Finite Group Databasebeta

← #10722365

Subgroups — #10722365

Order 768.

1926 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#877C211yesmaximal, characteristic
2C2×#4975C211yesmaximal, characteristic, complemented
3C2×#4942C212yesmaximal
4C2×#4650C211yesmaximal, characteristic, complemented
5C3×#1439C211yesmaximal, characteristic
6#71566C211yesmaximal, characteristic, complemented
7#71547C212yesmaximal
8#71546C211yesmaximal, characteristic
9#69410C211yesmaximal, characteristic, complemented
10#69409C212yesmaximal, complemented
11Dic3⋊#95C212yesmaximal, complemented
12#14340—31nomaximal, Sylow
13C24×Dic3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C2×#877—24no—
28C2×#877—24no—
29C2×#877C418yescomplemented
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4975—24no—
40#4942—28no—
41#4650—24no—
42C22×#117—31no—
43C2×#384—31no—
44C2×#382—31no—
45C2×#380—32no—
46#1441—32no—
47#1441—31no—
48#1441—31no—
49#1439—31no—
50#1439—32no—
51#1439D311yescharacteristic
52C4⋊#95—32no—
53C25×C3C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—22no—
59C23×Dic3—22no—
60C23×Dic3—22no—
61C23×Dic3—22no—
62C23×Dic3C2×C414yes—
63C23×Dic3D414yes—
64C23×Dic3Q814yes—
65C23×Dic3C2312yes—
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2311yescharacteristic
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—42no—
77C2×C3×C22⋊C4—42no—
78C2×C3×C22⋊C4C2×C414yes—
79C2×#217—22no—
80C2×#217—22no—
81C2×#217—24no—
82C2×#217—42no—
83C2×#217—42no—
84C2×#217C2×C414yes—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—42no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—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⋊C4—42no—
100C2×Dic3⋊C4C2×C414yes—
101C2×Dic3⋊C4C2×C414yes—
102C2×C4×Dic3—22no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3C2×C414yes—
105C2×C4×Dic3C2×C414yes—
106C2×C4×Dic3D414yes—
107C2×C4×Dic3D414yes—
108#877—232no—
109C3×#95—22no—
110C3×#95—42no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—42no—
116C3⋊#95—42no—
117C3⋊#95—42no—
118C24×C4—31no—
119C2×#117—38no—
120C2×#117—64no—
121C2×#117—64no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4C2×D311yescharacteristic
129C22×C22⋊C4—31no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4C2×D311yescharacteristic
132C22×C22⋊C4C2×D312yes—
133C22×C42—31no—
134C22×C42—31no—
135#384—64no—
136#382—64no—
137#380—68no—
138C2×#95—32no—
139C2×#95—31no—
140C2×#95—32no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95C2×D311yescharacteristic
146C2×#95C2×D312yes—
147C24×C3—21no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—21no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3C22×C412yes—
156C24×C3C2×D412yes—
157C24×C3C2×Q812yes—
158C24×C3C2411yescharacteristic
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
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—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—22no—
179C22×C3×C4C22×C412yes—
180C22×C3×C4C22×C412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—28no—
190C22×Dic3—28no—
191C22×Dic3—28no—
192C22×Dic3—28no—
193C22×Dic3—28no—
194C22×Dic3—28no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
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—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3C4⋊C4116yescomplemented
214C22×Dic3C22×C412yes—
215C22×Dic3C22×C412yes—
216C22×Dic3C22×C412yes—
217C22×Dic3C22×C412yes—
218C22×Dic3C2×D412yes—
219C22×Dic3C2×D412yes—
220C22×Dic3C2×D412yes—
221C22×Dic3C2×D412yes—
222C22×Dic3C2×D412yes—
