Finite Group Databasebeta

← #10945124

Subgroups — #10945124

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
1C3×#2861C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic, complemented
3C2×#5057C211yesmaximal, characteristic
4C42×D12C211yesmaximal, characteristic, complemented
5#71738C211yesmaximal, characteristic, complemented
6#71721C211yesmaximal, characteristic, complemented
7#69687C211yesmaximal, characteristic, complemented
8#69687C211yesmaximal, characteristic, complemented
9#69667C211yesmaximal, characteristic, complemented
10#69666C211yesmaximal, characteristic, complemented
11#69657C211yesmaximal, characteristic
12#69657C211yesmaximal, characteristic
13#69637C211yesmaximal, characteristic, complemented
14#69636C211yesmaximal, characteristic, complemented
15#69392C211yesmaximal, characteristic, complemented
16#33028—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C42×D3—21no—
25C3×#391C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#388—21no—
30C3×#388—21no—
31C3×#388—21no—
32C3×#388—21no—
33C3×C43C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40#5066—24no—
41#5057—24no—
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C4×(C2×D3)⋊C4—21no—
47C4×(C2×D3)⋊C4—21no—
48C4×(C2×D3)⋊C4—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478—31no—
63C2×#463—31no—
64#2861D311yescharacteristic, complemented
65C42×D4—31no—
66#1989—31no—
67#1983—31no—
68#1928—31no—
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×C3×D4C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C3×#125—22no—
84C3×#125—22no—
85C3×#124—22no—
86C3×#124—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42D411yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#215—22no—
105C2×#215—22no—
106C2×#215—22no—
107C2×#213—22no—
108C2×#213—22no—
109C2×#213—22no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C4×D12—22no—
124C4×D12—22no—
125C4×D12—22no—
126C4×D12—22no—
127C4×D12—22no—
128C4×D12D414yes—
129C42×D3—44no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134#835—22no—
135#835—22no—
136#835—22no—
137#832—22no—
138#832—22no—
139#832—22no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#809—22no—
144#809—22no—
145#808—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#125C2×D311yescharacteristic
152C2×#124C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C22×C42—61no—
157#478—64no—
158#463—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171#391C2×D311yescharacteristic
172#391C2×D311yescharacteristic
173#390C2×D311yescharacteristic
174#390C2×D311yescharacteristic
175#388—21no—
176#388—21no—
177#388—21no—
178#388—21no—
179C4×C4⋊C4—31no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4—31no—
182C4×C22⋊C4—61no—
183C4×C22⋊C4—61no—
184C4×C22⋊C4—61no—
185C43C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193#336—31no—
194C24×C3—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—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—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—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248C3×C42—22no—
249C3×C42—22no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—41no—
255C3×C42—41no—
256C3×C42C2×D412yes—
257C3×C42#4112yes—
258#215—44no—
259#213—44no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3C2×D412yes—
267C4⋊Dic3#4112yes—
268C4⋊Dic3#4112yes—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4C22×D311yescharacteristic
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×C8—31no—
299C22×C8—31no—
300#125—22no—
301#125—22no—
302#124—22no—
303#124—22no—
304C4×D4—34no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42—124no—
330C2×C42#6711yescharacteristic
331C2×C42#6711yescharacteristic
332C2×C42#6711yescharacteristic
333C2×C42#6711yescharacteristic
334C2×C42C22×D311yescharacteristic
335C2×C42C22×D311yescharacteristic
336C2×C42C22×D311yescharacteristic
337#107—62no—
338#106—62no—
339C4⋊C8—62no—
340C4⋊C8—62no—
341#103—62no—
342#103—62no—
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—21no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3C22×D411yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—22no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4#12112yes—
403C2×C3×C4#12112yes—
404C2×C3×C4#13512yes—
405C2×C3×C4AGL(1,ℤ/8)12yes—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421D12—22no—
422D12—22no—
423D12—22no—
424D12#12414yes—
425C4×D3—48no—
426C4×D3—48no—
427C4×D3—48no—
428C3⋊C8—41no—
429C3⋊C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C24—21no—
433C24—61no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—32no—
447C2×D4—32no—
448C2×D4—32no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—64no—
456C22×C4—64no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—64no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C23×D311yescharacteristic
474SD16—124no—
475D8—124no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
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—32no—
494C4⋊C4—32no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—64no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—41no—
517C42—122no—
518C42—122no—
519C42—122no—
520C42D3×D412yes—
521C42#23712yes—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3C2×#12111yescharacteristic
530C22×C3#52511yescharacteristic
531C22×C3#52511yescharacteristic
532C22×C3C2×#13511yescharacteristic
533C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57511yescharacteristic
536C2×D3—24no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—42no—
543C3×C4—22no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—81no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4C2×#12411yescharacteristic
553C3×C4#52911yescharacteristic
554C3×C4#52911yescharacteristic
555C3×C4#53211yescharacteristic
556Dic3—42no—
557Dic3—42no—
558Dic3—42no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—62no—
567C23—62no—
568C23—62no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23C22×#6711yescharacteristic
574Q8—41no—
575Q8—41no—
576Q8—41no—
577Q8—41no—
578D4—34no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—62no—
584D4—62no—
585D4—62no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—128no—
613C2×C4—128no—
614C2×C4—128no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4#93812yes—
622C2×C4#93812yes—
623C2×C4#94912yes—
624C2×C4#95012yes—
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×#23711yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C8—121no—
632C8—121no—
633C8—121no—
634C8—121no—
635C2×C3—161no—
636C2×C3#285011yescharacteristic
637C2×C3#303511yescharacteristic
638C2×C3#303611yescharacteristic
639C2×C3#310211yescharacteristic
640C2×C3#310511yescharacteristic
641C2×C3#311111yescharacteristic
642C2×C3#311411yescharacteristic
643D3—44no—
644C22—64no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—122no—
657C22—122no—
658C22C2×#93811yescharacteristic
659C22C2×#94911yescharacteristic
660C22C2×#95011yescharacteristic
661C22#582211yescharacteristic
662C22#582211yescharacteristic
663C22#582811yescharacteristic
664C22#582811yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—41no—
670C4—41no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—122no—
675C4—122no—
676C4—122no—
677C4#566811yescharacteristic
678C4D3×#12411yescharacteristic
679C4#568011yescharacteristic
680C4#568011yescharacteristic
681C3#3302811yescharacteristic, complemented, Sylow
682C2—124no—
683C2—161no—
684C2#8300811yescharacteristic
685C2#8314011yescharacteristic
686C2#8314111yescharacteristic
687C2#8314211yescharacteristic
688C2#8314311yescharacteristic
689C2#8329611yescharacteristic
690C2#8329711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2861C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2861D3
hypocenterC3#33028
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ #2861

Complements

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

NH ≅ G/Ntype
1. C3×#2861C2semidirect
2. C2×#5066C2semidirect
4. C42×D12C2semidirect
5. #71738C2semidirect
6. #71721C2semidirect
7. #69687C2semidirect
8. #69687C2semidirect
9. #69667C2semidirect
10. #69666C2semidirect
13. #69637C2semidirect
14. #69636C2semidirect
15. #69392C2semidirect
64. #2861D3semidirect
681. C3#33028semidirect

Maximal subgroups

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

subgroupindex
C3×#28612
C2×#50662
C2×#50572
C42×D122
#717382
#717212
#696872
#696672
#696662
#696572
#696372
#696362
#693922
#330283