Finite Group Databasebeta

← #11025037

Subgroups — #11025037

Order 768.

2364 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×C3×#532C211yesmaximal, characteristic, complemented
2C2×#5731C211yesmaximal, characteristic, complemented
3#84097C212yesmaximal, complemented
4#84089C212yesmaximal, complemented
5#83227C211yesmaximal, characteristic, complemented
6#83189C211yesmaximal, characteristic, complemented
7#83180C211yesmaximal, characteristic, complemented
8#82995C212yesmaximal, complemented
9#82963C212yesmaximal, complemented
10#82696C214yesmaximal, complemented
11#82695C214yesmaximal, complemented
12#82370C214yesmaximal, complemented
13#82366C214yesmaximal, complemented
14#82074C211yesmaximal, characteristic, complemented
15#82065C211yesmaximal, characteristic, complemented
16#37701—31nomaximal, Sylow
17C22×C3×#41C2212yes—
18C3×#532—28no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#127C2211yescharacteristic
21C2×C3×#124C2212yes—
22C2×C3×#121C2214yes—
23C2×C3×C4×D4C2214yes—
24C22×C3×C42C2211yescharacteristic
25C2×#946C2212yes—
26C2×Q8×Dic3C2212yes—
27C2×Q8×Dic3C2212yes—
28C2×#938C2212yes—
29C2×C4×#67C2211yescharacteristic
30#5731—28no—
31C2×#898C2214yes—
32C2×#889C2214yes—
33C2×#872C2211yescharacteristic
34C2×#870C2211yescharacteristic
35#5229C2212yes—
36#5223C2214yes—
37#5223C2214yes—
38#5222C2214yes—
39#5222C2214yes—
40#5222C2214yes—
41#5222C2214yes—
42#5221C2214yes—
43#5221C2214yes—
44#5221C2212yes—
45#5221C2212yes—
46#5221C2212yes—
47#5213C2212yes—
48#5213C2214yes—
49#5212C2212yes—
50#5212C2214yes—
51#5210C2212yes—
52#5210C2214yes—
53#5210C2214yes—
54C4×#217C2211yescharacteristic
55C4×#217C2214yes—
56#4980C2214yes—
57#4980C2214yes—
58#4961C2214yes—
59#4957C2214yes—
60#4953C2214yes—
61#4949C2214yes—
62#4948C2214yes—
63#4945C2214yes—
64#4944C2214yes—
65#4944C2214yes—
66#4940C2212yes—
67C4×(C2×D3)⋊C4C2211yescharacteristic
68#4936C2211yescharacteristic
69#4933C2212yes—
70#4933C2214yes—
71#4932C2211yescharacteristic
72C4×Dic3⋊C4C2211yescharacteristic
73#4928C2212yes—
74C2×#541—31no—
75C2×#532D311yescharacteristic, complemented
76#2851—31no—
77#2714—34no—
78#2711—34no—
79#2641—31no—
80#2619—31no—
81#2467—34no—
82#2460—34no—
83#2448—32no—
84#2426—32no—
85#2407—32no—
86#2341—32no—
87#2327—31no—
88#2290—31no—
89C2×C3×#41—28no—
90C2×C3×#41—28no—
91C2×C3×#41—44no—
92C22×C3×Q8C2312yes—
93C22×C3×D4C2312yes—
94C22×C3×D4C2314yes—
95C22×C3×D4C2314yes—
96C23×C3×C4C2314yes—
97C23×C3×C4C2311yescharacteristic
98C23×C3×C4C2312yes—
99C22×#67C2311yescharacteristic
100C22×#67C2314yes—
101C22×C4×D3C2311yescharacteristic
102C3×C4⋊Q8D418yescomplemented
103C3×#127—24no—
104C3×#124—28no—
105C3×#121—216no—
106C3×C4×D4—216no—
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C42—24no—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2312yes—
113C2×C3×C42C2311yescharacteristic
114C2×#217C2311yescharacteristic
115C2×#217C2314yes—
116C2×#217C2314yes—
117C2×#217C2314yes—
118C2×#217C2314yes—
119C2×#217C2314yes—
120C2×#217C2314yes—
121C2×#217C2314yes—
