Finite Group Databasebeta

← #11069556

Subgroups — #11069556

Order 768.

2370 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5697C211yesmaximal, characteristic, complemented
2C2×#5535C211yesmaximal, characteristic, complemented
3C3×#2629C211yesmaximal, characteristic, complemented
4#83056C211yesmaximal, characteristic, complemented
5#82225C211yesmaximal, characteristic, complemented
6#77979C214yesmaximal, complemented
7#77764C214yesmaximal, complemented
8#77439C214yesmaximal, complemented
9#77284C214yesmaximal, complemented
10#76213C212yesmaximal, complemented
11#76202C212yesmaximal, complemented
12#74344C211yesmaximal, characteristic, complemented
13#74319C211yesmaximal, characteristic, complemented
14#73935C212yesmaximal, complemented
15#73891C212yesmaximal, complemented
16#39378—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#930C2212yes—
20C2×C4×#67C2211yescharacteristic
21#5697—28no—
22#5535—28no—
23C2×#895C2214yes—
24C2×#895C2212yes—
25C2×#894C2214yes—
26C2×#894C2214yes—
27C2×#893C2212yes—
28C2×#893C2212yes—
29C2×#893C2214yes—
30C2×D3×C4⋊C4C2214yes—
31C2×#884C2214yes—
32C2×#884C2214yes—
33C2×#874C2211yescharacteristic
34C2×C4×D12C2212yes—
35C2×#872C2211yescharacteristic
36C2×#870C2211yescharacteristic
37C3×#398C2214yes—
38C3×#396C2214yes—
39C3×#389C2212yes—
40C3×#384C2211yescharacteristic
41C3×#380C2212yes—
42C4×#217C2211yescharacteristic
43#4994C2214yes—
44#4992C2214yes—
45#4991C2214yes—
46#4990C2212yes—
47#4989C2214yes—
48#4988C2214yes—
49#4988C2212yes—
50#4987C2212yes—
51#4984C2212yes—
52#4981C2214yes—
53#4977C2212yes—
54#4976C2212yes—
55#4976C2214yes—
56#4976C2214yes—
57#4975C2212yes—
58#4961C2214yes—
59#4959C2214yes—
60#4957C2212yes—
61#4956C2214yes—
62#4945C2212yes—
63C4×(C2×D3)⋊C4C2211yescharacteristic
64C4×Dic3⋊C4C2211yescharacteristic
65C42×Dic3C2211yescharacteristic
66#4674C2214yes—
67#4672C2214yes—
68#4668C2214yes—
69#4667C2214yes—
70#4667C2214yes—
71#4665C2214yes—
72#4664C2214yes—
73#4664C2214yes—
74C2×#540—31no—
75C2×#540—31no—
76#2859—31no—
77#2701—34no—
78#2701—34no—
79#2629—31no—
80#2629D311yescharacteristic, complemented
81#2537—32no—
82#2486—34no—
83#2420—32no—
84#2420—32no—
85#2416—34no—
86D4×C4⋊C4—32no—
87C4×#127—31no—
88#2293—31no—
89C23×C3×C4C2311yescharacteristic
90C22×#67C2311yescharacteristic
91C22×#67C2314yes—
92C22×D12C2312yes—
93C22×D12C2314yes—
94C22×D12C2314yes—
95C22×C4×D3C2314yes—
96C22×C4×D3C2311yescharacteristic
97C22×C4×D3C2314yes—
98C22×C4×D3C2314yes—
99C3×#125—24no—
100C3×#117—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C22⋊C4C2312yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×#217C2311yescharacteristic
112C2×#217C2312yes—
113#930—28no—
114C4×#67—24no—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2311yescharacteristic
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×C4⋊Dic3C2312yes—
127C2×C4⋊Dic3C2314yes—
128C2×Dic3⋊C4C2311yescharacteristic
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2311yescharacteristic
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2311yescharacteristic
139#895—216no—
140#895—28no—
141#894—216no—
142#894—216no—
143#893—28no—
