Finite Group Databasebeta

← #11166408

Subgroups — #11166408

Order 768.

2054 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5616C211yesmaximal, characteristic
2C2×#5500C211yesmaximal, characteristic, complemented
3C3×#2504C211yesmaximal, characteristic, complemented
4#82678C211yesmaximal, characteristic, complemented
5#82196C211yesmaximal, characteristic, complemented
6#77647C212yesmaximal, complemented
7#77547C212yesmaximal, complemented
8#77538C212yesmaximal, complemented
9#77537C212yesmaximal, complemented
10#76449C212yesmaximal, complemented
11#76323C212yesmaximal, complemented
12#74811C214yesmaximal, complemented
13#74794C214yesmaximal, complemented
14#74180C212yesmaximal, complemented
15#74162C212yesmaximal, complemented
16#73583C211yesmaximal, characteristic, complemented
17#73459C211yesmaximal, characteristic, complemented
18#42965—31nomaximal, Sylow
19C2×C3×#121C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×#939C2212yes—
22C2×#938C2211yescharacteristic
23C2×#936C2212yes—
24C2×D4×Dic3C2211yescharacteristic
25C2×#929C2212yes—
26C2×C4×#67C2211yescharacteristic
27C2×#926C2211yescharacteristic
28C2×#926C2211yescharacteristic
29C2×#924C2214yes—
30#5616—28no—
31C2×#892C2211yescharacteristic
32#5500—28no—
33C2×#890C2212yes—
34C2×#890C2212yes—
35C2×#888C2212yes—
36C2×#888C2211yescharacteristic
37C2×#888C2212yes—
38C2×#885C2212yes—
39C2×#878C2214yes—
40C2×#877C2212yes—
41C3×#397C2212yes—
42C3×#397C2212yes—
43C3×#390C2212yes—
44C3×#386C2214yes—
45C3×#382C2212yes—
46C3×C4×C22⋊C4C2211yescharacteristic
47#5222C2214yes—
48#5221C2212yes—
49#5212C2211yescharacteristic
50#5211C2212yes—
51#4992C2212yes—
52#4987C2212yes—
53#4983C2212yes—
54#4972C2212yes—
55#4972C2212yes—
56#4961C2212yes—
57#4961C2214yes—
58#4957C2214yes—
59#4953C2212yes—
60#4953C2212yes—
61#4952C2214yes—
62#4951C2214yes—
63#4949C2212yes—
64#4948C2214yes—
65#4947C2214yes—
66#4946C2214yes—
67#4944C2214yes—
68C4×(C2×D3)⋊C4C2212yes—
69C4×C4⋊Dic3C2212yes—
70C4×C4⋊Dic3C2212yes—
71#4928C2212yes—
72#4677C2211yescharacteristic
73#4676C2214yes—
74#4674C2211yescharacteristic
75#4673C2214yes—
76#4661C2211yescharacteristic
77#4659C2211yescharacteristic
78#4657C2214yes—
79#4656C2214yes—
80#4652C2214yes—
81#4650C2212yes—
82#4648C2212yes—
83#4647C2214yes—
84#4646C2214yes—
85#4646C2214yes—
86C2×#555—31no—
87C2×#544—31no—
88#2820—32no—
89#2700—34no—
90#2681—34no—
91#2640—32no—
92#2624—31no—
93#2621—32no—
94#2611—32no—
95#2550—32no—
96#2504D311yescharacteristic, complemented
97#2490—32no—
98#2486—31no—
99#2452—32no—
100#2439—32no—
101#2395—31no—
102#2346—31no—
103C22×C3×D4C2311yescharacteristic
104C22×C3×D4C2312yes—
105C23×C3×C4C2311yescharacteristic
106C23×C3×C4C2312yes—
107C22×#67C2311yescharacteristic
108C22×#67C2312yes—
109C23×Dic3C2312yes—
110C22×C4×D3C2311yescharacteristic
111C22×Dic6C2312yes—
112C22×Dic6C2314yes—
113C3×#121—24no—
