Finite Group Databasebeta

← #72258708

Subgroups — #72258708

Order 1536.

1189 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 1189 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(#126⋊C3)C212yesmaximal, complemented
2C2×C4⋊(#126⋊C3)C211yesmaximal, characteristic, complemented
3C2×C4×#126⋊C3C211yesmaximal, characteristic, complemented
4#14711⋊C3C212yesmaximal, complemented
5C4⋊C4×C42⋊C3C211yesmaximal, characteristic, complemented
6#616936C311yesmaximal, characteristic, complemented, Sylow
7C22×#126⋊C3C2212yes—
8C22×#126⋊C3C2211yescharacteristic
9A4×#121—41nomaximal
10C4⋊(#126⋊C3)—24no—
11C4⋊(#126⋊C3)—22no—
12C4×#126⋊C3—22no—
13C2×#383⋊C3C2211yescharacteristic
14C2×C4×C42⋊C3C2212yes—
15C2×C4×C42⋊C3C2211yescharacteristic
16C2×#2860C2×C311yescharacteristic, complemented
17C2×#2860C2×C312yescomplemented
18C2×C4×#126C2×C311yescharacteristic, complemented
19#14993—32no—
20#14986—32no—
21#14711C2×C312yescomplemented
22#14046—31no—
23#14022—31no—
24#13998—31no—
25#13974—31no—
26C42×C4⋊C4C2×C311yescharacteristic, complemented
27C2×D4×A4—41no—
28C2×D4×A4—42no—
29C22×C4×A4—41no—
30C2×#126⋊C3—21no—
31C2×#126⋊C3—22no—
32C2×#126⋊C3—22no—
33C2×#126⋊C3—21no—
34C2×#126⋊C3—22no—
35C2×#126⋊C3D412yescomplemented
36A4×C4⋊C4—41no—
37A4×C22⋊C4—42no—
38C22×C42⋊C3C2311yescharacteristic
39#383⋊C3—21no—
40C4×C42⋊C3—22no—
41C4×C42⋊C3D412yescomplemented
42C22×#126C22×C312yes—
43C22×#126C22×C311yescharacteristic
44C22×#121A411yescharacteristic
45#2860—22no—
46#2860—24no—
47#2860—64no—
48#2860—62no—
49C4×#126—22no—
50C4×#126—62no—
51C2×#390—32no—
52C2×#390—31no—
53C2×#390—31no—
54C2×#389—31no—
55C2×#389—32no—
56C2×#389—31no—
57C2×#388—32no—
58C2×#388—32no—
59C2×#388—32no—
60C2×#388—31no—
61C2×#387—31no—
62C2×#383C22×C311yescharacteristic
63C2×#380—31no—
64C2×C4×C4⋊C4—31no—
65C2×C4×C4⋊C4—31no—
66C2×C4×C4⋊C4—31no—
67C2×C4×C22⋊C4—31no—
68C2×C43C22×C312yes—
69C2×C43C22×C311yescharacteristic
70#1443—32no—
71#1442—32no—
72#1441—32no—
73#1440—32no—
74#1440—32no—
75#1439—32no—
76#1439—32no—
77#1438—32no—
78#1437—32no—
79#1437—32no—
80#1436—32no—
81#1436—32no—
82C4⋊#95—31no—
83C4⋊#95—31no—
84C4×#95—32no—
85C23×A4—41no—
86C23×A4—42no—
87D4×A4—82no—
88D4×A4—84no—
89C2×C4×A4—41no—
90C2×C4×A4—41no—
91C2×C4×A4—42no—
92C2×C4×A4—82no—
93C3×#121—161no—
94#126⋊C3—22no—
95#126⋊C3—42no—
96C2×C42⋊C3C2×D411yescharacteristic
97C2×C42⋊C3C2×D411yescharacteristic
98C2×C42⋊C3#4111yescharacteristic
99C23×D4C2×A411yescharacteristic
100C23×D4C2×A412yes—
101C24×C4C2×A411yescharacteristic
102C2×#126—21no—
103C2×#126—22no—
104C2×#126—22no—
105C2×#126—21no—
106C2×#126—22no—
107C2×#126—32no—
108C2×#126—61no—
109C2×#126—61no—
110C2×#126—62no—
111C2×#126—62no—
112C2×#126—62no—
113C2×#126C3×D412yescomplemented
114C2×#121—61no—
115C2×#121—62no—
116C2×#121—61no—
117C22×C4⋊C4—31no—
118C22×C4⋊C4—31no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4C2×A411yescharacteristic
124C22×C22⋊C4—31no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4C2×A412yes—
132C22×C42—31no—
133C22×C42—31no—
134C22×C42—31no—
135C22×C42—32no—
136C22×C42—32no—
137C22×C42C23×C311yescharacteristic
138#390—62no—
139#390—62no—
140#390—64no—
141#390—64no—
142#390—64no—
143#389—62no—
144#389—62no—
145#389—64no—
146#389—64no—
147#389—61no—
148#389—62no—
149#389—61no—
150#388—62no—
151#388—64no—
152#388—62no—
153#388—64no—
154#388—64no—
155#388—64no—
156#388—64no—
157#388—64no—
158#387—62no—
159#387—62no—
160#383—21no—
161#383—61no—
162#380—64no—
163C4×C4⋊C4—62no—
164C4×C4⋊C4—62no—
165C4×C4⋊C4—61no—
166C4×C4⋊C4—62no—
167C4×C4⋊C4—61no—
