Finite Group Databasebeta

← #11233652

Subgroups — #11233652

Order 768.

2498 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×#5713C211yesmaximal, characteristic, complemented
3#84090C214yesmaximal, complemented
4#83178C212yesmaximal, complemented
5#83176C212yesmaximal, complemented
6#83172C211yesmaximal, characteristic, complemented
7#82694C214yesmaximal, complemented
8#82687C214yesmaximal, complemented
9#82655C214yesmaximal, complemented
10#82480C211yesmaximal, characteristic, complemented
11#82467C211yesmaximal, characteristic, complemented
12#82388C211yesmaximal, characteristic, complemented
13#82359C212yesmaximal, complemented
14D4×Dic3⋊C4C212yesmaximal, complemented
15#82328C211yesmaximal, characteristic, complemented
16#45354—31nomaximal, Sylow
17C22×C3×#41C2212yes—
18C3×#550—28no—
19C2×C3×#127C2212yes—
20C2×C3×#127C2211yescharacteristic
21C2×C3×#121C2212yes—
22C2×C3×#121C2214yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2212yes—
26C2×#936C2211yescharacteristic
27C2×#936C2212yes—
28C2×#936C2212yes—
29C2×#935C2214yes—
30C2×#935C2214yes—
31C2×D4×Dic3C2212yes—
32C2×D4×Dic3C2212yes—
33C2×D4×Dic3C2212yes—
34C22×Dic3⋊C4C2214yes—
35#5713—28no—
36C2×C4×Dic6C2211yescharacteristic
37C2×C4×Dic6C2211yescharacteristic
38#5230C2212yes—
39#5230C2212yes—
40#5222C2214yes—
41#5222C2214yes—
42#5222C2214yes—
43#5222C2214yes—
44#5221C2211yescharacteristic
45#5221C2212yes—
46#5221C2212yes—
47#5220C2214yes—
48#5220C2214yes—
49#5220C2214yes—
50#5220C2214yes—
51#5220C2214yes—
52#5211C2214yes—
53#5210C2214yes—
54#5210C2214yes—
55#5210C2214yes—
56#5210C2214yes—
57#4985C2212yes—
58#4983C2212yes—
59#4981C2212yes—
60#4979C2212yes—
61#4978C2214yes—
62#4975C2211yescharacteristic
63#4973C2211yescharacteristic
64#4972C2212yes—
65#4952C2214yes—
66#4952C2214yes—
67#4949C2212yes—
68#4947C2212yes—
69#4946C2214yes—
70#4944C2214yes—
71#4942C2214yes—
72#4936C2211yescharacteristic
73#4932C2211yescharacteristic
74C4×Dic3⋊C4C2212yes—
75C4×Dic3⋊C4C2212yes—
76#4929C2212yes—
77#4928C2212yes—
78C2×#559—31no—
79C2×#550D311yescharacteristic
80#2708—34no—
81#2686—34no—
82#2581—31no—
83#2580—31no—
84#2544—32no—
85#2447—32no—
86#2445—32no—
87#2443—34no—
88#2406—34no—
89#2399—31no—
90D4×C4⋊C4—32no—
91#2328—31no—
92#2326—31no—
93C2×C3×#41—28no—
94C2×C3×#41—48no—
95C22×C3×Q8C2311yescharacteristic
96C22×C3×D4C2311yescharacteristic
97C22×C3×D4C2314yes—
98C22×C3×D4C2312yes—
99C22×C3×D4C2314yes—
100C22×C3×D4C2312yes—
101C22×C3×D4C2312yes—
102C23×C3×C4C2314yes—
103C23×C3×C4C2312yes—
104C23×Dic3C2314yes—
105C23×Dic3C2314yes—
106C22×Dic6C2311yescharacteristic
107C22×Dic6C2311yescharacteristic
108C3×#127—28no—
109C3×#127D418yes—
110C3×#121—28no—
111C3×#121—216no—
112C3×C4×Q8—24no—
113C3×C4×D4—28no—
114C3×C4×D4—24no—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C42C2312yes—
121C2×C3×C42C2311yescharacteristic
122C2×#217C2314yes—
123C2×#217C2314yes—
124C2×#217C2314yes—
125C2×#217C2314yes—
126C2×#217C2314yes—
127C2×#217C2314yes—
128C2×#217C2314yes—
129C2×#217C2314yes—
130C2×#217C2314yes—
131C2×#217C2314yes—
132#936—28no—
133#936—28no—
134#936—24no—
135#935—216no—
136#935—48no—
137D4×Dic3—28no—
138D4×Dic3—28no—
139D4×Dic3—28no—
140C2×C4⋊Dic3C2311yescharacteristic
