Finite Group Databasebeta

← #10727098

Subgroups — #10727098

Order 768.

2242 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic, complemented
3C2×#4989C211yesmaximal, characteristic
4C2×#4987C212yesmaximal
5C2×D3×#95C211yesmaximal, characteristic
6#72134C211yesmaximal, characteristic, complemented
7#71568C212yesmaximal, complemented
8#71565C211yesmaximal, characteristic, complemented
9(C2×D3)⋊#95C212yesmaximal
10#69419C211yesmaximal, characteristic, complemented
11#69411C212yesmaximal, complemented
12#14758—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C23×C4×D3C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C2×#946—24no—
19C2×#946—28no—
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27C3×#395—24no—
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4989—24no—
40#4987—28no—
41D3×#95—24no—
42C22×C22⋊Q8—31no—
43C2×#395D311yescharacteristic, complemented
44C2×#385—31no—
45C2×#380—32no—
46C22×#95—31no—
47#1446—31no—
48#1444—31no—
49#1444—32no—
50#1441—32no—
51#1441—31no—
52C22⋊#95—32no—
53C24×D3C2311yescharacteristic
54C22×C3×Q8—24no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—42no—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2312yes—
68C23×Dic3C2311yescharacteristic
69C22×C4×D3—24no—
70C22×C4×D3—24no—
71C22×C4×D3—24no—
72C22×C4×D3—22no—
73C22×C4×D3—24no—
74C22×C4×D3—22no—
75C22×C4×D3—24no—
76C22×C4×D3—42no—
77C22×C4×D3—42no—
78C22×C4×D3—42no—
79C22×C4×D3D414yes—
80C22×C4×D3Q818yescomplemented
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—42no—
87#946—232no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4D418yes—
95C2×(C2×D3)⋊C4Q814yescomplemented
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—24no—
98C2×C4⋊Dic3—24no—
99C2×C4⋊Dic3—24no—
100C2×C4⋊Dic3—42no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—42no—
106C3×#95—24no—
107C3×#95D414yes—
108C3⋊#95—22no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—22no—
115C3⋊#95—42no—
116C3⋊#95—42no—
117C23×Q8C2×D311yescharacteristic
118C24×C4—31no—
119C24×C4—32no—
120C2×C22⋊Q8—68no—
121C2×C22⋊Q8—64no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C4⋊C4C2×D312yes—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—31no—
131#395—24no—
132#385—64no—
133#380—68no—
134C2×#95—32no—
135C2×#95—31no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—31no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—64no—
144C2×#95C2×D312yes—
145C2×#95C2×D311yescharacteristic
146C24×C3C2411yescharacteristic
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—22no—
150C23×D3—22no—
151C23×D3—21no—
152C23×D3—21no—
153C23×D3C2×D412yes—
154C23×D3C2×D414yes—
155C23×D3C2×Q812yes—
156C23×D3C2×Q814yes—
157C23×D3#4112yes—
158C2×C3×Q8—28no—
159C2×C3×Q8—48no—
160C2×C3×Q8—48no—
161C2×C3×Q8—44no—
162C2×C3×Q8—44no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D414yes—
190C22×C3×C4C2×Q812yes—
191C22×C3×C4C2×Q814yes—
192C22×C3×C4#4112yes—
193C22×C3×C4#4114yes—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—21no—
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—21no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3C2×D412yes—
220C22×Dic3C2×D414yes—
221C22×Dic3C2×Q812yes—
222C22×Dic3C2×Q814yes—
223C2×C4×D3—28no—
224C2×C4×D3—216no—
225C2×C4×D3—28no—
226C2×C4×D3—216no—
227C2×C4×D3—44no—
228C2×C4×D3—48no—
229C2×C4×D3—48no—
230C2×C4×D3—48no—
231C2×C4×D3—48no—
232C2×C4×D3—48no—
233C2×C4×D3—48no—
234C2×C4×D3—48no—
235C3×C4⋊C4—48no—
236C3×C4⋊C4—44no—
237(C2×D3)⋊C4—216no—
238(C2×D3)⋊C4—28no—
239(C2×D3)⋊C4—44no—
240C4⋊Dic3—28no—
241C4⋊Dic3—48no—
242Dic3⋊C4—216no—
243Dic3⋊C4—44no—
244C25—31no—
245C22×Q8—24no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—42no—
250C23×C4—32no—
251C23×C4—31no—
252C23×C4—31no—
253C23×C4—32no—