168C4×C4⋊C4—64no—
169C4×C22⋊C4—62no—
170C4×C22⋊C4—62no—
171C2×#95—31no—
172C2×#95—32no—
173C2×#95—32no—
174C2×#95—32no—
175C2×#95—31no—
176C2×#95—32no—
177C2×#95—32no—
178C2×#95—31no—
179C2×#95—32no—
180C2×#95—32no—
181C2×#95—31no—
182C2×#95—32no—
183C2×#95—32no—
184C43—22no—
185C43—32no—
186C43—62no—
187C43C3×D412yescomplemented
188C22×A4—42no—
189C22×A4—41no—
190C22×A4—81no—
191C22×A4—81no—
192C22×A4—82no—
193C22×A4—82no—
194C22×A4—82no—
195C2×C3×D4—161no—
196C2×C3×D4—162no—
197C22×C3×C4—161no—
198C4×A4—42no—
199C4×A4—82no—
200C4×A4—81no—
201C3×C4⋊C4—161no—
202C3×C22⋊C4—162no—
203C42⋊C3#12111yescharacteristic, complemented
204C25C22×A412yes—
205C25C22×A411yescharacteristic
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—61no—
209C22×D4—62no—
210C22×D4—61no—
211C22×D4—62no—
212C22×D4—62no—
213C22×D4—122no—
214C23×C4—22no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—32no—
218C23×C4—31no—
219C23×C4—32no—
220C23×C4—31no—
221C23×C4—61no—
222C23×C4—61no—
223C23×C4—62no—
224C23×C4C22×A412yes—
225C23×C4C22×A411yescharacteristic
226C23×C4C22×A411yescharacteristic
227#126—22no—
228#126—42no—
229#126—64no—
230#126—62no—
231#126—62no—
232#126—62no—
233#126—124no—
234#126—122no—
235#126—122no—
236#126—122no—
237#121—41no—
238#121—122no—
239#121—121no—
240#121—122no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—122no—
269C2×C4⋊C4—121no—
270C2×C4⋊C4—121no—
271C2×C22⋊C4—32no—
272C2×C22⋊C4—32no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4—64no—
289C2×C22⋊C4—64no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—64no—
292C2×C22⋊C4—122no—
293C2×C22⋊C4—122no—
294C2×C22⋊C4—122no—
295C2×C22⋊C4—122no—
296C2×C22⋊C4—121no—
297C2×C22⋊C4—121no—
298C2×C22⋊C4—122no—
299C2×C22⋊C4—122no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—32no—
304C2×C42—32no—
305C2×C42—32no—
306C2×C42—32no—
307C2×C42—61no—
308C2×C42—62no—
309C2×C42—61no—
310C2×C42—62no—
311C2×C42—62no—
312C2×C42—62no—
313C2×C42—62no—
314C2×C42—62no—
315C2×C42—62no—
316C2×C42—64no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42—62no—
320C2×C42—62no—
321C2×C42—121no—
322C2×C42—121no—
323C2×C42—122no—
324C2×C42—122no—
325C2×C42C2×C3×D411yescharacteristic
326C2×C42C2×C3×D411yescharacteristic
327C2×C42C3×#4111yescharacteristic
328#95—61no—
329#95—61no—
330#95—62no—
331#95—61no—
332#95—61no—
333#95—62no—
334#95—62no—
335#95—64no—
336#95—62no—
337#95—64no—
338#95—64no—
339#95—64no—
340#95—64no—
341#95—62no—
342#95—62no—
343#95—122no—
344#95—122no—
345C23×C3—161no—
346C23×C3—162no—
347C2×A4—41no—
348C2×A4—41no—
349C2×A4—41no—
350C2×A4—82no—
351C2×A4—162no—
352C3×D4—322no—
353C3×D4—324no—
354C2×C3×C4—161no—
355C2×C3×C4—161no—
356C2×C3×C4—162no—
357C2×C3×C4—322no—
358C24—21no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—61no—
364C24—62no—
365C24—61no—
366C24—61no—
367C24—61no—
368C24—62no—
369C24—62no—
370C24—62no—
371C24D4×A412yes—
372C24C23×A411yescharacteristic
373C2×D4—42no—
374C2×D4—41no—
375C2×D4—122no—
376C2×D4—122no—
377C2×D4—121no—
378C2×D4—122no—
379C2×D4—122no—
380C2×D4—122no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—242no—
384C2×D4—242no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—32no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—32no—
397C22×C4—32no—
398C22×C4—34no—
399C22×C4—32no—
400C22×C4—32no—
401C22×C4—41no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—121no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4D4×A412yes—
447C4⋊C4—32no—
448C4⋊C4—32no—
449C4⋊C4—31no—
450C4⋊C4—62no—
451C4⋊C4—62no—
452C4⋊C4—64no—
