Finite Group Databasebeta

← #10804448

Subgroups — #10804448

Order 768.

1538 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5605C211yesmaximal, characteristic
2#74004C211yesmaximal, characteristic, complemented
3#73567C211yesmaximal, characteristic, complemented
4C3×#1964C211yesmaximal, characteristic, complemented
5#71686C211yesmaximal, characteristic, complemented
6#70035C214yesmaximal, complemented
7#70030C214yesmaximal, complemented
8#69528C211yesmaximal, characteristic, complemented
9#69456C211yesmaximal, characteristic, complemented
10#21795—31nomaximal, Sylow
11C22×D3×D4C2212yes—
12C2×C3×#121C2211yescharacteristic
13C2×#939C2212yes—
14C2×#937C412yescomplemented
15C2×#930C2211yescharacteristic
16C22×(C2×D3)⋊C4C2211yescharacteristic
17#5605—22no—
18#5605—22no—
19#5605—22no—
20#5605—22no—
21C2×#894C412yescomplemented
22C2×#883C412yescomplemented
23C2×#882C412yescomplemented
24C2×C3×#99C2214yescomplemented
25C3×#380C2211yescharacteristic, complemented
26C3×C4×C22⋊C4C2211yescharacteristic, complemented
27C2×#834C2214yescomplemented
28#5212C2211yescharacteristic, complemented
29C4×#217C2211yescharacteristic, complemented
30#4990—22no—
31#4955C2212yes—
32#4955C2212yes—
33C2×#806C2214yescomplemented
34C2×#806C2214yescomplemented
35#4942C2212yescomplemented
36Dic3×C22⋊C4C2212yescomplemented
37#4940—22no—
38#4670—22no—
39#4669—22no—
40C2×#534—31no—
41#2312—31no—
42#2312—31no—
43#1964—34no—
44#1964—31no—
45#1964—31no—
46#1964—31no—
47#1964D311yescharacteristic, complemented
48#1781—34no—
49C24×D3C2×C412yes—
50C24×D3C2311yescharacteristic
51C22×C3×D4C2×C411yescharacteristic
52C22×C3×D4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C22×#67C2×C412yes—
55C22×#67C2×C412yes—
56C22×#67C2312yes—
57C23×Dic3C2311yescharacteristic
58C2×D3×D4—24no—
59C2×D3×D4—24no—
60C2×D3×D4—44no—
61C2×D3×D4—44no—
62C22×D12C2×C412yes—
63C22×D12C2×C411yescharacteristic
64C22×C4×D3C2×C412yes—
65C3×#121—21no—
66C3×#121—21no—
67C3×#121—21no—
68C3×#121—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2×C411yescharacteristic
71C2×C3×C22⋊C4C2×C412yes—
72C2×C3×C22⋊C4C2312yes—
73C2×C3×C22⋊C4C2312yes—
74C2×C3×C42—21no—
75C2×#217C2×C412yes—
76C2×#217C2312yes—
77C2×#217C2312yes—
78#939—24no—
79#939—24no—
80#937—24no—
81#937—24no—
82#930—21no—
83#930—21no—
84#930—21no—
85#930—21no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4C2×C412yes—
91C2×(C2×D3)⋊C4C2×C412yes—
92C2×(C2×D3)⋊C4C2×C412yes—
93C2×(C2×D3)⋊C4C2×C412yes—
94C2×(C2×D3)⋊C4D412yes—
95C2×(C2×D3)⋊C4D412yes—
96C2×(C2×D3)⋊C4D412yes—
97C2×(C2×D3)⋊C4D412yes—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2×C411yescharacteristic
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4C2×C412yes—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—22no—
104#894—24no—
105#894—24no—
106#883—24no—
107#883—24no—
108#882—24no—
109#882—24no—
110C3×#99—44no—
111C3×#95—21no—
112C3×#95—21no—
113#834—44no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#806—44no—
119#806—44no—
120C23×D4—32no—
121#534—62no—
122#534—62no—
123#534—62no—
124#534—62no—
125C2×#123—32no—
126C2×#123—32no—
127C2×#121—32no—
128C2×#121—31no—
129C2×#121C2×D311yescharacteristic
130C2×#120—32no—
131C2×#120—32no—
132C22×C22⋊C4—31no—
133C2×#99—34no—
134C2×#99—34no—
135C2×#99—34no—
136C2×#99C2×D314yescomplemented
137#388—62no—
138#388—62no—
139#388—62no—
140#388—62no—
141#380—31no—
142#380—32no—
143#380C2×D311yescharacteristic, complemented
144#379—32no—
145#379—32no—
146C4×C22⋊C4—31no—
147C4×C22⋊C4—32no—
148C4×C22⋊C4C2×D311yescharacteristic, complemented
149C24×C3C22×C412yes—
150C24×C3C2411yescharacteristic
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—24no—
154C23×D3—44no—
155C23×D3—81no—
156C23×D3—81no—
157C23×D3—82no—
158C23×D3—82no—
159C23×D3C22×C412yes—
160C23×D3C22×C412yes—
161C23×D3C2×D412yes—
162C23×D3C2×D412yes—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—81no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—81no—
173C22×C3×C4C22×C412yes—
