Finite Group Databasebeta

← #11233336

Subgroups — #11233336

Order 768.

2410 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×#5543C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic, complemented
3C3×#2535C211yesmaximal, characteristic
4#82647C212yesmaximal, complemented
5#82636C212yesmaximal, complemented
6#82348C211yesmaximal, characteristic, complemented
7#82328C211yesmaximal, characteristic, complemented
8#77573C214yesmaximal, complemented
9#76681C214yesmaximal, complemented
10#74827C214yesmaximal, complemented
11#74689C214yesmaximal, complemented
12#74191C211yesmaximal, characteristic, complemented
13#74154C211yesmaximal, characteristic, complemented
14#73916C212yesmaximal, complemented
15#73899C212yesmaximal, complemented
16#45337—31nomaximal, Sylow
17C23×Dic6C2212yes—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×#936C2211yescharacteristic
22C2×D4×Dic3C2212yes—
23C2×D4×Dic3C2212yes—
24C2×D4×Dic3C2214yes—
25C22×C4⋊Dic3C2212yes—
26C2×#924C2212yes—
27C2×#924C2212yes—
28#5543—28no—
29D4×Dic6—28no—
30C2×#890C2212yes—
31C2×#888C2212yes—
32C2×#878C2214yes—
33C2×#878C2214yes—
34C2×#878C2214yes—
35C2×#869C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37C2×C4×Dic6C2211yescharacteristic
38C3×#397C2212yes—
39C3×#392C2212yes—
40C3×#386C2214yes—
41C3×#382C2211yescharacteristic
42C3×#380C2212yes—
43#5222C2214yes—
44#5222C2212yes—
45#5221C2211yescharacteristic
46#5220C2214yes—
47#5220C2214yes—
48#5210C2214yes—
49C4×#217C2212yes—
50C4×#217C2212yes—
51#4985C2212yes—
52#4984C2212yes—
53#4975C2211yescharacteristic
54Dic3×C4⋊C4C2212yes—
55#4973C2211yescharacteristic
56#4973C2212yes—
57#4972C2212yes—
58#4953C2212yes—
59#4952C2214yes—
60#4951C2214yes—
61#4950C2214yes—
62#4947C2214yes—
63#4947C2212yes—
64#4946C2214yes—
65#4944C2214yes—
66#4944C2214yes—
67#4942C2214yes—
68#4932C2211yescharacteristic
69#4659C2212yes—
70#4658C2214yes—
71#4657C2214yes—
72#4656C2214yes—
73#4656C2212yes—
74#4653C2214yes—
75#4653C2214yes—
76#4646C2214yes—
77#4646C2214yes—
78C2×#554—31no—
79C2×D4×Q8—31no—
80#2728—34no—
81#2690—34no—
82#2581—31no—
83#2535D311yescharacteristic
84#2492—32no—
85#2463—34no—
86#2440—32no—
87#2427—32no—
88#2409—34no—
89#2398—31no—
90#2361—32no—
91#2331—31no—
92#2326—31no—
93C22×C3×D4C2311yescharacteristic
94C22×C3×D4C2314yes—
95C22×C3×D4C2311yescharacteristic
96C23×C3×C4C2312yes—
97C23×C3×C4C2312yes—
98C23×Dic3C2314yes—
99C23×Dic3C2314yes—
100C22×Dic6—28no—
101C22×Dic6—48no—
102C22×Dic6C2311yescharacteristic
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2314yes—
105C22×Dic6C2312yes—
106C3×#121—28no—
107C3×C4×D4—24no—
108C3×C4×D4—24no—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4C2314yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2312yes—
115C2×C3×C22⋊C4C2314yes—
116C2×C3×C42C2311yescharacteristic
117C2×#217C2314yes—
118C2×#217C2312yes—
119C2×#217C2314yes—
120C2×#217C2312yes—
121C2×#217C2312yes—
122C2×#217C2312yes—
123C2×#217C2314yes—
124C2×#217C2314yes—
125#936—24no—
126D4×Dic3—28no—
127D4×Dic3—28no—
