Finite Group Databasebeta

← #10795118

Subgroups — #10795118

Order 768.

1866 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67⋊C4C211yesmaximal, characteristic
2#73817C212yesmaximal, complemented
3C3×#1819C211yesmaximal, characteristic, complemented
4#71542C211yesmaximal, characteristic, complemented
5#69498C212yesmaximal
6#68959C214yesmaximal, complemented
7#68952C214yesmaximal, complemented
8#20722—31nomaximal, Sylow
9C23×#67C2211yescharacteristic
10C24×Dic3C2211yescharacteristic
11C23×C4×D3C411yescharacteristic, complemented
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×(C2×D3)⋊C4C411yescharacteristic, complemented
14C22×Dic3⋊C4C411yescharacteristic, complemented
15C22×C4×Dic3C411yescharacteristic, complemented
16C2×#67⋊C4—22no—
17C2×#67⋊C4—22no—
18C2×#67⋊C4—22no—
19C2×#67⋊C4—22no—
20C2×#67⋊C4C412yescomplemented
21C2×#67⋊C4C412yescomplemented
22C2×#67⋊C4C2212yes—
23C2×#67⋊C4C2212yes—
24C2×#877—22no—
25C3×#379C2212yes—
26#5247C2212yes—
27C2×#806C2218yes—
28#4943—22no—
29#4942—22no—
30#4942C2214yescomplemented
31Dic3×C22⋊C4—22no—
32Dic3×C22⋊C4C2214yescomplemented
33C3×#342C2214yescomplemented
34#4526C2214yescomplemented
35C22×C4×D4—31no—
36#2280—32no—
37#1819—31no—
38#1819D311yescharacteristic, complemented
39#1799—34no—
40#1781—32no—
41C4×#99—34no—
42C25×C3C2311yescharacteristic
43C24×D3C2×C411yescharacteristic
44C23×C3×C4C2×C411yescharacteristic
45C23×C3×C4C2×C411yescharacteristic
46C22×#67—22no—
47C22×#67—22no—
48C22×#67—28no—
49C22×#67C2×C412yes—
50C22×#67C2312yes—
51C23×Dic3—44no—
52C23×Dic3C2×C411yescharacteristic
53C23×Dic3C2×C411yescharacteristic
54C23×Dic3C2×C411yescharacteristic
55C23×Dic3C2×C411yescharacteristic
56C23×Dic3C2×C411yescharacteristic
57C23×Dic3D412yes—
58C23×Dic3D412yes—
59C23×Dic3D412yes—
60C23×Dic3D412yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C22×C4×D3—21no—
64C22×C4×D3—21no—
65C22×C4×D3—21no—
66C22×C4×D3—21no—
67C22×C4×D3—44no—
68C22×C4×D3C2×C412yes—
69C22×C4×D3C2×C412yes—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4C2×C412yes—
75C2×C3×C22⋊C4C2314yes—
76C2×C3×C22⋊C4C2312yes—
77C2×#217—22no—
78C2×#217—22no—
79C2×#217C2314yes—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4C2×C412yes—
83C2×(C2×D3)⋊C4C2×C412yes—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4C2×C412yes—
88C2×Dic3⋊C4C2×C412yes—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3C2×C412yes—
95C2×C4×Dic3C2×C412yes—
96#67⋊C4—24no—
97#67⋊C4—24no—
98#67⋊C4—24no—
99#67⋊C4—24no—
100#67⋊C4—24no—
101#67⋊C4—24no—
102#67⋊C4—24no—
103#67⋊C4—24no—
104#877—48no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107#806—216no—
108C23×D4—31no—
109C24×C4—31no—
110C24×C4—31no—
111C2×C4×D4—32no—
112C2×C4×D4—32no—
113C2×C4×D4—32no—
114C2×C4×D4—32no—
115C2×C4×D4—62no—
116C2×C4×D4—62no—
117C2×C4×D4—62no—
118C2×C4×D4—62no—
119C2×#117—62no—
120C22×C4⋊C4—31no—
121C22×C22⋊C4—31no—
122C22×C22⋊C4C2×D311yescharacteristic
123C22×C42—31no—
124C2×#99—38no—
125#381—62no—
126#380—34no—
127#380—62no—
128#379—32no—
129#379C2×D312yes—
130C4×C22⋊C4—34no—
131C4×C22⋊C4—62no—
132#342—34no—
133#342C2×D314yescomplemented
134C24×C3—24no—
135C24×C3—42no—
136C24×C3—42no—
137C24×C3C22×C411yescharacteristic
138C24×C3C22×C411yescharacteristic
139C24×C3C2×D412yes—
140C24×C3C2×D412yes—
141C24×C3C2411yescharacteristic
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—24no—
145C23×D3—42no—
146C23×D3—42no—
147C23×D3C22×C412yes—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—24no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4C22×C412yes—
156C22×C3×C4C22×C412yes—
157C2×#67—216no—
158C2×#67—216no—
159C2×#67—24no—
160C2×#67—24no—
161C2×#67—24no—
162C2×#67—24no—
163C2×#67—24no—
164C2×#67#4118yes—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—28no—
168C22×Dic3—24no—
