Finite Group Databasebeta

← #10751744

Subgroups — #10751744

Order 768.

1794 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#924C211yesmaximal, characteristic
2C2×#5210C211yesmaximal, characteristic
3C2×#4661C211yesmaximal, characteristic, complemented
4C2×#4656C211yesmaximal, characteristic, complemented
5C3×C22⋊#95C211yesmaximal, characteristic
6#71546C212yesmaximal
7#71546C212yesmaximal
8#69424C211yesmaximal, characteristic, complemented
9#69423C211yesmaximal, characteristic, complemented
10#69420C212yesmaximal, complemented
11#69411C212yesmaximal, complemented
12#16606—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×#217C2211yescharacteristic
17C22×#217C2212yes—
18C22×#217C2211yescharacteristic
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2212yes—
22C2×#924—24no—
23C2×#924—24no—
24C2×#924—24no—
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31#5210—24no—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4661—24no—
41#4656—24no—
42C22×C22⋊Q8—31no—
43C2×#398—31no—
44C2×#395—31no—
45C2×#385—31no—
46#1446—31no—
47#1445—31no—
48#1444—32no—
49#1441—32no—
50#1439—32no—
51#1439—32no—
52C22⋊#95D311yescharacteristic
53C25×C3C2311yescharacteristic
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—42no—
57C23×C3×C4—42no—
58C23×C3×C4D414yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2312yes—
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—42no—
72C22×Dic6—42no—
73C22×Dic6D414yes—
74C2×C3×C22⋊C4—42no—
75C2×C3×C22⋊C4—42no—
76C2×C3×C22⋊C4Q818yescomplemented
77C2×#217—22no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—22no—
82C2×#217—22no—
83C2×#217—22no—
84C2×#217—42no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—42no—
91C2×C4⋊Dic3—42no—
92C2×C4⋊Dic3—42no—
93C2×C4⋊Dic3D414yes—
94#924—232no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—42no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—42no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C3⋊#95—42no—
112C3⋊#95—42no—
113C3⋊#95—42no—
114C23×Q8—31no—
115C24×C4C2×D311yescharacteristic
116C2×C22⋊Q8—64no—
117C2×C22⋊Q8—64no—
118C2×C22⋊Q8—64no—
119C22×C4⋊C4—31no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4—31no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—31no—
127C22×C22⋊C4—31no—
128C22×C22⋊C4C2×D312yes—
129#398—64no—
130#395—64no—
131#385—64no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—31no—
136C2×#95—31no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95C2×D311yescharacteristic
141C2×#95C2×D311yescharacteristic
142C2×#95C2×D312yes—
143C24×C3—21no—
144C24×C3—21no—
145C24×C3—21no—
146C24×C3—21no—
147C24×C3—22no—
148C24×C3—21no—
149C24×C3—22no—
150C24×C3—21no—
151C24×C3—22no—
152C24×C3C2×D412yes—
153C24×C3C2×Q814yes—
154C24×C3C2411yescharacteristic
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
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—28no—
174C22×C3×C4—44no—
175C22×C3×C4—44no—
176C22×C3×C4—48no—
177C22×C3×C4—48no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×Q814yes—
183C22×C3×C4C2×Q814yes—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—21no—
215C22×Dic3—22no—
216C22×Dic3C2×D414yes—
217C22×Dic3C2×D414yes—
218C22×Dic3#4112yes—
219C22×Dic3#4112yes—
220C2×Dic6—28no—
221C2×Dic6—48no—
222C2×Dic6—44no—
223C2×Dic6—44no—
224C2×Dic6—48no—
225C3×C22⋊C4—48no—
226#217—216no—
227#217—28no—
228#217—44no—
229C4⋊Dic3—28no—
230C4⋊Dic3—48no—
231C4⋊Dic3—48no—
232Dic3⋊C4—216no—
233Dic3⋊C4—44no—
234Dic3⋊C4—44no—
235C25C22×D311yescharacteristic
236C22×Q8—34no—
237C22×Q8—62no—
238C22×Q8—62no—
239C22×Q8—122no—
240C22×Q8—122no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—32no—
244C23×C4—31no—
245C23×C4—32no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4D1214yes—
252C23×C4C22×D312yes—
253C23×C4C22×D312yes—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D311yescharacteristic
256C22⋊Q8—632no—
257C2×C4⋊C4—34no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—122no—
271C2×C4⋊C4—122no—
272C2×C4⋊C4—122no—
273C2×C4⋊C4—122no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—64no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—122no—
284C2×C22⋊C4Dic618yescomplemented
285#95—22no—
286#95—22no—
287#95—42no—
288#95—64no—
289#95—64no—
290#95—64no—
291#95—62no—
292#95—62no—
293#95—122no—
294#95—122no—
295#95—122no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3C4⋊Q814yes—
319C23×C3C22×D411yescharacteristic
320C23×C3C22×D411yescharacteristic
321C23×C3C22×D411yescharacteristic
322C23×C3C22×Q812yes—
323C23×C3C2×#4111yescharacteristic
324C23×C3C2×#4111yescharacteristic
325C23×C3H(5,𝔽2)12yes—
326C23×C3H(5,𝔽2)12yes—
327C23×C3H(5,𝔽2)11yescharacteristic
328C23×C32−1+411yescharacteristic
329C23×C32−1+412yes—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—416no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
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#12014yes—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—22no—
