Finite Group Databasebeta

← #10732204

Subgroups — #10732204

Order 768.

2446 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#896C212yesmaximal
2C22×#888C211yesmaximal, characteristic, complemented
3C2×#4646C212yesmaximal, complemented
4C2×#4666C211yesmaximal, characteristic
5C3×#1444C211yesmaximal, characteristic, complemented
6#71570C211yesmaximal, characteristic
7#71568C212yesmaximal, complemented
8#69428C212yesmaximal
9#69427C211yesmaximal, characteristic
10#69420C211yesmaximal, characteristic, complemented
11Dic3⋊#95C211yesmaximal, characteristic, complemented
12#15178—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×Dic6C2212yes—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C2×#896—28no—
28C2×#896—28no—
29C2×#896—28no—
30C2×#888—28no—
31C2×#888—24no—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2211yescharacteristic
40C2×C3⋊#95C2212yes—
41#4666—24no—
42#4646—28no—
43C22×C4⋊Q8—31no—
44C22×C22⋊Q8—32no—
45C2×#391—32no—
46C2×C4×C22⋊C4—31no—
47#1444—31no—
48#1444—32no—
49#1444D311yescharacteristic, complemented
50#1439—31no—
51#1439—32no—
52#1439—31no—
53C4⋊#95—31no—
54C24×D3C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C22×C4×D3—22no—
66C22×C4×D3—24no—
67C22×C4×D3—24no—
68C22×C4×D3—22no—
69C22×C4×D3D414yes—
70C22×Dic6—24no—
71C22×Dic6—24no—
72C22×Dic6—24no—
73C22×Dic6—24no—
74C22×Dic6—24no—
75C22×Dic6—24no—
76C22×Dic6D418yes—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4Q814yescomplemented
93C2×(C2×D3)⋊C4Q814yescomplemented
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4D418yes—
105C2×C4×Dic3—24no—
106C2×C4×Dic3—24no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3D414yes—
109#896—264no—
110#888—232no—
111C3×#95—24no—
112C3×#95—22no—
113C3×#95—22no—
114C3⋊#95—24no—
115C3⋊#95—22no—
116C3⋊#95—24no—
117C3⋊#95—24no—
118C3⋊#95—22no—
119C3⋊#95—24no—
120C23×Q8—32no—
121C24×C4—31no—
122C2×C4⋊Q8—64no—
123C2×C4⋊Q8—68no—
124C2×C22⋊Q8—68no—
125C2×C22⋊Q8—68no—
126C2×C22⋊Q8—68no—
127C22×C4⋊C4—32no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4C2×D312yes—
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C22⋊C4—31no—
134C22×C22⋊C4—31no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C42—31no—
138C22×C42—31no—
139#391—68no—
140C4×C22⋊C4—64no—
141C2×#95—32no—
142C2×#95—31no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—31no—
147C2×#95C2×D311yescharacteristic
148C2×#95C2×D312yes—
149C2×#95C2×D311yescharacteristic
150C24×C3C2411yescharacteristic
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—21no—
154C23×D3—21no—
155C23×D3—21no—
156C23×D3—21no—
157C23×D3—22no—
158C23×D3—22no—
159C23×D3C2×D412yes—
160C23×D3C2×Q812yes—
161C23×D3C2×Q812yes—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
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—21no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×Q812yes—
189C22×C3×C4C2×Q812yes—
190C22×C3×C4#4112yes—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3C2×D414yes—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D412yes—
212C22×Dic3C2×D414yes—
213C22×Dic3C2×D412yes—
214C22×Dic3C2×D412yes—
215C22×Dic3C2×D414yes—
216C22×Dic3C2×Q812yes—
217C22×Dic3C2×Q812yes—
218C22×Dic3#4112yes—
219C22×Dic3#4112yes—
220C2×C4×D3—216no—
221C2×C4×D3—28no—
222C2×C4×D3—44no—
223C2×C4×D3—48no—
224C2×C4×D3—44no—
225C2×Dic6—216no—
226C2×Dic6—216no—
227C2×Dic6—216no—
228C2×Dic6—48no—
229C2×Dic6—48no—
230C2×Dic6—48no—
231C2×Dic6—48no—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—216no—
234(C2×D3)⋊C4—216no—
235(C2×D3)⋊C4—216no—
236(C2×D3)⋊C4—28no—
237(C2×D3)⋊C4—28no—
238C4⋊Dic3—28no—
239Dic3⋊C4—216no—
240Dic3⋊C4—216no—
241Dic3⋊C4—216no—
242C4×Dic3—28no—
243C4×Dic3—44no—
244C25—31no—
245C22×Q8—38no—
246C22×Q8—64no—
247C22×Q8—64no—
