Finite Group Databasebeta

← #10720116

Subgroups — #10720116

Order 768.

2366 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C4⋊C4C211yesmaximal, characteristic
2C2×#4989C212yesmaximal
3C2×#4976C212yesmaximal, complemented
4C2×#4667C211yesmaximal, characteristic
5C2×#4666C211yesmaximal, characteristic
6C3×#1444C211yesmaximal, characteristic, complemented
7#71567C211yesmaximal, characteristic, complemented
8(C2×D3)⋊#95C212yesmaximal
9#69420C211yesmaximal, characteristic, complemented
10#69412C211yesmaximal, characteristic, complemented
11Dic3⋊#95C212yesmaximal, complemented
12#14130—31nomaximal, Sylow
13C23×C4×D3C2212yes—
14C23×C4×D3C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C2×D3×C4⋊C4—24no—
27C2×D3×C4⋊C4—28no—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38#4989—28no—
39#4976—28no—
40#4667C418yescomplemented
41#4666—24no—
42C23×C4⋊C4—31no—
43C2×#389—32no—
44C2×#385—32no—
45C2×#380—31no—
46C2×C4×C22⋊C4—31no—
47#1444—31no—
48#1444D311yescharacteristic, complemented
49#1442—31no—
50#1442—31no—
51C4⋊#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—24no—
65C22×C4×D3—24no—
66C22×C4×D3—24no—
67C22×C4×D3—24no—
68C22×C4×D3—24no—
69C22×C4×D3—24no—
70C22×C4×D3—24no—
71C22×C4×D3—42no—
72C22×C4×D3—42no—
73C22×C4×D3C2×C414yes—
74C22×C4×D3D414yes—
75C22×C4×D3Q814yes—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—42no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4C2×C418yes—
88C2×(C2×D3)⋊C4D414yes—
89C2×(C2×D3)⋊C4Q814yes—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—42no—
99C2×Dic3⋊C4C2×C418yescomplemented
100C2×Dic3⋊C4D418yes—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3—24no—
105D3×C4⋊C4—232no—
106C3×#95—22no—
107C3×#95—42no—
108C3×#95C2×C414yescomplemented
109C3⋊#95—22no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—42no—
115C3⋊#95C2×C414yescomplemented
116C24×C4—31no—
117C24×C4—32no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—68no—
123C22×C4⋊C4—64no—
124C22×C4⋊C4C2×D311yescharacteristic
125C22×C4⋊C4C2×D312yes—
126C22×C22⋊C4—31no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—31no—
129C22×C42—31no—
130C22×C42—31no—
131#389—68no—
132#385—68no—
133#380—38no—
134C4×C22⋊C4—64no—
135C2×#95—31no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95C2×D311yescharacteristic
143C2×#95C2×D312yes—
144C2×#95C2×D311yescharacteristic
145C24×C3C2411yescharacteristic
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—22no—
149C23×D3—22no—
150C23×D3C22×C412yes—
151C23×D3C2×D412yes—
152C23×D3C2×D412yes—
153C23×D3C2×Q812yes—
154C23×D3C2×Q812yes—
155C23×D3#4112yes—
156C23×D3#4112yes—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
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×C4C22×C414yes—
180C22×C3×C4C22×C412yes—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×Q812yes—
185C22×C3×C4C2×Q812yes—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3C22×C414yes—
207C22×Dic3C22×C412yes—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×D414yes—
210C22×Dic3C2×D412yes—
211C22×Dic3C2×D414yes—
212C22×Dic3C2×Q812yes—
213C22×Dic3C2×Q812yes—
214C22×Dic3#4114yes—
215C2×C4×D3—216no—
216C2×C4×D3—28no—
217C2×C4×D3—216no—
218C2×C4×D3—28no—
219C2×C4×D3—216no—
220C2×C4×D3—48no—
221C2×C4×D3—48no—
222C2×C4×D3—48no—
223C2×C4×D3—48no—
224C2×C4×D3—48no—
225C2×C4×D3—48no—
226C2×C4×D3C4⋊C4116yescomplemented
227C3×C4⋊C4—28no—
228C3×C4⋊C4—48no—
229(C2×D3)⋊C4—216no—
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—28no—
232C4⋊Dic3—28no—
233Dic3⋊C4—216no—
234Dic3⋊C4—216no—
235Dic3⋊C4—48no—
236C4×Dic3—44no—
237C4×Dic3—44no—
238C25—31no—
239C23×C4—31no—
