Finite Group Databasebeta

← #11069466

Subgroups — #11069466

Order 768.

2354 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5562C212yesmaximal, complemented
2C3×#2349C211yesmaximal, characteristic, complemented
3#82416C211yesmaximal, characteristic, complemented
4#82406C212yesmaximal, complemented
5C4×#940C211yesmaximal, characteristic, complemented
6#76251C212yesmaximal, complemented
7#76215C212yesmaximal, complemented
8#74538C214yesmaximal, complemented
9#74511C214yesmaximal, complemented
10#74239C211yesmaximal, characteristic, complemented
11#74220C211yesmaximal, characteristic, complemented
12#74022C214yesmaximal, complemented
13#73770C211yesmaximal, characteristic, complemented
14#73617C211yesmaximal, characteristic, complemented
15#73349C214yesmaximal, complemented
16#39372—31nomaximal, Sylow
17C2×C3×C4×D4C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C2×#940C2211yescharacteristic
20C2×#938C2212yes—
21C2×#930C2214yes—
22C2×#929C2214yes—
23C2×#929C2214yes—
24C2×C4×#67C2214yes—
25#5562—216no—
26C2×#895C2212yes—
27C2×#894C2214yes—
28C2×#890C2212yes—
29C2×#884C2214yes—
30C2×#883C2214yes—
31C2×C4×D12C2212yes—
32C2×C4×D12C2211yescharacteristic
33C2×C4×D12C2211yescharacteristic
34C2×#872C2212yes—
35C3×#389C2212yes—
36C3×#385C2214yes—
37C3×#382C2211yescharacteristic
38C3×#380C2212yes—
39C3×C4×C4⋊C4C2211yescharacteristic
40C2×C3×#95C2212yes—
41#5221C2211yescharacteristic
42#5213C2214yes—
43#5210C2214yes—
44#4994C2214yes—
45#4989C2214yes—
46#4989C2214yes—
47#4988C2212yes—
48#4988C2211yescharacteristic
49#4987C2212yes—
50#4987C2212yes—
51#4983C2214yes—
52#4975C2211yescharacteristic
53#4975C2212yes—
54#4972C2212yes—
55#4960C2212yes—
56#4959C2214yes—
57#4957C2214yes—
58#4953C2212yes—
59#4952C2214yes—
60C4×(C2×D3)⋊C4C2212yes—
61#4936C2211yescharacteristic
62C42×Dic3C2211yescharacteristic
63#4928C2212yes—
64#4676C2214yes—
65#4675C2214yes—
66#4674C2212yes—
67#4672C2214yes—
68#4671C2212yes—
69#4669C2212yes—
70#4669C2212yes—
71#4667C2212yes—
72#4665C2214yes—
73D3×#95C2212yes—
74#4657C2214yes—
75#4647C2214yes—
76#4647C2214yes—
77C2×#540—32no—
78#2859—31no—
79#2724—34no—
80#2682—34no—
81#2629—32no—
82#2535—31no—
83#2535—31no—
84#2425—34no—
85#2425—34no—
86#2422—32no—
87#2422—32no—
88#2349—31no—
89#2349D311yescharacteristic, complemented
90C4×#127—31no—
91#2293—31no—
92C22×C3×D4C2311yescharacteristic
93C23×C3×C4C2314yes—
94C23×C3×C4C2312yes—
95C22×#67C2314yes—
96C22×#67C2314yes—
97C22×D12C2311yescharacteristic
98C22×D12C2314yes—
99C22×D12C2311yescharacteristic
100C22×C4×D3C2312yes—
101C22×C4×D3C2312yes—
102C22×C4×D3C2314yes—
103C3×C4×D4—28no—
104C3×C4×D4—24no—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C42C2312yes—
112C2×C3×C42C2311yescharacteristic
113C2×#217C2314yes—
114C2×#217C2314yes—
115#940D418yescomplemented
116#938—28no—
117#930—216no—
118#929—216no—
119#929—48no—
120C4×#67—48no—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2311yescharacteristic
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2311yescharacteristic
147#895—28no—
148#894—48no—
149#890—28no—
150#884—216no—
151#883—216no—
152C4×D12—28no—