122C2×#217C2312yes—
123C2×#217C2314yes—
124#946—28no—
125Q8×Dic3—28no—
126Q8×Dic3—44no—
127#938—44no—
128C4×#67—24no—
129C2×(C2×D3)⋊C4C2311yescharacteristic
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2314yes—
136C2×C4⋊Dic3C2314yes—
137C2×Dic3⋊C4C2311yescharacteristic
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2312yes—
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2311yescharacteristic
146C2×C4×Dic3C2314yes—
147C2×C4×Dic3C2314yes—
148#898—216no—
149#889—216no—
150#872—24no—
151#870—24no—
152C3⋊#95C2314yes—
153C3⋊#95C2312yes—
154C3⋊#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C22×#41C2×D312yes—
163#541—68no—
164#532—28no—
165C2×C4⋊Q8C2×D311yescharacteristic
166C2×#127C2×D311yescharacteristic
167C2×#126—34no—
168C2×#125—34no—
169C2×#125—31no—
170C2×#124C2×D312yes—
171C2×C22⋊Q8—32no—
172C2×#121—32no—
173C2×#121C2×D314yes—
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×D4—31no—
177C2×C4×D4C2×D314yes—
178C2×#117—31no—
179C22×C42C2×D311yescharacteristic
180#399—34no—
181#397—34no—
182#396—34no—
183#394—34no—
184#394—34no—
185#394—34no—
186#394—34no—
187#394—34no—
188#392—34no—
189#392—34no—
190#391—32no—
191#390—32no—
192#390—32no—
193#390—34no—
194#390—34no—
195#390—32no—
196#390—34no—
197#389—32no—
198#388—34no—
199#388—34no—
200#388—34no—
201#388—32no—
202#387—34no—
203#387—34no—
204#387—34no—
205#387—32no—
206#386—34no—
207#386—32no—
208#385—34no—
209#385—34no—
210#385—32no—
211#382—31no—
212#380—32no—
213#380—34no—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—34no—
217C4×C22⋊C4—31no—
218#376—31no—
219C24×C3C2414yes—
220C24×C3C2411yescharacteristic
221C23×D3C2411yescharacteristic
222C3×#41—432no—
223C2×C3×Q8—24no—
224C2×C3×Q8—24no—
225C2×C3×Q8C2×D418yes—
226C2×C3×D4—28no—
227C2×C3×D4—28no—
228C2×C3×D4—28no—
229C2×C3×D4—24no—
230C2×C3×D4—24no—
231C2×C3×D4—24no—
232C2×C3×D4—44no—
233C2×C3×D4—44no—
234C2×C3×D4—44no—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—24no—
238C22×C3×C4—28no—
239C22×C3×C4—28no—
240C22×C3×C4—42no—
241C22×C3×C4—42no—
242C22×C3×C4—44no—
243C22×C3×C4#4114yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2414yes—
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2411yescharacteristic
249C2×#67—22no—
250C2×#67—42no—
251C2×#67—44no—
252C2×#67—44no—
253C2×#67—44no—
254C22×Dic3C2411yescharacteristic
255C22×Dic3C2411yescharacteristic
256C22×Dic3C2414yes—
257C22×Dic3C2412yes—
258C22×Dic3C2412yes—
259C22×Dic3C2414yes—
260C22×Dic3C2411yescharacteristic
261C2×C4×D3—22no—
262C2×C4×D3—22no—
263C2×C4×D3—22no—
264C3×C4⋊C4C2×D4116yes—
265C3×C22⋊C4—28no—
266C3×C22⋊C4—28no—
267C3×C42—24no—
268C3×C42—22no—
269C3×C42C2×D414yes—
270#217—22no—
271#217—44no—
272#217—44no—
273#217—44no—
274#217—44no—
275#217—44no—
276#217—44no—
277#217—44no—
278#217—42no—
279#217—44no—
280(C2×D3)⋊C4—24no—
281(C2×D3)⋊C4—28no—
282(C2×D3)⋊C4—22no—
283(C2×D3)⋊C4—42no—
284C4⋊Dic3—24no—
285C4⋊Dic3—28no—
286C4⋊Dic3—24no—
287C4⋊Dic3—44no—
288Dic3⋊C4—28no—
289Dic3⋊C4—24no—
290Dic3⋊C4—28no—
291Dic3⋊C4—22no—
292Dic3⋊C4—42no—