114C2×C3×C4⋊C4—24no—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C22⋊C4C2314yes—
121C2×C3×C22⋊C4C2314yes—
122C2×C3×C42C2312yes—
123C2×#217C2312yes—
124C2×#217C2312yes—
125C2×#217C2311yescharacteristic
126C2×#217C2312yes—
127C2×#217C2312yes—
128C2×#217C2314yes—
129C2×#217C2314yes—
130C2×#217C2312yes—
131#939—28no—
132#938—24no—
133#936—28no—
134D4×Dic3—24no—
135#929—44no—
136C4×#67—24no—
137C2×(C2×D3)⋊C4C2312yes—
138C2×(C2×D3)⋊C4C2311yescharacteristic
139C2×(C2×D3)⋊C4C2312yes—
140C2×(C2×D3)⋊C4C2314yes—
141C2×(C2×D3)⋊C4C2312yes—
142#926—24no—
143#926D418yes—
144C2×C4⋊Dic3C2314yes—
145C2×C4⋊Dic3C2312yes—
146C2×C4⋊Dic3C2311yescharacteristic
147C2×C4⋊Dic3C2311yescharacteristic
148C2×C4⋊Dic3C2312yes—
149C2×C4⋊Dic3C2312yes—
150C2×C4⋊Dic3C2311yescharacteristic
151C2×C4⋊Dic3C2311yescharacteristic
152#924—216no—
153C2×Dic3⋊C4C2314yes—
154C2×Dic3⋊C4C2312yes—
155C2×Dic3⋊C4C2314yes—
156C2×Dic3⋊C4C2314yes—
157C2×Dic3⋊C4C2311yescharacteristic
158C2×Dic3⋊C4C2314yes—
159C2×Dic3⋊C4C2314yes—
160C2×C4×Dic3C2312yes—
161C2×C4×Dic3C2311yescharacteristic
162C2×C4×Dic3C2312yes—
163C2×C4×Dic3C2314yes—
164C2×C4×Dic3C2312yes—
165#892—24no—
166#890—28no—
167#890—28no—
168#888—28no—
169#888—24no—
170#888—28no—
171#885—28no—
172#878—48no—
173#877—28no—
174C3×#95C2314yes—
175C3×#95C2314yes—
176C3×#95C2311yescharacteristic
177C3×#95C2311yescharacteristic
178C3⋊#95C2314yes—
179C3⋊#95C2312yes—
180C3⋊#95C2312yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2312yes—
184C3⋊#95C2312yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2312yes—
188#555—68no—
189#544—68no—
190C2×C4⋊Q8—32no—
191C2×C4⋊Q8—32no—
192C2×C4⋊Q8—31no—
193C2×#125—32no—
194C2×#125—32no—
195C2×#124—32no—
196C2×#124—32no—
197C2×#123—32no—
198C2×C22⋊Q8—34no—
199C2×C22⋊Q8—34no—
200C2×#121—32no—
201C2×#121—31no—
202C2×#121C2×D311yescharacteristic
203C2×C4×D4—31no—
204C2×C4×D4—31no—
205C2×#117—31no—
206C2×#117—31no—
207C2×#117—31no—
208C2×#117—32no—
209C22×C4⋊C4C2×D311yescharacteristic
210#400—31no—
211#399—32no—
212#399—31no—
213#398—32no—
214#398—34no—
215#398—31no—
216#397—32no—
217#397—32no—
218#397—34no—
219#397—32no—
220#397—31no—
221#397C2×D312yes—
222#397C2×D312yes—
223#396—34no—
224#396—34no—
225#395—34no—
226#394—34no—
227#394—34no—
228#394—32no—
229#393—34no—
230#393—34no—
231#391—34no—
232#391—34no—
233#390—32no—
234#390—34no—
235#390C2×D312yes—
236#389—32no—
237#389—32no—
238#389—32no—
239#389—34no—
240#388—34no—
241#388—34no—
242#387—34no—
243#386C2×D314yes—
244#385—34no—
245#382—32no—
246#382C2×D312yes—
247#381—32no—
248#380—31no—
249#380—32no—
250C4×C4⋊C4—32no—
251C4×C4⋊C4—32no—
252C4×C22⋊C4—32no—
253C4×C22⋊C4C2×D311yescharacteristic
254#376—32no—
255C24×C3C2412yes—
256C24×C3C2411yescharacteristic
257C23×D3C2411yescharacteristic
258C2×C3×D4—24no—
259C2×C3×D4—22no—
260C2×C3×D4—24no—
261C2×C3×D4—44no—
262C22×C3×C4—24no—
263C22×C3×C4—24no—
264C22×C3×C4—42no—
265C22×C3×C4—42no—
266C22×C3×C4C2×D414yes—
