Finite Group Databasebeta

← #11025135

Subgroups — #11025135

Order 768.

2356 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×#5719C211yesmaximal, characteristic, complemented
2C2×D3×#125C211yesmaximal, characteristic, complemented
3C3×#2641C211yesmaximal, characteristic, complemented
4#83192C211yesmaximal, characteristic, complemented
5#83065C211yesmaximal, characteristic, complemented
6#77117C214yesmaximal, complemented
7#77031C214yesmaximal, complemented
8#77027C214yesmaximal, complemented
9#76898C214yesmaximal, complemented
10#76449C212yesmaximal, complemented
11#76439C212yesmaximal, complemented
12#76433C212yesmaximal, complemented
13#76325C212yesmaximal, complemented
14#73414C211yesmaximal, characteristic, complemented
15#73390C211yesmaximal, characteristic, complemented
16#37706—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×#939C2214yes—
20C2×#938C2212yes—
21C2×#938C2212yes—
22C2×#938C2214yes—
23C2×#935C2214yes—
24C2×D4×Dic3C2212yes—
25#5719—28no—
26D3×#125—28no—
27C2×D3×C4⋊C4C2214yes—
28C2×D3×C4⋊C4C2212yes—
29C2×#890C2212yes—
30C2×#889C2214yes—
31C2×#884C2214yes—
32C2×#67⋊C4C2214yes—
33C2×C4×D12C2211yescharacteristic
34C2×#872C2211yescharacteristic
35C2×C42×D3C2211yescharacteristic
36C2×#870C2211yescharacteristic
37C3×#394C2214yes—
38C3×#394C2214yes—
39C3×#390C2212yes—
40C3×#390C2212yes—
41C3×#376C2211yescharacteristic
42#5221C2212yes—
43#5221C2214yes—
44#4994C2214yes—
45#4987C2212yes—
46#4982C2214yes—
47#4972C2212yes—
48#4961C2214yes—
49#4959C2214yes—
50#4957C2214yes—
51#4956C2214yes—
52#4956C2214yes—
53#4952C2214yes—
54#4951C2214yes—
55#4950C2214yes—
56#4949C2214yes—
57#4948C2214yes—
58#4948C2214yes—
59#4946C2214yes—
60#4944C2214yes—
61#4942C2214yes—
62C4×(C2×D3)⋊C4C2211yescharacteristic
63C4×(C2×D3)⋊C4C2211yescharacteristic
64#4935C2211yescharacteristic
65C4×C4⋊Dic3C2211yescharacteristic
66C4×C4⋊Dic3C2212yes—
67#4928C2212yes—
68#4670C2212yes—
69#4668C2212yes—
70#4665C2212yes—
71#4655C2214yes—
72#4653C2212yes—
73#4647C2214yes—
74C2×#540—31no—
75C22×#125—31no—
76#2856—31no—
77#2711—34no—
78#2707—34no—
79#2641D311yescharacteristic, complemented
80#2629—31no—
81#2490—32no—
82#2454—32no—
83#2443—34no—
84#2416—34no—
85#2416—32no—
86#2361—32no—
87#2330—31no—
88#2290—31no—
89C22×C3×D4C2312yes—
90C22×C3×D4C2314yes—
91C22×#67C2314yes—
92C22×#67C2314yes—
93C23×Dic3C2314yes—
94C22×D12C2311yescharacteristic
95C22×C4×D3C2311yescharacteristic
96C22×C4×D3C2312yes—
97C22×C4×D3C2314yes—
98C22×C4×D3C2311yescharacteristic
99C3×#127—24no—
100C3×#125—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C22⋊C4C2314yes—
104C2×C3×C22⋊C4C2314yes—
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×#217C2314yes—
111C2×#217C2314yes—
112C2×#217C2314yes—
113C2×#217C2314yes—
114C2×#217C2314yes—
115#939—216no—
116#938—28no—
117#938—28no—
118#938—48no—
119#935—216no—
120D4×Dic3—28no—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2312yes—
123C2×(C2×D3)⋊C4C2311yescharacteristic
124C2×(C2×D3)⋊C4C2311yescharacteristic
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2312yes—
129C2×C4⋊Dic3C2314yes—