153C4×D12—24no—
154C4×D12—24no—
155#872—28no—
156C3×#95C2312yes—
157C3×#95C2312yes—
158C3×#95C2314yes—
159C3×#95C2311yescharacteristic
160C3×#95C2311yescharacteristic
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2312yes—
167C3⋊#95C2312yes—
168#540—616no—
169C2×#127—31no—
170C2×#125—32no—
171C2×#123—34no—
172C2×#123—34no—
173C2×#123—34no—
174C2×#123—34no—
175C2×#121—32no—
176C2×#121—34no—
177C2×#121—32no—
178C2×#121—34no—
179C2×C4×D4—34no—
180C2×C4×D4—31no—
181C2×C4×D4—32no—
182C2×C4×D4—31no—
183C2×C4×D4C2×D311yescharacteristic
184C2×C4×D4C2×D312yes—
185C2×#117—32no—
186#398—34no—
187#398—34no—
188#397—34no—
189#397—32no—
190#397—32no—
191#397—34no—
192#396—34no—
193#396—34no—
194#394—34no—
195#394—34no—
196#392—32no—
197#392—32no—
198#390—31no—
199#390—31no—
200#390—32no—
201#390—34no—
202#389—32no—
203#389—32no—
204#389—34no—
205#389—34no—
206#389C2×D312yes—
207#388—32no—
208#388—32no—
209#386—34no—
210#386—34no—
211#385—34no—
212#385—34no—
213#385—34no—
214#385C2×D314yes—
215#384—31no—
216#384—32no—
217#382—31no—
218#382C2×D311yescharacteristic
219#380—32no—
220#380—32no—
221#380—32no—
222#380C2×D312yes—
223C4×C4⋊C4C2×D311yescharacteristic
224C4×C22⋊C4—32no—
225C2×#95—32no—
226C2×#95C2×D312yes—
227C43—31no—
228C24×C3C2412yes—
229C23×D3C2412yes—
230C23×D3C2412yes—
231C2×C3×D4—22no—
232C2×C3×D4—22no—
233C2×C3×D4C2×D414yes—
234C22×C3×C4—28no—
235C22×C3×C4—24no—
236C22×C3×C4—44no—
237C22×C3×C4—44no—
238C22×C3×C4—44no—
239C22×C3×C4C2412yes—
240C22×C3×C4C2412yes—
241C22×C3×C4C2414yes—
242C22×C3×C4C2412yes—
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C2×#67—28no—
247C2×#67—44no—
248C2×#67—44no—
249C2×#67—44no—
250C2×#67—44no—
251C2×#67C2×D4116yes—
252C22×Dic3C2412yes—
253C22×Dic3C2414yes—
254C22×Dic3C2412yes—
255C22×Dic3C2412yes—
256C22×Dic3C2412yes—
257C2×D12—22no—
258C2×D12—22no—
259C2×D12—22no—
260C2×D12—28no—
261C2×D12—42no—
262C2×D12—44no—
263C2×D12—44no—
264C2×D12C2×D414yes—
265C2×C4×D3—24no—
266C2×C4×D3—24no—
267C2×C4×D3—24no—
268C2×C4×D3—28no—
269C2×C4×D3—24no—
270C2×C4×D3—44no—
271C2×C4×D3—44no—
272C2×C4×D3—44no—
273C3×C4⋊C4—24no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—44no—
276C3×C4⋊C4—42no—
277C3×C22⋊C4—28no—
278C3×C22⋊C4—24no—
279C3×C42—24no—
280C3×C42#4114yes—
281#217—28no—
282#217—28no—
283(C2×D3)⋊C4—28no—
284(C2×D3)⋊C4—28no—
285(C2×D3)⋊C4—24no—
286(C2×D3)⋊C4—24no—
287(C2×D3)⋊C4—24no—
288(C2×D3)⋊C4—24no—
289(C2×D3)⋊C4—24no—
290(C2×D3)⋊C4—28no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—44no—
293(C2×D3)⋊C4—44no—
294C4⋊Dic3—24no—
295C4⋊Dic3—22no—
296C4⋊Dic3—22no—
297C4⋊Dic3—28no—
298Dic3⋊C4—24no—
299Dic3⋊C4—24no—
300Dic3⋊C4—28no—
301Dic3⋊C4—28no—
302Dic3⋊C4—44no—
303Dic3⋊C4—44no—
304Dic3⋊C4—44no—
305C4×Dic3—22no—
306C4×Dic3—24no—
307C4×Dic3—42no—
308C4×Dic3C2×D414yes—
309C22×D4—34no—
310C22×D4—34no—
311C22×D4—31no—
312C22×D4—31no—
313C22×D4—34no—
314C22×D4C22×D311yescharacteristic
315C23×C4—32no—
316C23×C4—32no—
317C23×C4—34no—
318C23×C4C22×D312yes—
319C23×C4C22×D314yes—
320#127—38no—
321#125—68no—
322#123—616no—
323#123—616no—
324#123—128no—
325#123—128no—
326#121—616no—
327#121—68no—
