Finite Group Databasebeta

← #10944017

Subgroups — #10944017

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×Q8C211yesmaximal, characteristic
2C2×#5031C211yesmaximal, characteristic, complemented
3C2×#4849C211yesmaximal, characteristic, complemented
4C2×#4809C212yesmaximal
5#74376C211yesmaximal, characteristic
6C3×#1803C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic
8#70239C212yesmaximal, complemented
9#69588C211yesmaximal, characteristic, complemented
10#68908C212yesmaximal, complemented
11#68878C212yesmaximal
12#32906—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×#216C2212yes—
17C22×Dic12C2212yes—
18C4×D3×Q8—24no—
19C4×D3×Q8—24no—
20C2×#897C2212yes—
21C2×D3×C4⋊C4—21no—
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×#887—21no—
24C2×#887C2211yescharacteristic
25C2×C42×D3—21no—
26C2×C42×D3C2211yescharacteristic
27C2×#869C2211yescharacteristic
28C2×C4×Dic6—21no—
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C3×#384C2211yescharacteristic
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2212yes—
35#5031—24no—
36#4987—21no—
37#4987—21no—
38#4984—21no—
39#4984—21no—
40#4973C2212yes—
41C2×#810C2212yes—
42C2×#810C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4849—24no—
45#4809—28no—
46C3×#340C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C22×C4×Q8—31no—
50C2×#462—31no—
51C2×#462—31no—
52C2×#451—32no—
53#2857—31no—
54#1878—32no—
55#1878—32no—
56#1872—32no—
57#1803—31no—
58#1803D311yescharacteristic, complemented
59#1797—31no—
60C22×C3×Q8C2311yescharacteristic
61C2×D3×Q8—22no—
62C2×D3×Q8—22no—
63C2×D3×Q8—22no—
64C2×D3×Q8—22no—
65C2×D3×Q8—22no—
66C2×D3×Q8D414yes—
67C22×C4×D3—21no—
68C22×C4×D3—21no—
69C22×C4×D3D411yescharacteristic
70C22×C4×D3D411yescharacteristic
71C22×C4×D3C2311yescharacteristic
72C22×Dic6—21no—
73C22×Dic6C2312yes—
74C22×Dic6C2311yescharacteristic
75C22×C3×C8C2312yes—
76C3×C4×Q8—22no—
77C3×C4×Q8D414yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C42—21no—
84C2×C3×C42C2311yescharacteristic
85Q8×Dic3—22no—
86Q8×Dic3—22no—
87C2×#216—24no—
88C2×#216—24no—
89C2×#216—24no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106C2×Dic12—24no—
107C2×Dic12—24no—
108C2×Dic12—24no—
109#897—24no—
110#897—24no—
111D3×C4⋊C4—28no—
112D3×C4⋊C4—22no—
113D3×C4⋊C4—22no—
114#887—28no—
115#887—22no—
116#887—22no—
117C42×D3—28no—
118C42×D3—22no—
119C42×D3—22no—
120#869—22no—
121#869—22no—
122C4×Dic6—28no—
123C4×Dic6—22no—
124C4×Dic6D414yes—
125C3×C4⋊C8—22no—
126C3×#103—24no—
127C3⋊#95—22no—
128C3⋊#95—22no—
129(C2×D3)⋊C8—24no—
130#816—24no—
131#810—24no—
132#810—24no—
133C4⋊(C3⋊C8)—22no—
134C23×Q8—31no—
135C22×Q16—32no—
136C22×Q16—32no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—32no—
139C2×C4×Q8—31no—
140C2×C4×Q8—31no—
141C2×C4×Q8—31no—
142C2×C4×Q8—61no—
143C2×C4×Q8—61no—
144C2×C4×Q8—64no—
145C2×C4×Q8—64no—
146C2×C4×Q8C2×D311yescharacteristic
147C22×C4⋊C4—31no—
148C22×C4⋊C4—61no—
149C22×C42—31no—
150C22×C42—61no—
151#462—64no—
152#462—64no—
153#451—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#103C2×D312yes—
160C2×C22⋊C8—32no—
161#391—32no—
162#389—61no—
163#389—61no—
164#384C2×D311yescharacteristic
165#380—61no—
166#380—61no—
167#340—32no—
168#340C2×D312yes—
169#336—32no—
170C23×D3C2×D411yescharacteristic
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8C2×D412yes—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8#4112yes—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189D3×Q8—28no—
190D3×Q8—28no—
191D3×Q8—28no—
192D3×Q8—28no—
193C2×C4×D3—24no—
194C2×C4×D3—24no—
