Finite Group Databasebeta

← #10732231

Subgroups — #10732231

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×#893C211yesmaximal, characteristic
2C2×#4667C212yesmaximal, complemented
3C2×#4667C212yesmaximal, complemented
4C2×#4647C211yesmaximal, characteristic, complemented
5C3×#1442C211yesmaximal, characteristic, complemented
6#71567C211yesmaximal, characteristic, complemented
7#69428C212yesmaximal, complemented
8#69427C212yesmaximal, complemented
9#69426C211yesmaximal, characteristic
10#69426C211yesmaximal, characteristic
11Dic3⋊#95C211yesmaximal, characteristic, complemented
12#15181—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C2×#893—28no—
28C2×#893—24no—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4667—28no—
39#4667—28no—
40#4647—24no—
41C22×C4×D4—31no—
42C2×#389—31no—
43C2×#380—32no—
44C2×#380—32no—
45#1442—31no—
46#1442D311yescharacteristic, complemented
47#1439—32no—
48#1439—32no—
49#1438—31no—
50#1438—31no—
51C4⋊#95—31no—
52C24×D3C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C22×D12—24no—
64C22×D12—24no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12Q814yes—
68C22×C4×D3—24no—
69C22×C4×D3—24no—
70C22×C4×D3—24no—
71C22×C4×D3—24no—
72C22×C4×D3—24no—
73C22×C4×D3—24no—
74C22×C4×D3D418yes—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—42no—
87C2×(C2×D3)⋊C4—42no—
88C2×(C2×D3)⋊C4—42no—
89C2×(C2×D3)⋊C4—42no—
90C2×(C2×D3)⋊C4—42no—
91C2×(C2×D3)⋊C4—42no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—42no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—42no—
103C2×Dic3⋊C4D414yescomplemented
104C2×C4×Dic3—24no—
105C2×C4×Dic3D414yes—
106#893—232no—
107C3×#95—22no—
108C3×#95—24no—
109C3×#95—24no—
110C3×#95—22no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—22no—
114C3⋊#95—24no—
115C3⋊#95—42no—
116C23×D4—31no—
117C24×C4—32no—
118C2×C4×D4—64no—
119C2×C4×D4—68no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4C2×D311yescharacteristic
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4—32no—
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×#95C2×D311yescharacteristic
141C2×#95C2×D311yescharacteristic
142C2×#95C2×D312yes—
143C2×#95C2×D312yes—
144C24×C3C2411yescharacteristic
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—22no—
149C23×D3—22no—
150C23×D3—22no—
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—22no—
154C23×D3—22no—
155C23×D3—22no—
156C23×D3—22no—
157C23×D3C2×D414yes—
158C23×D3C2×Q814yes—
159C23×D3#4114yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
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—21no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×Q812yes—
188C22×C3×C4#4112yes—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
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—21no—
207C22×Dic3—22no—
208C22×Dic3—21no—
209C22×Dic3C2×D412yes—
210C22×Dic3C2×D412yes—
211C22×Dic3C2×D414yes—
212C22×Dic3C2×D412yes—
213C22×Dic3C2×D412yes—
214C22×Dic3#4112yes—
215C22×Dic3#4114yes—
216C2×D12—28no—
217C2×D12—28no—
218C2×D12—216no—
219C2×D12—216no—
220C2×D12—28no—
221C2×C4×D3—216no—
222C2×C4×D3—216no—
223C2×C4×D3—216no—
224C2×C4×D3—48no—
225C2×C4×D3—48no—
226C2×C4×D3—48no—
227C2×C4×D3—48no—
228C3×C4⋊C4—28no—
229(C2×D3)⋊C4—216no—
230(C2×D3)⋊C4—48no—
231(C2×D3)⋊C4—48no—
232(C2×D3)⋊C4—48no—
233(C2×D3)⋊C4—48no—
234C4⋊Dic3—44no—
235Dic3⋊C4—28no—
236Dic3⋊C4—44no—
237Dic3⋊C4—44no—
238Dic3⋊C4—48no—
239C4×Dic3—28no—
240C25—32no—
241C22×D4—34no—
242C22×D4—64no—
243C22×D4—64no—
244C22×D4—62no—
245C22×D4—62no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—38no—
250C23×C4—31no—
251C23×C4—32no—
252C23×C4—64no—
253C23×C4—64no—
254C23×C4—64no—
