Finite Group Databasebeta

← #10968240

Subgroups — #10968240

Order 768.

1978 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
2#84031C212yesmaximal
3#82117C212yesmaximal
4#82093C212yesmaximal
5#82074C212yesmaximal, complemented
6C4×#924C212yesmaximal, complemented
7#73542C214yesmaximal, complemented
8#73536C214yesmaximal, complemented
9#73528C214yesmaximal, complemented
10#73498C214yesmaximal, complemented
11#73361C211yesmaximal, characteristic, complemented
12#73361C211yesmaximal, characteristic, complemented
13#73334C212yesmaximal, complemented
14#35012—31nomaximal, Sylow
15C24×C3×C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C2×#924C2212yes—
21C2×#924C2212yes—
22C2×#924C2212yes—
23C2×#924C2212yes—
24C2×#870C2212yescomplemented
25C2×C4×Dic6C2212yescomplemented
26C3×C4×C22⋊C4C2214yes—
27C3×C4×C22⋊C4C2214yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30#5247C2214yes—
31#5212C2212yes—
32#5212C2212yes—
33#5212C2212yes—
34#5212C2212yes—
35#5212C2212yes—
36#5212C2212yes—
37#5210C2214yes—
38#5210C2214yes—
39#5210C2214yes—
40#5210C2214yes—
41C4×#217C2214yes—
42#4951C2214yes—
43#4951C2214yes—
44#4951C2214yes—
45#4951C2214yes—
46#4949C2214yes—
47#4948C2214yes—
48#4948C2214yes—
49#4947C2214yes—
50#4936C2212yescomplemented
51C4×C4⋊Dic3C2212yescomplemented
52#4933C2214yescomplemented
53#4933C2214yescomplemented
54#4933C2214yescomplemented
55C4×Dic3⋊C4C2214yescomplemented
56#4929C2212yescomplemented
57#4928C2212yescomplemented
58#4928C2212yescomplemented
59#4928C2212yescomplemented
60#4661C2214yescomplemented
61#4661C2214yescomplemented
62#4656C2214yescomplemented
63#4656C2214yescomplemented
64#4655C2214yescomplemented
65#4654C2214yescomplemented
66#4647C2214yescomplemented
67#4646C2214yescomplemented
68#2851—32no—
69#2848—32no—
70#2659—31no—
71#2659—31no—
72#2626—32no—
73#2574—34no—
74#2556—34no—
75#2522—34no—
76#2465—34no—
77#2329—32no—
78#2312—32no—
79C4×C22⋊Q8—32no—
80C2×C4×C22⋊C4D311yescharacteristic
81C25×C3C2311yescharacteristic
82C23×C3×C4—24no—
83C23×C3×C4—42no—
84C23×C3×C4—42no—
85C23×C3×C4C2311yescharacteristic
86C23×C3×C4C2312yes—
87C23×C3×C4C2312yes—
88C23×C3×C4C2311yescharacteristic
89C23×C3×C4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C22×Dic6C2312yes—
94C22×Dic6C2312yes—
95C2×C3×C22⋊C4—24no—
96C2×C3×C22⋊C4Q818yescomplemented
97C2×C3×C22⋊C4C2314yes—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C42—42no—
100C2×C3×C42—42no—
101C2×C3×C42C2312yes—
102C2×C3×C42C2312yes—
103C2×C3×C42C2312yes—
104C2×C3×C42C2312yes—
105C2×#217C2314yes—
106C2×#217C2314yes—
107C2×#217C2314yes—
108C2×#217C2314yes—
109C2×#217C2314yes—
110C2×#217C2314yes—
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2312yes—
113C2×C4⋊Dic3C2312yes—
114C2×C4⋊Dic3C2312yes—
115C2×C4⋊Dic3C2312yes—
116C2×C4⋊Dic3C2312yes—
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2312yes—
121#924—28no—
122#924—28no—
123#924—44no—
124#924—44no—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×C4×Dic3C2314yes—
132C2×C4×Dic3C2314yes—