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—42no—
172C22×Dic3—42no—
173C22×Dic3—44no—
174C22×Dic3—44no—
175C22×Dic3—42no—
176C22×Dic3—42no—
177C22×Dic3—42no—
178C22×Dic3—44no—
179C22×Dic3C22⋊C412yes—
180C22×Dic3C22⋊C412yes—
181C22×Dic3C22⋊C411yescharacteristic
182C22×Dic3C22⋊C411yescharacteristic
183C22×Dic3C22⋊C411yescharacteristic
184C22×Dic3C22⋊C412yes—
185C22×Dic3C22⋊C412yes—
186C22×Dic3C22⋊C412yes—
187C22×Dic3C22⋊C412yes—
188C22×Dic3C22⋊C411yescharacteristic
189C22×Dic3C22×C411yescharacteristic
190C22×Dic3C22×C411yescharacteristic
191C22×Dic3C22×C412yes—
192C22×Dic3C22×C411yescharacteristic
193C22×Dic3C22×C411yescharacteristic
194C22×Dic3C2×D412yes—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×D412yes—
197C22×Dic3C2×D412yes—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—44no—
203C2×C4×D3—44no—
204C2×C4×D3—44no—
205C2×C4×D3—44no—
206C2×C4×D3—44no—
207C2×C4×D3—44no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—44no—
212C3×C22⋊C4—44no—
213#217—28no—
214#217—44no—
215#217—44no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220Dic3⋊C4—48no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C4×Dic3—22no—
224C4×Dic3—22no—
225C4×Dic3—48no—
226C25—31no—
227C25C22×D311yescharacteristic
228C22×D4—32no—
229C22×D4—32no—
230C22×D4—68no—
231C22×D4—62no—
232C22×D4—62no—
233C23×C4—31no—
234C23×C4—32no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—32no—
239C23×C4—32no—
240C23×C4—32no—
241C23×C4—31no—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4—32no—
246C23×C4—61no—
247C23×C4—61no—
248C23×C4—61no—
249C23×C4—61no—
250C23×C4—124no—
251C23×C4—124no—
252C23×C4C4×D311yescharacteristic
253C23×C4C4×D311yescharacteristic
254C4×D4—64no—
255C4×D4—64no—
256C4×D4—64no—
257C4×D4—64no—
258C4×D4—64no—
259C4×D4—64no—
260C4×D4—64no—
261C4×D4—64no—
262#117—128no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—64no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—32no—
273C2×C22⋊C4—32no—
274C2×C22⋊C4—34no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4C4×D312yes—
280C2×C22⋊C4C22×D314yes—
281C2×C22⋊C4C22×D312yes—
282C2×C42—32no—
283C2×C42—32no—
284C2×C42—61no—
285C2×C42—61no—
286C2×C42—61no—
287C2×C42—61no—
288C2×C42—64no—
289#99—616no—
290#95—64no—
291#95—64no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3—416no—
307C23×C3C2×C22⋊C412yes—
308C23×C3C2×C22⋊C411yescharacteristic
309C23×C3C2×C22⋊C411yescharacteristic
310C23×C3C23×C411yescharacteristic
311C23×C3C22×D412yes—
312C23×C3C2×#4114yes—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—22no—
316C22×D3—22no—
317C22×D3—42no—
318C22×D3—416no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3#11714yescomplemented
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4—44no—
335C2×C3×C4—44no—
336#67—216no—
337#67—216no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—44no—
360C2×Dic3—44no—
361C2×Dic3—44no—
362C2×Dic3#9914yescomplemented
363C2×Dic3C2×C22⋊C411yescharacteristic
364C2×Dic3C2×C22⋊C411yescharacteristic
365C2×Dic3C2×C22⋊C412yes—
366C2×Dic3C2×C22⋊C412yes—
367C2×Dic3C2×C22⋊C411yescharacteristic
368C2×Dic3C2×C22⋊C411yescharacteristic
369C2×Dic3#11714yescomplemented
370C4×D3—44no—
371C4×D3—44no—
372C4×D3—44no—
373C4×D3—44no—
374C24—24no—
375C24—32no—
376C24—42no—
377C24—42no—
378C24—64no—
379C24—61no—
380C24—61no—
381C24—122no—
382C24—122no—
383C24C2×C4×D311yescharacteristic
384C24C2×C4×D311yescharacteristic
385C24D3×D412yes—
386C24D3×D412yes—
387C24C23×D311yescharacteristic
388C2×D4—38no—
389C2×D4—64no—
390C2×D4—64no—
391C2×D4—616no—
392C2×D4—616no—
393C2×D4—64no—
394C2×D4—64no—
395C2×D4—64no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—68no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—64no—
432C22×C4—124no—
433C22×C4—124no—
434C22×C4—124no—
435C22×C4—124no—
436C22×C4—124no—