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C2×#67—22no—
181C2×#67—22no—
182C2×#67—24no—
183C2×#67—24no—
184C2×#67—24no—
185C2×#67—24no—
186C2×#67—44no—
187C2×#67—82no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—81no—
191C22×Dic3C22×C412yes—
192C22×Dic3C22×C412yes—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195D3×D4—48no—
196D3×D4—48no—
197D3×D4—416no—
198C2×D12—24no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—44no—
202C2×D12—81no—
203C2×C4×D3—24no—
204C2×C4×D3—44no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—81no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—22no—
213C3×C42—81no—
214#217—24no—
215#217—22no—
216#217—24no—
217#217—22no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—82no—
222(C2×D3)⋊C4—82no—
223(C2×D3)⋊C4—82no—
224(C2×D3)⋊C4—82no—
225(C2×D3)⋊C4C22⋊C414yes—
226(C2×D3)⋊C4C22⋊C414yes—
227(C2×D3)⋊C4C22⋊C414yes—
228(C2×D3)⋊C4C22⋊C414yes—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—81no—
232Dic3⋊C4—24no—
233Dic3⋊C4—82no—
234C4×Dic3—82no—
235C4×Dic3—81no—
236C25—31no—
237C25—32no—
238C22×D4—32no—
239C22×D4—32no—
240C22×D4—32no—
241C22×D4—32no—
242C22×D4—31no—
243C22×D4—64no—
244C22×D4—64no—
245C22×D4—124no—
246C22×D4—124no—
247C22×D4C4×D311yescharacteristic
248C22×D4C22×D312yes—
249C23×C4—31no—
250C23×C4—32no—
251C23×C4C22×D311yescharacteristic
252#123—64no—
253#123—64no—
254#123—64no—
255#123—64no—
256#121—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#121—64no—
261#121—64no—
262#121—61no—
263#121—61no—
264#121—61no—
265#121—61no—
266#120—64no—
267#120—64no—
268#120—64no—
269#120—64no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4C4×D311yescharacteristic
276C2×C22⋊C4—32no—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—32no—
279C2×C22⋊C4—32no—
280C2×C22⋊C4—32no—
281C2×C22⋊C4—32no—
282C2×C22⋊C4—32no—
283C2×C22⋊C4—32no—
284C2×C22⋊C4—32no—
285C2×C22⋊C4—32no—
286C2×C22⋊C4—32no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4C4×D312yes—
292C2×C22⋊C4C22×D312yes—
293C2×C22⋊C4C22×D312yes—
294C2×C42—21no—
295C2×C42—62no—
296C2×C42—61no—
297#99—44no—
298#99—124no—
299#99—124no—
300#99—124no—
301#95—21no—
302#95—21no—
303#95—62no—
304#95—62no—
305#95—61no—
306#95—61no—
307C23×C3—22no—
308C23×C3—24no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3C23×C411yescharacteristic
314C23×C3C22×D411yescharacteristic
315C23×C3C22×D411yescharacteristic
316C23×C3H(5,𝔽2)12yes—
317C23×C3H(5,𝔽2)12yes—
318C22×D3—24no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—24no—
322C22×D3—22no—
323C22×D3—24no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C22×D3—48no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—84no—
334C22×D3—84no—
335C22×D3—84no—
336C22×D3—84no—
337C22×D3—816no—
338C22×D3C2×C22⋊C414yes—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—44no—
342C3×D4—81no—
343C3×D4—81no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
359C2×C3×C4C2×C22⋊C411yescharacteristic
360C2×C3×C4C2×C22⋊C411yescharacteristic
361C2×C3×C4C2×C22⋊C411yescharacteristic
362#67—48no—
363#67—42no—
364#67—42no—
365#67—44no—
366#67—84no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3C2×C22⋊C414yes—
379D12—48no—
380D12—81no—
381D12—81no—
382C4×D3—48no—
383C24—32no—
384C24—32no—
385C24—32no—
386C24—32no—
387C24—64no—
388C24—62no—
389C24—62no—
390C24—124no—
391C24—241no—
392C24—241no—
393C24—242no—
394C24—242no—
395C24C2×C4×D312yes—
396C24C23×D311yescharacteristic
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—64no—
404C2×D4—64no—
405C2×D4—64no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—64no—
409C2×D4—64no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—81no—
413C2×D4—124no—
414C2×D4—124no—
415C2×D4—128no—
416C2×D4—128no—
417C2×D4—1216no—
418C2×D4—241no—
419C2×D4—242no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—64no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—81no—
430C22×C4—124no—
431C22×C4—241no—
432C22×C4C2×C4×D312yes—
433C22×C4C2×C4×D311yescharacteristic