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2314yes—
143D3×C4⋊C4—28no—
144D3×C4⋊C4—216no—
145#890—28no—
146#889—216no—
147#884—48no—
148#67⋊C4—48no—
149C4×D12—24no—
150#872—24no—
151C42×D3—24no—
152#870D418yescomplemented
153C3×#95C2312yes—
154C3×#95C2312yes—
155C3⋊#95C2314yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162#540—68no—
163C2×#127C2×D311yescharacteristic
164C2×#125—34no—
165C2×#125—32no—
166C2×#125—31no—
167C2×#125—68no—
168C2×#125C2×D311yescharacteristic
169C2×#123—34no—
170C2×#121—32no—
171C2×#121—34no—
172C2×#121—32no—
173C2×#121—34no—
174C2×#121—34no—
175C2×C4×D4—32no—
176C2×C4×D4—34no—
177C2×C4×D4—31no—
178C2×#117—31no—
179C22×C4⋊C4—32no—
180C22×C4⋊C4—34no—
181C22×C42—31no—
182#399—34no—
183#398—34no—
184#397—34no—
185#396—34no—
186#396—34no—
187#396—34no—
188#394—34no—
189#394—34no—
190#394C2×D314yes—
191#394C2×D314yes—
192#390—34no—
193#390—32no—
194#390—34no—
195#390C2×D312yes—
196#390C2×D312yes—
197#389—32no—
198#389—32no—
199#389—34no—
200#388—34no—
201#388—34no—
202#388—32no—
203#387—34no—
204#387—32no—
205#386—34no—
206#386—34no—
207#386—32no—
208#385—34no—
209#385—34no—
210#385—32no—
211#384—31no—
212#380—32no—
213#380—34no—
214C4×C4⋊C4—32no—
215C4×C4⋊C4—31no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—31no—
218#376C2×D311yescharacteristic
219C24×C3C2414yes—
220C23×D3C2411yescharacteristic
221C23×D3C2411yescharacteristic
222C2×C3×D4—24no—
223C2×C3×D4—28no—
224C2×C3×D4—24no—
225C2×C3×D4—24no—
226C2×C3×D4—44no—
227C2×C3×D4—44no—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2414yes—
232C22×C3×C4C2412yes—
233C22×C3×C4C2411yescharacteristic
234C2×#67—28no—
235C2×#67—28no—
236C2×#67—28no—
237C2×#67—28no—
238C2×#67—44no—
239C2×#67—44no—
240C22×Dic3—28no—
241C22×Dic3—44no—
242C22×Dic3—44no—
243C22×Dic3C2411yescharacteristic
244C22×Dic3C2412yes—
245C22×Dic3C2414yes—
246C22×Dic3C2412yes—
247C22×Dic3C2414yes—
248C22×Dic3C2411yescharacteristic
249C2×D12—22no—
250C2×D12—24no—
251C2×C4×D3—22no—
252C2×C4×D3—22no—
253C2×C4×D3—28no—
254C2×C4×D3—24no—
255C2×C4×D3—24no—
256C2×C4×D3—22no—
257C2×C4×D3—22no—
258C2×C4×D3—22no—
259C2×C4×D3—28no—
260C2×C4×D3—44no—
261C2×C4×D3—42no—
262C2×C4×D3#4114yes—
263C3×C4⋊C4—28no—
264C3×C4⋊C4—24no—
265C3×C22⋊C4—44no—
266C3×C22⋊C4—44no—
267C3×C22⋊C4—44no—
268C3×C22⋊C4—44no—
269C3×C42—22no—
270C3×C42—22no—
271C3×C42C2×D414yes—
272#217—28no—
273#217—28no—
274#217—28no—
275#217—44no—
276#217—44no—
277(C2×D3)⋊C4—22no—
278(C2×D3)⋊C4—24no—
279(C2×D3)⋊C4—24no—
280(C2×D3)⋊C4—22no—
281(C2×D3)⋊C4—44no—
282(C2×D3)⋊C4—42no—
283C4⋊Dic3—24no—
284C4⋊Dic3—28no—
285C4⋊Dic3—22no—
286C4⋊Dic3—42no—
287C4⋊Dic3—44no—
288C4⋊Dic3C2×D418yes—
289Dic3⋊C4—24no—
290Dic3⋊C4—28no—
291Dic3⋊C4—28no—
292Dic3⋊C4—44no—
293Dic3⋊C4—44no—
294Dic3⋊C4C2×D4116yes—
295C4×Dic3—24no—
296C4×Dic3—22no—
297C4×Dic3—22no—
298C4×Dic3—44no—
299C22×D4—34no—
300C22×D4—34no—
301C22×D4—31no—
302C22×D4C22×D312yes—
303C22×D4C22×D314yes—
304C23×C4—31no—
305C23×C4—34no—
306C23×C4—31no—
307C23×C4—34no—
308C23×C4—32no—
309#127—24no—
310#125—24no—
311#125—38no—
312#125—616no—
