Finite Group Databasebeta

← #10720202

Subgroups — #10720202

Order 768.

2526 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#897C211yesmaximal, characteristic
2C22×D3×C4⋊C4C211yesmaximal, characteristic
3C2×#4973C211yesmaximal, characteristic, complemented
4C2×#4929C211yesmaximal, characteristic, complemented
5C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
6C3×C4⋊#95C211yesmaximal, characteristic, complemented
7#71563C211yesmaximal, characteristic, complemented
8(C2×D3)⋊#95C214yesmaximal
9#69411C214yesmaximal, complemented
10#14139—31nomaximal, Sylow
11C23×C4×D3C2212yes—
12C23×C4×D3C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C2×#897—24no—
24C2×#897—24no—
25C2×#897C418yescomplemented
26C2×D3×C4⋊C4—24no—
27C2×D3×C4⋊C4—24no—
28C2×D3×C4⋊C4—24no—
29C2×C3×#95C2214yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2214yes—
32C2×C3⋊#95C2214yes—
33C2×C3⋊#95C2214yes—
34#4973—24no—
35C4×(C2×D3)⋊C4—24no—
36#4929—24no—
37C22×C22⋊Q8—31no—
38C23×C4⋊C4—31no—
39C2×#391—31no—
40C2×#391—31no—
41C2×C4×C22⋊C4—31no—
42#1441—34no—
43C4⋊#95—31no—
44C4⋊#95D311yescharacteristic, complemented
45C22⋊#95—34no—
46C24×D3C2311yescharacteristic
47C23×C3×C4C2312yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×Dic3C2312yes—
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2311yescharacteristic
55C22×C4×D3—22no—
56C22×C4×D3—22no—
57C22×C4×D3—22no—
58C22×C4×D3—22no—
59C22×C4×D3—24no—
60C22×C4×D3—24no—
61C22×C4×D3—24no—
62C22×C4×D3—28no—
63C22×C4×D3—44no—
64C22×C4×D3C2×C414yes—
65C22×C4×D3D414yes—
66C22×C4×D3D414yes—
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6C2×C414yescomplemented
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2×C414yescomplemented
79C2×C3×C4⋊C4D414yes—
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4C2×C418yes—
88C2×(C2×D3)⋊C4Q818yescomplemented
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3C2×C418yescomplemented
94C2×C4⋊Dic3D414yes—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101#897—232no—
102D3×C4⋊C4—232no—
103C3×#95—44no—
104C3⋊#95—28no—
105C3⋊#95—28no—
106C3⋊#95—24no—
107C3⋊#95—44no—
108C23×Q8—31no—
109C24×C4—31no—
110C24×C4—32no—
111C2×C22⋊Q8—38no—
112C2×C22⋊Q8—64no—
113C2×C22⋊Q8—64no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4—64no—
118C22×C4⋊C4—64no—
119C22×C4⋊C4—64no—
120C22×C4⋊C4C2×D311yescharacteristic
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C22⋊C4—32no—
123C22×C22⋊C4—32no—
124C22×C42—31no—
125C22×C42C2×D311yescharacteristic
126#391—64no—
127#391—64no—
128C4×C22⋊C4—64no—
129C2×#95—34no—
130C2×#95—34no—
131C2×#95—32no—
132C2×#95—34no—
133C2×#95C2×D314yes—
134C24×C3C2411yescharacteristic
135C23×D3—21no—
136C23×D3—21no—
137C23×D3—21no—
138C23×D3—21no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3C22×C412yes—
142C23×D3C2×D412yes—
143C23×D3C2×D412yes—
144C23×D3C2×Q814yes—
145C23×D3#4114yes—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4C22×C414yes—
161C22×C3×C4C22×C412yes—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×Q814yes—
168C22×C3×C4#4112yes—
169C22×C3×C4#4112yes—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—24no—
187C22×Dic3—22no—
188C22×Dic3—24no—
189C22×Dic3—22no—
190C22×Dic3C22×C414yes—
191C22×Dic3C22×C412yes—
192C22×Dic3C2×D414yes—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×Q814yes—
196C2×C4×D3—28no—
197C2×C4×D3—28no—
198C2×C4×D3—28no—
199C2×C4×D3—28no—
200C2×C4×D3—28no—
201C2×C4×D3—28no—
202C2×C4×D3—216no—
203C2×C4×D3—416no—
204C2×C4×D3—416no—
205C2×C4×D3—48no—
206C2×C4×D3—48no—
207C2×C4×D3C22⋊C4116yes—
208C2×Dic6—28no—
209C2×Dic6—28no—
210C2×Dic6—28no—
211C2×Dic6—44no—
212C2×Dic6—44no—
213C2×Dic6—44no—
214C2×Dic6—44no—
215C3×C4⋊C4—28no—
216C3×C4⋊C4—28no—
217C3×C42—44no—
218(C2×D3)⋊C4—216no—
219(C2×D3)⋊C4—216no—
220C4⋊Dic3—28no—
221C4⋊Dic3—216no—
