Finite Group Databasebeta

← #10727948

Subgroups — #10727948

Order 768.

2046 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#400C211yesmaximal, characteristic, complemented
3C2×#4987C211yesmaximal, characteristic
4C2×#4985C211yesmaximal, characteristic, complemented
5#71565C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#95C212yesmaximal
7(C2×D3)⋊#95C212yesmaximal
8(C2×D3)⋊#95C211yesmaximal, characteristic
9#69415C212yesmaximal, complemented
10#69414C212yesmaximal, complemented
11#69410C211yesmaximal, characteristic, complemented
12#14822—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C2×D3×C4⋊C4—24no—
25C2×D3×C4⋊C4—24no—
26C2×D3×C4⋊C4—24no—
27C3×#400—24no—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4987—24no—
41#4985—24no—
42C23×C4⋊C4—31no—
43C2×#400—31no—
44C2×#400D311yescharacteristic, complemented
45C2×#380—31no—
46#1442—32no—
47#1442—32no—
48#1441—31no—
49#1441—31no—
50C22⋊#95—31no—
51C22⋊#95—32no—
52C22⋊#95—32no—
53C24×D3C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—22no—
70C22×C4×D3—22no—
71C22×C4×D3—24no—
72C22×C4×D3—22no—
73C22×C4×D3—22no—
74C22×C4×D3—22no—
75C22×C4×D3—22no—
76C22×C4×D3—24no—
77C22×C4×D3—42no—
78C22×C4×D3—42no—
79C22×C4×D3D414yes—
80C22×C4×D3Q814yes—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4D414yes—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—42no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4D414yes—
104D3×C4⋊C4—232no—
105C3×#95—22no—
106C3×#95—24no—
107C3×#95—24no—
108C3⋊#95—22no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—42no—
115C3⋊#95—42no—
116C3⋊#95—42no—
117C24×C4—31no—
118C24×C4—31no—
119C24×C4—31no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—64no—
125C22×C4⋊C4—64no—
126C22×C4⋊C4—64no—
127C22×C4⋊C4C2×D311yescharacteristic
128C22×C4⋊C4C2×D311yescharacteristic
129C22×C22⋊C4—32no—
130C22×C22⋊C4—32no—
131#400—24no—
132#400—64no—
133#380—64no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95C2×D311yescharacteristic
144C2×#95C2×D312yes—
145C2×#95C2×D312yes—
146C24×C3C2411yescharacteristic
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3—21no—
151C23×D3—22no—
152C23×D3—22no—
153C23×D3C2×D412yes—
154C23×D3C2×Q812yes—
155C23×D3#4112yes—
156C23×D3#4114yes—
157C23×D3#4114yes—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×Q812yes—
188C22×C3×C4#4112yes—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3C2×D412yes—
216C22×Dic3C2×D412yes—
217C22×Dic3C2×D412yes—
218C22×Dic3C2×Q812yes—
219C22×Dic3#4112yes—
220C2×C4×D3—28no—
221C2×C4×D3—28no—
222C2×C4×D3—28no—
223C2×C4×D3—28no—
224C2×C4×D3—28no—
225C2×C4×D3—28no—
226C2×C4×D3—28no—
227C2×C4×D3—28no—
228C2×C4×D3—28no—
229C2×C4×D3—28no—
230C2×C4×D3—44no—
231C2×C4×D3—48no—
232C2×C4×D3—48no—
233C2×C4×D3—48no—
234C2×C4×D3—48no—
235C2×C4×D3—44no—
236C2×C4×D3—44no—
237C3×C4⋊C4—28no—
238C3×C4⋊C4—44no—
239(C2×D3)⋊C4—48no—
240(C2×D3)⋊C4—48no—
241C4⋊Dic3—28no—
242C4⋊Dic3—44no—
243Dic3⋊C4—28no—
244Dic3⋊C4—28no—
245C25—31no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—34no—
250C23×C4—34no—
251C23×C4—32no—
252C23×C4—32no—
253C23×C4—62no—
254C23×C4—62no—
255C23×C4—62no—
256C23×C4—62no—
257C23×C4—62no—
258C23×C4—62no—
259C23×C4—64no—
260C23×C4—64no—