313#125—68no—
314#123—616no—
315#121—68no—
316#121—616no—
317#121—68no—
318#121—128no—
319#121—128no—
320C4×D4—64no—
321C4×D4—68no—
322C4×D4—128no—
323#117—64no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—68no—
337C2×C4⋊C4—616no—
338C2×C4⋊C4C22×D312yes—
339C2×C4⋊C4C22×D314yes—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—31no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4C22×D314yes—
352C2×C22⋊C4C22×D314yes—
353C2×C22⋊C4C22×D314yes—
354C2×C22⋊C4C22×D314yes—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—32no—
358C2×C42—34no—
359C2×C42—64no—
360C2×C42C22×D311yescharacteristic
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×D312yes—
371#95C22×D312yes—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3—24no—
375C23×C3—24no—
376C23×C3—24no—
377C23×C3—24no—
378C23×C3—24no—
379C23×C3C2511yescharacteristic
380C22×D3—21no—
381C22×D3—21no—
382C22×D3—22no—
383C22×D3—21no—
384C22×D3—21no—
385C22×D3—22no—
386C22×D3—21no—
387C22×D3—22no—
388C22×D3C2×#4112yes—
389C22×D3H(5,𝔽2)14yes—
390C3×D4—48no—
391C3×D4—88no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—24no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—24no—
404C2×C3×C4C22×D414yes—
405C2×C3×C4C22×D412yes—
406C2×C3×C4C2×#4112yes—
407C2×C3×C42−1+412yes—
408#67—416no—
409#67—88no—
410C2×Dic3—21no—
411C2×Dic3—24no—
412C2×Dic3—21no—
413C2×Dic3—24no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—24no—
419C2×Dic3—24no—
420C2×Dic3—22no—
421C2×Dic3—24no—
422C2×Dic3—22no—
423C2×Dic3—21no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—88no—
427C2×Dic3C22×D418yes—
428C2×Dic3C2×#4112yes—
429D12—44no—
430C4×D3—28no—
431C4×D3—416no—
432C4×D3—48no—
433C4×D3—44no—
434C24—31no—
435C24—31no—
436C24C23×D314yes—
437C2×D4—24no—
438C2×D4—28no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—62no—
444C2×D4—64no—
445C2×D4—68no—
446C2×D4—68no—
447C2×D4—68no—
448C2×D4—68no—
449C2×D4—124no—
450C2×D4—124no—
451C22×C4—31no—
452C22×C4—34no—
453C22×C4—34no—
454C22×C4—31no—
455C22×C4—34no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—68no—
459C22×C4—68no—
460C22×C4—64no—
461C22×C4—68no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—64no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—122no—
469C22×C4—124no—
470C22×C4—124no—
471C22×C4—124no—
472C22×C4C23×D312yes—
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D311yescharacteristic
475C22×C4C23×D314yes—
476C22×C4C23×D312yes—
477C22×C4C23×D311yescharacteristic
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—38no—
481C4⋊C4—316no—
482C4⋊C4—68no—
483C4⋊C4—68no—
484C4⋊C4—68no—
485C4⋊C4—62no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—124no—
489C4⋊C4—124no—
490C4⋊C4—124no—
491C4⋊C4—122no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—62no—
497C22⋊C4—62no—
498C22⋊C4—64no—
499C22⋊C4—68no—
500C22⋊C4—68no—
501C22⋊C4—68no—
502C22⋊C4—64no—
503C22⋊C4—124no—
504C22⋊C4—122no—
505C22⋊C4—124no—
506C22⋊C4—124no—
507C42—22no—
508C42—22no—
509C42—64no—
510C42—62no—
511C42—62no—
512C42—124no—
513C42D3×D414yes—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3—44no—