222Dic3⋊C4—216no—
223C4×Dic3—44no—
224C25—31no—
225C22×Q8—34no—
226C22×Q8—62no—
227C22×Q8—62no—
228C22×Q8—62no—
229C22×Q8—62no—
230C22×Q8—62no—
231C22×Q8—62no—
232C23×C4—31no—
233C23×C4—34no—
234C23×C4—34no—
235C23×C4—34no—
236C23×C4—32no—
237C23×C4—32no—
238C23×C4—32no—
239C23×C4—64no—
240C23×C4—64no—
241C23×C4—68no—
242C23×C4—62no—
243C23×C4—62no—
244C23×C4—62no—
245C23×C4—62no—
246C23×C4—64no—
247C23×C4—124no—
248C23×C4C22×D312yes—
249C23×C4C22×D312yes—
250C23×C4C22×D312yes—
251C23×C4C22×D311yescharacteristic
252C22⋊Q8—632no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—38no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—632no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4C4×D314yescomplemented
268C2×C4⋊C4D1214yes—
269C2×C22⋊C4—38no—
270C2×C22⋊C4—38no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—64no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—62no—
279C2×C42—62no—
280C2×C42—62no—
281#95—44no—
282#95—64no—
283#95—68no—
284#95—68no—
285#95—124no—
286C23×C3C23×C411yescharacteristic
287C23×C3C22×D411yescharacteristic
288C23×C3C22×D411yescharacteristic
289C23×C3C22×D411yescharacteristic
290C23×C3C22×D411yescharacteristic
291C23×C3C22×Q812yes—
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4112yes—
295C23×C3H(5,𝔽2)12yes—
296C23×C32−1+412yes—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—24no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3C2×C22⋊C414yes—
316C22×D3C4×Q818yes—
317C22×D3C22⋊Q818yes—
318C22×D3#12414yes—
319C22×D3C4⋊Q814yes—
320C2×C3×C4—24no—
321C2×C3×C4—28no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—28no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—44no—
333C2×C3×C4—42no—
334C2×C3×C4—44no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4C2×C22⋊C414yes—
338C2×C3×C4C4×D414yes—
339C2×C3×C4C4×D414yes—
340C2×C3×C4#12014yes—
341C2×C3×C4#12114yes—
342C2×C3×C4#12114yes—
343C2×C3×C4#12314yes—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—28no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—44no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3C2×C22⋊C414yes—
370C4×D3—232no—
371C4×D3—432no—
372Dic6—416no—
373C24—32no—
374C24—32no—
375C24—32no—
376C24—34no—
377C24—34no—
378C24—61no—
379C24—61no—
380C24—61no—
381C24—62no—
382C24—61no—
383C24—62no—
384C24C23×D311yescharacteristic
385C2×Q8—68no—
386C2×Q8—68no—
387C2×Q8—68no—
388C2×Q8—124no—
389C2×Q8—124no—
390C2×Q8—124no—
391C2×Q8—124no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—316no—
410C22×C4—34no—
411C22×C4—34no—
412C22×C4—34no—
413C22×C4—616no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—68no—
419C22×C4—68no—
420C22×C4—68no—
421C22×C4—68no—
422C22×C4—68no—
423C22×C4—68no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—64no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—1216no—
441C22×C4—1216no—
442C22×C4—128no—
443C22×C4—128no—
444C22×C4C2×C4×D314yes—
445C22×C4C2×C4×D312yes—
446C22×C4C2×D1214yes—
447C22×C4C2×D1212yes—
448C22×C4#23512yes—
449C22×C4D3×D412yes—
450C22×C4D3×D412yes—
451C22×C4D3×D412yes—
452C22×C4D3×Q814yes—
453C22×C4#23912yes—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—616no—
457C4⋊C4—616no—
458C4⋊C4—68no—
459C22⋊C4—616no—
460C22⋊C4—616no—
461C42—44no—
462C42—124no—
463C22×C3C22×C22⋊C411yescharacteristic
464C22×C3C2×C4×D411yescharacteristic
465C22×C3C2×C4×D411yescharacteristic
466C22×C3C2×C4×Q812yes—
467C22×C3#52012yes—
468C22×C3C2×#12011yescharacteristic
469C22×C3C2×#12111yescharacteristic
470C22×C3C2×#12111yescharacteristic
471C22×C3C2×C22⋊Q812yes—
472C22×C3C2×#12311yescharacteristic
473C22×C3C2×#12411yescharacteristic
474C22×C3C2×C4⋊Q811yescharacteristic
475C22×C3#53511yescharacteristic
476C22×C3#53611yescharacteristic
477C22×C3#53712yes—
478C22×C3#54612yes—
479C22×C3#54612yes—
480C22×C3D4×Q812yes—