128D4×Dic3—48no—
129C2×C4⋊Dic3—28no—
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2314yes—
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136#924—28no—
137#924—28no—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2311yescharacteristic
146C2×C4×Dic3C2311yescharacteristic
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2314yes—
149#890—28no—
150#888—28no—
151#878—216no—
152#878—216no—
153#878—48no—
154#869D418yescomplemented
155C4×Dic6—24no—
156C4×Dic6—24no—
157C3×#95C2312yes—
158C3×#95C2314yes—
159C3×#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2312yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2312yes—
172C23×Q8—32no—
173#554—68no—
174D4×Q8—68no—
175C2×C4⋊Q8—32no—
176C2×C4⋊Q8—31no—
177C2×#125—32no—
178C2×#124—31no—
179C2×C22⋊Q8—32no—
180C2×C22⋊Q8—32no—
181C2×C22⋊Q8—34no—
182C2×C22⋊Q8—34no—
183C2×C22⋊Q8—34no—
184C2×#121C2×D312yes—
185C2×C4×Q8—31no—
186C2×C4×Q8—31no—
187C2×C4×D4—32no—
188C2×C4×D4—32no—
189C2×C4×D4—34no—
190C2×C4×D4C2×D311yescharacteristic
191C2×C4×D4C2×D311yescharacteristic
192C22×C4⋊C4—32no—
193#400—32no—
194#400—32no—
195#400—34no—
196#398—34no—
197#397—34no—
198#397—32no—
199#397C2×D312yes—
200#395—32no—
201#395—34no—
202#394—32no—
203#394—34no—
204#393—32no—
205#393—34no—
206#392C2×D312yes—
207#391—32no—
208#391—31no—
209#391—34no—
210#391—34no—
211#390—31no—
212#389—32no—
213#389—32no—
214#388—34no—
215#388—34no—
216#388—34no—
217#387—34no—
218#387—34no—
219#386—34no—
220#386—34no—
221#386C2×D314yes—
222#385—34no—
223#385—34no—
224#385—34no—
225#384—31no—
226#382C2×D311yescharacteristic
227#380—34no—
228#380C2×D312yes—
229C4×C4⋊C4—32no—
230C4×C22⋊C4—32no—
231C4×C22⋊C4—32no—
232#376—31no—
233C24×C3C2412yes—
234C24×C3C2412yes—
235C2×C3×D4—24no—
236C2×C3×D4—28no—
237C2×C3×D4—28no—
238C2×C3×D4—22no—
239C2×C3×D4—22no—
240C2×C3×D4—22no—
241C2×C3×D4—44no—
242C2×C3×D4#4114yes—
243C22×C3×C4—24no—
244C22×C3×C4—24no—
245C22×C3×C4—24no—
246C22×C3×C4—24no—
247C22×C3×C4—44no—
248C22×C3×C4C2414yes—
249C22×C3×C4C2412yes—
250C22×C3×C4C2412yes—
251C22×C3×C4C2411yescharacteristic
252C22×C3×C4C2411yescharacteristic
253C22×C3×C4C2411yescharacteristic
254C22×Dic3—28no—
255C22×Dic3—28no—
256C22×Dic3—28no—
257C22×Dic3—44no—
258C22×Dic3—44no—
259C22×Dic3—44no—
260C22×Dic3C2412yes—
261C22×Dic3C2412yes—
262C22×Dic3C2412yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2414yes—
266C22×Dic3C2412yes—
267C2×Dic6—22no—
268C2×Dic6—24no—
269C2×Dic6—22no—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6—28no—
273C2×Dic6—28no—
274C2×Dic6—432no—
275C2×Dic6—44no—
276C2×Dic6—44no—
277C2×Dic6C2×D418yes—
278C3×C4⋊C4—22no—
279C3×C4⋊C4—22no—
280C3×C4⋊C4—24no—
281C3×C22⋊C4—28no—
282C3×C22⋊C4—24no—
283C3×C22⋊C4—24no—
284C3×C22⋊C4—44no—
285C3×C42C2×D414yes—
286#217—28no—
287#217—24no—
288#217—24no—
289#217—28no—
290#217—28no—
291#217—24no—
292#217—24no—
293#217—44no—
294C4⋊Dic3—22no—
295C4⋊Dic3—22no—
296C4⋊Dic3—28no—
297C4⋊Dic3—28no—
298C4⋊Dic3C2×D418yes—
299C4⋊Dic3C2×D418yes—
300Dic3⋊C4—24no—
301Dic3⋊C4—24no—
302Dic3⋊C4—28no—
