Finite Group Databasebeta

← #11166391

Subgroups — #11166391

Order 768.

1980 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5639C211yesmaximal, characteristic
2C2×#5500C211yesmaximal, characteristic, complemented
3C3×#2502C211yesmaximal, characteristic, complemented
4#82787C211yesmaximal, characteristic, complemented
5#82196C211yesmaximal, characteristic
6#77926C212yesmaximal, complemented
7#77837C212yesmaximal, complemented
8#77818C212yesmaximal, complemented
9#77816C212yesmaximal, complemented
10#76630C212yesmaximal, complemented
11#76509C212yesmaximal, complemented
12#74631C214yesmaximal
13#74521C214yesmaximal, complemented
14#74336C212yesmaximal, complemented
15#74326C212yesmaximal, complemented
16#73583C211yesmaximal, characteristic
17#73459C211yesmaximal, characteristic, complemented
18#42964—31nomaximal, Sylow
19C2×C3×C22⋊Q8C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×#946C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×#929C2212yes—
24C2×C4×#67C2211yescharacteristic
25C2×#926C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C2×#924C2214yes—
28#5639—28no—
29C2×#896C2214yes—
30C2×#892C2212yes—
31C2×D3×C4⋊C4C2211yescharacteristic
32#5500—28no—
33C2×#890C2212yes—
34C2×#890C2211yescharacteristic
35C2×#890C2212yes—
36C2×#889C2212yes—
37C2×#888C2212yes—
38C2×#888C2212yes—
39C2×#883C2212yes—
40C2×#879C2212yes—
41C3×#398C2212yes—
42C3×#398C2212yes—
43C3×#391C2212yes—
44C3×#385C2214yes—
45C3×#384C2212yes—
46C3×C4×C22⋊C4C2211yescharacteristic
47#5229C2212yes—
48#5212C2211yescharacteristic
49#5211C2212yes—
50#4992C2212yes—
51#4989C2214yes—
52#4987C2212yes—
53#4986C2212yes—
54#4984C2214yes—
55#4983C2212yes—
56#4983C2214yes—
57#4982C2214yes—
58#4982C2214yes—
59#4981C2212yes—
60#4979C2214yes—
61#4975C2214yes—
62#4972C2212yes—
63#4972C2212yes—
64#4961C2212yes—
65#4953C2212yes—
66#4952C2214yes—
67#4944C2214yes—
68C4×(C2×D3)⋊C4C2212yes—
69C4×C4⋊Dic3C2212yes—
70C4×C4⋊Dic3C2212yes—
71#4928C2212yes—
72#4677C2211yescharacteristic
73#4675C2214yes—
74#4674C2211yescharacteristic
75#4670C2214yes—
76#4661C2211yescharacteristic
77#4660C2214yes—
78#4659C2211yescharacteristic
79#4657C2214yes—
80#4656C2214yes—
81#4654C2214yes—
82#4650C2214yes—
83#4650C2212yes—
84#4648C2212yes—
85#4646C2214yes—
86C2×#552—31no—
87C2×#544—31no—
88#2822—32no—
89#2698—34no—
90#2680—34no—
91#2660—32no—
92#2625—31no—
93#2621—32no—
94#2611—32no—
95#2552—32no—
96#2502D311yescharacteristic, complemented
97#2489—32no—
98#2486—31no—
99#2451—32no—
100#2440—32no—
101#2395—31no—
102#2346—31no—
103C22×C3×Q8C2311yescharacteristic
104C23×C3×C4C2311yescharacteristic
105C23×C3×C4C2312yes—
106C22×#67C2311yescharacteristic
107C22×C4×D3C2311yescharacteristic
108C22×C4×D3C2312yes—
109C22×Dic6C2314yes—