254C23×C4—34no—
255C23×C4—38no—
256C23×C4—31no—
257C23×C4—64no—
258C23×C4—64no—
259C23×C4—64no—
260C23×C4—62no—
261C23×C4—64no—
262C23×C4—62no—
263C23×C4—64no—
264C23×C4—122no—
265C23×C4—122no—
266C23×C4—122no—
267C23×C4C22×D311yescharacteristic
268C23×C4C22×D312yes—
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D311yescharacteristic
271C23×C4C22×D312yes—
272C22⋊Q8—632no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—64no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—64no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—64no—
287C2×C4⋊C4—122no—
288C2×C4⋊C4—122no—
289C2×C22⋊C4—38no—
290C2×C22⋊C4—34no—
291C2×C22⋊C4—64no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—62no—
297#95—24no—
298#95—64no—
299#95—62no—
300#95—64no—
301#95—62no—
302#95—64no—
303#95—62no—
304#95—64no—
305#95—122no—
306#95—122no—
307#95#6714yes—
308C23×C3C22×D411yescharacteristic
309C23×C3C22×D412yes—
310C23×C3C22×D411yescharacteristic
311C23×C3C22×Q812yes—
312C23×C3C22×Q811yescharacteristic
313C23×C3C2×#4111yescharacteristic
314C23×C3C2×#4112yes—
315C23×C3C2×#4111yescharacteristic
316C23×C3H(5,𝔽2)12yes—
317C23×C3H(5,𝔽2)11yescharacteristic
318C23×C32−1+411yescharacteristic
319C22×D3—22no—
320C22×D3—24no—
321C22×D3—22no—
322C22×D3—24no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—24no—
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—22no—
329C22×D3—24no—
330C22×D3—24no—
331C22×D3—24no—
332C22×D3—22no—
333C22×D3—24no—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—24no—
337C22×D3#12014yes—
338C22×D3C22⋊Q814yes—
339C22×D3C22⋊Q818yes—
340C22×D3C4⋊Q814yes—
341C22×D3C4⋊Q818yes—
342C3×Q8—416no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4#12114yes—
373C2×C3×C4C22⋊Q818yescomplemented
374C2×Dic3—22no—
375C2×Dic3—24no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—22no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—41no—
386C2×Dic3—42no—
387C2×Dic3—41no—
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—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C4×D3—416no—
404C4×D3—432no—
405C24—34no—
406C24—32no—
407C24—32no—
408C24—34no—
409C24—32no—
410C24—61no—
411C24—62no—
412C24—62no—
413C24—61no—
414C24—61no—
415C24—61no—
416C24C23×D311yescharacteristic
417C2×Q8—28no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8—44no—
421C2×Q8—44no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—34no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—34no—
449C22×C4—68no—
450C22×C4—616no—
451C22×C4—62no—
452C22×C4—61no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—61no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—62no—
471C22×C4—68no—
472C22×C4—616no—
473C22×C4—61no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—128no—
479C22×C4—124no—
480C22×C4—128no—
481C22×C4—128no—
482C22×C4—128no—
483C22×C4—128no—
484C22×C4—128no—
485C22×C4—128no—
486C22×C4D3×D414yes—
487C22×C4D3×D412yes—
488C22×C4#23712yes—
489C22×C4D3×Q812yes—
490C22×C4D3×Q814yes—
491C22×C4#23914yes—
492C22×C4C2×#6712yes—
493C22×C4C2×#6714yes—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—616no—
497C4⋊C4—68no—
498C4⋊C4—128no—
499C4⋊C4—124no—
500C22⋊C4—616no—
501C22⋊C4—68no—
502C22⋊C4—124no—
503C22×C3C2×#12011yescharacteristic
504C22×C3C2×#12111yescharacteristic
505C22×C3C2×C22⋊Q812yes—
506C22×C3C2×C22⋊Q811yescharacteristic
507C22×C3C2×C22⋊Q812yes—
508C22×C3C2×C4⋊Q811yescharacteristic
509C22×C3C2×C4⋊Q812yes—
510C22×C3#53211yescharacteristic
511C22×C3#53312yes—
512C22×C3#53411yescharacteristic
513C22×C3#53711yescharacteristic
514C22×C3#54312yes—
515C22×C3#54412yes—
516C22×C3D4212yes—
517C22×C3#54611yescharacteristic
518C22×C3#54612yes—
519C22×C3#54612yes—
520C22×C3#54712yes—
521C22×C3D4×Q811yescharacteristic