261C23×C4—62no—
262C23×C4—62no—
263C23×C4—62no—
264C23×C4—62no—
265C23×C4—62no—
266C23×C4—122no—
267C23×C4—122no—
268C23×C4C22×D312yes—
269C23×C4C22×D312yes—
270C23×C4C22×D311yescharacteristic
271C23×C4C22×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—34no—
279C2×C4⋊C4—632no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—122no—
290C2×C4⋊C4#6714yes—
291C2×C22⋊C4—64no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—64no—
301#95—64no—
302#95—64no—
303#95—64no—
304#95—62no—
305#95—64no—
306#95—122no—
307#95—122no—
308#95—122no—
309C23×C3C22×D411yescharacteristic
310C23×C3C22×D411yescharacteristic
311C23×C3C22×D411yescharacteristic
312C23×C3C22×Q811yescharacteristic
313C23×C3C2×#4111yescharacteristic
314C23×C3C2×#4111yescharacteristic
315C23×C3C2×#4111yescharacteristic
316C23×C3C2×#4112yes—
317C23×C3C2×#4112yes—
318C23×C3H(5,𝔽2)12yes—
319C23×C32−1+412yes—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—24no—
329C22×D3—24no—
330C22×D3—24no—
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—24no—
334C22×D3—24no—
335C22×D3—24no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3C22⋊Q814yes—
339C22×D3#12314yes—
340C22×D3#12414yes—
341C22×D3#12518yes—
342C22×D3#12618yes—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4#12014yes—
373C2×C3×C4C22⋊Q814yes—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—24no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3C22⋊Q814yes—
405C4×D3—232no—
406C4×D3—416no—
407C4×D3—416no—
408C24—32no—
409C24—32no—
410C24—34no—
411C24—32no—
412C24—34no—
413C24—61no—
414C24—61no—
415C24—61no—
416C24—62no—
417C24—62no—
418C24—61no—
419C24C23×D311yescharacteristic
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—68no—
450C22×C4—68no—
451C22×C4—68no—
452C22×C4—68no—
453C22×C4—68no—
454C22×C4—68no—
455C22×C4—68no—
456C22×C4—68no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—68no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—68no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—124no—
486C22×C4—128no—
487C22×C4—128no—
488C22×C4—128no—
489C22×C4—128no—
490C22×C4—124no—
491C22×C4—124no—
492C22×C4D3×D412yes—
493C22×C4D3×D412yes—
494C22×C4D3×Q812yes—
495C22×C4#23912yes—
496C22×C4C2×#6712yes—
497C22×C4C2×#6712yes—
498C22×C4C2×#6712yes—
499C4⋊C4—28no—
500C4⋊C4—44no—
501C4⋊C4—68no—
502C4⋊C4—68no—
503C4⋊C4—68no—
504C4⋊C4—124no—
505C22⋊C4—128no—
506C22⋊C4—128no—
507C22×C3C2×#12011yescharacteristic
508C22×C3C2×C22⋊Q811yescharacteristic
509C22×C3C2×C22⋊Q811yescharacteristic
510C22×C3C2×C22⋊Q811yescharacteristic
511C22×C3C2×#12311yescharacteristic
512C22×C3#52511yescharacteristic
513C22×C3#52511yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3C2×#12411yescharacteristic
516C22×C3C2×#12512yes—
517C22×C3C2×#12612yes—
518C22×C3#53411yescharacteristic
519C22×C3#53611yescharacteristic
520C22×C3#53912yes—
521C22×C3#54412yes—
522C22×C3#54612yes—
523C22×C3#54612yes—
524C22×C3#54812yes—
525C22×C3#54812yes—
526C22×C3#55212yes—
527C22×C3#55212yes—
528C22×C3#55312yes—
529C22×C3#55312yes—
530C2×D3—24no—
531C2×D3—24no—
532C2×D3—24no—
533C2×D3—24no—
534C2×D3—24no—
535C2×D3—28no—
536C2×D3—28no—
537C2×D3—28no—
538C2×D3—28no—
539C2×D3—24no—
540C2×D3#40018yescomplemented