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2311yescharacteristic
143C2×Dic3⋊C4—48no—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2312yes—
150C2×Dic3⋊C4C2312yes—
151C2×Dic3⋊C4C2314yes—
152C2×Dic3⋊C4C2314yes—
153C2×C4×Dic3C2311yescharacteristic
154C2×C4×Dic3C2312yes—
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2311yescharacteristic
157C4×Dic6—24no—
158C4×Dic6—24no—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2312yes—
163C3⋊#95C2312yes—
164C3⋊#95C2312yes—
165C3⋊#95C2312yes—
166C3⋊#95C2312yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C22×#41C2×D312yes—
173#559—68no—
174#550—28no—
175C2×#127C2×D312yes—
176C2×#127C2×D311yescharacteristic
177C2×#124—31no—
178C2×#124—32no—
179C2×#124—32no—
180C2×#123—34no—
181C2×#123—34no—
182C2×#121C2×D314yes—
183C2×#121C2×D312yes—
184C2×C4×Q8—31no—
185C2×C4×Q8—31no—
186C2×C4×Q8C2×D311yescharacteristic
187C2×C4×D4—32no—
188C2×C4×D4—32no—
189C2×C4×D4—32no—
190C2×C4×D4C2×D311yescharacteristic
191C2×C4×D4C2×D312yes—
192C22×C4⋊C4—34no—
193#400—32no—
194#399—32no—
195#398—32no—
196#398—32no—
197#397—34no—
198#397—34no—
199#395—32no—
200#395—32no—
201#395—32no—
202#394—34no—
203#394—34no—
204#394—34no—
205#394—34no—
206#393—32no—
207#391—31no—
208#391—32no—
209#390—32no—
210#390—31no—
211#390—32no—
212#389—32no—
213#389—32no—
214#388—34no—
215#387—34no—
216#386—34no—
217#386—34no—
218#386—34no—
219#386—34no—
220#386—34no—
221#385—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#385—34no—
226#384—31no—
227#382—31no—
228#381—34no—
229#380—34no—
230C4×C4⋊C4—32no—
231C4×C4⋊C4—32no—
232#376—31no—
233C24×C3C2412yes—
234C24×C3C2414yes—
235C3×#41—432no—
236C2×C3×Q8—22no—
237C2×C3×Q8—24no—
238C2×C3×D4—28no—
239C2×C3×D4—28no—
240C2×C3×D4—22no—
241C2×C3×D4—24no—
242C2×C3×D4—22no—
243C2×C3×D4—28no—
244C2×C3×D4—24no—
245C2×C3×D4—22no—
246C2×C3×D4—44no—
247C2×C3×D4—44no—
248C2×C3×D4—44no—
249C2×C3×D4—44no—
250C2×C3×D4—44no—
251C2×C3×D4C2×D418yes—
252C2×C3×D4C2×D418yes—
253C2×C3×D4C2×D418yes—
254C22×C3×C4—28no—
255C22×C3×C4—24no—
256C22×C3×C4—28no—
257C22×C3×C4—44no—
258C22×C3×C4—44no—
259C22×C3×C4C2412yes—
260C22×C3×C4C2412yes—
261C22×C3×C4C2411yescharacteristic
262C22×C3×C4C2411yescharacteristic
263C22×C3×C4C2412yes—
264C22×C3×C4C2411yescharacteristic
265C22×Dic3—28no—
266C22×Dic3—28no—
267C22×Dic3—44no—
268C22×Dic3—44no—
269C22×Dic3—44no—
270C22×Dic3—44no—
271C22×Dic3C2412yes—
272C22×Dic3C2412yes—
273C22×Dic3C2414yes—
274C22×Dic3C2412yes—
275C22×Dic3C2412yes—
276C22×Dic3C2412yes—
277C22×Dic3C2412yes—
278C2×Dic6—22no—
279C2×Dic6—22no—
280C2×Dic6—22no—
281C2×Dic6—22no—
282C2×Dic6—42no—
283C3×C4⋊C4—22no—
284C3×C4⋊C4—24no—
285C3×C4⋊C4—22no—
286C3×C22⋊C4—28no—
287C3×C22⋊C4—24no—
288C3×C42—24no—
289C3×C42C2×D414yes—
290#217—28no—
291#217—28no—
292#217—28no—
293#217—28no—
294#217—28no—
295#217—44no—
296#217—44no—
297#217—44no—
298#217—44no—
299#217—44no—
300C4⋊Dic3—22no—
301C4⋊Dic3—22no—
302C4⋊Dic3—24no—
303Dic3⋊C4—24no—
304Dic3⋊C4—24no—
305Dic3⋊C4—24no—
306Dic3⋊C4—28no—
307Dic3⋊C4—24no—
308Dic3⋊C4—44no—
309Dic3⋊C4—44no—
310Dic3⋊C4—42no—
311Dic3⋊C4—42no—
312C4×Dic3—24no—
313C4×Dic3—24no—
314C4×Dic3—22no—
315C4×Dic3#4114yescomplemented
316C2×#41—28no—
317C2×#41—48no—
318C22×Q8—31no—
319C22×Q8—31no—
