Finite Group Databasebeta

← #10945136

Subgroups — #10945136

Order 768.

984 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4891C211yesmaximal, characteristic, complemented
2C2×#4819C211yesmaximal, characteristic
3#76606C211yesmaximal, characteristic, complemented
4C4×#893C211yesmaximal, characteristic, complemented
5C3×#1928C211yesmaximal, characteristic, complemented
6#70623C211yesmaximal, characteristic, complemented
7#70276C211yesmaximal, characteristic, complemented
8#70241C211yesmaximal, characteristic, complemented
9#70230C211yesmaximal, characteristic, complemented
10#69687C211yesmaximal, characteristic, complemented
11#69463C211yesmaximal, characteristic, complemented
12#68897C211yesmaximal, characteristic
13#68801C211yesmaximal, characteristic, complemented
14#68673C211yesmaximal, characteristic
15#68605C211yesmaximal, characteristic, complemented
16#33028—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×D24C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C42×D3—21no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4988C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43#4939C2211yescharacteristic
44C4×(C2×D3)⋊C4—21no—
45C42×Dic3C2211yescharacteristic
46#4891—24no—
47#4819—24no—
48#4670—21no—
49#4670—21no—
50#4669—21no—
51#4669—21no—
52#4666—21no—
53#4666—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478—31no—
63C2×#463—31no—
64#2861—31no—
65C42×D4—31no—
66#1989—31no—
67#1983—31no—
68#1928D311yescharacteristic, complemented
69#1927—31no—
70#1926—31no—
71#1925—31no—
72#1879—31no—
73#1879—31no—
74#1877—31no—
75#1877—31no—
76#1831—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90#947—22no—
91#947—22no—
92C2×#215—22no—
93C2×#215—22no—
94C2×#215—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×D24—22no—
119C2×D24—22no—
120C2×D24—22no—
121#893—22no—
122#893—22no—
123#893—22no—
124#893D414yes—
125#890—22no—
126#890—22no—
127C4×D12—22no—
128C4×D12—22no—
129C42×D3—44no—
130C3×#107—22no—
131C3×#103—22no—
132C3×#103—22no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136#835—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#821—22no—
142#821—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145#809—22no—
146#809—22no—
147#809—22no—
148#808—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#125—31no—
152C2×#124—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C22×C42—61no—
157#478—64no—
158#463—64no—
159C2×#107C2×D311yescharacteristic
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171#391—31no—
172#391C2×D311yescharacteristic
173#390—31no—
174#390—31no—
175#388—61no—
176#388—61no—
177#388—61no—
178#388—61no—
179C4×C4⋊C4—31no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4C2×D311yescharacteristic
182C4×C22⋊C4—61no—
183C4×C22⋊C4—61no—
184C4×C22⋊C4—61no—
185C43—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192#336—31no—
193#336C2×D311yescharacteristic
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×D412yes—
225C2×D12#4112yes—
226C2×D12#4112yes—
227C2×C4×D3—24no—
228C2×C4×D3—24no—
229C2×C4×D3—24no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2×D412yes—
247C3×C4⋊C4#4112yes—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—42no—
251#215—44no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3#4112yes—
266Dic3⋊C4—42no—
267C4×Dic3—22no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—22no—
271C4×Dic3—41no—
272C4×Dic3—41no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275C4×Dic3C2×D412yes—
276C4×Dic3#4112yes—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283D24—44no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—31no—
287C23×C4—61no—
288C23×C4—61no—
289C23×C4—61no—
290C2×SD16—62no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×D8—62no—
294C2×D8—62no—
295C2×D8—62no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300#125—62no—
301#125—62no—
302#124—62no—
303#124—62no—
304C4×D4—34no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—21no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—61no—
334C2×C42—61no—
335C2×C42—124no—
336C2×C42C22×D311yescharacteristic
337#107—22no—
338#106—62no—
339C4⋊C8—62no—
340C4⋊C8—62no—
341#103—22no—
342#103—22no—
343#103—62no—
344#102—62no—
345#102—62no—
346#102—62no—