267C22×C3×C4C2412yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2412yes—
270C22×C3×C4C2414yes—
271C22×C3×C4C2411yescharacteristic
272C22×C3×C4C2411yescharacteristic
273C2×#67—22no—
274C2×#67—24no—
275C2×#67—22no—
276C2×#67—41no—
277C2×#67—44no—
278C22×Dic3—24no—
279C22×Dic3—24no—
280C22×Dic3—42no—
281C22×Dic3C2411yescharacteristic
282C22×Dic3C2414yes—
283C22×Dic3C2411yescharacteristic
284C22×Dic3C2412yes—
285C22×Dic3C2412yes—
286C22×Dic3C2412yes—
287C22×Dic3C2412yes—
288C22×Dic3C2411yescharacteristic
289C2×C4×D3—22no—
290C2×C4×D3—24no—
291C2×Dic6—24no—
292C2×Dic6—28no—
293C2×Dic6—44no—
294C2×Dic6—44no—
295C2×Dic6—44no—
296C3×C4⋊C4—24no—
297C3×C4⋊C4—24no—
298C3×C4⋊C4—22no—
299C3×C22⋊C4—24no—
300C3×C22⋊C4—44no—
301C3×C22⋊C4—42no—
302C3×C22⋊C4—44no—
303C3×C42—24no—
304#217—24no—
305#217—22no—
306#217—24no—
307#217—24no—
308#217—44no—
309#217—42no—
310#217—44no—
311#217C2×D418yes—
312(C2×D3)⋊C4—22no—
313(C2×D3)⋊C4—24no—
314(C2×D3)⋊C4—42no—
315(C2×D3)⋊C4—42no—
316(C2×D3)⋊C4—44no—
317C4⋊Dic3—22no—
318C4⋊Dic3—22no—
319C4⋊Dic3—24no—
320C4⋊Dic3—28no—
321C4⋊Dic3—24no—
322C4⋊Dic3—42no—
323C4⋊Dic3C2×D414yes—
324C4⋊Dic3C2×D414yes—
325Dic3⋊C4—22no—
326Dic3⋊C4—24no—
327Dic3⋊C4—28no—
328Dic3⋊C4—28no—
329Dic3⋊C4—28no—
330Dic3⋊C4—44no—
331Dic3⋊C4—44no—
332C4×Dic3—24no—
333C4×Dic3—22no—
334C4×Dic3—24no—
335C4×Dic3—44no—
336C4×Dic3C2×D418yes—
337C22×Q8—34no—
338C22×Q8—32no—
339C22×D4—31no—
340C22×D4—32no—
341C22×D4C22×D311yescharacteristic
342C22×D4C22×D312yes—
343C23×C4—32no—
344C23×C4—31no—
345C23×C4C22×D311yescharacteristic
346C23×C4C22×D312yes—
347C4⋊Q8—68no—
348C4⋊Q8—68no—
349C4⋊Q8—64no—
350#125—68no—
351#125—68no—
352#124—68no—
353#124—68no—
354#123—124no—
355C22⋊Q8—616no—
356C22⋊Q8—128no—
357#121—24no—
358#121—68no—
359#121—64no—
360C4×D4—64no—
361C4×D4—64no—
362#117—38no—
363#117—68no—
364#117—64no—
365#117—64no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—31no—
369C2×C4⋊C4—31no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—31no—
372C2×C4⋊C4—31no—
373C2×C4⋊C4—32no—
374C2×C4⋊C4—34no—
375C2×C4⋊C4—34no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—34no—
378C2×C4⋊C4—32no—
379C2×C4⋊C4—34no—
380C2×C4⋊C4—32no—
381C2×C4⋊C4—34no—
382C2×C4⋊C4C22×D312yes—
383C2×C4⋊C4C22×D312yes—
384C2×C4⋊C4C22×D311yescharacteristic
385C2×C22⋊C4—32no—
386C2×C22⋊C4—32no—
387C2×C22⋊C4—32no—
388C2×C22⋊C4—31no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4—32no—
391C2×C22⋊C4—31no—
392C2×C22⋊C4—34no—
393C2×C22⋊C4—32no—
394C2×C22⋊C4—34no—
395C2×C22⋊C4—34no—
396C2×C22⋊C4—32no—
397C2×C22⋊C4—32no—
398C2×C22⋊C4C22×D312yes—
399C2×C22⋊C4C22×D314yes—
400C2×C22⋊C4C22×D312yes—
401C2×C22⋊C4C22×D314yes—
402C2×C42—32no—
403C2×C42—32no—
404C2×C42—32no—
405C2×C42—31no—
406C2×C42—34no—
407C2×C42C22×D312yes—
408#95—32no—
409#95—34no—
410#95—32no—
411#95—32no—
412#95—34no—
413#95—32no—
414#95—32no—
415#95—34no—
416#95—34no—
417#95—34no—