293Dic3⋊C4—44no—
294C4×Dic3—22no—
295C4×Dic3—28no—
296C4×Dic3—22no—
297C4×Dic3—22no—
298C4×Dic3—44no—
299C2×#41—28no—
300C2×#41—28no—
301C2×#41—44no—
302C22×Q8C22×D312yes—
303C22×D4—31no—
304C22×D4—34no—
305C22×D4C22×D312yes—
306C22×D4C22×D314yes—
307C22×D4C22×D314yes—
308C23×C4—31no—
309C23×C4C22×D314yes—
310C23×C4C22×D311yescharacteristic
311C23×C4C22×D312yes—
312C4⋊Q8#6718yescomplemented
313#127—24no—
314#126—616no—
315#125—616no—
316#125—64no—
317#124—28no—
318C22⋊Q8—68no—
319#121—216no—
320#121—124no—
321C4×Q8—68no—
322C4×Q8—124no—
323C4×D4—216no—
324C4×D4—64no—
325#117—64no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—34no—
336C2×C4⋊C4C22×D314yes—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—31no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
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×C42—24no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—34no—
358C2×C42—34no—
359C2×C42C22×D312yes—
360C2×C42C22×D311yescharacteristic
361C2×C42C22×D311yescharacteristic
362#95—32no—
363#95—32no—
364#95—32no—
365#95—34no—
366#95—32no—
367#95—32no—
368#95—34no—
369#95—34no—
370#95—34no—
371#95—34no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—24no—
377C23×C3—24no—
378C23×C3—24no—
379C23×C3—24no—
380C23×C3—24no—
381C23×C3—24no—
382C23×C3—24no—
383C23×C3C2×#4112yes—
384C23×C3C2511yescharacteristic
385C22×D3—21no—
386C22×D3—21no—
387C22×D3—21no—
388C22×D3—22no—
389C22×D3—22no—
390C3×Q8—216no—
391C3×D4—416no—
392C3×D4—416no—
393C3×D4—48no—
394C2×C3×C4—21no—
395C2×C3×C4—24no—
396C2×C3×C4—24no—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—28no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—416no—
405C2×C3×C4—48no—
406C2×C3×C4C22×D418yes—
407C2×C3×C4C22×D414yes—
408C2×C3×C4C22×D412yes—
409C2×C3×C4C2×#4112yes—
410C2×C3×C4C2×#4112yes—
411C2×C3×C4H(5,𝔽2)12yes—
412#67—44no—
413#67—88no—
414C2×Dic3—22no—
415C2×Dic3—24no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—24no—
422C2×Dic3—21no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—24no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—24no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—24no—
433C4×D3—44no—
434C24—31no—
435C24C23×D314yes—
436C24C23×D311yescharacteristic
437#41—432no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8C2×#6718yes—
441C2×D4—28no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—62no—
451C2×D4—122no—
452C2×D4—124no—
453C2×D4—124no—
454C2×D4—124no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—24no—
459C22×C4—28no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—34no—
463C22×C4—32no—
464C22×C4—32no—
465C22×C4—34no—
466C22×C4—31no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4#23714yes—
474C22×C4C23×D312yes—
475C22×C4C23×D311yescharacteristic
476C22×C4C23×D312yes—
477C22×C4C23×D314yes—
478C22×C4C23×D311yescharacteristic
479C4⋊C4—62no—
480C4⋊C4—64no—
481C4⋊C4—64no—
482C4⋊C4—64no—
483C4⋊C4—68no—
484C4⋊C4—68no—
485C4⋊C4—68no—
486C4⋊C4—124no—
487C4⋊C4—122no—
488C4⋊C4—124no—