144#893—28no—
145#893—48no—
146D3×C4⋊C4—48no—
147#884—216no—
148#884—48no—
149#874D418yescomplemented
150C4×D12—28no—
151#872—24no—
152#870—24no—
153C3×#95C2314yes—
154C3×#95C2314yes—
155C3⋊#95C2312yes—
156C3⋊#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162#540—68no—
163#540—68no—
164C2×#127—31no—
165C2×#125—31no—
166C2×#125C2×D311yescharacteristic
167C2×#123—34no—
168C2×#123—34no—
169C2×#121—32no—
170C2×#121—34no—
171C2×#121—32no—
172C2×#121—34no—
173C2×#121—34no—
174C2×C4×D4—31no—
175C2×C4×D4—32no—
176C2×C4×D4—32no—
177C2×C4×D4—34no—
178C2×C4×D4—32no—
179C2×#117—31no—
180C2×#117C2×D311yescharacteristic
181C22×C4⋊C4—34no—
182#398—34no—
183#398C2×D314yes—
184#397—34no—
185#397—34no—
186#396—34no—
187#396C2×D314yes—
188#394—34no—
189#394—34no—
190#394—34no—
191#394—34no—
192#390—32no—
193#390—34no—
194#390—32no—
195#389—34no—
196#389—34no—
197#389—32no—
198#389—32no—
199#389C2×D312yes—
200#388—32no—
201#388—32no—
202#386—34no—
203#386—34no—
204#385—34no—
205#385—34no—
206#384—32no—
207#384C2×D311yescharacteristic
208#382—32no—
209#381—34no—
210#381—34no—
211#380—32no—
212#380—34no—
213#380—34no—
214#380C2×D312yes—
215C4×C4⋊C4—31no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—31no—
218C43—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C23×D3C2414yes—
222C22×C3×C4—22no—
223C22×C3×C4—22no—
224C22×C3×C4—22no—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2414yes—
227C22×C3×C4C2412yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C2×#67—28no—
232C2×#67—22no—
233C2×#67—24no—
234C2×#67—44no—
235C2×#67—44no—
236C22×Dic3C2412yes—
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2412yes—
239C22×Dic3C2414yes—
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2411yescharacteristic
242C2×D12—24no—
243C2×D12—24no—
244C2×D12—28no—
245C2×D12—44no—
246C2×D12—44no—
247C2×D12—44no—
248C2×D12—44no—
249C2×D12C2×D418yes—
250C2×D12C2×D4116yes—
251C2×C4×D3—22no—
252C2×C4×D3—22no—
253C2×C4×D3—22no—
254C2×C4×D3—28no—
255C2×C4×D3—28no—
256C2×C4×D3—28no—
257C2×C4×D3—28no—
258C2×C4×D3—28no—
259C2×C4×D3—44no—
260C2×C4×D3—44no—
261C2×C4×D3—44no—
262C2×C4×D3—44no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—28no—
266C3×C4⋊C4—24no—
267C3×C4⋊C4—44no—
268C3×C4⋊C4—44no—
269C3×C22⋊C4—24no—
270C3×C22⋊C4—44no—
271C3×C42—22no—
272C3×C42C2×D414yes—
273#217—22no—
274#217—42no—
275(C2×D3)⋊C4—28no—
276(C2×D3)⋊C4—28no—
277(C2×D3)⋊C4—28no—
278(C2×D3)⋊C4—28no—
279(C2×D3)⋊C4—24no—
280(C2×D3)⋊C4—28no—
281(C2×D3)⋊C4—22no—
282(C2×D3)⋊C4—44no—
283(C2×D3)⋊C4—44no—
284(C2×D3)⋊C4—44no—
285(C2×D3)⋊C4—42no—
286C4⋊Dic3—24no—
287C4⋊Dic3—44no—
288Dic3⋊C4—28no—
289Dic3⋊C4—24no—
290Dic3⋊C4—22no—
291Dic3⋊C4—28no—
292Dic3⋊C4—24no—
293Dic3⋊C4—44no—
294C4×Dic3—22no—
295C4×Dic3—22no—
296C4×Dic3—24no—
297C4×Dic3—42no—
298C4×Dic3#4114yes—
299C22×D4—31no—
300C22×D4—34no—
301C22×D4—32no—
302C22×D4—34no—
303C22×D4—34no—
304C23×C4—34no—
305C23×C4—31no—
306C23×C4—34no—
307C23×C4—34no—
308C23×C4C22×D311yescharacteristic
309#127—38no—
310#125—24no—
311#125—64no—
312#123—616no—
313#123—616no—
314#121—616no—
315#121—616no—
316#121—68no—
317#121—68no—
318#121—128no—
319C4×D4—68no—