453C4⋊C4—64no—
454C4⋊C4—62no—
455C4⋊C4—62no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—124no—
463C4⋊C4—124no—
464C4⋊C4—124no—
465C4⋊C4#126⋊C311yescharacteristic, complemented
466C22⋊C4—42no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C22⋊C4—122no—
474C22⋊C4—122no—
475C22⋊C4—122no—
476C22⋊C4—128no—
477C22⋊C4—128no—
478C22⋊C4—128no—
479C22⋊C4—124no—
480C22⋊C4—244no—
481C42—31no—
482C42—31no—
483C42—31no—
484C42—32no—
485C42—62no—
486C42—62no—
487C42—62no—
488C42—62no—
489C42—122no—
490C42—122no—
491C42—122no—
492C42—122no—
493C42—122no—
494C42—122no—
495C42—122no—
496C42—122no—
497C42—124no—
498C42—124no—
499C42C3×#12111yescharacteristic, complemented
500C22×C3—162no—
501C22×C3—161no—
502C22×C3—321no—
503C22×C3—321no—
504C22×C3—322no—
505C22×C3—322no—
506C22×C3—322no—
507A4—41no—
508C3×C4—162no—
509C3×C4—322no—
510C3×C4—321no—
511C23—22no—
512C23—31no—
513C23—31no—
514C23—31no—
515C23—31no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—62no—
520C23—62no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—122no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—122no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—122no—
541C23C2×D4×A411yescharacteristic
542C23C2×D4×A411yescharacteristic
543C23A4×#4111yescharacteristic
544D4—84no—
545D4—82no—
546D4—242no—
547D4—244no—
548C2×C4—32no—
549C2×C4—31no—
550C2×C4—32no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—32no—
558C2×C4—41no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—82no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—122no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—122no—
599C2×C4—121no—
600C2×C4—122no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—122no—
611C2×C4—121no—
612C2×C4—122no—
613C2×C4—242no—
614C2×C4C2×#126⋊C312yes—
615C2×C4C2×#126⋊C311yescharacteristic
616C2×C3—161no—
617C2×C3—161no—
618C2×C3—161no—
619C2×C3—322no—
620C2×C3—642no—
621C22—31no—
622C22—31no—
623C22—31no—
624C22—31no—
625C22—31no—
626C22—31no—
627C22—31no—
628C22—31no—
629C22—31no—
630C22—31no—
631C22—31no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—122no—
639C22—122no—
640C22—241no—
641C22—241no—
642C22—242no—
643C22—242no—
644C22—242no—
645C22—242no—
646C22A4×#12111yescharacteristic
647C22C22×#126⋊C311yescharacteristic
648C4—22no—
649C4—32no—
650C4—61no—
651C4—61no—
652C4—61no—
653C4—61no—
654C4—62no—
655C4—81no—
656C4—122no—
657C4—124no—
658C4—122no—
659C4—242no—
660C4—241no—
661C4C4⋊(#126⋊C3)12yescomplemented
662C3—161noSylow
663C2—31no—
664C2—31no—
665C2—31no—
666C2—31no—
667C2—82no—
668C2—162no—
669C2D4×#126⋊C311yescharacteristic
670C2#1171001011yescharacteristic
671C2C2×C4⋊(#126⋊C3)11yescharacteristic
subgroupG/H
centerC22C22×#126⋊C3
commutator, cosocleC22×C42C23×C3
Frattini, socleC24C23×A4
Fitting#616936C3
hypercenterC4⋊C4#126⋊C3
hypocenterC42C3×#121
LCSG ▷ C22×C42 ▷ C42
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(#126⋊C3)C2semidirect
2. C2×C4⋊(#126⋊C3)C2semidirect
3. C2×C4×#126⋊C3C2semidirect
4. #14711⋊C3C2semidirect
5. C4⋊C4×C42⋊C3C2semidirect
6. #616936C3semidirect
16. C2×#2860C2×C3semidirect
17. C2×#2860C2×C3semidirect
18. C2×C4×#126C2×C3semidirect
21. #14711C2×C3semidirect
26. C42×C4⋊C4C2×C3semidirect
35. C2×#126⋊C3D4semidirect
41. C4×C42⋊C3D4semidirect
113. C2×#126C3×D4semidirect
187. C43C3×D4semidirect
203. C42⋊C3#121semidirect
465. C4⋊C4#126⋊C3semidirect
499. C42C3×#121semidirect
661. C4C4⋊(#126⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(#126⋊C3)2
C2×C4×#126⋊C32
#14711⋊C32
C4⋊C4×C42⋊C32
#6169363
A4×#1214