248C22×Q8—64no—
249C22×Q8—64no—
250C22×Q8—64no—
251C22×Q8—64no—
252C23×C4—31no—
253C23×C4—31no—
254C23×C4—32no—
255C23×C4—32no—
256C23×C4—31no—
257C23×C4—34no—
258C23×C4—64no—
259C23×C4—64no—
260C23×C4—62no—
261C23×C4—62no—
262C23×C4C22×D312yes—
263C23×C4C22×D312yes—
264C23×C4C22×D311yescharacteristic
265C23×C4C22×D311yescharacteristic
266C23×C4C22×D311yescharacteristic
267C4⋊Q8—632no—
268C22⋊Q8—664no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—38no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—64no—
277C2×C4⋊C4—64no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—64no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—62no—
285C2×C22⋊C4—34no—
286C2×C22⋊C4—34no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4—64no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—64no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—62no—
297C2×C42—34no—
298C2×C42—64no—
299C2×C42—62no—
300C2×C42—64no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—64no—
305#95—62no—
306#95—64no—
307#95—64no—
308#95—62no—
309#95—64no—
310C23×C3C22×D411yescharacteristic
311C23×C3C22×D412yes—
312C23×C3C22×D411yescharacteristic
313C23×C3C22×D412yes—
314C23×C3C22×Q811yescharacteristic
315C23×C3C22×Q811yescharacteristic
316C23×C3C2×#4111yescharacteristic
317C23×C3C2×#4111yescharacteristic
318C23×C3C2×#4111yescharacteristic
319C23×C32−1+412yes—
320C23×C32−1+412yes—
321C22×D3—24no—
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—41no—
334C22×D3—42no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3C4⋊Q814yescomplemented
344C2×C3×C4—22no—
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—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4C22⋊Q814yescomplemented
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—22no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—41no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3#12014yes—
397C2×Dic3#12018yes—
398C2×Dic3#12118yes—
399C2×Dic3#12114yes—
400C2×Dic3C22⋊Q814yescomplemented
401C2×Dic3#12714yes—
402C4×D3—416no—
403Dic6—432no—
404C24—32no—
405C24—32no—
406C24—32no—
407C24—61no—
408C24—62no—
409C24—62no—
410C24—61no—
411C24—62no—
412C24—61no—
413C24—61no—
414C24—62no—
415C24C23×D311yescharacteristic
416C2×Q8—616no—
417C2×Q8—616no—
418C2×Q8—616no—
419C2×Q8—128no—
420C2×Q8—128no—
421C2×Q8—128no—
422C2×Q8—128no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
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—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—32no—
446C22×C4—34no—
447C22×C4—34no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—32no—
452C22×C4—32no—
453C22×C4—34no—
454C22×C4—34no—
455C22×C4—32no—
456C22×C4—616no—
457C22×C4—68no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—61no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—124no—
477C22×C4—124no—
478C22×C4—128no—
479C22×C4#23512yes—
480C22×C4D3×D414yes—
481C22×C4D3×D414yes—
482C22×C4D3×D412yes—
483C22×C4D3×D412yes—
484C22×C4D3×Q812yes—
485C22×C4D3×Q812yes—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—616no—
489C4⋊C4—616no—
490C4⋊C4—616no—
491C4⋊C4—68no—
492C22⋊C4—68no—
493C22⋊C4—616no—
494C22⋊C4—616no—
495C22⋊C4—68no—
496C42—68no—
497C42—124no—
498C22×C3C2×#12011yescharacteristic
499C22×C3C2×#12012yes—
500C22×C3C2×#12111yescharacteristic
501C22×C3C2×#12112yes—
502C22×C3C2×C22⋊Q811yescharacteristic
503C22×C3C2×C22⋊Q811yescharacteristic
504C22×C3C2×#12711yescharacteristic
505C22×C3C2×C4⋊Q811yescharacteristic
506C22×C3#53211yescharacteristic
507C22×C3#53311yescharacteristic
508C22×C3#53311yescharacteristic
509C22×C3#53611yescharacteristic
510C22×C3#53611yescharacteristic
511C22×C3#53612yes—
512C22×C3#54112yes—
513C22×C3#54712yes—
514C22×C3#54712yes—
515C22×C3#54812yes—