434C22×C4C2×C4×D311yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4D3×D411yescharacteristic
438C22×C4D3×D411yescharacteristic
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—64no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—81no—
445C4⋊C4—242no—
446C4⋊C4—241no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—34no—
453C22⋊C4—34no—
454C22⋊C4—34no—
455C22⋊C4—34no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C22⋊C4—62no—
459C22⋊C4—62no—
460C22⋊C4—62no—
461C22⋊C4—64no—
462C22⋊C4—62no—
463C22⋊C4—242no—
464C22⋊C4—242no—
465C22⋊C4—242no—
466C22⋊C4—242no—
467C42—81no—
468C42—241no—
469C42—242no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—84no—
482C22×C3C22×C22⋊C411yescharacteristic
483C22×C3#51812yes—
484C22×C3#53511yescharacteristic
485C22×C3#53511yescharacteristic
486C22×C3#53512yes—
487C2×D3—24no—
488C2×D3—24no—
489C2×D3—24no—
490C2×D3—48no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—82no—
495C2×D3—84no—
496C2×D3—82no—
497C2×D3—816no—
498C2×D3—84no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—82no—
505C3×C4—82no—
506Dic3—24no—
507Dic3—42no—
508Dic3—82no—
509Dic3—82no—
510C23—22no—
511C23—24no—
512C23—34no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—64no—
518C23—62no—
519C23—62no—
520C23—64no—
521C23—62no—
522C23—64no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23—128no—
532C23—244no—
533C23—244no—
534C23—244no—
535C23—244no—
536C23—2416no—
537C23C22×C4×D311yescharacteristic
538C23C2×D3×D411yescharacteristic
539C23C2×D3×D411yescharacteristic
540C23#100412yes—
541C23#100912yes—
542D4—42no—
543D4—42no—
544D4—44no—
545D4—81no—
546D4—81no—
547D4—128no—
548D4—128no—
549D4—122no—
550D4—122no—
551D4—124no—
552D4—241no—
553D4—241no—
554D4—244no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—34no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—128no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4D3×C22⋊C411yescharacteristic
583C2×C4D3×C22⋊C411yescharacteristic
584C2×C4D3×C22⋊C411yescharacteristic
585C2×C4D3×C22⋊C411yescharacteristic
586C2×C3—21no—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—82no—
591C2×C3#230911yescharacteristic
592C2×C3#288311yescharacteristic
593C2×C3#288311yescharacteristic
594D3—44no—
595D3—82no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—64no—
608C22—64no—
609C22—64no—
610C22—84no—
611C22—128no—
612C22—122no—
613C22—122no—
614C22—122no—
615C22—242no—
616C22—244no—
617C22—242no—
618C22—244no—
619C22—2416no—
620C22C2×D3×C22⋊C411yescharacteristic
621C22#554512yes—
622C22#560712yes—
623C22#567411yescharacteristic
624C22#571511yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—64no—
631C4—82no—
632C4—82no—
633C4—122no—
634C4—242no—
635C4—242no—
636C3#2179511yescharacteristic, complemented, Sylow
637C2—21no—
638C2—22no—
639C2—21no—
640C2—22no—
641C2—82no—
642C2—124no—
643C2—242no—
644C2#7642811yescharacteristic
645C2#7868811yescharacteristic
646C2#7869411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1964C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5605C2
cosocleC24×C3C24
hypercenter#1964D3
hypocenterC3#21795
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #74004C2semidirect
3. #73567C2semidirect
4. C3×#1964C2semidirect
5. #71686C2semidirect
6. #70035C2semidirect
7. #70030C2semidirect
8. #69528C2semidirect
9. #69456C2semidirect
14. C2×#937C4semidirect
21. C2×#894C4semidirect
22. C2×#883C4semidirect
23. C2×#882C4semidirect
24. C2×C3×#99C22semidirect
25. C3×#380C22semidirect
26. C3×C4×C22⋊C4C22semidirect
27. C2×#834C22semidirect
28. #5212C22semidirect
29. C4×#217C22semidirect
33. C2×#806C22semidirect
34. C2×#806C22semidirect
35. #4942C22semidirect
36. Dic3×C22⋊C4C22semidirect
47. #1964D3semidirect
136. C2×#99C2×D3semidirect
143. #380C2×D3semidirect
148. C4×C22⋊C4C2×D3semidirect
636. C3#21795semidirect

Maximal subgroups

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

subgroupindex
C2×#56052
#740042
#735672
C3×#19642
#716862
#700352
#700302
#695282
#694562
#217953