520C22×C3—44no—
521C22×C3C23×D411yescharacteristic
522C22×C3C22×#4111yescharacteristic
523C22×C3C2×H(5,𝔽2)12yes—
524C22×C3C2×2−1+411yescharacteristic
525C22×C3#58512yes—
526C2×D3—22no—
527C2×D3—24no—
528C2×D3—22no—
529C2×D3—24no—
530C2×D3—41no—
531C2×D3—41no—
532C2×D3—42no—
533C2×D3—41no—
534C2×D3—42no—
535C2×D3#54014yescomplemented
536C3×C4—24no—
537C3×C4—41no—
538C3×C4—44no—
539C3×C4—41no—
540C3×C4—42no—
541C3×C4#54814yes—
542Dic3—22no—
543Dic3—28no—
544Dic3—44no—
545Dic3—41no—
546Dic3—42no—
547Dic3—42no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—32no—
556C23—34no—
557C23—61no—
558C23—61no—
559C23—62no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—62no—
564C23—62no—
565C23C24×D311yescharacteristic
566D4—48no—
567D4—88no—
568D4—124no—
569D4—1216no—
570D4—248no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—38no—
584C2×C4—32no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—68no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—64no—
595C2×C4—61no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—64no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—128no—
603C2×C4—1216no—
604C2×C4—124no—
605C2×C4—248no—
606C2×C4C2×D3×D414yes—
607C2×C4C2×D3×D412yes—
608C2×C4D3×#4112yes—
609C2×C4#101012yes—
610C2×C3—84no—
611C2×C3C2×#54011yescharacteristic
612C2×C3C2×#54811yescharacteristic
613C2×C3#348812yes—
614C2×C3#351211yescharacteristic
615C2×C3#355912yes—
616D3—24no—
617D3—81no—
618C22—34no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—62no—
627C22—64no—
628C22—62no—
629C22—64no—
630C22—121no—
631C22—122no—
632C22—121no—
633C22—121no—
634C22—122no—
635C22C22×D3×D411yescharacteristic
636C22C2×D3×#4111yescharacteristic
637C22C2×#101011yescharacteristic
638C22D3×H(5,𝔽2)12yes—
639C22#596612yes—
640C4—24no—
641C4—44no—
642C4—41no—
643C4—41no—
644C4—42no—
645C4—62no—
646C4—68no—
647C4—121no—
648C4—122no—
649C4—122no—
650C4—124no—
651C4#567714yes—
652C3#3770611yescharacteristic, complemented, Sylow
653C2—64no—
654C2—84no—
655C2—241no—
656C2C2×#567711yescharacteristic
657C2#8633112yes—
658C2#8646911yescharacteristic
659C2D3×#54011yescharacteristic
660C2#8649412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2641C2
hypercenter#2641D3
hypocenterC3#37706
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2641

Complements

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

NH ≅ G/Ntype
1. C2×#5719C2semidirect
2. C2×D3×#125C2semidirect
3. C3×#2641C2semidirect
4. #83192C2semidirect
5. #83065C2semidirect
6. #77117C2semidirect
7. #77031C2semidirect
8. #77027C2semidirect
9. #76898C2semidirect
10. #76449C2semidirect
11. #76439C2semidirect
12. #76433C2semidirect
13. #76325C2semidirect
14. #73414C2semidirect
15. #73390C2semidirect
79. #2641D3semidirect
152. #870D4semidirect
535. C2×D3#540semidirect
652. C3#37706semidirect

Maximal subgroups

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

subgroupindex
C2×#57192
C2×D3×#1252
C3×#26412
#831922
#830652
#771172
#770312
#770272
#768982
#764492
#764392
#764332
#763252
#734142
#733902
#377063