418#95C22×D311yescharacteristic
419#95C22×D311yescharacteristic
420#95C22×D314yes—
421#95C22×D314yes—
422C23×C3—21no—
423C23×C3—22no—
424C23×C3—22no—
425C23×C3—21no—
426C23×C3—22no—
427C23×C3—22no—
428C23×C3—24no—
429C23×C3—24no—
430C23×C3—22no—
431C23×C3C22×D412yes—
432C23×C3C2511yescharacteristic
433C22×D3—21no—
434C22×D3—21no—
435C22×D3—22no—
436C22×D3—21no—
437C22×D3—22no—
438C3×D4—44no—
439C3×D4—84no—
440C2×C3×C4—22no—
441C2×C3×C4—28no—
442C2×C3×C4—22no—
443C2×C3×C4—24no—
444C2×C3×C4—24no—
445C2×C3×C4—22no—
446C2×C3×C4—22no—
447C2×C3×C4—22no—
448C2×C3×C4—22no—
449C2×C3×C4—24no—
450C2×C3×C4—22no—
451C2×C3×C4—22no—
452C2×C3×C4—24no—
453C2×C3×C4—84no—
454C2×C3×C4C22×D412yes—
455C2×C3×C4C22×D412yes—
456C2×C3×C42−1+414yes—
457#67—44no—
458#67—84no—
459C2×Dic3—21no—
460C2×Dic3—22no—
461C2×Dic3—24no—
462C2×Dic3—22no—
463C2×Dic3—21no—
464C2×Dic3—21no—
465C2×Dic3—22no—
466C2×Dic3—22no—
467C2×Dic3—21no—
468C2×Dic3—24no—
469C2×Dic3—22no—
470C2×Dic3—24no—
471C2×Dic3—22no—
472C2×Dic3—24no—
473C2×Dic3—22no—
474C2×Dic3—22no—
475C2×Dic3—22no—
476C2×Dic3—22no—
477C2×Dic3—48no—
478C2×Dic3C22×D414yes—
479C2×Dic3C22×D414yes—
480C2×Dic3H(5,𝔽2)12yes—
481C4×D3—44no—
482Dic6—88no—
483Dic6—84no—
484C24—31no—
485C24C23×D312yes—
486C24C23×D311yescharacteristic
487C2×Q8—68no—
488C2×Q8—64no—
489C2×Q8—124no—
490C2×Q8—124no—
491C2×Q8—124no—
492C2×D4—24no—
493C2×D4—22no—
494C2×D4—24no—
495C2×D4—44no—
496C2×D4—62no—
497C2×D4—64no—
498C2×D4—62no—
499C2×D4—121no—
500C2×D4—124no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—32no—
504C22×C4—32no—
505C22×C4—31no—
506C22×C4—31no—
507C22×C4—34no—
508C22×C4—31no—
509C22×C4—32no—
510C22×C4—32no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—64no—
514C22×C4—64no—
515C22×C4—62no—
516C22×C4—64no—
517C22×C4—122no—
518C22×C4D3×D414yes—
519C22×C4C23×D312yes—
520C22×C4C23×D312yes—
521C22×C4C23×D312yes—
522C22×C4C23×D314yes—
523C22×C4C23×D311yescharacteristic
524C22×C4C23×D311yescharacteristic
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—34no—
529C4⋊C4—34no—
530C4⋊C4—64no—
531C4⋊C4—68no—
532C4⋊C4—68no—
533C4⋊C4—68no—
534C4⋊C4—64no—
535C4⋊C4—68no—
536C4⋊C4—62no—
537C4⋊C4—62no—
538C4⋊C4—62no—
539C4⋊C4—64no—
540C4⋊C4—124no—
541C4⋊C4—124no—
542C4⋊C4—122no—
543C22⋊C4—24no—
544C22⋊C4—38no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—64no—
549C22⋊C4—64no—
550C22⋊C4—64no—
551C22⋊C4—64no—
552C22⋊C4—62no—
553C22⋊C4—62no—
554C22⋊C4—124no—
555C22⋊C4—122no—
556C22⋊C4—122no—
557C22⋊C4—124no—
558C22⋊C4—122no—
559C22⋊C4—124no—
560C42—24no—
561C42—38no—
562C42—64no—
563C42—64no—
564C42—62no—
565C42—124no—
566C22×C3—22no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—42no—
570C22×C3—42no—
571C22×C3—42no—
572C22×C3—42no—
573C22×C3—42no—
574C22×C3—44no—
575C22×C3—44no—
576C22×C3C23×D411yescharacteristic
577C22×C3C2×H(5,𝔽2)11yescharacteristic
578C22×C3C2×2−1+412yes—
579C22×C3#58512yes—
580C22×C3#58511yescharacteristic