328#121—616no—
329#121—68no—
330C4×D4—28no—
331C4×D4—24no—
332C4×D4—68no—
333C4×D4—64no—
334C4×D4—64no—
335C4×D4—128no—
336#117—68no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4C22×D312yes—
349C2×C4⋊C4C22×D311yescharacteristic
350C2×C4⋊C4C22×D314yes—
351C2×C4⋊C4C22×D312yes—
352C2×C22⋊C4—34no—
353C2×C22⋊C4—34no—
354C2×C22⋊C4—32no—
355C2×C22⋊C4—32no—
356C2×C22⋊C4—34no—
357C2×C22⋊C4—34no—
358C2×C22⋊C4—32no—
359C2×C22⋊C4—32no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4—32no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4C22×D314yes—
366C2×C22⋊C4C22×D312yes—
367C2×C42—31no—
368C2×C42—32no—
369C2×C42—31no—
370C2×C42—32no—
371C2×C42C22×D312yes—
372C2×C42C22×D311yescharacteristic
373#95—32no—
374#95—34no—
375#95—32no—
376#95—32no—
377#95—34no—
378#95—34no—
379#95—32no—
380#95C22×D311yescharacteristic
381#95C22×D311yescharacteristic
382#95C22×D312yes—
383#95C22×D314yes—
384#95C22×D312yes—
385C23×C3—22no—
386C23×C3—24no—
387C23×C3—22no—
388C23×C3—24no—
389C23×C3C22×D414yes—
390C23×C3C2511yescharacteristic
391C22×D3—22no—
392C22×D3—24no—
393C22×D3—22no—
394C22×D3—24no—
395C22×D3—22no—
396C22×D3—22no—
397C22×D3—22no—
398C22×D3—22no—
399C22×D3—22no—
400C22×D3—22no—
401C22×D3—22no—
402C22×D3C22×D414yes—
403C3×D4—28no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—24no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—24no—
411C2×C3×C4—22no—
412C2×C3×C4—24no—
413C2×C3×C4—22no—
414C2×C3×C4—24no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—84no—
418C2×C3×C4—88no—
419C2×C3×C4C22×D412yes—
420C2×C3×C4C2×#4112yes—
421C2×C3×C4C2×#4112yes—
422C2×C3×C4C2×#4112yes—
423C2×C3×C42−1+412yes—
424#67—416no—
425#67—88no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—24no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—24no—
435C2×Dic3—24no—
436C2×Dic3—22no—
437C2×Dic3—24no—
438C2×Dic3C22×D414yes—
439C2×Dic3H(5,𝔽2)14yes—
440D12—28no—
441D12—44no—
442D12—88no—
443C4×D3—48no—
444C4×D3—84no—
445C4×D3—88no—
446C24—32no—
447C24—32no—
448C24C23×D312yes—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—316no—
452C2×D4—34no—
453C2×D4—68no—
454C2×D4—68no—
455C2×D4—62no—
456C2×D4—62no—
457C2×D4—62no—
458C2×D4—122no—
459C2×D4—124no—
460C2×D4—124no—
461C2×D4—124no—
462C2×D4—124no—
463C2×D4—124no—
464C2×D4—124no—
465C2×D4D3×D414yes—
466C22×C4—28no—
467C22×C4—24no—
468C22×C4—32no—
469C22×C4—32no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—34no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—68no—
477C22×C4—64no—
478C22×C4—64no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4—124no—
484C22×C4C23×D312yes—
485C22×C4C23×D312yes—
486C22×C4C23×D312yes—
487C22×C4C23×D314yes—
488C22×C4C23×D311yescharacteristic
489C22×C4C23×D311yescharacteristic
490C22×C4C23×D311yescharacteristic
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—68no—
498C4⋊C4—68no—
499C4⋊C4—68no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—64no—
503C4⋊C4—124no—
504C4⋊C4—124no—
505C4⋊C4—124no—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—64no—
509C22⋊C4—64no—
510C22⋊C4—68no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—68no—
