Finite Group Databasebeta

← #10718896

Subgroups — #10718896

Order 768.

2006 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
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
2C2×#5210C211yesmaximal, characteristic
3C22×C3⋊#95C211yesmaximal, characteristic
4C2×#4950C211yesmaximal, characteristic
5C2×#4950C211yesmaximal, characteristic
6#71544C211yesmaximal, characteristic
7#71544C211yesmaximal, characteristic
8#71536C212yesmaximal, complemented
9#71535C212yesmaximal, complemented
10#69415C212yesmaximal, complemented
11#69411C212yesmaximal, complemented
12#14044—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C42C2212yes—
19C22×#217C2211yescharacteristic
20C22×#217C2211yescharacteristic
21C22×C4⋊Dic3C2212yes—
22C22×C4⋊Dic3C2212yes—
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C3×C4×C22⋊C4—24no—
26C2×C3×#95C2212yes—
27#5210—24no—
28C2×C3⋊#95—24no—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4950—24no—
40#4950C418yescomplemented
41C2×#385—31no—
42C2×#385—31no—
43C2×#385—31no—
44C2×C4×C22⋊C4D311yescharacteristic
45C22×#95—31no—
46#1442—32no—
47#1441—32no—
48#1436—32no—
49C4⋊#95—32no—
50C22⋊#95—31no—
51C22⋊#95—31no—
52C25×C3C2311yescharacteristic
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—24no—
56C23×C3×C4—42no—
57C23×C3×C4D414yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—22no—
63C23×Dic3—22no—
64C23×Dic3—22no—
65C23×Dic3—24no—
66C23×Dic3—22no—
67C23×Dic3—22no—
68C23×Dic3—42no—
69C23×Dic3—42no—
70C23×Dic3—42no—
71C23×Dic3C2×C414yes—
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2312yes—
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2311yescharacteristic
77C23×Dic3C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4C2×C414yes—
81C2×C3×C22⋊C4D414yes—
82C2×C3×C22⋊C4Q814yes—
83C2×C3×C22⋊C4Q814yes—
84C2×C3×C42—24no—
85C2×C3×C42—24no—
86C2×C3×C42—42no—
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—22no—
90C2×#217—22no—
91C2×#217—22no—
92C2×#217C2×C414yes—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—42no—
96C2×C4⋊Dic3—42no—
97C2×C4⋊Dic3—42no—
98C2×C4⋊Dic3C2×C418yescomplemented
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—42no—
103C3×#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—24no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C3⋊#95—42no—
113C3⋊#95C2×C418yescomplemented
114C24×C4—31no—
115C24×C4—31no—
116C24×C4C2×D311yescharacteristic
117C22×C4⋊C4—32no—
118C22×C4⋊C4—31no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—31no—
121C22×C22⋊C4—31no—
122C22×C22⋊C4—31no—
123C22×C22⋊C4C2×D311yescharacteristic
124C22×C22⋊C4C2×D311yescharacteristic
125C22×C42C2×D312yes—
126#385—38no—
127#385—64no—
128#385—64no—
129C4×C22⋊C4—24no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—31no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—64no—
141C2×#95C2×D312yes—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3C22×C412yes—
147C24×C3C2×D412yes—
148C24×C3C2×D412yes—
149C24×C3C2×Q812yes—
150C24×C3C2×Q812yes—
151C24×C3#4112yes—
152C24×C3#4112yes—
153C24×C3C2411yescharacteristic
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—28no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—28no—
173C22×C3×C4—44no—
174C22×C3×C4—48no—
175C22×C3×C4—48no—
176C22×C3×C4C22×C414yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×Q814yes—
181C22×C3×C4C2×Q814yes—
182C22×C3×C4#4114yes—
183C22×Dic3—28no—
184C22×Dic3—28no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
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—22no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—48no—