437C22×C4—124no—
438C22×C4—122no—
439C22×C4—124no—
440C22×C4—122no—
441C22×C4—124no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—124no—
445C22×C4—122no—
446C22×C4C2×C4×D312yes—
447C22×C4C2×C4×D312yes—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—128no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—28no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C22⋊C4—68no—
459C22⋊C4—124no—
460C22⋊C4—124no—
461C42—62no—
462C42—62no—
463C42—62no—
464C42—62no—
465C42—128no—
466C22×C3—28no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—22no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—44no—
479C22×C3—48no—
480C22×C3—48no—
481C22×C3—44no—
482C22×C3C2×#9912yes—
483C22×C3#41012yes—
484C22×C3C22×C22⋊C411yescharacteristic
485C22×C3C2×#11712yes—
486C22×C3#52514yes—
487C2×D3—24no—
488C2×D3—44no—
489C2×D3—48no—
490C2×D3—48no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—44no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—84no—
497Dic3—21no—
498Dic3—21no—
499Dic3—22no—
500Dic3—22no—
501Dic3—24no—
502Dic3—84no—
503Dic3C2×#9912yes—
504Dic3C2×#9912yes—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—34no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—416no—
521C23—64no—
522C23—64no—
523C23—62no—
524C23—62no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—1216no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23D3×C22⋊C412yes—
533C23D3×C22⋊C411yescharacteristic
534C23D3×C22⋊C411yescharacteristic
535C23C22×C4×D311yescharacteristic
536C23C2×D3×D412yes—
537C23D3×#4114yes—
538D4—616no—
539D4—616no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—31no—
543C2×C4—31no—
544C2×C4—34no—
545C2×C4—31no—
546C2×C4—31no—
547C2×C4—32no—
548C2×C4—32no—
549C2×C4—34no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—124no—
574C2×C4—124no—
575C2×C4—124no—
576C2×C4—124no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—124no—
585C2×C4—124no—
586C2×C4—124no—
587C2×C3—24no—
588C2×C3—22no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3—42no—
592C2×C3#228011yescharacteristic
593C2×C3C22×#9911yescharacteristic
594C2×C3C2×#41011yescharacteristic
595D3—44no—
596C22—28no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—21no—
601C22—21no—
602C22—22no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—44no—
609C22—48no—
610C22—48no—
611C22—44no—
612C22—64no—
613C22—124no—
614C22—128no—
615C22—128no—
616C22—124no—
617C22—122no—
618C22—122no—
619C22#474012yes—
620C22D3×#9912yes—
621C22C2×D3×C22⋊C411yescharacteristic
622C22D3×#11712yes—
623C22#565614yes—
624C4—32no—
625C4—32no—
626C4—42no—
627C4—42no—
628C4—64no—
629C4—61no—
630C4—61no—
631C4—62no—
632C4—62no—
633C4—84no—
634C4—244no—
635C3#2072211yescharacteristic, complemented, Sylow
636C2—24no—
637C2—22no—
638C2—22no—
639C2—22no—
640C2—42no—
641C2—124no—
642C2#7405011yescharacteristic
643C2C2×#474011yescharacteristic
644C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1819C2
socleC22×C3C22×C22⋊C4
prime radicalC23×#67C22
cosocleC24×C3C24
hypercenter#1819D3
hypocenterC3#20722
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #73817C2semidirect
3. C3×#1819C2semidirect
4. #71542C2semidirect
6. #68959C2semidirect
7. #68952C2semidirect
11. C23×C4×D3C4semidirect
13. C22×(C2×D3)⋊C4C4semidirect
14. C22×Dic3⋊C4C4semidirect
15. C22×C4×Dic3C4semidirect
20. C2×#67⋊C4C4semidirect
21. C2×#67⋊C4C4semidirect
30. #4942C22semidirect
32. Dic3×C22⋊C4C22semidirect
33. C3×#342C22semidirect
34. #4526C22semidirect
38. #1819D3semidirect
133. #342C2×D3semidirect
324. C22×D3#117semidirect
362. C2×Dic3#99semidirect
369. C2×Dic3#117semidirect
635. C3#20722semidirect

Maximal subgroups

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

subgroupindex
C22×#67⋊C42
#738172
C3×#18192
#715422
#694982
#689592
#689522
#207223