133#870—44no—
134C4×Dic6—44no—
135C3×#95C2314yes—
136C3×#95C2314yes—
137C3⋊#95C2314yes—
138C3⋊#95C2314yes—
139C3⋊#95C2314yes—
140C3⋊#95C2314yes—
141C3⋊#95C2314yes—
142C3⋊#95C2314yes—
143C3⋊#95C2314yes—
144C3⋊#95C2314yes—
145C3⋊#95C2314yes—
146C3⋊#95C2314yes—
147C24×C4C2×D311yescharacteristic
148C2×#125—32no—
149C2×C22⋊Q8—32no—
150C2×C22⋊Q8—32no—
151C2×C22⋊Q8—32no—
152C2×C22⋊Q8—32no—
153C2×C4×Q8—32no—
154C22×C22⋊C4C2×D311yescharacteristic
155C22×C22⋊C4C2×D311yescharacteristic
156C22×C42C2×D311yescharacteristic
157C22×C42C2×D311yescharacteristic
158#399—34no—
159#398—34no—
160#398—34no—
161#396—34no—
162#396—34no—
163#395—34no—
164#395—34no—
165#393—34no—
166#391—32no—
167#391—34no—
168#389—32no—
169#389—32no—
170#389—32no—
171#389—34no—
172#388—34no—
173#388—34no—
174#388—34no—
175#388—34no—
176#387—34no—
177#387—34no—
178#387—34no—
179#387—34no—
180#387—34no—
181#385—34no—
182#385—34no—
183#385—34no—
184#385—34no—
185#382—32no—
186#380—32no—
187#380—32no—
188#380—32no—
189#380—32no—
190#380—32no—
191#380—32no—
192#379—34no—
193C4×C4⋊C4—34no—
194C4×C4⋊C4—32no—
195C4×C22⋊C4—34no—
196C4×C22⋊C4C2×D314yes—
197C4×C22⋊C4C2×D314yes—
198C2×#95C2×D311yescharacteristic
199C2×#95C2×D311yescharacteristic
200C24×C3—22no—
201C24×C3—22no—
202C24×C3—41no—
203C24×C3—41no—
204C24×C3—41no—
205C24×C3—41no—
206C24×C3C2×Q814yes—
207C24×C3C2412yes—
208C24×C3C2411yescharacteristic
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—41no—
218C22×C3×C4—41no—
219C22×C3×C4—41no—
220C22×C3×C4—41no—
221C22×C3×C4—41no—
222C22×C3×C4—41no—
223C22×C3×C4—42no—
224C22×C3×C4—42no—
225C22×C3×C4—41no—
226C22×C3×C4—41no—
227C22×C3×C4—41no—
228C22×C3×C4—41no—
229C22×C3×C4—416no—
230C22×C3×C4C2×Q814yes—
231C22×C3×C4C2×Q814yes—
232C22×C3×C4#4114yes—
233C22×C3×C4#4114yes—
234C22×C3×C4C2412yes—
235C22×C3×C4C2412yes—
236C22×C3×C4C2412yes—
237C22×C3×C4C2412yes—
238C22×C3×C4C2412yes—
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2411yescharacteristic
241C22×Dic3C2414yes—
242C22×Dic3C2414yes—
243C22×Dic3C2414yes—
244C22×Dic3C2414yes—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—42no—
248C2×Dic6—42no—
249C2×Dic6—42no—
250C2×Dic6—42no—
251C3×C22⋊C4—28no—
252C3×C22⋊C4C2×Q8116yes—
253C3×C42—24no—
254C3×C42—24no—
255C3×C42—42no—
256C3×C42—42no—
257#217—28no—
258#217—28no—
259#217—44no—
260#217—44no—
261#217—44no—
262#217—44no—
263C4⋊Dic3—24no—
264C4⋊Dic3—24no—
265C4⋊Dic3—42no—
266C4⋊Dic3—42no—
267C4⋊Dic3—42no—
268C4⋊Dic3—42no—
269C4⋊Dic3—42no—
270C4⋊Dic3—42no—
271C4⋊Dic3—42no—
272C4⋊Dic3—42no—
273Dic3⋊C4—28no—
274Dic3⋊C4—28no—
275Dic3⋊C4—28no—
276Dic3⋊C4—28no—
277Dic3⋊C4—44no—
278Dic3⋊C4—44no—
279C4×Dic3—44no—
280C4×Dic3—44no—
281C25C22×D311yescharacteristic
282C22×Q8—32no—
283C22×Q8—32no—
284C23×C4—24no—
285C23×C4—42no—
286C23×C4—42no—
287C23×C4C22×D311yescharacteristic
288C23×C4C22×D312yes—
289C23×C4C22×D312yes—
290C23×C4C22×D311yescharacteristic
291C23×C4C22×D311yescharacteristic
292C23×C4C22×D311yescharacteristic
293C23×C4C22×D311yescharacteristic