320C4×D4—68no—
321C4×D4—64no—
322C4×D4—68no—
323C4×D4—128no—
324#117—24no—
325#117—64no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—128no—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D312yes—
337C2×C4⋊C4C22×D312yes—
338C2×C4⋊C4C22×D314yes—
339C2×C4⋊C4C22×D314yes—
340C2×C4⋊C4C22×D314yes—
341C2×C22⋊C4—31no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4—34no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4C22×D312yes—
355C2×C22⋊C4C22×D314yes—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—32no—
359C2×C42—31no—
360C2×C42—32no—
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—32no—
366#95—34no—
367#95—34no—
368#95—34no—
369#95—34no—
370#95C22×D314yes—
371#95C22×D314yes—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—22no—
375C23×C3—21no—
376C23×C3—22no—
377C23×C3C2511yescharacteristic
378C22×D3—24no—
379C22×D3—24no—
380C22×D3—24no—
381C22×D3—24no—
382C22×D3—24no—
383C22×D3—24no—
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—22no—
387C22×D3—21no—
388C22×D3—22no—
389C22×D3C22×D418yes—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—24no—
395C2×C3×C4—21no—
396C2×C3×C4—24no—
397C2×C3×C4—24no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—24no—
401C2×C3×C4—21no—
402C2×C3×C4—24no—
403C2×C3×C4—22no—
404C2×C3×C4—44no—
405C2×C3×C4C22×D414yes—
406C2×C3×C4C22×D412yes—
407C2×C3×C4C2×#4112yes—
408C2×C3×C42−1+412yes—
409#67—44no—
410#67—88no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—24no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—24no—
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—22no—
422C2×Dic3—24no—
423C2×Dic3—22no—
424C2×Dic3C2×#4112yes—
425C2×Dic3C2×#4112yes—
426C2×Dic3H(5,𝔽2)14yes—
427D12—216no—
428D12—416no—
429D12—88no—
430C4×D3—44no—
431C4×D3—416no—
432C4×D3—88no—
433C4×D3—88no—
434C24—34no—
435C24—31no—
436C24C23×D311yescharacteristic
437C2×D4—38no—
438C2×D4—316no—
439C2×D4—68no—
440C2×D4—62no—
441C2×D4—64no—
442C2×D4—68no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—124no—
446C2×D4—124no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—124no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—32no—
457C22×C4—31no—
458C22×C4—32no—
459C22×C4—34no—
460C22×C4—68no—
461C22×C4—68no—
462C22×C4—68no—
463C22×C4—68no—
464C22×C4—68no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—124no—
469C22×C4—124no—
470C22×C4—124no—
471C22×C4—124no—
472C22×C4C23×D311yescharacteristic
473C22×C4C23×D312yes—
474C22×C4C23×D314yes—
475C22×C4C23×D314yes—
476C22×C4C23×D312yes—
477C22×C4C23×D311yescharacteristic
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—64no—
485C4⋊C4—68no—
486C4⋊C4—62no—
487C4⋊C4—64no—
488C4⋊C4—64no—
489C4⋊C4—68no—
490C4⋊C4—124no—
491C4⋊C4—124no—
492C22⋊C4—24no—
493C22⋊C4—44no—
494C22⋊C4—68no—
495C22⋊C4—62no—
496C22⋊C4—68no—
497C22⋊C4—68no—
498C22⋊C4—68no—
499C22⋊C4—68no—
500C22⋊C4—62no—
501C22⋊C4—64no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C42—22no—
508C42—34no—
509C42—64no—
510C42—62no—
511C42—62no—
512C42—122no—
513C42D3×D414yes—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—42no—
517C22×C3—41no—
518C22×C3—42no—