541C3×C4—24no—
542C3×C4—24no—
543C3×C4—42no—
544C3×C4—82no—
545C3×C4—82no—
546Dic3—24no—
547Dic3—42no—
548Dic3—42no—
549Dic3—82no—
550Dic3—82no—
551C23—34no—
552C23—34no—
553C23—38no—
554C23—38no—
555C23—34no—
556C23—62no—
557C23—64no—
558C23—62no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23—64no—
564C23—64no—
565C23—64no—
566C23—62no—
567C23—62no—
568C23—62no—
569C23—64no—
570C23—64no—
571C23—62no—
572C23—62no—
573C23—62no—
574C23C2×D3×D411yescharacteristic
575C23C2×D3×D411yescharacteristic
576C23C2×D3×Q811yescharacteristic
577C23C2×#23911yescharacteristic
578C23D3×#4112yes—
579C23D3×#4112yes—
580C23D3×#4111yescharacteristic
581C23D3×#4111yescharacteristic
582C23#100912yes—
583C23#101012yes—
584C23C22×#6711yescharacteristic
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—34no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—632no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—64no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—1216no—
628C2×C4—1216no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4#93714yes—
649C2×C4#94614yes—
650C2×C3C2×#40011yescharacteristic
651C2×C3#240511yescharacteristic
652C2×C3#243611yescharacteristic
653C2×C3#244212yes—
654C2×C3#244212yes—
655C2×C3#248112yes—
656C2×C3#248112yes—
657C2×C3#248411yescharacteristic
658C2×C3#259111yescharacteristic
659C2×C3#262711yescharacteristic
660C2×C3#284811yescharacteristic
661D3—28no—
662C22—38no—
663C22—64no—
664C22—64no—
665C22—64no—
666C22—64no—
667C22—68no—
668C22—68no—
669C22—68no—
670C22—68no—
671C22—64no—
672C22—64no—
673C22D3×C22⋊Q811yescharacteristic
674C22D3×C22⋊Q811yescharacteristic
675C22#562711yescharacteristic
676C22#562711yescharacteristic
677C22D3×#12311yescharacteristic
678C22#565311yescharacteristic
679C22#565412yes—
680C22D3×#12411yescharacteristic
681C22#567511yescharacteristic
682C22#567712yes—
683C22D3×#12512yes—
684C22#568912yes—
685C22#569412yes—
686C22#569512yes—
687C22D3×#12612yes—
688C22#570712yes—
689C22#570812yes—
690C22#570912yes—
691C22C2×#93711yescharacteristic
692C22C2×#94611yescharacteristic
693C22#582811yescharacteristic
694C22#582912yes—
695C22#583112yes—
696C4—24no—
697C4—24no—
698C4—42no—
699C4—64no—
700C4—82no—
701C4—82no—
702C4—122no—
703C4—122no—
704C4—242no—
705C4—242no—
706C3#1482211yescharacteristic, complemented, Sylow
707C2—68no—
708C2#7813512yes—
709C2#7814512yes—
710C2D3×#40011yescharacteristic
711C2#7834511yescharacteristic
712C2#7835811yescharacteristic
713C2#7836011yescharacteristic
714C2#8283411yescharacteristic
715C2#8289311yescharacteristic
716C2#8296111yescharacteristic
717C2#8302812yes—
718C2#8308212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#400C2
prime radicalC24×D3C23
hypercenterC2×#400D3
hypocenterC3#14822
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
2. C2×C3×#400C2semidirect
4. C2×#4985C2semidirect
5. #71565C2semidirect
9. #69415C2semidirect
10. #69414C2semidirect
11. #69410C2semidirect
44. C2×#400D3semidirect
540. C2×D3#400semidirect
706. C3#14822semidirect

Maximal subgroups

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

subgroupindex
C22×D3×C4⋊C42
C2×C3×#4002
C2×#49872
C2×#49852
#715652
(C2×D3)⋊#952
#694152
#694142
#694102
#148223