195C2×C4×D3—24no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—24no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—41no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3Q1614yescomplemented
209C2×C4×D3C2×D412yes—
210C2×C4×D3#4112yes—
211C2×Dic6—24no—
212C2×Dic6—24no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—24no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6C2×D412yes—
224C2×Dic6C2×D412yes—
225C2×Dic6#4112yes—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4C2×D412yes—
236C3×C42—24no—
237C3×C42C2×D412yes—
238C3×C42C2×D412yes—
239#216—48no—
240(C2×D3)⋊C4—44no—
241C4⋊Dic3—24no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—42no—
245C4⋊Dic3—44no—
246C4⋊Dic3C2×D412yes—
247Dic3⋊C4—24no—
248Dic3⋊C4—24no—
249Dic3⋊C4—24no—
250C4×Dic3—24no—
251C4×Dic3—24no—
252C4×Dic3—22no—
253C4×Dic3—24no—
254C4×Dic3—22no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260Dic12—48no—
261C22×Q8—34no—
262C22×Q8—31no—
263C22×Q8—32no—
264C22×Q8—61no—
265C22×Q8—62no—
266C22×Q8—62no—
267C22×Q8—62no—
268C22×Q8—62no—
269C22×Q8—62no—
270C22×Q8C22×D311yescharacteristic
271C23×C4—31no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—61no—
275C23×C4—61no—
276C2×Q16—64no—
277C2×Q16—64no—
278C2×Q16—64no—
279C2×Q16—64no—
280C2×Q16—64no—
281C2×Q16—64no—
282C22×C8—32no—
283C22×C8C22×D312yes—
284C4⋊Q8—62no—
285C4⋊Q8—62no—
286C22⋊Q8—64no—
287C22⋊Q8—64no—
288C4×Q8—22no—
289C4×Q8—34no—
290C4×Q8—68no—
291C4×Q8—62no—
292C4×Q8—68no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×Q8D1214yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—68no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D312yes—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—62no—
315C2×C42—21no—
316C2×C42—31no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—68no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—62no—
324C2×C42—62no—
325C2×C42C22×D311yescharacteristic
326C4⋊C8—22no—
327C4⋊C8—62no—
328#103—24no—
329#103—64no—
330#103—64no—
331#103—64no—
332C22⋊C8—64no—
333#95—62no—
334#95—62no—
335C23×C3C22×D411yescharacteristic
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3#12112yes—
341C22×D3C2×Q1612yes—
342C22×D3#13712yes—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8#12114yes—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4#12012yes—
359C2×C3×C4#12012yes—
360C2×C3×C4C2×Q1612yes—
361C2×C3×C4#13712yes—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3#12112yes—
385C2×Dic3C2×Q1612yes—
386C4×D3—216no—
387C4×D3—22no—
388C4×D3—22no—
389C4×D3—22no—
390C4×D3—22no—
391C4×D3—28no—
392C4×D3—44no—
393C4×D3—48no—
394Dic6—216no—
395Dic6—22no—
396Dic6—22no—
397Dic6—22no—
398Dic6—42no—
399Dic6—42no—
400Dic6—42no—
401Dic6—42no—
402Dic6#12114yes—
403C3×C8—42no—
404C3⋊C8—42no—
405C24—31no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—32no—
410C2×Q8—32no—
411C2×Q8—32no—
412C2×Q8—64no—
413C2×Q8—64no—
414C2×Q8—64no—
415C2×Q8—68no—
416C2×Q8—68no—
417C2×Q8—68no—
418C2×Q8—68no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—62no—
423C2×Q8—62no—
424C2×Q8—62no—
425C2×Q8—62no—
426C2×Q8—62no—
427C2×Q8—62no—
428C2×Q8C2×D1212yes—
429C2×Q8D3×D412yes—
430C2×Q8#23912yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—34no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—31no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—62no—
444C22×C4—64no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—121no—
456C22×C4—122no—
457C22×C4—122no—
458C22×C4D3×D411yescharacteristic
459C22×C4D3×D411yescharacteristic
460C22×C4C23×D311yescharacteristic
461Q16—128no—
462Q16—128no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—62no—
467C2×C8—62no—