519C22×C3C23×D411yescharacteristic
520C22×C3C22×#4111yescharacteristic
521C22×C3C2×H(5,𝔽2)12yes—
522C22×C3C2×2−1+411yescharacteristic
523C22×C3#58512yes—
524C2×D3—28no—
525C2×D3—28no—
526C2×D3—28no—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—44no—
530C2×D3—44no—
531C2×D3—41no—
532C2×D3—41no—
533C2×D3—42no—
534C2×D3—41no—
535C2×D3—42no—
536C3×C4—24no—
537C3×C4—44no—
538C3×C4—41no—
539C3×C4—42no—
540C3×C4—44no—
541C3×C4#54714yes—
542Dic3—22no—
543Dic3—41no—
544Dic3—42no—
545Dic3—44no—
546Dic3—42no—
547Dic3#54014yescomplemented
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—38no—
554C23—64no—
555C23—64no—
556C23—64no—
557C23—64no—
558C23—64no—
559C23—64no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—62no—
564C23—62no—
565C23C24×D311yescharacteristic
566D4—616no—
567D4—1216no—
568D4—124no—
569D4—248no—
570D4—248no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—32no—
586C2×C4—34no—
587C2×C4—32no—
588C2×C4—44no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—61no—
592C2×C4—64no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—64no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—64no—
602C2×C4—1216no—
603C2×C4—124no—
604C2×C4—248no—
605C2×C4—248no—
606C2×C4C2×D3×D414yes—
607C2×C4C2×D3×D412yes—
608C2×C4#100712yes—
609C2×C4D3×#4112yes—
610C2×C3—81no—
611C2×C3C2×#54011yescharacteristic
612C2×C3C2×#54711yescharacteristic
613C2×C3#349612yes—
614C2×C3#351211yescharacteristic
615C2×C3#354712yes—
616D3—48no—
617D3—81no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—68no—
624C22—68no—
625C22—68no—
626C22—124no—
627C22—124no—
628C22—124no—
629C22—124no—
630C22—122no—
631C22—122no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22C22×D3×D411yescharacteristic
636C22C2×#100711yescharacteristic
637C22C2×D3×#4111yescharacteristic
638C22D3×H(5,𝔽2)12yes—
639C22#596812yes—
640C4—24no—
641C4—34no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—41no—
646C4—62no—
647C4—122no—
648C4—122no—
649C4—124no—
650C4—121no—
651C4#563114yes—
652C3#3937811yescharacteristic, complemented, Sylow
653C2—81no—
654C2—128no—
655C2—241no—
656C2C2×#563111yescharacteristic
657C2#8638212yes—
658C2D3×#54011yescharacteristic
659C2#8649711yescharacteristic
660C2#8658912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2629C2
hypercenter#2629D3
hypocenterC3#39378
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2629

Complements

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

NH ≅ G/Ntype
1. C2×#5697C2semidirect
2. C2×#5535C2semidirect
3. C3×#2629C2semidirect
4. #83056C2semidirect
5. #82225C2semidirect
6. #77979C2semidirect
7. #77764C2semidirect
8. #77439C2semidirect
9. #77284C2semidirect
10. #76213C2semidirect
11. #76202C2semidirect
12. #74344C2semidirect
13. #74319C2semidirect
14. #73935C2semidirect
15. #73891C2semidirect
80. #2629D3semidirect
149. #874D4semidirect
547. Dic3#540semidirect
652. C3#39378semidirect

Maximal subgroups

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

subgroupindex
C2×#56972
C2×#55352
C3×#26292
#830562
#822252
#779792
#777642
#774392
#772842
#762132
#762022
#743442
#743192
#739352
#738912
#393783