Finite Group Databasebeta

← #10732227

Subgroups — #10732227

Order 768.

2314 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic
2C2×#4942C212yesmaximal, complemented
3C2×#4942C212yesmaximal, complemented
4C2×#4647C211yesmaximal, characteristic, complemented
5C3×#1438C211yesmaximal, characteristic
6#72131C211yesmaximal, characteristic
7#71547C212yesmaximal, complemented
8#71546C212yesmaximal, complemented
9#69415C211yesmaximal, characteristic, complemented
10#69412C211yesmaximal, characteristic, complemented
11Dic3⋊#95C211yesmaximal, characteristic, complemented
12#15181—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×Dic3C2212yes—
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C22×#217C2212yes—
20C2×D4×Dic3—28no—
21C2×D4×Dic3—24no—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2211yescharacteristic
28C22×C4×Dic3C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4942—28no—
39#4942—28no—
40#4647—24no—
41C22×C4×D4—31no—
42C2×#389—31no—
43C2×#380—32no—
44C2×#380—32no—
45#1442—31no—
46#1442—31no—
47#1439—32no—
48#1439—32no—
49#1438—31no—
50#1438D311yescharacteristic
51C4⋊#95—31no—
52C25×C3C2312yes—
53C22×C3×D4—24no—
54C22×C3×D4—24no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4Q814yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—24no—
63C23×Dic3—24no—
64C23×Dic3—24no—
65C23×Dic3—24no—
66C23×Dic3—24no—
67C23×Dic3—24no—
68C23×Dic3D418yes—
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2312yes—
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2311yescharacteristic
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—42no—
79C2×C3×C22⋊C4—42no—
80C2×C3×C22⋊C4—42no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—24no—
86C2×#217—24no—
87C2×#217—42no—
88C2×#217—42no—
89C2×#217—42no—
90D4×Dic3—232no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—42no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—42no—
104C2×Dic3⋊C4D414yescomplemented
105C2×C4×Dic3—24no—
106C2×C4×Dic3D414yes—
107C3×#95—22no—
108C3×#95—22no—
109C3⋊#95—22no—
110C3⋊#95—24no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—42no—
116C23×D4C2×D311yescharacteristic
117C24×C4—32no—
118C2×C4×D4—64no—
119C2×C4×D4—68no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—31no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4C2×D312yes—
130C22×C22⋊C4C2×D312yes—
131C22×C42—31no—
132#389—64no—
133#380—68no—
134#380—68no—
135C2×#95—31no—
136C2×#95—31no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95—32no—
142C2×#95C2×D311yescharacteristic
143C2×#95C2×D311yescharacteristic
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3C2×D414yes—
157C24×C3C2×Q814yes—
158C24×C3#4114yes—
159C24×C3C2411yescharacteristic
160C2×C3×D4—28no—
161C2×C3×D4—28no—
162C2×C3×D4—216no—
163C2×C3×D4—216no—
164C2×C3×D4—28no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×Q812yes—
186C22×C3×C4#4112yes—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—216no—
202C22×Dic3—216no—
203C22×Dic3—216no—
204C22×Dic3—21no—
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—21no—
214C22×Dic3—22no—
215C22×Dic3—21no—
216C22×Dic3—48no—
217C22×Dic3—48no—
218C22×Dic3—48no—
219C22×Dic3—48no—
220C22×Dic3C2×D412yes—
221C22×Dic3C2×D414yes—
222C22×Dic3C2×D412yes—
223C22×Dic3C2×D414yes—
224C22×Dic3C2×D412yes—
225C22×Dic3C2×D412yes—
226C22×Dic3#4112yes—
227C22×Dic3#4114yes—
228C3×C22⋊C4—48no—
229C3×C22⋊C4—48no—
230#217—216no—
231#217—48no—
232#217—48no—
233C4⋊Dic3—28no—
234C4⋊Dic3—44no—
235Dic3⋊C4—28no—