366C2×Dic3—41no—
367C2×Dic3—42no—
368C2×Dic3—41no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—41no—
378C2×Dic3—42no—
379C2×Dic3—41no—
380C2×Dic3—42no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—42no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—42no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391Dic6—416no—
392C24—21no—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—22no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—22no—
401C24C2×Dic614yes—
402C24C2×D1212yes—
403C24C23×D311yescharacteristic
404C2×Q8—68no—
405C2×Q8—128no—
406C2×Q8—124no—
407C2×Q8—124no—
408C2×Q8—128no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
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—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—34no—
429C22×C4—34no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—61no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—61no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—61no—
467C22×C4—62no—
468C22×C4C2×Dic614yes—
469C22×C4C2×Dic614yes—
470C22×C4C2×D1212yes—
471C22×C4C2×D1212yes—
472C22×C4D3×D412yes—
473C22×C4D3×D412yes—
474C4⋊C4—616no—
475C4⋊C4—68no—
476C4⋊C4—124no—
477C4⋊C4—128no—
478C4⋊C4—128no—
479C4⋊C4—124no—
480C22⋊C4—48no—
481C22⋊C4—616no—
482C22⋊C4—68no—
483C22⋊C4—124no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—24no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—42no—
495C22×C3C2×#12011yescharacteristic
496C22×C3#52511yescharacteristic
497C22×C3C2×C4⋊Q811yescharacteristic
498C22×C3#53411yescharacteristic
499C22×C3#53411yescharacteristic
500C22×C3#53412yes—
501C22×C3#53512yes—
502C22×C3#53611yescharacteristic
503C22×C3#53711yescharacteristic
504C22×C3#53912yes—
505C22×C3#54212yes—
506C22×C3#54312yes—
507C22×C3#54412yes—
508C22×C3#54611yescharacteristic
509C22×C3#54711yescharacteristic
510C22×C3#54811yescharacteristic
511C22×C3D4×Q812yes—
512C22×C3D4×Q812yes—
513C22×C3#55011yescharacteristic
514C22×C3#55412yes—
515C22×C3#55412yes—
516C22×C3#56212yes—
517C22×C3#56312yes—
518C3×C4—24no—
519C3×C4—42no—
520C3×C4—44no—
521C3×C4—44no—
522Dic3—44no—
523Dic3—42no—
524Dic3—44no—
525Dic3—42no—
526Dic3—81no—
527Dic3—81no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23#86914yes—
551C23C22×Dic612yes—
552C23C22×D1211yescharacteristic
553C23C2×D3×D411yescharacteristic
554C23C2×D3×D411yescharacteristic
555C23#100412yes—
556C23#100412yes—
557C23#100411yescharacteristic
558C23#100711yescharacteristic
559C23D3×#4111yescharacteristic
560C23D3×#4111yescharacteristic
561C23#101012yes—
562Q8—1216no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—416no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—121no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—121no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—121no—
611C2×C4—122no—
612C2×C4—121no—
613C2×C4—122no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4#88214yes—
624C2×C3—44no—
625C2×C3#240711yescharacteristic
626C2×C3#243411yescharacteristic
627C2×C3#258311yescharacteristic
628C2×C3#261311yescharacteristic
629C2×C3#263811yescharacteristic
630C2×C3#265911yescharacteristic
631C2×C3#269212yes—
632C2×C3#272012yes—
633C2×C3#272912yes—
634C2×C3#273212yes—
635C2×C3#281411yescharacteristic
636C22—24no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22C2×#86911yescharacteristic
648C22#548312yes—
649C22C2×#88211yescharacteristic
650C22#549311yescharacteristic
651C22#549311yescharacteristic
652C22#550012yes—
653C22#551011yescharacteristic
654C22#552411yescharacteristic
655C22#552611yescharacteristic
656C22D4×Dic612yes—
657C22D4×Dic612yes—
658C22#553912yes—
659C22#553912yes—
660C22#558912yes—
661C22#559212yes—
662C22#559911yescharacteristic
663C22#561211yescharacteristic
664C22#562112yes—
665C22#563111yescharacteristic
666C22#563311yescharacteristic
667C22#564612yes—
668C22#566012yes—
669C22#566212yes—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—42no—
674C4—122no—
675C4—124no—
676C4—122no—
677C4—124no—
678C4—241no—
679C4—241no—
680C3#1660611yescharacteristic, complemented, Sylow
681C2—44no—
682C2#7336111yescharacteristic
683C2#7349011yescharacteristic
684C2#7354711yescharacteristic
685C2#7385511yescharacteristic
686C2#7401511yescharacteristic
687C2#7404711yescharacteristic
688C2#7409111yescharacteristic
689C2#7453412yes—
690C2#7454012yes—
691C2#7480312yes—
692C2#7483012yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C22⋊#95C2
prime radicalC25×C3C23
hypercenterC22⋊#95D3
hypocenterC3#16606
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#4661C2semidirect
4. C2×#4656C2semidirect
8. #69424C2semidirect
9. #69423C2semidirect
10. #69420C2semidirect
11. #69411C2semidirect
76. C2×C3×C22⋊C4Q8semidirect
284. C2×C22⋊C4Dic6semidirect
680. C3#16606semidirect

Maximal subgroups

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

subgroupindex
C22×#9242
C2×#52102
C2×#46612
C2×#46562
C3×C22⋊#952
#715462
#694242
#694232
#694202
#694112
#166063