240C23×C4—34no—
241C23×C4—34no—
242C23×C4—34no—
243C23×C4—32no—
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—32no—
247C23×C4—64no—
248C23×C4—64no—
249C23×C4—64no—
250C23×C4—64no—
251C23×C4—64no—
252C23×C4—64no—
253C23×C4—64no—
254C23×C4—122no—
255C23×C4—122no—
256C23×C4C22×D312yes—
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D312yes—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—38no—
266C2×C4⋊C4—38no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—632no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—64no—
277C2×C4⋊C4—122no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—34no—
280C2×C22⋊C4—34no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—62no—
287C2×C42—62no—
288C2×C42—62no—
289C2×C42—64no—
290C2×C42—64no—
291#95—22no—
292#95—34no—
293#95—42no—
294#95—64no—
295#95—64no—
296#95—64no—
297#95—62no—
298#95—64no—
299#95—122no—
300#95C4×D314yescomplemented
301C23×C3C23×C411yescharacteristic
302C23×C3C22×D411yescharacteristic
303C23×C3C22×D412yes—
304C23×C3C22×D411yescharacteristic
305C23×C3C22×Q811yescharacteristic
306C23×C3C22×Q811yescharacteristic
307C23×C3C2×#4112yes—
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4111yescharacteristic
310C23×C3H(5,𝔽2)12yes—
311C23×C32−1+412yes—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—22no—
315C22×D3—24no—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—22no—
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×D3C2×C4⋊C414yes—
329C22×D3C4×D414yes—
330C22×D3C4×Q814yes—
331C22×D3#12114yes—
332C22×D3C22⋊Q814yes—
333C22×D3#12514yes—
334C22×D3C4⋊Q814yes—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
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×C4C2×C4⋊C414yes—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—41no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3C2×C4⋊C414yes—
389C2×Dic3C4×D418yes—
390C2×Dic3#12014yes—
391C2×Dic3C22⋊Q818yes—
392C2×Dic3#12314yes—
393C4×D3—232no—
394C4×D3—432no—
395C24—32no—
396C24—32no—
397C24—32no—
398C24—32no—
399C24—32no—
400C24—32no—
401C24—32no—
402C24—62no—
403C24—62no—
404C24—62no—
405C24—62no—
406C24C23×D311yescharacteristic
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—34no—
432C22×C4—32no—
433C22×C4—34no—
434C22×C4—316no—
435C22×C4—34no—
436C22×C4—32no—
437C22×C4—34no—
438C22×C4—32no—
439C22×C4—68no—
440C22×C4—68no—
441C22×C4—616no—
442C22×C4—616no—
443C22×C4—616no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—128no—
465C22×C4—128no—
466C22×C4—128no—
467C22×C4—128no—
468C22×C4—128no—
469C22×C4—128no—
470C22×C4C2×C4×D314yes—
471C22×C4C2×C4×D312yes—
472C22×C4D3×D414yes—
473C22×C4D3×D412yes—
474C22×C4D3×D412yes—
475C22×C4D3×Q812yes—
476C22×C4D3×Q812yes—
477C4⋊C4—28no—
478C4⋊C4—48no—
479C4⋊C4—616no—
480C4⋊C4—616no—
481C4⋊C4—68no—
482C4⋊C4—128no—
483C22⋊C4—616no—
484C22⋊C4—68no—
485C22⋊C4—68no—
486C42—124no—
487C42—124no—
488C22×C3C22×C4⋊C411yescharacteristic
489C22×C3C2×C4×D412yes—
490C22×C3C2×C4×D411yescharacteristic
491C22×C3C2×C4×Q811yescharacteristic
492C22×C3#52012yes—
493C22×C3C2×#12011yescharacteristic
494C22×C3C2×#12111yescharacteristic
495C22×C3C2×C22⋊Q811yescharacteristic
496C22×C3C2×C22⋊Q812yes—
497C22×C3C2×#12311yescharacteristic
498C22×C3C2×#12511yescharacteristic
499C22×C3C2×C4⋊Q811yescharacteristic
500C22×C3#53511yescharacteristic
501C22×C3#53611yescharacteristic
502C22×C3#54412yes—
503C22×C3D4212yes—
504C22×C3#54612yes—
505C22×C3#54712yes—
506C22×C3#54812yes—
507C22×C3D4×Q812yes—
508C22×C3#55212yes—
509C22×C3#55312yes—
510C22×C3#55412yes—
511C2×D3—24no—
512C2×D3—28no—
513C2×D3—28no—
514C2×D3—24no—
515C2×D3—24no—