110C3×C22⋊Q8—24no—
111C2×C3×C4⋊C4—24no—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2314yes—
117C2×C3×C4⋊C4C2314yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C42C2312yes—
121C2×#217C2312yes—
122C2×#217C2311yescharacteristic
123C2×#217C2312yes—
124C2×#217C2312yes—
125#946—24no—
126Q8×Dic3—24no—
127#929—44no—
128C4×#67—24no—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2311yescharacteristic
134#926—24no—
135#926D418yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2311yescharacteristic
139C2×C4⋊Dic3C2314yes—
140C2×C4⋊Dic3C2311yescharacteristic
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2312yes—
143C2×C4⋊Dic3C2312yes—
144C2×C4⋊Dic3C2311yescharacteristic
145C2×C4⋊Dic3C2311yescharacteristic
146#924—216no—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2311yescharacteristic
151C2×Dic3⋊C4C2312yes—
152C2×Dic3⋊C4C2312yes—
153C2×Dic3⋊C4C2314yes—
154C2×Dic3⋊C4C2314yes—
155C2×Dic3⋊C4C2314yes—
156C2×C4×Dic3C2312yes—
157C2×C4×Dic3C2314yes—
158C2×C4×Dic3C2311yescharacteristic
159C2×C4×Dic3C2312yes—
160C2×C4×Dic3C2312yes—
161#896—48no—
162#892—28no—
163D3×C4⋊C4—24no—
164#890—24no—
165#890—28no—
166#890—28no—
167#889—28no—
168#888—28no—
169#888—28no—
170#883—28no—
171#879—28no—
172C3×#95C2314yes—
173C3×#95C2311yescharacteristic
174C3×#95C2311yescharacteristic
175C3×#95C2314yes—
176C3⋊#95C2312yes—
177C3⋊#95C2314yes—
178C3⋊#95C2314yes—
179C3⋊#95C2312yes—
180C3⋊#95C2314yes—
181C3⋊#95C2312yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2312yes—
186C3⋊#95C2312yes—
187C3⋊#95C2314yes—
188#552—68no—
189#544—68no—
190C2×C4⋊Q8—32no—
191C2×C4⋊Q8—32no—
192C2×#126—32no—
193C2×#125—32no—
194C2×#125—32no—
195C2×#125—32no—
196C2×#125—31no—
197C2×#123—32no—
198C2×#123—32no—
199C2×C22⋊Q8—31no—
200C2×C22⋊Q8—34no—
201C2×C22⋊Q8—34no—
202C2×C22⋊Q8C2×D311yescharacteristic
203C2×C4×Q8—31no—
204C2×C4×D4—31no—
205C2×#117—31no—
206C2×#117—31no—
207C2×#117—32no—
208C22×C4⋊C4—31no—
209C22×C4⋊C4C2×D311yescharacteristic
210#400—32no—
211#400—31no—
212#400—34no—
213#399—31no—
214#398—32no—
215#398—34no—
216#398—34no—
217#398—32no—
218#398—34no—
219#398—34no—
220#398—31no—
221#398C2×D312yes—
222#398C2×D312yes—
223#397—32no—
224#397—34no—
225#397—32no—
226#397—31no—
227#397—34no—
228#395—34no—
229#395—34no—
230#394—32no—
231#391—32no—
232#391—34no—
233#391C2×D312yes—
234#389—34no—
235#389—32no—
236#389—32no—
237#389—32no—
238#388—34no—
239#388—34no—
240#387—34no—
241#385—34no—
242#385C2×D314yes—
243#384—34no—
244#384C2×D312yes—
245#382—32no—
246#382—34no—
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×C3C2411yescharacteristic
256C23×D3C2411yescharacteristic
257C2×C3×Q8—22no—
258C2×C3×Q8—24no—
259C22×C3×C4—24no—
260C22×C3×C4—24no—
261C22×C3×C4—42no—