489C4⋊C4C2×#67116yes—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C22⋊C4—64no—
495C22⋊C4—68no—
496C22⋊C4—122no—
497C22⋊C4—124no—
498C22⋊C4—122no—
499C22⋊C4—124no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—124no—
506C42—24no—
507C42—22no—
508C42—62no—
509C42—62no—
510C42—62no—
511C42—68no—
512C42—124no—
513C42C2×#6714yes—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—44no—
520C22×C3—44no—
521C22×C3—44no—
522C22×C3—44no—
523C22×C3—44no—
524C22×C3—44no—
525C22×C3—44no—
526C22×C3C23×D411yescharacteristic
527C22×C3C22×#4111yescharacteristic
528C22×C3C2×H(5,𝔽2)11yescharacteristic
529C22×C3#58512yes—
530C22×C3#58512yes—
531C2×D3—41no—
532C2×D3—41no—
533C2×D3—42no—
534C2×D3—41no—
535C2×D3—42no—
536C3×C4—22no—
537C3×C4—28no—
538C3×C4—24no—
539C3×C4—42no—
540C3×C4#54614yes—
541Dic3—41no—
542Dic3—41no—
543Dic3—42no—
544Dic3—44no—
545Dic3—41no—
546Dic3—42no—
547Dic3—44no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—61no—
560C23—61no—
561C23—62no—
562C23—62no—
563C23—61no—
564C23C2×#23712yes—
565C23C24×D311yescharacteristic
566Q8—216no—
567D4—416no—
568D4—416no—
569D4—48no—
570D4—124no—
571D4—248no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—416no—
583C2×C4—48no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—61no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—62no—
601C2×C4—64no—
602C2×C4—61no—
603C2×C4—124no—
604C2×C4C2×#23712yes—
605C2×C4C2×#23712yes—
606C2×C4#100412yes—
607C2×C4C22×#6714yes—
608C2×C4C22×#6718yes—
609C2×C4C22×#6712yes—
610C2×C3—42no—
611C2×C3—84no—
612C2×C3#345511yescharacteristic
613C2×C3C2×#54611yescharacteristic
614C2×C3#348712yes—
615C2×C3#351211yescharacteristic
616C2×C3#356212yes—
617D3—81no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—121no—
631C22—121no—
632C22—121no—
633C22—122no—
634C22—122no—
635C22C22×#23711yescharacteristic
636C22C2×#100411yescharacteristic
637C22#596412yes—
638C22#596412yes—
639C22C23×#6711yescharacteristic
640C4—22no—
641C4—28no—
642C4—24no—
643C4—42no—
644C4—121no—
645C4—121no—
646C4—121no—
647C4—122no—
648C4—124no—
649C4—124no—
650C4—122no—
651C4#580614yes—
652C3#3770111yescharacteristic, complemented, Sylow
653C2—42no—
654C2—84no—
655C2—241no—
656C2#8622711yescharacteristic
657C2#8623611yescharacteristic
658C2#8625612yes—
659C2C2×#580611yescharacteristic
660C2#8727412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#532C2
hypercenterC2×#532D3
hypocenterC3#37701
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C2×#5731C2semidirect
3. #84097C2semidirect
4. #84089C2semidirect
5. #83227C2semidirect
6. #83189C2semidirect
7. #83180C2semidirect
8. #82995C2semidirect
9. #82963C2semidirect
10. #82696C2semidirect
11. #82695C2semidirect
12. #82370C2semidirect
13. #82366C2semidirect
14. #82074C2semidirect
15. #82065C2semidirect
75. C2×#532D3semidirect
102. C3×C4⋊Q8D4semidirect
312. C4⋊Q8#67semidirect
652. C3#37701semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#57312
#840972
#840892
#832272
#831892
#831802
#829952
#829632
#826962
#826952
#823702
#823662
#820742
#820652
#377013