481C22×C3D4×Q812yes—
482C22×C3#55212yes—
483C22×C3#55212yes—
484C22×C3#55412yes—
485C22×C3#55412yes—
486C2×D3—24no—
487C2×D3—28no—
488C2×D3—28no—
489C2×D3—28no—
490C2×D3—28no—
491C2×D3—24no—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—24no—
495C2×D3—24no—
496C2×D3#39118yescomplemented
497C3×C4—44no—
498C3×C4—44no—
499C3×C4—44no—
500C3×C4#38618yes—
501Dic3—24no—
502Dic3—44no—
503Dic3—44no—
504Dic3—44no—
505C23—34no—
506C23—34no—
507C23—34no—
508C23—38no—
509C23—38no—
510C23—64no—
511C23—62no—
512C23—64no—
513C23—62no—
514C23—62no—
515C23—64no—
516C23—62no—
517C23—62no—
518C23—62no—
519C23—62no—
520C23—62no—
521C23—64no—
522C23—64no—
523C23—64no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23—64no—
528C23C22×C4×D311yescharacteristic
529C23C22×D1211yescharacteristic
530C23C2×#23511yescharacteristic
531C23C2×D3×D411yescharacteristic
532C23C2×D3×D411yescharacteristic
533C23C2×D3×D411yescharacteristic
534C23#100412yes—
535C23C2×D3×Q812yes—
536C23C2×#23911yescharacteristic
537C23D3×#4112yes—
538C23#101012yes—
539Q8—1216no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—34no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—632no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—68no—
572C2×C4—64no—
573C2×C4—1232no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—124no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4C4×D1214yes—
586C2×C4D3×C22⋊C414yes—
587C2×C4#88214yes—
588C2×C4#88414yes—
589C2×C4#89314yes—
590C2×C4#89414yes—
591C2×C4#89514yes—
592C2×C3C2×#38611yescharacteristic
593C2×C3C2×#39111yescharacteristic
594C2×C3#231111yescharacteristic
595C2×C3#236012yes—
596C2×C3#236012yes—
597C2×C3#241712yes—
598C2×C3#241712yes—
599C2×C3#243411yescharacteristic
600C2×C3#243511yescharacteristic
601C2×C3#250211yescharacteristic
602C2×C3#250311yescharacteristic
603D3—28no—
604C22—38no—
605C22—64no—
606C22—64no—
607C22—68no—
608C22—64no—
609C22—64no—
610C22—68no—
611C22—68no—
612C22—68no—
613C22—64no—
614C22—64no—
615C22C2×C4×D1211yescharacteristic
616C22C2×D3×C22⋊C411yescharacteristic
617C22C2×#88211yescharacteristic
618C22C2×#88411yescharacteristic
619C22C2×#89311yescharacteristic
620C22C2×#89411yescharacteristic
621C22C2×#89511yescharacteristic
622C22#551112yes—
623C22#552912yes—
624C22#554612yes—
625C22#555412yes—
626C22C4×D3×Q812yes—
627C22Q8×D1212yes—
628C22#560812yes—
629C22#561212yes—
630C22D3×C22⋊Q812yes—
631C22#563212yes—
632C22D3×#12411yescharacteristic
633C22#567511yescharacteristic
634C22#569412yes—
635C22#571511yescharacteristic
636C22D3×C4⋊Q811yescharacteristic
637C22#573012yes—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—64no—
642C4—124no—
643C4—124no—
644C4—124no—
645C4#466518yes—
646C3#1413911yescharacteristic, complemented, Sylow
647C2—68no—
648C2C2×#466511yescharacteristic
649C2#7377312yes—
650C2#7388812yes—
651C2#7416711yescharacteristic
652C2#7432911yescharacteristic
653C2#7620512yes—
654C2#7621212yes—
655C2#7629411yescharacteristic
656C2#7631511yescharacteristic
657C2#7649911yescharacteristic
658C2D3×#39111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C4⋊#95C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenterC4⋊#95D3
hypocenterC3#14139
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#4973C2semidirect
4. C2×#4929C2semidirect
6. C3×C4⋊#95C2semidirect
7. #71563C2semidirect
9. #69411C2semidirect
25. C2×#897C4semidirect
44. C4⋊#95D3semidirect
73. C22×Dic6C2×C4semidirect
78. C2×C3×C4⋊C4C2×C4semidirect
88. C2×(C2×D3)⋊C4Q8semidirect
93. C2×C4⋊Dic3C2×C4semidirect
267. C2×C4⋊C4C4×D3semidirect
496. C2×D3#391semidirect
646. C3#14139semidirect

Maximal subgroups

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

subgroupindex
C22×#8972
C22×D3×C4⋊C42
C2×#49732
C2×#49292
C2×C4×(C2×D3)⋊C42
C3×C4⋊#952
#715632
(C2×D3)⋊#952
#694112
#141393