294C23×C4C22×D311yescharacteristic
295#125—124no—
296C22⋊Q8—68no—
297C22⋊Q8—68no—
298C22⋊Q8—124no—
299C22⋊Q8—124no—
300C4×Q8—124no—
301C2×C4⋊C4—34no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—34no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—34no—
319C2×C22⋊C4—34no—
320C2×C22⋊C4—34no—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4Dic618yescomplemented
325C2×C22⋊C4C22×D314yes—
326C2×C22⋊C4C22×D314yes—
327C2×C42—34no—
328C2×C42—34no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42C22×D312yes—
332C2×C42C22×D312yes—
333C2×C42C22×D312yes—
334C2×C42C22×D312yes—
335#95—34no—
336#95—34no—
337#95—34no—
338#95—34no—
339#95—34no—
340#95—34no—
341#95—34no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95C22×D314yes—
346#95C22×D314yes—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—28no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—44no—
359C23×C3—44no—
360C23×C3—44no—
361C23×C3—44no—
362C23×C3—44no—
363C23×C3—44no—
364C23×C3C22×Q814yes—
365C23×C3C22×Q812yes—
366C23×C3C2×#4112yes—
367C23×C3C2×#4112yes—
368C23×C3C2511yescharacteristic
369C2×C3×C4—28no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—44no—
386C2×C3×C4—44no—
387C2×C3×C4—44no—
388C2×C3×C4—44no—
389C2×C3×C4—48no—
390C2×C3×C4—48no—
391C2×C3×C4—44no—
392C2×C3×C4C22×Q814yes—
393C2×C3×C4C22×Q814yes—
394C2×C3×C4C2×#4114yes—
395C2×C3×C4C2×#4114yes—
396C2×C3×C4H(5,𝔽2)12yes—
397C2×C3×C4H(5,𝔽2)12yes—
398C2×Dic3—24no—
399C2×Dic3—24no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3—24no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—24no—
407C2×Dic3—24no—
408C2×Dic3—24no—
409C2×Dic3—24no—
410Dic6—48no—
411Dic6—84no—
412C24—22no—
413C24—22no—
414C24—41no—
415C24—41no—
416C24—41no—
417C24—41no—
418C24C2×Dic614yes—
419C24C23×D312yes—
420C24C23×D311yescharacteristic
421C2×Q8—64no—
422C2×Q8—64no—
423C2×Q8—122no—
424C2×Q8—122no—
425C2×Q8—122no—
426C2×Q8—122no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—34no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—34no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—416no—
452C22×C4C2×Dic614yes—
453C22×C4C2×Dic614yes—
454C22×C4#23514yes—
455C22×C4#23514yes—
456C22×C4C23×D312yes—
457C22×C4C23×D312yes—
458C22×C4C23×D312yes—
459C22×C4C23×D312yes—
460C22×C4C23×D312yes—
461C22×C4C23×D311yescharacteristic
462C22×C4C23×D311yescharacteristic
463C4⋊C4—68no—
464C4⋊C4—68no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—68no—
468C4⋊C4—68no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—124no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—124no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C22⋊C4—28no—
480C22⋊C4—68no—
481C22⋊C4—68no—
482C22⋊C4—124no—
483C22⋊C4—124no—
484C22⋊C4—124no—
485C22⋊C4—124no—
486C22⋊C4C2×Dic6116yes—
487C42—24no—
488C42—24no—
489C42—42no—
490C42—42no—
491C42—124no—
492C42—124no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—24no—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—416no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3#53314yes—