255C23×C4—64no—
256C23×C4—64no—
257C23×C4—64no—
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D312yes—
260C23×C4C22×D311yescharacteristic
261C23×C4C22×D312yes—
262C23×C4C22×D311yescharacteristic
263C4×D4—632no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—34no—
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—64no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—122no—
288C2×C22⋊C4—122no—
289C2×C22⋊C4—122no—
290C2×C22⋊C4—122no—
291C2×C22⋊C4—122no—
292C2×C22⋊C4—122no—
293C2×C42—34no—
294C2×C42—64no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—64no—
300#95—62no—
301#95—64no—
302#95—62no—
303#95—122no—
304C23×C3C22×D412yes—
305C23×C3C22×D411yescharacteristic
306C23×C3C22×D411yescharacteristic
307C23×C3C22×Q811yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4111yescharacteristic
310C23×C3C2×#4112yes—
311C23×C3C2×#4112yes—
312C23×C3H(5,𝔽2)12yes—
313C23×C3H(5,𝔽2)11yescharacteristic
314C23×C32−1+411yescharacteristic
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D3—24no—
320C22×D3—24no—
321C22×D3—24no—
322C22×D3—24no—
323C22×D3—24no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—24no—
329C22×D3—24no—
330C22×D3—24no—
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3C22⋊Q818yes—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—42no—
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—42no—
368C2×C3×C4—42no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4C22⋊Q814yes—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—24no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—41no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—41no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—41no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3#12014yes—
402C2×Dic3#12114yes—
403C2×Dic3#12118yes—
404C2×Dic3#12318yes—
405C2×Dic3#12414yes—
406D12—232no—
407C4×D3—432no—
408C24—34no—
409C24—34no—
410C24—34no—
411C24—62no—
412C24—62no—
413C24—62no—
414C24—62no—
415C24—62no—
416C24—62no—
417C24—62no—
418C24—62no—
419C24—62no—
420C24—62no—
421C24—62no—
422C24—62no—
423C24C23×D311yescharacteristic
424C2×D4—616no—
425C2×D4—68no—
426C2×D4—68no—
427C2×D4—616no—
428C2×D4—68no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—32no—
454C22×C4—34no—
455C22×C4—32no—
456C22×C4—32no—
457C22×C4—34no—
458C22×C4—32no—
459C22×C4—32no—
460C22×C4—616no—
461C22×C4—616no—
462C22×C4—616no—
463C22×C4—61no—
464C22×C4—62no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—61no—
482C22×C4—62no—
483C22×C4—128no—
484C22×C4—128no—
485C22×C4—128no—
486C22×C4—128no—
487C22×C4#23512yes—
488C22×C4D3×D412yes—
489C22×C4D3×D414yes—
490C22×C4D3×D412yes—
491C22×C4D3×Q812yes—
492C4⋊C4—28no—
493C4⋊C4—68no—
494C4⋊C4—124no—
495C4⋊C4—128no—
496C4⋊C4—124no—
497C4⋊C4—124no—
498C22⋊C4—616no—
499C22⋊C4—128no—
500C22⋊C4—128no—
501C22⋊C4—128no—
502C22⋊C4—128no—
503C42—68no—
504C22×C3C2×#12011yescharacteristic
505C22×C3C2×#12112yes—
506C22×C3C2×#12111yescharacteristic
507C22×C3C2×C22⋊Q811yescharacteristic
508C22×C3C2×C22⋊Q812yes—
509C22×C3C2×#12312yes—
510C22×C3#52511yescharacteristic
511C22×C3#52512yes—
512C22×C3C2×#12411yescharacteristic
513C22×C3#53511yescharacteristic
514C22×C3#53512yes—
515C22×C3#53711yescharacteristic
516C22×C3#53712yes—
517C22×C3#54612yes—
518C22×C3D4×Q811yescharacteristic
519C22×C3#55011yescharacteristic
520C22×C3#55212yes—
521C22×C3#55312yes—
522C22×C3#55312yes—
523C22×C3#55412yes—
524C22×C3#55411yescharacteristic
525C22×C3#55512yes—
526C22×C3#55611yescharacteristic
527C2×D3—28no—
528C2×D3—28no—
529C2×D3—28no—
530C2×D3—44no—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—44no—
534C2×D3—44no—
535C2×D3—44no—