581C2×D3—41no—
582C2×D3—41no—
583C2×D3—41no—
584C2×D3—42no—
585C2×D3—42no—
586C3×C4—22no—
587C3×C4—44no—
588C3×C4—42no—
589C3×C4—42no—
590C3×C4—42no—
591C3×C4#53614yes—
592Dic3—24no—
593Dic3—22no—
594Dic3—24no—
595Dic3—41no—
596Dic3—42no—
597Dic3—41no—
598Dic3—42no—
599Dic3—44no—
600C23—22no—
601C23—22no—
602C23—21no—
603C23—21no—
604C23—22no—
605C23—24no—
606C23—24no—
607C23—22no—
608C23—22no—
609C23—61no—
610C23—62no—
611C23—62no—
612C23—61no—
613C23—61no—
614C23C2×D3×D412yes—
615C23C24×D311yescharacteristic
616Q8—248no—
617Q8—244no—
618D4—44no—
619D4—84no—
620D4—124no—
621D4—244no—
622C2×C4—22no—
623C2×C4—28no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—34no—
636C2×C4—34no—
637C2×C4—32no—
638C2×C4—61no—
639C2×C4—64no—
640C2×C4—62no—
641C2×C4—61no—
642C2×C4—62no—
643C2×C4—61no—
644C2×C4—61no—
645C2×C4—62no—
646C2×C4—62no—
647C2×C4—64no—
648C2×C4—62no—
649C2×C4—64no—
650C2×C4—62no—
651C2×C4—62no—
652C2×C4—62no—
653C2×C4—64no—
654C2×C4—62no—
655C2×C4—62no—
656C2×C4—84no—
657C2×C4—128no—
658C2×C4—124no—
659C2×C4C2×D3×D412yes—
660C2×C4C2×D3×D412yes—
661C2×C4#101014yes—
662C2×C3—42no—
663C2×C3—82no—
664C2×C3C2×#53611yescharacteristic
665C2×C3#348811yescharacteristic
666C2×C3#350011yescharacteristic
667C2×C3#354712yes—
668C2×C3#356512yes—
669D3—81no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22—44no—
679C22—42no—
680C22—121no—
681C22—121no—
682C22—122no—
683C22—122no—
684C22—121no—
685C22C22×D3×D411yescharacteristic
686C22C2×#101012yes—
687C22#596412yes—
688C22D3×H(5,𝔽2)11yescharacteristic
689C22#596611yescharacteristic
690C4—22no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—62no—
696C4—64no—
697C4—64no—
698C4—122no—
699C4—121no—
700C4—122no—
701C4—124no—
702C4—121no—
703C4#564814yes—
704C3#4296511yescharacteristic, complemented, Sylow
705C2—42no—
706C2—82no—
707C2—241no—
708C2C2×#564811yescharacteristic
709C2#8607611yescharacteristic
710C2#8633111yescharacteristic
711C2#8689612yes—
712C2#8690412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2504C2
prime radicalC2×#5616C2
hypercenter#2504D3
hypocenterC3#42965
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2504

Complements

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

NH ≅ G/Ntype
2. C2×#5500C2semidirect
3. C3×#2504C2semidirect
4. #82678C2semidirect
5. #82196C2semidirect
6. #77647C2semidirect
7. #77547C2semidirect
8. #77538C2semidirect
9. #77537C2semidirect
10. #76449C2semidirect
11. #76323C2semidirect
12. #74811C2semidirect
13. #74794C2semidirect
14. #74180C2semidirect
15. #74162C2semidirect
16. #73583C2semidirect
17. #73459C2semidirect
96. #2504D3semidirect
704. C3#42965semidirect

Maximal subgroups

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

subgroupindex
C2×#56162
C2×#55002
C3×#25042
#826782
#821962
#776472
#775472
#775382
#775372
#764492
#763232
#748112
#747942
#741802
#741622
#735832
#734592
#429653