223C22×Dic3C2×D412yes—
224C22×Dic3C2×Q812yes—
225C22×Dic3C2×Q812yes—
226C22×Dic3#4112yes—
227C22×Dic3#4112yes—
228C3×C4⋊C4—44no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—48no—
231#217—28no—
232#217—48no—
233C4⋊Dic3—44no—
234Dic3⋊C4—28no—
235Dic3⋊C4—28no—
236Dic3⋊C4—48no—
237Dic3⋊C4—48no—
238Dic3⋊C4—48no—
239C4×Dic3—28no—
240C4×Dic3C22⋊C4116yescomplemented
241C25C22×D311yescharacteristic
242C23×C4—31no—
243C23×C4—34no—
244C23×C4—34no—
245C23×C4—34no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—32no—
250C23×C4—32no—
251C23×C4—62no—
252C23×C4—62no—
253C23×C4—62no—
254C23×C4—62no—
255C23×C4C22×D312yes—
256C23×C4C22×D312yes—
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259#117—632no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—122no—
277C2×C4⋊C4—122no—
278C2×C4⋊C4—122no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—34no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4—62no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4—122no—
289C2×C22⋊C4—122no—
290C2×C22⋊C4C4×D314yes—
291C2×C42—34no—
292C2×C42—34no—
293C2×C42—34no—
294C2×C42—34no—
295C2×C42—62no—
296C2×C42—62no—
297#95—22no—
298#95—42no—
299#95—64no—
300#95—64no—
301#95—64no—
302#95—62no—
303#95—122no—
304#95—122no—
305#95—122no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3C2×C4⋊C414yes—
329C23×C3C23×C411yescharacteristic
330C23×C3C22×D411yescharacteristic
331C23×C3C22×D411yescharacteristic
332C23×C3C22×D411yescharacteristic
333C23×C3C22×Q811yescharacteristic
334C23×C3C2×#4111yescharacteristic
335C23×C3C2×#4111yescharacteristic
336C23×C3H(5,𝔽2)12yes—
337C23×C3H(5,𝔽2)12yes—
338C23×C32−1+412yes—
339C23×C32−1+412yes—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
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×C22⋊C414yes—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—24no—
374C2×Dic3—232no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—42no—
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×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3C2×C22⋊C414yes—
395C2×Dic3C2×C22⋊C414yes—
396C2×Dic3C2×C4⋊C414yes—
397C2×Dic3C2×C4⋊C414yes—
398C2×Dic3#11714yes—
399C2×Dic3#12114yes—
400C2×Dic3#12114yes—
401C2×Dic3C22⋊Q814yes—
402C2×Dic3C22⋊Q814yes—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—22no—
407C24—21no—
408C24—21no—
409C24—22no—
410C24—22no—
411C24C2×C4×D312yes—
412C24D3×D412yes—
413C24D3×Q812yes—
414C24C23×D311yescharacteristic
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—32no—
442C22×C4—316no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—68no—
451C22×C4—68no—
452C22×C4—68no—
453C22×C4—68no—
454C22×C4—68no—
455C22×C4—68no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—62no—
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—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4C2×C4×D312yes—
481C22×C4C2×C4×D312yes—
482C22×C4D3×D412yes—
483C22×C4D3×D412yes—
484C4⋊C4—44no—
485C4⋊C4—68no—
486C4⋊C4—68no—
487C4⋊C4—124no—
488C4⋊C4—128no—
489C4⋊C4—128no—
490C4⋊C4—128no—
491C22⋊C4—28no—
492C22⋊C4—48no—
493C22⋊C4—68no—
494C22⋊C4—128no—
495C42—316no—
496C42—68no—
497C22×C3—28no—
498C22×C3—24no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—44no—
502C22×C3—42no—
503C22×C3—44no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—44no—
508C22×C3C22×C22⋊C411yescharacteristic
509C22×C3C22×C4⋊C411yescharacteristic
510C22×C3C2×#11711yescharacteristic
511C22×C3#52012yes—
512C22×C3#52012yes—
513C22×C3C2×#12111yescharacteristic