522C22×C3#55112yes—
523C22×C3#55211yescharacteristic
524C22×C3#55411yescharacteristic
525C22×C3#55412yes—
526C2×D3—24no—
527C2×D3—24no—
528C2×D3—24no—
529C2×D3—24no—
530C2×D3—24no—
531C2×D3—24no—
532C2×D3—28no—
533C2×D3—28no—
534C2×D3—28no—
535C2×D3—28no—
536C2×D3#39518yescomplemented
537C3×C4—24no—
538C3×C4—28no—
539C3×C4—44no—
540C3×C4—42no—
541C3×C4—81no—
542Dic3—42no—
543Dic3—42no—
544Dic3—44no—
545Dic3—44no—
546Dic3—81no—
547C23—38no—
548C23—34no—
549C23—34no—
550C23—34no—
551C23—38no—
552C23—64no—
553C23—62no—
554C23—64no—
555C23—62no—
556C23—62no—
557C23—64no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—62no—
562C23—64no—
563C23—64no—
564C23—62no—
565C23—62no—
566C23—62no—
567C23—64no—
568C23—62no—
569C23—64no—
570C23C2×D3×D411yescharacteristic
571C23C2×D3×D412yes—
572C23C2×#23711yescharacteristic
573C23#100412yes—
574C23C2×D3×Q812yes—
575C23C2×D3×Q811yescharacteristic
576C23C2×#23912yes—
577C23#100711yescharacteristic
578C23D3×#4111yescharacteristic
579C23#100911yescharacteristic
580C23C22×#6711yescharacteristic
581Q8—416no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—64no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—62no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—1232no—
623C2×C4—1216no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—121no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—121no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—121no—
638C2×C4—122no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—122no—
642C2×C4#93814yes—
643C2×C4#94618yescomplemented
644C2×C3C2×#39511yescharacteristic
645C2×C3#241011yescharacteristic
646C2×C3#241711yescharacteristic
647C2×C3#243011yescharacteristic
648C2×C3#243712yes—
649C2×C3#243812yes—
650C2×C3#244012yes—
651C2×C3#254912yes—
652C2×C3#256411yescharacteristic
653C2×C3#265911yescharacteristic
654C2×C3#285711yescharacteristic
655D3—28no—
656C22—38no—
657C22—68no—
658C22—64no—
659C22—64no—
660C22—68no—
661C22—64no—
662C22—64no—
663C22—68no—
664C22—68no—
665C22—64no—
666C22—64no—
667C22D3×#12011yescharacteristic
668C22#558911yescharacteristic
669C22#559112yes—
670C22D3×C22⋊Q811yescharacteristic
671C22D3×C22⋊Q812yes—
672C22#562911yescharacteristic
673C22#563012yes—
674C22#563412yes—
675C22#563511yescharacteristic
676C22#563712yes—
677C22#563911yescharacteristic
678C22D3×C4⋊Q812yes—
679C22D3×C4⋊Q811yescharacteristic
680C22#572511yescharacteristic
681C22#572612yes—
682C22#572812yes—
683C22#572912yes—
684C22#573012yes—
685C22C2×#93811yescharacteristic
686C22D4×#6712yes—
687C22C2×#94612yes—
688C22Q8×#6711yescharacteristic
689C22#582911yescharacteristic
690C4—28no—
691C4—24no—
692C4—42no—
693C4—44no—
694C4—81no—
695C4—124no—
696C4—122no—
697C4—124no—
698C4—122no—
699C4—241no—
700C3#1475811yescharacteristic, complemented, Sylow
701C2—68no—
702C2D3×#39511yescharacteristic
703C2#7725011yescharacteristic
704C2#7725112yes—
705C2#7725312yes—
706C2#7725411yescharacteristic
707C2#7725811yescharacteristic
708C2#8260511yescharacteristic
709C2#8275211yescharacteristic
710C2#8281612yes—
711C2#8323312yes—
712C2#8326911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#395C2
prime radicalC24×D3C23
hypercenterC2×#395D3
hypocenterC3#14758
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C2×C3×#395C2semidirect
6. #72134C2semidirect
7. #71568C2semidirect
8. #71565C2semidirect
10. #69419C2semidirect
11. #69411C2semidirect
43. C2×#395D3semidirect
80. C22×C4×D3Q8semidirect
95. C2×(C2×D3)⋊C4Q8semidirect
373. C2×C3×C4C22⋊Q8semidirect
536. C2×D3#395semidirect
643. C2×C4#946semidirect
700. C3#14758semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
C2×C3×#3952
C2×#49892
C2×#49872
C2×D3×#952
#721342
#715682
#715652
(C2×D3)⋊#952
#694192
#694112
#147583