262C22×C3×C4—42no—
263C22×C3×C4C2×D414yes—
264C22×C3×C4C2412yes—
265C22×C3×C4C2412yes—
266C22×C3×C4C2414yes—
267C22×C3×C4C2412yes—
268C22×C3×C4C2411yescharacteristic
269C22×C3×C4C2412yes—
270C22×C3×C4C2411yescharacteristic
271C2×#67—22no—
272C2×#67—22no—
273C2×#67—41no—
274C22×Dic3C2414yes—
275C22×Dic3C2411yescharacteristic
276C22×Dic3C2411yescharacteristic
277C22×Dic3C2412yes—
278C22×Dic3C2412yes—
279C22×Dic3C2411yescharacteristic
280C22×Dic3C2412yes—
281C22×Dic3C2412yes—
282C2×C4×D3—24no—
283C2×C4×D3—22no—
284C2×C4×D3—24no—
285C2×C4×D3—24no—
286C2×C4×D3—42no—
287C2×Dic6—28no—
288C2×Dic6—44no—
289C2×Dic6—44no—
290C3×C4⋊C4—24no—
291C3×C4⋊C4—24no—
292C3×C4⋊C4—24no—
293C3×C4⋊C4—22no—
294C3×C4⋊C4—44no—
295C3×C4⋊C4—44no—
296C3×C22⋊C4—24no—
297C3×C22⋊C4—24no—
298C3×C42—24no—
299#217—22no—
300#217—24no—
301#217—42no—
302#217C2×D418yes—
303(C2×D3)⋊C4—24no—
304(C2×D3)⋊C4—28no—
305(C2×D3)⋊C4—22no—
306(C2×D3)⋊C4—24no—
307(C2×D3)⋊C4—24no—
308C4⋊Dic3—22no—
309C4⋊Dic3—22no—
310C4⋊Dic3—24no—
311C4⋊Dic3—24no—
312C4⋊Dic3—24no—
313C4⋊Dic3—24no—
314C4⋊Dic3—28no—
315C4⋊Dic3—42no—
316C4⋊Dic3C2×D414yes—
317C4⋊Dic3C2×D414yes—
318Dic3⋊C4—22no—
319Dic3⋊C4—24no—
320Dic3⋊C4—24no—
321Dic3⋊C4—24no—
322Dic3⋊C4—28no—
323Dic3⋊C4—28no—
324Dic3⋊C4—44no—
325Dic3⋊C4—44no—
326Dic3⋊C4—44no—
327C4×Dic3—24no—
328C4×Dic3—22no—
329C4×Dic3—28no—
330C4×Dic3—24no—
331C4×Dic3C2×D418yes—
332C22×Q8—34no—
333C22×Q8C22×D311yescharacteristic
334C22×D4—31no—
335C23×C4—31no—
336C23×C4—32no—
337C23×C4C22×D311yescharacteristic
338C23×C4C22×D312yes—
339C4⋊Q8—68no—
340C4⋊Q8—68no—
341#126—68no—
342#125—68no—
343#125—68no—
344#125—68no—
345#125—64no—
346#123—68no—
347#123—124no—
348C22⋊Q8—24no—
349C22⋊Q8—616no—
350C22⋊Q8—64no—
351C22⋊Q8—128no—
352C4×Q8—64no—
353C4×D4—64no—
354#117—38no—
355#117—68no—
356#117—64no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—31no—
359C2×C4⋊C4—31no—
360C2×C4⋊C4—32no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4—31no—
364C2×C4⋊C4—31no—
365C2×C4⋊C4—32no—
366C2×C4⋊C4—34no—
367C2×C4⋊C4—32no—
368C2×C4⋊C4—34no—
369C2×C4⋊C4—34no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—32no—
373C2×C4⋊C4—34no—
374C2×C4⋊C4—34no—
375C2×C4⋊C4—34no—
376C2×C4⋊C4—32no—
377C2×C4⋊C4—64no—
378C2×C4⋊C4C22×D311yescharacteristic
379C2×C4⋊C4C22×D312yes—
380C2×C4⋊C4C22×D312yes—
381C2×C4⋊C4C22×D312yes—
382C2×C4⋊C4C22×D314yes—
383C2×C4⋊C4C22×D314yes—
384C2×C22⋊C4—32no—
385C2×C22⋊C4—32no—
386C2×C22⋊C4—31no—
387C2×C22⋊C4—31no—
388C2×C22⋊C4—32no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4—34no—
391C2×C22⋊C4—32no—
392C2×C22⋊C4—32no—
393C2×C22⋊C4C22×D312yes—
394C2×C22⋊C4C22×D312yes—
395C2×C42—32no—
396C2×C42—32no—
397C2×C42—32no—
398C2×C42—31no—
399C2×C42—34no—
400C2×C42C22×D312yes—
401#95—32no—