320C22×Q8C22×D311yescharacteristic
321C22×D4C22×D311yescharacteristic
322C22×D4C22×D314yes—
323C22×D4C22×D312yes—
324C22×D4C22×D314yes—
325C22×D4C22×D312yes—
326C22×D4C22×D312yes—
327C23×C4—34no—
328C23×C4—34no—
329C23×C4C22×D314yes—
330C23×C4C22×D312yes—
331#127—28no—
332#127#6718yes—
333#124—64no—
334#124—68no—
335#124—68no—
336#123—616no—
337#123—128no—
338#121—28no—
339#121—216no—
340C4×Q8—24no—
341C4×Q8—64no—
342C4×Q8—64no—
343C4×D4—28no—
344C4×D4—24no—
345C4×D4—68no—
346C4×D4—68no—
347C4×D4—68no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—32no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—32no—
356C2×C4⋊C4—34no—
357C2×C4⋊C4—34no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—128no—
361C2×C4⋊C4C22×D311yescharacteristic
362C2×C4⋊C4C22×D312yes—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—34no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—34no—
368C2×C22⋊C4—34no—
369C2×C22⋊C4—34no—
370C2×C22⋊C4—34no—
371C2×C22⋊C4—34no—
372C2×C22⋊C4—34no—
373C2×C22⋊C4—34no—
374C2×C22⋊C4C22×D314yes—
375C2×C22⋊C4C22×D312yes—
376C2×C42—31no—
377C2×C42—32no—
378C2×C42—32no—
379C2×C42—31no—
380C2×C42C22×D312yes—
381C2×C42C22×D311yescharacteristic
382#95—32no—
383#95—32no—
384#95—32no—
385#95—32no—
386#95—32no—
387#95—32no—
388#95—34no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95—32no—
393#95—32no—
394#95—34no—
395C23×C3—24no—
396C23×C3—24no—
397C23×C3—24no—
398C23×C3—24no—
399C23×C3—24no—
400C23×C3—24no—
401C23×C3—22no—
402C23×C3—22no—
403C23×C3—24no—
404C23×C3—22no—
405C23×C3—24no—
406C23×C3C22×D418yes—
407C23×C3C2511yescharacteristic
408C3×Q8—28no—
409C3×D4—216no—
410C3×D4—48no—
411C3×D4—48no—
412C3×D4—416no—
413C3×D4—44no—
414C3×D4—88no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—24no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—416no—
425C2×C3×C4—48no—
426C2×C3×C4C22×D412yes—
427C2×C3×C4C22×D412yes—
428C2×C3×C4C22×D412yes—
429C2×C3×C4C2×#4112yes—
430C2×C3×C4H(5,𝔽2)14yes—
431C2×C3×C42−1+412yes—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—24no—
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—24no—
442C2×Dic3—24no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3—22no—
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—24no—
451C2×Dic3—416no—
452C2×Dic3—88no—
453C2×Dic3C2×#4114yes—
454Dic6—44no—
455Dic6—44no—
456C24C23×D312yes—
457C24C23×D314yes—
458#41—432no—
459C2×Q8—24no—
460C2×Q8—22no—
461C2×Q8—62no—
462C2×Q8—62no—
463C2×Q8—62no—
464C2×Q8—62no—
465C2×Q8—122no—
466C2×D4—28no—
467C2×D4—28no—
468C2×D4—28no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—24no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—44no—
479C2×D4C2×#6718yes—
480C2×D4C2×#6718yes—
481C2×D4C2×#6718yes—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—24no—
485C22×C4—32no—
486C22×C4—32no—
487C22×C4—32no—
488C22×C4—34no—
489C22×C4—32no—
490C22×C4—32no—
491C22×C4—32no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—68no—
495C22×C4—68no—
496C22×C4—124no—
497C22×C4—124no—
498C22×C4—124no—
499C22×C4—124no—
500C22×C4C23×D312yes—
501C22×C4C23×D312yes—
502C22×C4C23×D312yes—
503C22×C4C23×D311yescharacteristic
504C22×C4C23×D311yescharacteristic
505C22×C4C23×D311yescharacteristic