508C22×C3C23×Q811yescharacteristic
509C22×C3C22×#4111yescharacteristic
510C22×C3C22×#4111yescharacteristic
511C22×C3C2×H(5,𝔽2)11yescharacteristic
512C22×C3C2×H(5,𝔽2)11yescharacteristic
513C22×C3#58511yescharacteristic
514C22×C3#58511yescharacteristic
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—24no—
520C3×C4—24no—
521C3×C4—42no—
522Dic3—44no—
523Dic3—44no—
524Dic3—44no—
525Dic3—44no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—28no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C22×Dic614yes—
544C23C22×Dic612yes—
545C23C2×#23512yes—
546C23C2×#23512yes—
547C23C24×D311yescharacteristic
548Q8—128no—
549Q8—244no—
550C2×C4—28no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4C22×Dic614yes—
586C2×C4C22×Dic614yes—
587C2×C4C2×#23514yes—
588C2×C4C2×#23514yes—
589C2×C4#100412yes—
590C2×C4#100412yes—
591C2×C3—24no—
592C2×C3—44no—
593C2×C3C2×#53311yescharacteristic
594C2×C3#345911yescharacteristic
595C2×C3#346011yescharacteristic
596C2×C3#349011yescharacteristic
597C2×C3#349311yescharacteristic
598C2×C3#350111yescharacteristic
599C2×C3#350111yescharacteristic
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—416no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22#547014yes—
615C22C23×Dic611yescharacteristic
616C22C22×#23511yescharacteristic
617C22C22×#23511yescharacteristic
618C22C2×#100411yescharacteristic
619C22C2×#100411yescharacteristic
620C22#596411yescharacteristic
621C22#596411yescharacteristic
622C4—24no—
623C4—24no—
624C4—24no—
625C4—22no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—124no—
630C4—124no—
631C4—124no—
632C4—124no—
633C3#3501211yescharacteristic, complemented, Sylow
634C2—24no—
635C2—44no—
636C2C2×#547011yescharacteristic
637C2#8569711yescharacteristic
638C2#8574611yescharacteristic
639C2#8576611yescharacteristic
640C2#8576611yescharacteristic
641C2#8577611yescharacteristic
642C2#8577711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×C4×C22⋊C4C2
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#35012
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. #82074C2semidirect
6. C4×#924C2semidirect
7. #73542C2semidirect
8. #73536C2semidirect
9. #73528C2semidirect
10. #73498C2semidirect
11. #73361C2semidirect
12. #73361C2semidirect
13. #73334C2semidirect
24. C2×#870C22semidirect
25. C2×C4×Dic6C22semidirect
50. #4936C22semidirect
51. C4×C4⋊Dic3C22semidirect
52. #4933C22semidirect
53. #4933C22semidirect
54. #4933C22semidirect
55. C4×Dic3⋊C4C22semidirect
56. #4929C22semidirect
57. #4928C22semidirect
58. #4928C22semidirect
59. #4928C22semidirect
60. #4661C22semidirect
61. #4661C22semidirect
62. #4656C22semidirect
63. #4656C22semidirect
64. #4655C22semidirect
65. #4654C22semidirect
66. #4647C22semidirect
67. #4646C22semidirect
96. C2×C3×C22⋊C4Q8semidirect
324. C2×C22⋊C4Dic6semidirect
633. C3#35012semidirect

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
#840312
#821172
#820932
#820742
C4×#9242
#735422
#735362
#735282
#734982
#733612
#733342
#350123