216C22×Dic3—48no—
217C22×Dic3—48no—
218C22×Dic3—44no—
219C22×Dic3—44no—
220C22×Dic3—44no—
221C22×Dic3—48no—
222C22×Dic3—44no—
223C22×Dic3C22×C412yes—
224C22×Dic3C22×C414yes—
225C22×Dic3C22×C412yes—
226C22×Dic3#4114yes—
227C3×C22⋊C4—28no—
228C3×C22⋊C4C4⋊C4116yescomplemented
229C3×C42—48no—
230#217—28no—
231#217—44no—
232C4⋊Dic3—48no—
233C4⋊Dic3—48no—
234Dic3⋊C4—44no—
235Dic3⋊C4—44no—
236C25C22×D311yescharacteristic
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—32no—
243C23×C4—32no—
244C23×C4—34no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—42no—
248C23×C4—64no—
249C23×C4—62no—
250C23×C4—62no—
251C23×C4—62no—
252C23×C4—62no—
253C23×C4—62no—
254C23×C4—122no—
255C23×C4—122no—
256C23×C4—122no—
257C23×C4D1214yes—
258C23×C4C22×D312yes—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D312yes—
261C23×C4C22×D311yescharacteristic
262C2×C4⋊C4—38no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4—122no—
269C2×C4⋊C4—122no—
270C2×C4⋊C4—122no—
271C2×C4⋊C4—122no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—34no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4Dic614yes—
281C2×C22⋊C4Dic614yes—
282C2×C22⋊C4C4×D314yes—
283C2×C22⋊C4#6714yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—42no—
287#95—24no—
288#95—38no—
289#95—64no—
290#95—64no—
291#95—64no—
292#95—62no—
293#95—62no—
294#95—64no—
295#95—122no—
296#95—122no—
297#95—122no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—24no—
303C23×C3—22no—
304C23×C3—22no—
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×C3C2×C4⋊C414yes—
315C23×C3C4×D414yes—
316C23×C3C4×Q814yes—
317C23×C3#12114yes—
318C23×C3C22⋊Q814yes—
319C23×C3#12514yes—
320C23×C3C4⋊Q814yes—
321C23×C3C23×C411yescharacteristic
322C23×C3C22×D411yescharacteristic
323C23×C3C22×D411yescharacteristic
324C23×C3C22×Q811yescharacteristic
325C23×C3C22×Q811yescharacteristic
326C23×C3C2×#4111yescharacteristic
327C23×C3C2×#4111yescharacteristic
328C23×C3C2×#4112yes—
329C23×C3C2×#4112yes—
330C23×C3H(5,𝔽2)12yes—
331C23×C3H(5,𝔽2)12yes—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—416no—
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×C4C2×C4⋊C418yes—
356C2×C3×C4#12314yes—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—416no—
368C2×Dic3—416no—
369C2×Dic3—42no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—42no—
375C2×Dic3—41no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24C2×Dic612yes—
398C24C2×Dic612yes—
399C24C2×C4×D312yes—
400C24C2×D1212yes—
401C24#23512yes—
402C24#23512yes—
403C24C2×#6712yes—
404C24C23×D311yescharacteristic
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—32no—
425C22×C4—34no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—68no—
432C22×C4—68no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—128no—
464C22×C4—128no—
465C22×C4—128no—
466C22×C4—124no—
467C22×C4—124no—
468C22×C4—124no—
469C22×C4—128no—
470C22×C4—124no—
471C22×C4C2×Dic614yes—
472C22×C4C2×Dic614yes—
473C22×C4C2×C4×D314yes—
474C22×C4C2×D1212yes—
475C22×C4C2×D1212yes—
476C22×C4#23714yes—
477C22×C4C2×#6714yes—
478C4⋊C4—128no—
479C4⋊C4—128no—
480C4⋊C4—124no—
481C4⋊C4—124no—
482C22⋊C4—28no—
483C22⋊C4—68no—
484C22⋊C4—124no—
485C22⋊C4Dic3⋊C4116yescomplemented
486C42—48no—
487C22×C3—24no—
488C22×C3—24no—
489C22×C3—28no—
490C22×C3—28no—
491C22×C3—24no—
492C22×C3—24no—
493C22×C3—28no—
494C22×C3—24no—
495C22×C3—28no—
496C22×C3—24no—
497C22×C3#38918yes—
498C22×C3C22×C4⋊C411yescharacteristic
499C22×C3C2×C4×D411yescharacteristic
500C22×C3C2×C4×Q811yescharacteristic
501C22×C3C4×#4112yes—
502C22×C3#51812yes—
503C22×C3C2×#12111yescharacteristic