236Dic3⋊C4—44no—
237Dic3⋊C4—44no—
238Dic3⋊C4—48no—
239C4×Dic3—28no—
240C25C22×D312yes—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4Dic614yes—
246C23×C4—31no—
247C23×C4—38no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—32no—
251C23×C4—31no—
252C23×C4—32no—
253C23×C4—64no—
254C23×C4—64no—
255C23×C4—64no—
256C23×C4—64no—
257C23×C4—64no—
258C23×C4—64no—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D312yes—
261C23×C4C22×D311yescharacteristic
262C23×C4C22×D311yescharacteristic
263C4×D4—632no—
264C2×C4⋊C4—34no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—122no—
277C2×C4⋊C4—122no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4—64no—
289C2×C22⋊C4—64no—
290C2×C22⋊C4—122no—
291C2×C22⋊C4—122no—
292C2×C22⋊C4—122no—
293C2×C42—34no—
294C2×C42—64no—
295#95—22no—
296#95—22no—
297#95—64no—
298#95—64no—
299#95—64no—
300#95—64no—
301#95—62no—
302#95—62no—
303#95—122no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—24no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3C22⋊Q818yes—
339C23×C3C22×D412yes—
340C23×C3C22×D411yescharacteristic
341C23×C3C22×D411yescharacteristic
342C23×C3C22×Q811yescharacteristic
343C23×C3C2×#4112yes—
344C23×C3C2×#4111yescharacteristic
345C23×C3C2×#4111yescharacteristic
346C23×C3C2×#4112yes—
347C23×C3H(5,𝔽2)12yes—
348C23×C3H(5,𝔽2)11yescharacteristic
349C23×C32−1+411yescharacteristic
350C3×D4—232no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
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⋊Q814yes—
373C2×Dic3—22no—
374C2×Dic3—24no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—24no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—41no—
385C2×Dic3—42no—
386C2×Dic3—41no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—432no—
391C2×Dic3—41no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—41no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—41no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3#12014yes—
404C2×Dic3#12114yes—
405C2×Dic3#12118yes—
406C2×Dic3#12318yes—
407C2×Dic3#12414yes—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24C2×Dic614yes—
421C24D3×D414yes—
422C24#23714yes—
423C24C23×D311yescharacteristic
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—216no—
427C2×D4—216no—
428C2×D4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—32no—
448C22×C4—34no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—34no—
452C22×C4—32no—
453C22×C4—34no—
454C22×C4—32no—
455C22×C4—616no—
456C22×C4—616no—
457C22×C4—616no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—61no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—61no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—61no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—61no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—61no—
484C22×C4—128no—
485C22×C4—128no—
486C22×C4—128no—
487C22×C4—128no—
488C22×C4C2×Dic612yes—
489C22×C4D3×D412yes—
490C22×C4D3×D412yes—
491C22×C4#23712yes—
492C4⋊C4—68no—
493C4⋊C4—68no—
494C4⋊C4—124no—
495C4⋊C4—128no—
496C4⋊C4—124no—
497C4⋊C4—124no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—616no—
501C22⋊C4—128no—
502C22⋊C4—128no—
503C42—68no—
504C22×C3—28no—
505C22×C3—28no—
506C22×C3—28no—
507C22×C3—44no—
508C22×C3—44no—
509C22×C3—44no—
510C22×C3—44no—
511C22×C3—44no—
512C22×C3—44no—
513C22×C3—44no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3C2×#12011yescharacteristic
520C22×C3C2×#12112yes—
521C22×C3C2×#12111yescharacteristic
522C22×C3C2×C22⋊Q811yescharacteristic
523C22×C3C2×C22⋊Q812yes—