347#102—62no—
348#102—62no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×Q8—41no—
374C3×Q8—41no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4#13512yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3#12112yes—
415C2×Dic3#12112yes—
416C2×Dic3AGL(1,ℤ/8)12yes—
417D12—22no—
418D12—22no—
419D12—22no—
420D12—41no—
421D12—41no—
422D12—41no—
423D12—41no—
424D12#12414yes—
425C4×D3—48no—
426C4×D3—48no—
427C4×D3—48no—
428C3×C8—41no—
429C3×C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C24—61no—
433C24—61no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×D4—32no—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D411yescharacteristic
473C22×C4C23×D311yescharacteristic
474SD16—124no—
475D8—124no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—32no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4#23512yes—
499C4⋊C4D3×D412yes—
500C22⋊C4—64no—
501C22⋊C4—64no—
502C22⋊C4—64no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C42—21no—
510C42—21no—
511C42—32no—
512C42—32no—
513C42—42no—
514C42—62no—
515C42—62no—
516C42—61no—
517C42—61no—
518C42—121no—
519C42—121no—
520C42—122no—
521C42—122no—
522C22×C3C2×#12111yescharacteristic
523C22×C3#52511yescharacteristic
524C22×C3#52511yescharacteristic
525C22×C3C2×#13511yescharacteristic
526C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57511yescharacteristic
529C2×D3—24no—
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C3×C4—22no—
544C3×C4—41no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4C2×#12411yescharacteristic
550C3×C4#52911yescharacteristic
551C3×C4#52911yescharacteristic
552C3×C4#53211yescharacteristic
553Dic3—22no—
554Dic3—22no—
555Dic3—22no—
556Dic3—41no—
557Dic3—42no—
558Dic3—81no—
559C23—62no—
560C23—62no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23C2×D3×D411yescharacteristic
574Q8—41no—
575Q8—41no—
576Q8—41no—
577Q8—41no—
578D4—34no—
579D4—62no—
580D4—62no—
581D4—62no—
582D4—121no—
583D4—121no—
584D4—121no—
585D4—121no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—32no—
594C2×C4—32no—
595C2×C4—32no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—62no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—128no—
617C2×C4—128no—
618C2×C4—128no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4#91912yes—
625C2×C4C2×#23511yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C8—41no—
632C8—41no—
633C8—121no—
634C8—121no—
635C2×C3#285011yescharacteristic
636C2×C3#303511yescharacteristic
637C2×C3#303611yescharacteristic
638C2×C3#310211yescharacteristic
639C2×C3#310511yescharacteristic
640C2×C3#311111yescharacteristic
641C2×C3#311411yescharacteristic
642D3—44no—
643D3—161no—
644C22—64no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22#554711yescharacteristic
659C22D3×#12111yescharacteristic
660C22#565611yescharacteristic
661C22C2×#91911yescharacteristic
662C22#576811yescharacteristic
663C22D3×AGL(1,ℤ/8)11yescharacteristic
664C22#577711yescharacteristic
665C4—22no—
666C4—41no—
667C4—42no—
668C4—42no—
669C4—62no—
670C4—62no—
671C4—62no—
672C4—81no—
673C4—81no—
674C4—122no—
675C4—121no—
676C4—241no—
677C4#552811yescharacteristic
678C4#554811yescharacteristic
679C4D3×#12411yescharacteristic
680C4#570611yescharacteristic
681C3#3302811yescharacteristic, complemented, Sylow
682C2—124no—
683C2—481no—
684C2#7595711yescharacteristic
685C2#7974611yescharacteristic
686C2#7978211yescharacteristic
687C2#8047511yescharacteristic
688C2#8053411yescharacteristic
689C2#8145311yescharacteristic
690C2#8149411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1928C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1928D3
hypocenterC3#33028
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
1. C2×#4891C2semidirect
3. #76606C2semidirect
4. C4×#893C2semidirect
5. C3×#1928C2semidirect
6. #70623C2semidirect
7. #70276C2semidirect
8. #70241C2semidirect
9. #70230C2semidirect
10. #69687C2semidirect
11. #69463C2semidirect
13. #68801C2semidirect
15. #68605C2semidirect
68. #1928D3semidirect
681. C3#33028semidirect

Maximal subgroups

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

subgroupindex
C2×#48912
C2×#48192
#766062
C4×#8932
C3×#19282
#706232
#702762
#702412
#702302
#696872
#694632
#688972
#688012
#686732
#686052
#330283