504C22×C3C2×C22⋊Q811yescharacteristic
505C22×C3C2×#12311yescharacteristic
506C22×C3#52511yescharacteristic
507C22×C3C2×#12511yescharacteristic
508C22×C3#52912yes—
509C22×C3C2×C4⋊Q811yescharacteristic
510C22×C3#53411yescharacteristic
511C22×C3#53511yescharacteristic
512C22×C3#53812yes—
513C22×C3#54612yes—
514C22×C3#54612yes—
515C22×C3#55112yes—
516C22×C3#55112yes—
517C22×C3#55412yes—
518C22×C3#55412yes—
519C22×C3#55412yes—
520C22×C3#55412yes—
521C3×C4—28no—
522C3×C4—24no—
523C3×C4—42no—
524C3×C4—44no—
525Dic3—42no—
526Dic3—44no—
527Dic3—44no—
528Dic3—42no—
529Dic3—81no—
530Dic3—81no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23C4×Dic614yes—
548C23#86914yes—
549C23#87014yes—
550C23C4×D1214yes—
551C23C2×Dic3⋊C414yes—
552C23#92414yes—
553C23#93014yes—
554C23C22×Dic611yescharacteristic
555C23C22×Dic611yescharacteristic
556C23C22×C4×D311yescharacteristic
557C23C22×D1211yescharacteristic
558C23C2×#23511yescharacteristic
559C23C2×#23511yescharacteristic
560C23C2×#23712yes—
561C23#100412yes—
562C23D3×#4112yes—
563C23#100912yes—
564C23C22×#6711yescharacteristic
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—1216no—
599C2×C4—1216no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—121no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—121no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—122no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—122no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4#88614yes—
625C2×C4C2×Dic3⋊C418yes—
626C2×C3—28no—
627C2×C3C2×#38911yescharacteristic
628C2×C3C4×#12311yescharacteristic
629C2×C3#232111yescharacteristic
630C2×C3#234912yes—
631C2×C3#236812yes—
632C2×C3#241111yescharacteristic
633C2×C3#241211yescharacteristic
634C2×C3#245012yes—
635C2×C3#250211yescharacteristic
636C2×C3#250512yes—
637C2×C3#258311yescharacteristic
638C22—28no—
639C22—24no—
640C22—28no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22#492818yes—
649C22C2×C4×Dic611yescharacteristic
650C22C2×#86911yescharacteristic
651C22C2×#87011yescharacteristic
652C22C2×C4×D1211yescharacteristic
653C22C2×#88611yescharacteristic
654C22#549311yescharacteristic
655C22#549512yes—
656C22#552411yescharacteristic
657C22#552511yescharacteristic
658C22#552712yes—
659C22C4×#23712yes—
660C22#553812yes—
661C22#553912yes—
662C22#553912yes—
663C22#554312yes—
664C22#554312yes—
665C22#554512yes—
666C22#556012yes—
667C22C22×Dic3⋊C411yescharacteristic
668C22C2×#92411yescharacteristic
669C22C2×#93011yescharacteristic
670C22#580612yes—
671C22#582912yes—
672C4—28no—
673C4—24no—
674C4—44no—
675C4—42no—
676C4—124no—
677C4—122no—
678C4—124no—
679C4—122no—
680C4—241no—
681C4—241no—
682C3#1404411yescharacteristic, complemented, Sylow
683C2—28no—
684C2C4×#88611yescharacteristic
685C2#7347412yes—
686C2#7351711yescharacteristic
687C2#7352012yes—
688C2#7354711yescharacteristic
689C2#7356911yescharacteristic
690C2C2×#492811yescharacteristic
691C2#8210811yescharacteristic
692C2#8227511yescharacteristic
693C2#8233412yes—
694C2#8236112yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×C4×C22⋊C4C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#14044
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #71536C2semidirect
9. #71535C2semidirect
10. #69415C2semidirect
11. #69411C2semidirect
40. #4950C4semidirect
98. C2×C4⋊Dic3C2×C4semidirect
113. C3⋊#95C2×C4semidirect
228. C3×C22⋊C4C4⋊C4semidirect
485. C22⋊C4Dic3⋊C4semidirect
682. C3#14044semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C42
C2×#52102
C22×C3⋊#952
C2×#49502
#715442
#715362
#715352
#694152
#694112
#140443