514C22×C3C2×#12111yescharacteristic
515C22×C3C2×C22⋊Q811yescharacteristic
516C22×C3C2×C22⋊Q811yescharacteristic
517C22×C3#53511yescharacteristic
518C22×C3#53511yescharacteristic
519C22×C3#53512yes—
520C22×C3#53611yescharacteristic
521C22×C3#53612yes—
522C22×C3#53611yescharacteristic
523C22×C3#53712yes—
524C22×C3#53712yes—
525C22×C3#54012yes—
526C22×C3#54112yes—
527C22×C3#54212yes—
528C22×C3#54212yes—
529C22×C3#54412yes—
530C22×C3#54412yes—
531C3×C4—24no—
532C3×C4—42no—
533C3×C4—82no—
534C3×C4—82no—
535Dic3—24no—
536Dic3—24no—
537Dic3—42no—
538Dic3—82no—
539Dic3—82no—
540Dic3#38018yes—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23D3×C4⋊C414yes—
564C23C22×C4×D311yescharacteristic
565C23C2×D3×D411yescharacteristic
566C23C2×D3×D411yescharacteristic
567C23C2×D3×D411yescharacteristic
568C23#100412yes—
569C23#100412yes—
570C23C2×D3×Q811yescharacteristic
571C23D3×#4111yescharacteristic
572C23D3×#4111yescharacteristic
573C23#101012yes—
574C23#101012yes—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—34no—
582C2×C4—34no—
583C2×C4—34no—
584C2×C4—34no—
585C2×C4—34no—
586C2×C4—34no—
587C2×C4—34no—
588C2×C4—34no—
589C2×C4—34no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—632no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—64no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—64no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4D3×C22⋊C414yes—
638C2×C3—44no—
639C2×C3C2×#38011yescharacteristic
640C2×C3#231111yescharacteristic
641C2×C3#231111yescharacteristic
642C2×C3#231412yes—
643C2×C3#231712yes—
644C2×C3#261911yescharacteristic
645C2×C3#262012yes—
646C2×C3#262111yescharacteristic
647C2×C3#262811yescharacteristic
648C2×C3#262811yescharacteristic
649C2×C3#262912yes—
650C22—24no—
651C22—28no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22C2×D3×C22⋊C411yescharacteristic
662C22C2×D3×C4⋊C411yescharacteristic
663C22D3×#11711yescharacteristic
664C22#554612yes—
665C22#554612yes—
666C22#559211yescharacteristic
667C22#559212yes—
668C22#559712yes—
669C22D3×#12111yescharacteristic
670C22D3×#12111yescharacteristic
671C22#560812yes—
672C22#561512yes—
673C22#562312yes—
674C22#562312yes—
675C22D3×C22⋊Q811yescharacteristic
676C22D3×C22⋊Q811yescharacteristic
677C22#564812yes—
678C22#564811yescharacteristic
679C22#566212yes—
680C22#566912yes—
681C22#567511yescharacteristic
682C22#570112yes—
683C22#571511yescharacteristic
684C4—24no—
685C4—38no—
686C4—42no—
687C4—64no—
688C4—64no—
689C4—82no—
690C4—82no—
691C4—122no—
692C4—242no—
693C4—242no—
694C3#1434011yescharacteristic, complemented, Sylow
695C2—44no—
696C2D3×#38011yescharacteristic
697C2#7438212yes—
698C2#7466211yescharacteristic
699C2#7545612yes—
700C2#7629411yescharacteristic
701C2#7669811yescharacteristic
702C2#7681011yescharacteristic
703C2#7697212yes—
704C2#7733212yes—
705C2#7753811yescharacteristic
706C2#7804911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1439C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1439D3
hypocenterC3#14340
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#4975C2semidirect
4. C2×#4650C2semidirect
6. #71566C2semidirect
9. #69410C2semidirect
10. #69409C2semidirect
11. Dic3⋊#95C2semidirect
29. C2×#877C4semidirect
213. C22×Dic3C4⋊C4semidirect
240. C4×Dic3C22⋊C4semidirect
694. C3#14340semidirect

Maximal subgroups

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

subgroupindex
C22×#8772
C2×#49752
C2×#49422
C2×#46502
C3×#14392
#715662
#715472
#715462
#694102
#694092
Dic3⋊#952
#143403