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—64no—
510C4⋊C4—68no—
511C4⋊C4—64no—
512C4⋊C4—64no—
513C4⋊C4—62no—
514C4⋊C4—62no—
515C4⋊C4—64no—
516C4⋊C4—64no—
517C4⋊C4—124no—
518C4⋊C4—124no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C22⋊C4—28no—
522C22⋊C4—24no—
523C22⋊C4—68no—
524C22⋊C4—68no—
525C22⋊C4—68no—
526C22⋊C4—68no—
527C22⋊C4—68no—
528C22⋊C4—124no—
529C22⋊C4—124no—
530C22⋊C4—124no—
531C22⋊C4—124no—
532C22⋊C4—124no—
533C42—24no—
534C42—34no—
535C42—62no—
536C42—64no—
537C42—64no—
538C42C2×#6714yes—
539C22×C3—28no—
540C22×C3—28no—
541C22×C3—28no—
542C22×C3—44no—
543C22×C3—44no—
544C22×C3—44no—
545C22×C3—44no—
546C22×C3—42no—
547C22×C3—42no—
548C22×C3—42no—
549C22×C3—44no—
550C22×C3—44no—
551C22×C3C23×D411yescharacteristic
552C22×C3C22×#4111yescharacteristic
553C22×C3C2×H(5,𝔽2)12yes—
554C22×C3C2×2−1+411yescharacteristic
555C22×C3#58512yes—
556C3×C4—22no—
557C3×C4—24no—
558C3×C4—22no—
559C3×C4—42no—
560C3×C4—42no—
561C3×C4#54714yes—
562Dic3—24no—
563Dic3—42no—
564Dic3—42no—
565Dic3—42no—
566Dic3—42no—
567Dic3—44no—
568Dic3—42no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—22no—
580C23C22×#6718yes—
581C23C24×D311yescharacteristic
582Q8—28no—
583Q8—124no—
584Q8—124no—
585D4—216no—
586D4—48no—
587D4—48no—
588D4—416no—
589D4—44no—
590D4—88no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—34no—
601C2×C4—416no—
602C2×C4—48no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—64no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—64no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—64no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—1216no—
623C2×C4—248no—
624C2×C4#100414yes—
625C2×C4D3×#4112yes—
626C2×C4#101012yes—
627C2×C4C22×#6712yes—
628C2×C4C22×#6712yes—
629C2×C4C22×#6712yes—
630C2×C3—48no—
631C2×C3—82no—
632C2×C3C2×#54711yescharacteristic
633C2×C3#348411yescharacteristic
634C2×C3#348511yescharacteristic
635C2×C3#350612yes—
636C2×C3#355412yes—
637C22—28no—
638C22—28no—
639C22—28no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22C2×#100412yes—
650C22C2×D3×#4111yescharacteristic
651C22C2×#101011yescharacteristic
652C22#596612yes—
653C22C23×#6711yescharacteristic
654C4—24no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—64no—
660C4—124no—
661C4—122no—
662C4—122no—
663C4—122no—
664C4—122no—
665C4—122no—
666C4#582414yes—
667C3#4535411yescharacteristic, complemented, Sylow
668C2—48no—
669C2—82no—
670C2#8678512yes—
671C2#8678912yes—
672C2#8723911yescharacteristic
673C2C2×#582411yescharacteristic
674C2#8728011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC2×C3×#550C2
hypercenterC2×#550D3
hypocenterC3#45354
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. C2×#5713C2semidirect
3. #84090C2semidirect
4. #83178C2semidirect
5. #83176C2semidirect
6. #83172C2semidirect
7. #82694C2semidirect
8. #82687C2semidirect
9. #82655C2semidirect
10. #82480C2semidirect
11. #82467C2semidirect
12. #82388C2semidirect
13. #82359C2semidirect
14. D4×Dic3⋊C4C2semidirect
15. #82328C2semidirect
315. C4×Dic3#41semidirect
667. C3#45354semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×#57132
#840902
#831782
#831762
#831722
#826942
#826872
#826552
#824802
#824672
#823882
#823592
D4×Dic3⋊C42
#823282
#453543