Finite Group Databasebeta

← #10840190

Subgroups — #10840190

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82777C211yesmaximal, characteristic
2C4⋊C4×#67C211yesmaximal, characteristic
3C3×#1872C211yesmaximal, characteristic, complemented
4#71695C211yesmaximal, characteristic, complemented
5#71554C211yesmaximal, characteristic, complemented
6#70268C211yesmaximal, characteristic, complemented
7#70243C211yesmaximal, characteristic, complemented
8#70221C211yesmaximal, characteristic, complemented
9#70220C211yesmaximal, characteristic, complemented
10#69940C211yesmaximal, characteristic, complemented
11#69917C211yesmaximal, characteristic
12#68889C211yesmaximal, characteristic, complemented
13#68871C211yesmaximal, characteristic, complemented
14#68849C211yesmaximal, characteristic, complemented
15#68797C211yesmaximal, characteristic, complemented
16#24979—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×#946C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×C4×#67—21no—
22C2×C4×#67C2211yescharacteristic
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5212C2211yescharacteristic
30#5212C2211yescharacteristic
31#5210—21no—
32C2×#830C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#4990—21no—
39#4989—21no—
40#4987C2211yescharacteristic
41#4987C2211yescharacteristic
42#4982—21no—
43#4977C2211yescharacteristic
44#4976—21no—
45#4975C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4961—21no—
49#4945—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2536—31no—
61D4×C4⋊C4—31no—
62#2077—31no—
63#2075—31no—
64#2036—31no—
65#2035—31no—
66#1955—31no—
67#1949—31no—
68#1872—31no—
69#1872D311yescharacteristic, complemented
70#1836—31no—
71#1824—31no—
72#1823—31no—
73#1803—31no—
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×#67D411yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—41no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—41no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217D411yescharacteristic
96#946—22no—
97#946—41no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100C4×#67—22no—
101C4×#67—22no—
102C4×#67—44no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4D411yescharacteristic
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122D3×C4⋊C4—22no—
123D3×C4⋊C4—41no—
124C3×#103—22no—
125C3×#103—41no—
126C3×C22⋊C8—41no—
127#835—22no—
128#835—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—41no—
137(C2×D3)⋊C8—41no—
138(C2×D3)⋊C8—41no—
139#817—41no—
140#815—22no—
141Dic3⋊C8—22no—
142#807—41no—
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8C2×D311yescharacteristic
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C22×C4⋊C4—31no—
149C22×C4⋊C4C2×D311yescharacteristic
150C2×#107—31no—
151C2×#106—31no—
152C2×C4⋊C8—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103C2×D311yescharacteristic
156C2×#103C2×D311yescharacteristic
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×C22⋊C8C2×D311yescharacteristic
161C2×#97—31no—
162#398—61no—
163#394—61no—
164#389—61no—
165#388—61no—
166#388—61no—
167#385—61no—
168#385—61no—
169#384—31no—
170#382—31no—
171#380—31no—
172#380—31no—
173#380—31no—
174#380—31no—
175C4×C4⋊C4—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340—31no—
184#340C2×D311yescharacteristic
185#340C2×D311yescharacteristic
186C24×C3C2×D411yescharacteristic
187C23×D3C2×D411yescharacteristic
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8#4112yes—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4—81no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—22no—
210C2×#67—22no—
211C2×#67—22no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—81no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4#4112yes—
242C3×C22⋊C4—42no—
243#217—22no—
244#217—42no—
245#217—42no—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259Dic3⋊C4—22no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—81no—
264C4×Dic3—42no—
265C4×Dic3—42no—
266C4×Dic3#4112yes—
267C4×Dic3#4112yes—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C22×Q8C22×D311yescharacteristic
275C22×D4—31no—
276C23×C4—21no—
277C23×C4—31no—
278C23×C4—61no—
279C23×C4C22×D311yescharacteristic
280C22×C8—31no—
281C22×C8—31no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C22⋊Q8—22no—
285C22⋊Q8—41no—
286C22⋊Q8—62no—
287C22⋊Q8—121no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—124no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—121no—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—31no—
314C2×C22⋊C4—31no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C42—31no—