402#95—34no—
403#95—32no—
404#95—32no—
405#95—34no—
406#95—32no—
407#95—32no—
408#95—34no—
409#95—34no—
410#95—34no—
411#95—34no—
412#95—34no—
413#95C22×D311yescharacteristic
414#95C22×D311yescharacteristic
415#95C22×D314yes—
416#95C22×D314yes—
417C23×C3—21no—
418C23×C3—22no—
419C23×C3—22no—
420C23×C3—21no—
421C23×C3C22×D412yes—
422C23×C3C2511yescharacteristic
423C22×D3—21no—
424C22×D3—22no—
425C22×D3—22no—
426C22×D3—21no—
427C22×D3H(5,𝔽2)12yes—
428C3×Q8—44no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—28no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—24no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—24no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—24no—
442C2×C3×C4—22no—
443C2×C3×C4—24no—
444C2×C3×C4—84no—
445C2×C3×C4C22×D412yes—
446C2×C3×C4C22×D412yes—
447C2×C3×C42−1+414yes—
448C2×C3×C42−1+414yes—
449#67—44no—
450C2×Dic3—22no—
451C2×Dic3—24no—
452C2×Dic3—22no—
453C2×Dic3—21no—
454C2×Dic3—21no—
455C2×Dic3—22no—
456C2×Dic3—22no—
457C2×Dic3—21no—
458C2×Dic3—21no—
459C2×Dic3—24no—
460C2×Dic3—22no—
461C2×Dic3—24no—
462C2×Dic3—22no—
463C2×Dic3—22no—
464C2×Dic3—24no—
465C2×Dic3—22no—
466C2×Dic3—22no—
467C2×Dic3—21no—
468C2×Dic3—22no—
469C2×Dic3C22×D414yes—
470C2×Dic3C22×D414yes—
471C4×D3—48no—
472C4×D3—44no—
473Dic6—88no—
474C24—31no—
475C24C23×D311yescharacteristic
476C2×Q8—22no—
477C2×Q8—24no—
478C2×Q8—68no—
479C2×Q8—124no—
480C2×Q8—124no—
481C2×D4—62no—
482C2×D4—62no—
483C2×D4—121no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—32no—
487C22×C4—32no—
488C22×C4—34no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—32no—
492C22×C4—32no—
493C22×C4—31no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—64no—
497C22×C4—64no—
498C22×C4—62no—
499C22×C4—64no—
500C22×C4—122no—
501C22×C4D3×D414yes—
502C22×C4C23×D312yes—
503C22×C4C23×D312yes—
504C22×C4C23×D314yes—
505C22×C4C23×D312yes—
506C22×C4C23×D311yescharacteristic
507C22×C4C23×D312yes—
508C22×C4C23×D311yescharacteristic
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—34no—
514C4⋊C4—34no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—68no—
521C4⋊C4—64no—
522C4⋊C4—68no—
523C4⋊C4—68no—
524C4⋊C4—64no—
525C4⋊C4—62no—
526C4⋊C4—62no—
527C4⋊C4—62no—
528C4⋊C4—64no—
529C4⋊C4—64no—
530C4⋊C4—124no—
531C4⋊C4—124no—
532C4⋊C4—124no—
533C4⋊C4—122no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—38no—
537C22⋊C4—64no—
538C22⋊C4—64no—
539C22⋊C4—68no—
540C22⋊C4—64no—
541C22⋊C4—62no—
542C22⋊C4—62no—
543C22⋊C4—64no—
544C22⋊C4—122no—
545C42—24no—
546C42—38no—
547C42—64no—
548C42—64no—
549C42—62no—
550C42—68no—
551C22×C3—22no—
552C22×C3—22no—
553C22×C3—22no—
554C22×C3—42no—
555C22×C3—42no—
556C22×C3C23×D411yescharacteristic
557C22×C3C2×H(5,𝔽2)11yescharacteristic
