112#924—22no—
113C2×Dic3⋊C4—21no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—22no—
116C22×C3⋊C8—41no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#830—22no—
131#810—22no—
132#810—22no—
133#810—41no—
134#810—41no—
135C4×C3⋊C8—22no—
136C4×C3⋊C8—41no—
137C23×C8—31no—
138C2×C4⋊Q8C2×D311yescharacteristic
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8—31no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4C2×D311yescharacteristic
143C2×#103—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×C22⋊C8—31no—
148C2×C4×C8—31no—
149C2×C4×C8—31no—
150#398—21no—
151#398—61no—
152#398—61no—
153#395—21no—
154#391—31no—
155#391—31no—
156#389C2×D311yescharacteristic
157#389C2×D311yescharacteristic
158#385—21no—
159#385—61no—
160#385—61no—
161#384—31no—
162#384—31no—
163#384C2×D311yescharacteristic
164#380C2×D311yescharacteristic
165C4×C22⋊C4C2×D311yescharacteristic
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#340—31no—
178#336—31no—
179#336—31no—
180C24×C3C2×D411yescharacteristic
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×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—81no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4#4112yes—
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—42no—
208C22×Dic3—81no—
209C22×Dic3—81no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4—42no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4—81no—
229C3×C4⋊C4#4112yes—
230C3×C22⋊C4—42no—
231C3×C22⋊C4SD1614yes—
232C3×C22⋊C4Q1614yes—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237#217—42no—
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—81no—
248C4⋊Dic3—81no—
249Dic3⋊C4—42no—
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—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—44no—
262C2×C3⋊C8#4112yes—
263C22×Q8—21no—
264C22×Q8—31no—
265C23×C4—21no—
266C23×C4—61no—
267C23×C4C22×D311yescharacteristic
268C22×C8—31no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8—62no—
273C22×C8—62no—
274C22×C8—121no—
275C4⋊Q8—22no—
276C4⋊Q8—41no—
277C22⋊Q8—44no—
278C22⋊Q8—62no—
279C22⋊Q8—62no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4#6711yescharacteristic
306C2×C22⋊C4#6711yescharacteristic
307C2×C42C22×D311yescharacteristic
308C2×C42C22×D311yescharacteristic
309#103—62no—
310#103—62no—
311#103—121no—
312#103—121no—
313C22⋊C8—62no—
314C4×C8—62no—
315C4×C8—121no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95#6711yescharacteristic
324#95#6711yescharacteristic
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3#12412yes—
330C23×C3C2×SD1612yes—
331C23×C3C2×Q1612yes—
332C23×C3C22×D411yescharacteristic
333C3×Q8—84no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
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—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
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—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—84no—
361C2×C3×C4#12412yes—
362C2×C3×C4C2×SD1612yes—
363C2×C3×C4C2×SD1612yes—
364C2×C3×C4C2×Q1612yes—
365C2×C3×C4C2×Q1612yes—
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C42−1+411yescharacteristic
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—41no—
383C2×Dic3—41no—
384C2×Dic3—84no—
385Dic6—41no—
386Dic6—41no—
387Dic6—41no—
388Dic6—41no—
389C3⋊C8—22no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C24C2×#6711yescharacteristic
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—81no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—81no—
418C22×C4—122no—
419C22×C4—241no—
420C22×C4—241no—
421C22×C4#23912yes—
422C22×C4#23912yes—
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428C22×C4C23×D311yescharacteristic
429C2×C8—32no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—124no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—81no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C4⋊C4#23712yes—
468C22⋊C4—42no—
469C22⋊C4—122no—
470C22⋊C4#21414yes—
471C22⋊C4#21614yes—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3#48714yes—
483C22×C3C2×#12411yescharacteristic
484C22×C3#54611yescharacteristic
485C22×C3#54711yescharacteristic
486C22×C3C22×SD1611yescharacteristic
487C22×C3C22×Q1611yescharacteristic
488C22×C3#57611yescharacteristic
489C22×C3#57711yescharacteristic
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#52911yescharacteristic
500C3×C4#53911yescharacteristic
501C3×C4#55111yescharacteristic
502C3×C4#55211yescharacteristic
503Dic3—81no—
504Dic3—81no—
505Dic3—81no—
506Dic3—81no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C2×#21412yes—
512C23C2×#21612yes—
513C23#94712yes—
514C23C22×#6711yescharacteristic
515Q8—84no—
516Q8—121no—
517Q8—121no—
518Q8—121no—
519Q8—121no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
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—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—244no—
560C2×C4C2×#21412yes—
561C2×C4C2×#21412yes—
562C2×C4C2×#21612yes—
563C2×C4C2×#21612yes—
564C2×C4#94712yes—
565C2×C4C2×#23711yescharacteristic
566C2×C4#100411yescharacteristic
567C2×C4C2×#23911yescharacteristic
568C2×C4C2×#23911yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4#101011yescharacteristic
571C8—62no—
572C8—121no—
573C8—121no—
574C8—121no—
575C2×C3—24no—
576C2×C3#248811yescharacteristic
577C2×C3C2×#48711yescharacteristic
578C2×C3#313611yescharacteristic
579C2×C3#329511yescharacteristic
580C2×C3#329911yescharacteristic
581C2×C3#333511yescharacteristic
582C2×C3#333811yescharacteristic
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22#509314yes—
590C22C22×#21411yescharacteristic
591C22#580611yescharacteristic
592C22C22×#21611yescharacteristic
593C22C2×#94711yescharacteristic
594C22#582411yescharacteristic
595C22#583511yescharacteristic
596C22#583711yescharacteristic
597C4—22no—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—241no—
607C4—241no—
608C4—241no—
609C4—241no—
610C4#562411yescharacteristic
611C4#563711yescharacteristic
612C4#564211yescharacteristic
613C4#564511yescharacteristic
614C3#2769711yescharacteristic, complemented, Sylow
615C2—24no—
616C2#8277111yescharacteristic
617C2C2×#509311yescharacteristic
618C2#8347011yescharacteristic
619C2#8348311yescharacteristic
620C2#8351711yescharacteristic
621C2#8355111yescharacteristic
622C2#8362311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2440C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2440D3
hypocenterC3#27697
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
3. #71794C2semidirect
4. #71787C2semidirect
7. #69614C2semidirect
8. #69614C2semidirect
9. #69612C2semidirect
10. #69612C2semidirect
11. #69589C2semidirect
12. #69574C2semidirect
614. C3#27697semidirect

Maximal subgroups

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

subgroupindex
C3×#24402
#735172
#717942
#717872
#715592
C2×#45292
#696142
#696122
#695892
#695742
#695142
#695022
C22⋊C4×C3⋊C82
#276973