516C22×C3#54812yes—
517C22×C3D4×Q812yes—
518C22×C3D4×Q812yes—
519C22×C3D4×Q812yes—
520C22×C3D4×Q812yes—
521C2×D3—24no—
522C2×D3—28no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—44no—
530C2×D3—42no—
531C2×D3—42no—
532C3×C4—24no—
533C3×C4—44no—
534C3×C4—42no—
535C3×C4—44no—
536C3×C4—42no—
537Dic3—24no—
538Dic3—44no—
539Dic3—44no—
540Dic3—42no—
541Dic3#39218yes—
542C23—34no—
543C23—64no—
544C23—64no—
545C23—62no—
546C23—64no—
547C23—64no—
548C23—62no—
549C23—62no—
550C23—62no—
551C23—64no—
552C23—62no—
553C23—62no—
554C23—64no—
555C23—122no—
556C23—121no—
557C23—122no—
558C23—122no—
559C23—121no—
560C23—121no—
561C23—122no—
562C23—122no—
563C23—121no—
564C23—122no—
565C23C2×#23511yescharacteristic
566C23C2×D3×D412yes—
567C23C2×D3×D412yes—
568C23C2×D3×D411yescharacteristic
569C23C2×D3×D411yescharacteristic
570C23C2×D3×Q811yescharacteristic
571C23C2×D3×Q811yescharacteristic
572C23#100712yes—
573C23D3×#4111yescharacteristic
574C23D3×#4111yescharacteristic
575C23#101012yes—
576Q8—1232no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—34no—
592C2×C4—34no—
593C2×C4—38no—
594C2×C4—34no—
595C2×C4—38no—
596C2×C4—34no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—64no—
612C2×C4—62no—
613C2×C4—64no—
614C2×C4—62no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—64no—
623C2×C4—1216no—
624C2×C4—122no—
625C2×C4—121no—
626C2×C4—122no—
627C2×C4—121no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4#89614yescomplemented
636C2×C3C2×#39211yescharacteristic
637C2×C3#240911yescharacteristic
638C2×C3#240911yescharacteristic
639C2×C3#242912yes—
640C2×C3#242912yes—
641C2×C3#243212yes—
642C2×C3#243412yes—
643C2×C3#254211yescharacteristic
644C2×C3#254211yescharacteristic
645C2×C3#262111yescharacteristic
646C2×C3#285711yescharacteristic
647D3—44no—
648C22—64no—
649C22—68no—
650C22—122no—
651C22—122no—
652C22—122no—
653C22—124no—
654C22—124no—
655C22—122no—
656C22—122no—
657C22—124no—
658C22—122no—
659C22#550712yes—
660C22C2×#89611yescharacteristic
661C22#554811yescharacteristic
662C22#555212yes—
663C22#557811yescharacteristic
664C22D3×#12011yescharacteristic
665C22D3×#12012yes—
666C22D3×#12111yescharacteristic
667C22D3×#12112yes—
668C22D3×C22⋊Q811yescharacteristic
669C22#562812yes—
670C22#563212yes—
671C22#563212yes—
672C22#563312yes—
673C22#564811yescharacteristic
674C22#565512yes—
675C22#568811yescharacteristic
676C22#569612yes—
677C22D3×#12711yescharacteristic
678C22D3×C4⋊Q811yescharacteristic
679C22#572411yescharacteristic
680C22#572712yes—
681C22#572712yes—
682C4—24no—
683C4—38no—
684C4—44no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—64no—
689C4—122no—
690C4—124no—
691C4—124no—
692C3#1517811yescharacteristic, complemented, Sylow
693C2—124no—
694C2#7433411yescharacteristic
695C2#7440511yescharacteristic
696C2#7441112yes—
697C2#7599612yes—
698C2#7628811yescharacteristic
699C2D3×#39211yescharacteristic
700C2#7717211yescharacteristic
701C2#7717312yes—
702C2#7774412yes—
703C2#7781811yescharacteristic
704C2#7803711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1444C2
prime radicalC24×D3C23
hypercenter#1444D3
hypocenterC3#15178
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C22×#888C2semidirect
3. C2×#4646C2semidirect
5. C3×#1444C2semidirect
7. #71568C2semidirect
10. #69420C2semidirect
11. Dic3⋊#95C2semidirect
49. #1444D3semidirect
92. C2×(C2×D3)⋊C4Q8semidirect
93. C2×(C2×D3)⋊C4Q8semidirect
343. C22×D3C4⋊Q8semidirect
372. C2×C3×C4C22⋊Q8semidirect
400. C2×Dic3C22⋊Q8semidirect
635. C2×C4#896semidirect
692. C3#15178semidirect

Maximal subgroups

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

subgroupindex
C22×#8962
C22×#8882
C2×#46462
C2×#46662
C3×#14442
#715702
#715682
#694282
#694272
#694202
Dic3⋊#952
#151783