303Dic3⋊C4—28no—
304Dic3⋊C4—44no—
305Dic3⋊C4—44no—
306C4×Dic3—24no—
307C4×Dic3—24no—
308C4×Dic3—22no—
309C4×Dic3—22no—
310C4×Dic3—44no—
311C22×Q8—32no—
312C22×Q8—31no—
313C22×Q8—31no—
314C22×Q8—34no—
315C22×Q8—68no—
316C22×Q8—128no—
317C22×D4C22×D311yescharacteristic
318C22×D4C22×D314yes—
319C22×D4C22×D311yescharacteristic
320C23×C4—34no—
321C23×C4—34no—
322C23×C4C22×D312yes—
323C23×C4C22×D312yes—
324C4⋊Q8—38no—
325C4⋊Q8—68no—
326#125—68no—
327#124—64no—
328C22⋊Q8—616no—
329C22⋊Q8—616no—
330C22⋊Q8—68no—
331C22⋊Q8—68no—
332C22⋊Q8—128no—
333#121—28no—
334C4×Q8—64no—
335C4×Q8—64no—
336C4×D4—24no—
337C4×D4—24no—
338C4×D4—68no—
339C4×D4—68no—
340C4×D4—128no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—34no—
353C2×C4⋊C4—68no—
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C4⋊C4C22×D312yes—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—34no—
359C2×C22⋊C4—34no—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—32no—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4C22×D312yes—
366C2×C22⋊C4C22×D312yes—
367C2×C22⋊C4C22×D314yes—
368C2×C22⋊C4C22×D314yes—
369C2×C42—32no—
370C2×C42—32no—
371C2×C42—31no—
372C2×C42—31no—
373C2×C42—34no—
374C2×C42C22×D311yescharacteristic
375#95—32no—
376#95—32no—
377#95—32no—
378#95—32no—
379#95—32no—
380#95—32no—
381#95—34no—
382#95—34no—
383#95—34no—
384#95—34no—
385#95—34no—
386#95—34no—
387#95C22×D314yes—
388#95C22×D312yes—
389#95C22×D312yes—
390C23×C3—22no—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—24no—
396C23×C3—22no—
397C23×C3—22no—
398C23×C3—22no—
399C23×C3—24no—
400C23×C3C2×#4114yes—
401C23×C32−1+414yes—
402C23×C3C2511yescharacteristic
403C3×D4—28no—
404C3×D4—28no—
405C3×D4—416no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—24no—
412C2×C3×C4—24no—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—48no—
417C2×C3×C4—48no—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D412yes—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C2×#4112yes—
422C2×C3×C42−1+412yes—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—24no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—416no—
442C2×Dic3—88no—
443C2×Dic3C22×D418yes—
444C2×Dic3H(5,𝔽2)14yes—
445Dic6—28no—
446Dic6—48no—
447Dic6—416no—
448Dic6—44no—
449C24C23×D312yes—
450C24C23×D312yes—
451C2×Q8—38no—
452C2×Q8—64no—
453C2×Q8—68no—
454C2×Q8—68no—
455C2×Q8—62no—
456C2×Q8—62no—
457C2×Q8—62no—
458C2×Q8—62no—
459C2×Q8—124no—
460C2×Q8—124no—
461C2×Q8—1232no—
462C2×D4—24no—
463C2×D4—28no—
464C2×D4—28no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—44no—
469C2×D4#23714yes—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—32no—
475C22×C4—34no—
476C22×C4—32no—
477C22×C4—32no—
478C22×C4—32no—
479C22×C4—32no—
480C22×C4—32no—
481C22×C4—44no—
482C22×C4—68no—
483C22×C4—68no—
484C22×C4—68no—
485C22×C4—124no—
486C22×C4—124no—
487C22×C4—124no—
488C22×C4C23×D314yes—
489C22×C4C23×D312yes—
490C22×C4C23×D312yes—
491C22×C4C23×D311yescharacteristic
492C22×C4C23×D311yescharacteristic
493C22×C4C23×D311yescharacteristic