536C2×D3—44no—
537C2×D3—44no—
538C2×D3—44no—
539C2×D3—44no—
540C2×D3—44no—
541C2×D3—44no—
542C3×C4—24no—
543C3×C4—44no—
544C3×C4—42no—
545C3×C4—81no—
546C3×C4—82no—
547Dic3—42no—
548Dic3—42no—
549Dic3—81no—
550Dic3—82no—
551Dic3#39418yes—
552C23—38no—
553C23—64no—
554C23—64no—
555C23—64no—
556C23—64no—
557C23—64no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23—64no—
564C23—64no—
565C23—64no—
566C23—64no—
567C23—64no—
568C23—64no—
569C23—64no—
570C23—64no—
571C23—122no—
572C23—122no—
573C23—122no—
574C23—122no—
575C23—122no—
576C23—122no—
577C23—122no—
578C23—122no—
579C23—122no—
580C23—122no—
581C23—122no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23C2×#23511yescharacteristic
588C23C2×D3×D412yes—
589C23C2×D3×D411yescharacteristic
590C23C2×D3×D411yescharacteristic
591C23C2×D3×Q811yescharacteristic
592C23#100711yescharacteristic
593C23D3×#4111yescharacteristic
594C23D3×#4112yes—
595C23D3×#4112yes—
596C23#100912yes—
597C23#100911yescharacteristic
598D4—632no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—38no—
609C2×C4—34no—
610C2×C4—34no—
611C2×C4—34no—
612C2×C4—38no—
613C2×C4—42no—
614C2×C4—41no—
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—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—64no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—1232no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—121no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—121no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—121no—
655C2×C4#89614yes—
656C2×C3C2×#39411yescharacteristic
657C2×C3#241412yes—
658C2×C3#241712yes—
659C2×C3#248512yes—
660C2×C3#248712yes—
661C2×C3#253511yescharacteristic
662C2×C3#253811yescharacteristic
663C2×C3#256211yescharacteristic
664C2×C3#256311yescharacteristic
665C2×C3#261511yescharacteristic
666C2×C3#284811yescharacteristic
667D3—48no—
668C22—68no—
669C22—68no—
670C22—68no—
671C22—124no—
672C22—124no—
673C22—124no—
674C22—124no—
675C22—124no—
676C22—124no—
677C22—124no—
678C22—124no—
679C22—124no—
680C22—124no—
681C22—124no—
682C22—124no—
683C22C2×#89611yescharacteristic
684C22#553212yes—
685C22#554712yes—
686C22#555311yescharacteristic
687C22#556811yescharacteristic
688C22D3×#12011yescharacteristic
689C22D3×#12111yescharacteristic
690C22D3×#12112yes—
691C22D3×C22⋊Q812yes—
692C22#562912yes—
693C22#562911yescharacteristic
694C22D3×#12312yes—
695C22#565412yes—
696C22#565611yescharacteristic
697C22D3×#12411yescharacteristic
698C22#567412yes—
699C22#567411yescharacteristic
700C22#569011yescharacteristic
701C22#569012yes—
702C22#569312yes—
703C22#570912yes—
704C22#570912yes—
705C22#572711yescharacteristic
706C4—24no—
707C4—38no—
708C4—44no—
709C4—42no—
710C4—81no—
711C4—82no—
712C4—122no—
713C4—122no—
714C4—241no—
715C4—242no—
716C3#1518111yescharacteristic, complemented, Sylow
717C2—128no—
718C2#7418211yescharacteristic
719C2#7465511yescharacteristic
720C2#7502112yes—
721C2#7549512yes—
722C2#7605911yescharacteristic
723C2#7680911yescharacteristic
724C2D3×#39411yescharacteristic
725C2#7751311yescharacteristic
726C2#7774212yes—
727C2#7814612yes—
728C2#7818011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1442C2
prime radicalC23×D12C22
hypercenter#1442D3
hypocenterC3#15181
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
2. C2×#4667C2semidirect
3. C2×#4667C2semidirect
4. C2×#4647C2semidirect
5. C3×#1442C2semidirect
6. #71567C2semidirect
7. #69428C2semidirect
8. #69427C2semidirect
11. Dic3⋊#95C2semidirect
46. #1442D3semidirect
103. 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×#8932
C2×#46672
C2×#46472
C3×#14422
#715672
#694282
#694272
#694262
Dic3⋊#952
#151813