516C2×D3—28no—
517C2×D3—28no—
518C2×D3—24no—
519C2×D3—24no—
520C2×D3—24no—
521C2×D3#38918yescomplemented
522C3×C4—24no—
523C3×C4—44no—
524C3×C4—44no—
525C3×C4—42no—
526C3×C4—42no—
527Dic3—44no—
528Dic3—44no—
529Dic3—42no—
530Dic3—42no—
531Dic3#38518yescomplemented
532C23—34no—
533C23—34no—
534C23—34no—
535C23—34no—
536C23—34no—
537C23—34no—
538C23—34no—
539C23—64no—
540C23—64no—
541C23—62no—
542C23—64no—
543C23—64no—
544C23—62no—
545C23—62no—
546C23—64no—
547C23—64no—
548C23—64no—
549C23—64no—
550C23—62no—
551C23—64no—
552C23—64no—
553C23—64no—
554C23—64no—
555C23C22×C4×D311yescharacteristic
556C23C2×D3×D412yes—
557C23C2×D3×D411yescharacteristic
558C23C2×D3×D411yescharacteristic
559C23#100412yes—
560C23C2×D3×Q811yescharacteristic
561C23C2×D3×Q811yescharacteristic
562C23#100712yes—
563C23D3×#4111yescharacteristic
564C23D3×#4112yes—
565C23D3×#4111yescharacteristic
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—34no—
581C2×C4—34no—
582C2×C4—38no—
583C2×C4—34no—
584C2×C4—38no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—632no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—62no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—1232no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—121no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4D3×C4⋊C414yes—
626C2×C3C2×#38511yescharacteristic
627C2×C3C2×#38911yescharacteristic
628C2×C3#231711yescharacteristic
629C2×C3#236012yes—
630C2×C3#236112yes—
631C2×C3#240811yescharacteristic
632C2×C3#240911yescharacteristic
633C2×C3#244012yes—
634C2×C3#244212yes—
635C2×C3#248611yescharacteristic
636C2×C3#248711yescharacteristic
637D3—28no—
638C22—38no—
639C22—64no—
640C22—64no—
641C22—68no—
642C22—64no—
643C22—64no—
644C22—68no—
645C22—68no—
646C22—64no—
647C22—64no—
648C22—68no—
649C22C2×D3×C4⋊C411yescharacteristic
650C22#550412yes—
651C22C4×D3×D412yes—
652C22C4×D3×D411yescharacteristic
653C22C4×D3×Q811yescharacteristic
654C22D3×#12011yescharacteristic
655C22#559012yes—
656C22D3×#12111yescharacteristic
657C22#561012yes—
658C22D3×C22⋊Q812yes—
659C22D3×C22⋊Q811yescharacteristic
660C22#563112yes—
661C22#563312yes—
662C22#563612yes—
663C22D3×#12311yescharacteristic
664C22#566112yes—
665C22D3×#12511yescharacteristic
666C22#569312yes—
667C22#571511yescharacteristic
668C22D3×C4⋊Q811yescharacteristic
669C22#572411yescharacteristic
670C22#572712yes—
671C22#572912yes—
672C4—24no—
673C4—38no—
674C4—44no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—122no—
679C4—124no—
680C4—124no—
681C4—122no—
682C3#1413011yescharacteristic, complemented, Sylow
683C2—68no—
684C2#7435911yescharacteristic
685C2#7439712yes—
686C2D3×#38511yescharacteristic
687C2#7598312yes—
688C2D3×#38911yescharacteristic
689C2#7666911yescharacteristic
690C2#7717211yescharacteristic
691C2#7717412yes—
692C2#7774612yes—
693C2#7776411yescharacteristic
694C2#7802911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1444C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenter#1444D3
hypocenterC3#14130
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
3. C2×#4976C2semidirect
6. C3×#1444C2semidirect
7. #71567C2semidirect
9. #69420C2semidirect
10. #69412C2semidirect
11. Dic3⋊#95C2semidirect
40. #4667C4semidirect
48. #1444D3semidirect
99. C2×Dic3⋊C4C2×C4semidirect
108. C3×#95C2×C4semidirect
115. C3⋊#95C2×C4semidirect
226. C2×C4×D3C4⋊C4semidirect
300. #95C4×D3semidirect
521. C2×D3#389semidirect
531. Dic3#385semidirect
682. C3#14130semidirect

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×#49892
C2×#49762
C2×#46672
C2×#46662
C3×#14442
#715672
(C2×D3)⋊#952
#694202
#694122
Dic3⋊#952
#141303