524C22×C3C2×#12312yes—
525C22×C3#52511yescharacteristic
526C22×C3#52512yes—
527C22×C3C2×#12411yescharacteristic
528C22×C3#53511yescharacteristic
529C22×C3#53512yes—
530C22×C3#53712yes—
531C22×C3#53711yescharacteristic
532C22×C3#54612yes—
533C22×C3D4×Q811yescharacteristic
534C22×C3#55011yescharacteristic
535C22×C3#55212yes—
536C22×C3#55312yes—
537C22×C3#55312yes—
538C22×C3#55412yes—
539C22×C3#55411yescharacteristic
540C22×C3#55512yes—
541C22×C3#55611yescharacteristic
542C3×C4—24no—
543C3×C4—42no—
544C3×C4—81no—
545C3×C4—82no—
546Dic3—42no—
547Dic3—44no—
548Dic3—42no—
549Dic3—81no—
550Dic3—82no—
551Dic3#39418yes—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23#87818yes—
587C23C22×Dic611yescharacteristic
588C23C2×D3×D412yes—
589C23C2×D3×D411yescharacteristic
590C23C2×D3×D411yescharacteristic
591C23C2×#23712yes—
592C23C2×#23711yescharacteristic
593C23#100412yes—
594C23#100411yescharacteristic
595C23D3×#4112yes—
596C23D3×#4111yescharacteristic
597C23#101011yescharacteristic
598D4—232no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—38no—
606C2×C4—34no—
607C2×C4—34no—
608C2×C4—34no—
609C2×C4—38no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—64no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—64no—
629C2×C4—64no—
630C2×C4—62no—
631C2×C4—64no—
632C2×C4—62no—
633C2×C4—1232no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—121no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—121no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—121no—
654C2×C4—122no—
655C2×C4#87814yes—
656C2×C3—48no—
657C2×C3C2×#39411yescharacteristic
658C2×C3#241412yes—
659C2×C3#241712yes—
660C2×C3#248512yes—
661C2×C3#248712yes—
662C2×C3#253511yescharacteristic
663C2×C3#253811yescharacteristic
664C2×C3#256211yescharacteristic
665C2×C3#256311yescharacteristic
666C2×C3#261511yescharacteristic
667C2×C3#284811yescharacteristic
668C22—28no—
669C22—28no—
670C22—28no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22C2×#87811yescharacteristic
684C22C2×#87812yes—
685C22D4×Dic611yescharacteristic
686C22#553911yescharacteristic
687C22#553912yes—
688C22#558612yes—
689C22#558611yescharacteristic
690C22D3×#12011yescharacteristic
691C22#559112yes—
692C22#559212yes—
693C22#559712yes—
694C22#560112yes—
695C22#560112yes—
696C22D3×#12111yescharacteristic
697C22D3×#12112yes—
698C22#560811yescharacteristic
699C22#561612yes—
700C22#565012yes—
701C22D3×#12312yes—
702C22#567111yescharacteristic
703C22D3×#12411yescharacteristic
704C22#571511yescharacteristic
705C22#571811yescharacteristic
706C4—24no—
707C4—38no—
708C4—42no—
709C4—82no—
710C4—81no—
711C4—124no—
712C4—122no—
713C4—122no—
714C4—242no—
715C4—241no—
716C3#1518111yescharacteristic, complemented, Sylow
717C2—48no—
718C2#7384611yescharacteristic
719C2#7469312yes—
720C2#7473812yes—
721C2#7634611yescharacteristic
722C2#7636111yescharacteristic
723C2#7667012yes—
724C2#7668611yescharacteristic
725C2#7669711yescharacteristic
726C2#7693312yes—
727C2#7695511yescharacteristic
728C2D3×#39411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#15181
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#4942C2semidirect
3. C2×#4942C2semidirect
4. C2×#4647C2semidirect
7. #71547C2semidirect
8. #71546C2semidirect
9. #69415C2semidirect
10. #69412C2semidirect
11. Dic3⋊#95C2semidirect
104. C2×Dic3⋊C4D4semidirect
716. C3#15181semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C2×#49422
C2×#46472
C3×#14382
#721312
#715472
#715462
#694152
#694122
Dic3⋊#952
#151813