321C2×C42—61no—
322C2×C42—61no—
323#107—121no—
324#106—121no—
325C4⋊C8—62no—
326#103—22no—
327#103—41no—
328#103—62no—
329#103—121no—
330C22⋊C8—41no—
331C22⋊C8—121no—
332C22⋊C8—121no—
333C22⋊C8—121no—
334#97—62no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3#13712yes—
349C23×C3C22×D411yescharacteristic
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3AGL(1,ℤ/8)12yes—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8—41no—
360C3×Q8—41no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—84no—
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387#67—28no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3#12312yes—
408C2×Dic3#12312yes—
409C4×D3—41no—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—84no—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—31no—
419C24D3×D411yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#23512yes—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—62no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—81no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4—241no—
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—62no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C4⋊C4—241no—
493C4⋊C4#23712yes—
494C22⋊C4—42no—
495C22⋊C4—62no—
496C22⋊C4—62no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C42—32no—
503C42—32no—
504C42—122no—
505C42—122no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3C2×#12311yescharacteristic
514C22×C3#54611yescharacteristic
515C22×C3#54611yescharacteristic
516C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3#57711yescharacteristic
519C22×C3#57811yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C3×C4—41no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4—81no—
531C3×C4—81no—
532C3×C4#52911yescharacteristic
533C3×C4#54011yescharacteristic
534C3×C4#55311yescharacteristic
535C3×C4#55511yescharacteristic
536Dic3—22no—
537Dic3—22no—
538Dic3—41no—
539Dic3—81no—
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—32no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23#91812yes—
557C23C2×D3×D411yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562D4—68no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—32no—
569C2×C4—32no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—84no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—244no—
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×#23711yescharacteristic
611C2×C4#100411yescharacteristic
612C2×C4#100411yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C8—41no—
616C8—41no—
617C8—121no—
618C8—121no—
619C2×C3—42no—
620C2×C3#242011yescharacteristic
621C2×C3#309211yescharacteristic
622C2×C3#309711yescharacteristic
623C2×C3#330811yescharacteristic
624C2×C3#331011yescharacteristic
625C2×C3#331211yescharacteristic
626C2×C3#331611yescharacteristic
627D3—42no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—62no—
636C22—62no—
637C22—62no—
638C22—121no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22#549011yescharacteristic
643C22#559111yescharacteristic
644C22D3×#12311yescharacteristic
645C22#576211yescharacteristic
646C22C2×#91811yescharacteristic
647C22D3×AGL(1,ℤ/8)11yescharacteristic
648C22#577911yescharacteristic
649C4—42no—
650C4—41no—
651C4—42no—
652C4—62no—
653C4—62no—
654C4—81no—
655C4—81no—
656C4—121no—
657C4—241no—
658C4—241no—
659C4—241no—
660C4—241no—
661C4#553811yescharacteristic
662C4#554011yescharacteristic
663C4#561511yescharacteristic
664C4#561611yescharacteristic
665C3#2497911yescharacteristic, complemented, Sylow
666C2—42no—
667C2—122no—
668C2#7469511yescharacteristic
669C2#7881111yescharacteristic
670C2#7889611yescharacteristic
671C2#8016111yescharacteristic
672C2#8019511yescharacteristic
673C2#8142911yescharacteristic
674C2#8146811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1872C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#1872D3
hypocenterC3#24979
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. C3×#1872C2semidirect
4. #71695C2semidirect
5. #71554C2semidirect
6. #70268C2semidirect
7. #70243C2semidirect
8. #70221C2semidirect
9. #70220C2semidirect
10. #69940C2semidirect
12. #68889C2semidirect
13. #68871C2semidirect
14. #68849C2semidirect
15. #68797C2semidirect
69. #1872D3semidirect
665. C3#24979semidirect

Maximal subgroups

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

subgroupindex
#827772
C4⋊C4×#672
C3×#18722
#716952
#715542
#702682
#702432
#702212
#702202
#699402
#699172
#688892
#688712
#688492
#687972
#249793