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—38no—
498C4⋊C4—38no—
499C4⋊C4—68no—
500C4⋊C4—68no—
501C4⋊C4—68no—
502C4⋊C4—68no—
503C4⋊C4—62no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—62no—
507C4⋊C4—124no—
508C4⋊C4—124no—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—68no—
514C22⋊C4—68no—
515C22⋊C4—64no—
516C22⋊C4—64no—
517C22⋊C4—68no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—124no—
521C42—64no—
522C42—64no—
523C42—62no—
524C42—62no—
525C42—124no—
526C42D3×D414yes—
527C22×C3—24no—
528C22×C3—24no—
529C22×C3—24no—
530C22×C3—28no—
531C22×C3—24no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—44no—
539C22×C3C23×D411yescharacteristic
540C22×C3C22×#4111yescharacteristic
541C22×C3C2×H(5,𝔽2)12yes—
542C22×C3C2×2−1+411yescharacteristic
543C22×C3C2×2−1+412yes—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—42no—
547C3×C4—42no—
548C3×C4—44no—
549C3×C4#54814yes—
550Dic3—24no—
551Dic3—28no—
552Dic3—42no—
553Dic3—42no—
554Dic3—42no—
555Dic3—42no—
556Dic3—42no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23C2×#23714yes—
568C23#101014yes—
569C23C24×D311yescharacteristic
570Q8—68no—
571Q8—128no—
572Q8—1216no—
573Q8—124no—
574D4—28no—
575D4—28no—
576D4—416no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—38no—
588C2×C4—34no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—62no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—1216no—
610C2×C4—248no—
611C2×C4C2×D3×D412yes—
612C2×C4C2×D3×D412yes—
613C2×C4C2×D3×D412yes—
614C2×C4C2×#23712yes—
615C2×C4#101012yes—
616C2×C3—44no—
617C2×C3—44no—
618C2×C3C2×#54811yescharacteristic
619C2×C3#348411yescharacteristic
620C2×C3#348611yescharacteristic
621C2×C3#352012yes—
622C2×C3#355212yes—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22C22×D3×D411yescharacteristic
636C22C22×#23711yescharacteristic
637C22C2×#101012yes—
638C22C2×#101011yescharacteristic
639C22D3×H(5,𝔽2)12yes—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—64no—
646C4—68no—
647C4—122no—
648C4—122no—
649C4—122no—
650C4—122no—
651C4—122no—
652C4#559814yes—
653C3#4533711yescharacteristic, complemented, Sylow
654C2—44no—
655C2—44no—
656C2C2×#559811yescharacteristic
657C2#8602211yescharacteristic
658C2#8633011yescharacteristic
659C2#8664112yes—
660C2#8689812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC3×#2535C2
hypercenter#2535D3
hypocenterC3#45337
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C2×#5543C2semidirect
2. C2×D4×Dic6C2semidirect
4. #82647C2semidirect
5. #82636C2semidirect
6. #82348C2semidirect
7. #82328C2semidirect
8. #77573C2semidirect
9. #76681C2semidirect
10. #74827C2semidirect
11. #74689C2semidirect
12. #74191C2semidirect
13. #74154C2semidirect
14. #73916C2semidirect
15. #73899C2semidirect
154. #869D4semidirect
653. C3#45337semidirect

Maximal subgroups

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

subgroupindex
C2×#55432
C2×D4×Dic62
C3×#25352
#826472
#826362
#823482
#823282
#775732
#766812
#748272
#746892
#741912
#741542
#739162
#738992
#453373