468C2×C8—62no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—32no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—64no—
477C4⋊C4—64no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—61no—
481C4⋊C4—124no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4C2×D1212yes—
485C22⋊C4—124no—
486C42—24no—
487C42—64no—
488C42—64no—
489C42—62no—
490C42—62no—
491C42—64no—
492C42—61no—
493C42—61no—
494C42C2×D1212yes—
495C42D3×D412yes—
496C22×C3C2×#12011yescharacteristic
497C22×C3C2×#12111yescharacteristic
498C22×C3#52511yescharacteristic
499C22×C3C22×Q1611yescharacteristic
500C22×C3C2×#13711yescharacteristic
501C22×C3C2×#13711yescharacteristic
502C22×C3#57511yescharacteristic
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3#46214yescomplemented
510C3×C4—24no—
511C3×C4—22no—
512C3×C4—42no—
513C3×C4—82no—
514C3×C4#45114yescomplemented
515C3×C4C2×#12111yescharacteristic
516C3×C4C2×#12111yescharacteristic
517C3×C4#52511yescharacteristic
518C3×C4#52511yescharacteristic
519Dic3—22no—
520Dic3—24no—
521Dic3—22no—
522Dic3—41no—
523Dic3—42no—
524Dic3—82no—
525C23—32no—
526C23—32no—
527C23—32no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23—61no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—34no—
537Q8—616no—
538Q8—62no—
539Q8—62no—
540Q8—62no—
541Q8—122no—
542Q8—122no—
543Q8—122no—
544Q8—122no—
545Q8#89514yes—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—616no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—68no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—124no—
577C2×C4—128no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4#88212yes—
584C2×C4#88212yes—
585C2×C4#91512yes—
586C2×C4D3×Q1612yes—
587C2×C4C22×D1211yescharacteristic
588C2×C4C2×D3×D411yescharacteristic
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×#23911yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C8—42no—
594C8—122no—
595C2×C3#284711yescharacteristic
596C2×C3C2×#45111yescharacteristic
597C2×C3#300711yescharacteristic
598C2×C3C2×#46211yescharacteristic
599C2×C3#303711yescharacteristic
600C2×C3#310411yescharacteristic
601C2×C3#311311yescharacteristic
602D3—24no—
603C22—34no—
604C22—62no—
605C22—62no—
606C22—62no—
607C22—62no—
608C22—62no—
609C22—62no—
610C22C2×#88211yescharacteristic
611C22D3×#12111yescharacteristic
612C22#562711yescharacteristic
613C22C2×#91511yescharacteristic
614C22C2×D3×Q1611yescharacteristic
615C22#577311yescharacteristic
616C22D3×#13711yescharacteristic
617C4—22no—
618C4—24no—
619C4—42no—
620C4—62no—
621C4—64no—
622C4—62no—
623C4—82no—
624C4—121no—
625C4—122no—
626C4—242no—
627C4#480914yescomplemented
628C4C2×#89511yescharacteristic
629C4#552411yescharacteristic
630C4D3×#12111yescharacteristic
631C4#562711yescharacteristic
632C3#3290611yescharacteristic, complemented, Sylow
633C2—64no—
634C2#7404811yescharacteristic
635C2C2×#480911yescharacteristic
636C2#7888511yescharacteristic
637C2#8038311yescharacteristic
638C2D3×#46211yescharacteristic
639C2#8099811yescharacteristic
640C2#8109211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1803C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1803D3
hypocenterC3#32906
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#5031C2semidirect
3. C2×#4849C2semidirect
6. C3×#1803C2semidirect
8. #70239C2semidirect
9. #69588C2semidirect
10. #68908C2semidirect
58. #1803D3semidirect
208. C2×C4×D3Q16semidirect
509. C2×D3#462semidirect
514. C3×C4#451semidirect
627. C4#4809semidirect
632. C3#32906semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×Q82
C2×#50312
C2×#48492
C2×#48092
#743762
C3×#18032
#704752
#702392
#695882
#689082
#688782
#329063