514C22⋊C4—68no—
515C22⋊C4—68no—
516C22⋊C4—68no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—124no—
520C22⋊C4—124no—
521C42—24no—
522C42—34no—
523C42—64no—
524C42—62no—
525C42—122no—
526C42#23514yes—
527C22×C3—24no—
528C22×C3—28no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—44no—
532C22×C3C23×D411yescharacteristic
533C22×C3C22×#4111yescharacteristic
534C22×C3C2×H(5,𝔽2)12yes—
535C22×C3C2×2−1+411yescharacteristic
536C22×C3#58512yes—
537C2×D3—24no—
538C2×D3—28no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—44no—
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—42no—
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—42no—
552C3×C4—44no—
553C3×C4—42no—
554C3×C4—42no—
555C3×C4#54714yes—
556Dic3—24no—
557Dic3—24no—
558Dic3—42no—
559Dic3—44no—
560Dic3—42no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—24no—
565C23—34no—
566C23—62no—
567C23—62no—
568C23—64no—
569C23—64no—
570C23—62no—
571C23—62no—
572C23—62no—
573C23—62no—
574C23—62no—
575C23—62no—
576C23—62no—
577C23C2×D3×D414yes—
578C23C24×D311yescharacteristic
579D4—28no—
580D4—68no—
581D4—124no—
582D4—1216no—
583D4—248no—
584D4—248no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—34no—
599C2×C4—34no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4—128no—
615C2×C4—244no—
616C2×C4—248no—
617C2×C4C2×#23512yes—
618C2×C4C2×#23512yes—
619C2×C4C2×#23512yes—
620C2×C4C2×D3×D412yes—
621C2×C4#100712yes—
622C2×C3—44no—
623C2×C3#345511yescharacteristic
624C2×C3C2×#54711yescharacteristic
625C2×C3#349612yes—
626C2×C3#350811yescharacteristic
627C2×C3#355912yes—
628D3—44no—
629D3—82no—
630C22—28no—
631C22—24no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—68no—
636C22—64no—
637C22—122no—
638C22—122no—
639C22—124no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—122no—
645C22—122no—
646C22—122no—
647C22C22×#23511yescharacteristic
648C22C22×D3×D411yescharacteristic
649C22C2×#100711yescharacteristic
650C22#596412yes—
651C22D3×H(5,𝔽2)12yes—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—64no—
659C4—64no—
660C4—122no—
661C4—124no—
662C4—122no—
663C4#550714yes—
664C3#3937211yescharacteristic, complemented, Sylow
665C2—44no—
666C2—124no—
667C2—242no—
668C2C2×#550711yescharacteristic
669C2#8583711yescharacteristic
670C2#8586211yescharacteristic
671C2#8666512yes—
672C2#8695712yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2349C2
hypercenter#2349D3
hypocenterC3#39372
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C2×#5562C2semidirect
2. C3×#2349C2semidirect
3. #82416C2semidirect
4. #82406C2semidirect
5. C4×#940C2semidirect
6. #76251C2semidirect
7. #76215C2semidirect
8. #74538C2semidirect
9. #74511C2semidirect
10. #74239C2semidirect
11. #74220C2semidirect
12. #74022C2semidirect
13. #73770C2semidirect
14. #73617C2semidirect
15. #73349C2semidirect
89. #2349D3semidirect
115. #940D4semidirect
664. C3#39372semidirect

Maximal subgroups

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

subgroupindex
C2×#55622
C3×#23492
#824162
#824062
C4×#9402
#762512
#762152
#745382
#745112
#742392
#742202
#740222
#737702
#736172
#733492
#393723