558C22×C3C2×2−1+412yes—
559C22×C3C2×2−1+412yes—
560C22×C3#58511yescharacteristic
561C2×D3—22no—
562C2×D3—24no—
563C2×D3—41no—
564C2×D3—41no—
565C2×D3—42no—
566C3×C4—22no—
567C3×C4—24no—
568C3×C4—42no—
569C3×C4—44no—
570C3×C4—42no—
571C3×C4—42no—
572C3×C4#53614yes—
573Dic3—24no—
574Dic3—24no—
575Dic3—41no—
576Dic3—42no—
577Dic3—41no—
578Dic3—41no—
579Dic3—42no—
580Dic3—44no—
581C23—22no—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—32no—
586C23—61no—
587C23—62no—
588C23—62no—
589C23—61no—
590C23C2×D3×D412yes—
591C23C24×D311yescharacteristic
592Q8—44no—
593Q8—248no—
594D4—124no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—34no—
611C2×C4—34no—
612C2×C4—64no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—64no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—64no—
625C2×C4—62no—
626C2×C4—64no—
627C2×C4—62no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—62no—
631C2×C4—84no—
632C2×C4—128no—
633C2×C4—124no—
634C2×C4C2×D3×D412yes—
635C2×C4C2×D3×D412yes—
636C2×C4#100714yes—
637C2×C4#101014yes—
638C2×C3—42no—
639C2×C3C2×#53611yescharacteristic
640C2×C3#348811yescharacteristic
641C2×C3#349911yescharacteristic
642C2×C3#354412yes—
643C2×C3#356312yes—
644D3—42no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—42no—
649C22—42no—
650C22—62no—
651C22—64no—
652C22—122no—
653C22—121no—
654C22—121no—
655C22C22×D3×D411yescharacteristic
656C22C2×#100712yes—
657C22C2×#101012yes—
658C22D3×H(5,𝔽2)11yescharacteristic
659C22#596611yescharacteristic
660C4—24no—
661C4—22no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—64no—
667C4—64no—
668C4—121no—
669C4—122no—
670C4—124no—
671C4—121no—
672C4—122no—
673C4—121no—
674C4#564814yes—
675C3#4296411yescharacteristic, complemented, Sylow
676C2—42no—
677C2—122no—
678C2C2×#564811yescharacteristic
679C2#8621411yescharacteristic
680C2#8633111yescharacteristic
681C2#8694312yes—
682C2#8695312yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2502C2
prime radicalC22×#67C23
hypercenter#2502D3
hypocenterC3#42964
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2502

Complements

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

NH ≅ G/Ntype
2. C2×#5500C2semidirect
3. C3×#2502C2semidirect
4. #82787C2semidirect
6. #77926C2semidirect
7. #77837C2semidirect
8. #77818C2semidirect
9. #77816C2semidirect
10. #76630C2semidirect
11. #76509C2semidirect
13. #74521C2semidirect
14. #74336C2semidirect
15. #74326C2semidirect
17. #73459C2semidirect
96. #2502D3semidirect
675. C3#42964semidirect

Maximal subgroups

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

subgroupindex
C2×#56392
C2×#55002
C3×#25022
#827872
#821962
#779262
#778372
#778182
#778162
#766302
#765092
#746312
#745212
#743362
#743262
#735832
#734592
#429643