Finite Group Databasebeta

← #11166387

Subgroups — #11166387

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×#5529C211yesmaximal, characteristic
2C2×#5519C211yesmaximal, characteristic, complemented
3C3×#2346C211yesmaximal, characteristic, complemented
4#76009C212yesmaximal, complemented
5#75999C212yesmaximal, complemented
6#75678C214yesmaximal, complemented
7#75668C214yesmaximal
8#74217C212yesmaximal, complemented
9#74193C212yesmaximal, complemented
10#74027C211yesmaximal, characteristic
11#73951C211yesmaximal, characteristic, complemented
12#73786C212yesmaximal, complemented
13#73717C212yesmaximal, complemented
14#73583C211yesmaximal, characteristic
15#73517C211yesmaximal, characteristic, complemented
16#73427C212yesmaximal, complemented
17#73410C212yesmaximal, complemented
18#42964—31nomaximal, Sylow
19C2×C3×#117C2211yescharacteristic
20C2×C4×#67C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C2×#924C2211yescharacteristic
24#5529—28no—
25#5519—28no—
26C2×#897C2211yescharacteristic
27C2×#897C2214yes—
28C2×D3×C4⋊C4C2211yescharacteristic
29C2×#890C2212yes—
30C2×#890C2211yescharacteristic
31C2×#888C2212yes—
32C2×#887C2211yescharacteristic
33C2×#886C2212yes—
34C2×#883C2212yes—
35C2×#879C2212yes—
36C2×#878C2214yes—
37C2×#872C2212yes—
38C2×#870C2212yes—
39C2×#870C2212yes—
40C2×#869C2212yes—
41C3×#389C2212yes—
42C3×#388C2214yes—
43C3×#382C2212yes—
44C3×#380C2211yescharacteristic
45C3×C4×C4⋊C4C2212yes—
46C3×C4×C22⋊C4C2211yescharacteristic
47C3×#376C2212yes—
48#4987C2212yes—
49#4984C2212yes—
50#4984C2212yes—
51#4981C2214yes—
52#4981C2214yes—
53#4973C2212yes—
54#4973C2214yes—
55#4961C2212yes—
56#4953C2212yes—
57#4953C2214yes—
58#4950C2214yes—
59#4943C2212yes—
60#4940C2214yes—
61C4×(C2×D3)⋊C4C2212yes—
62#4936C2214yes—
63#4935C2212yes—
64#4935C2214yes—
65C4×C4⋊Dic3C2212yes—
66#4929C2212yes—
67#4928C2214yes—
68#4677C2211yescharacteristic
69#4674C2212yes—
70#4674C2214yes—
71#4674C2211yescharacteristic
72#4668C2214yes—
73#4661C2212yes—
74#4661C2212yes—
75#4661C2212yes—
76#4661C2211yescharacteristic
77#4661C2214yes—
78#4661C2214yes—
79#4661C2212yes—
80#4659C2214yes—
81#4659C2211yescharacteristic
82#4659C2212yes—
83#4656C2214yes—
84#4656C2214yes—
85#4653C2214yes—
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#2502—31no—
97#2489—32no—
98#2486—31no—
99#2451—32no—
100#2440—32no—
101#2395—31no—
102#2346D311yescharacteristic, complemented
103C23×C3×C4C2311yescharacteristic
104C22×#67C2311yescharacteristic
105C23×Dic3C2312yes—
106C22×C4×D3C2311yescharacteristic
107C22×C4×D3C2312yes—
108C22×Dic6C2311yescharacteristic
109C22×Dic6C2314yes—
110C3×#117D418yes—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C22⋊C4C2312yes—
116C2×C3×C22⋊C4C2312yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C42C2314yes—
119C2×C3×C42C2312yes—
120C2×C3×C42C2312yes—
121C2×#217C2311yescharacteristic
122C2×#217C2312yes—
123C2×#217C2312yes—
124C4×#67—24no—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2311yescharacteristic
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2312yes—
130#926—24no—
131C2×C4⋊Dic3—24no—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2311yescharacteristic
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2314yes—
143C2×C4⋊Dic3C2314yes—
144C2×C4⋊Dic3C2314yes—
145C2×C4⋊Dic3C2314yes—
146#924—24no—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2312yes—
150C2×Dic3⋊C4C2311yescharacteristic
151C2×Dic3⋊C4C2312yes—
152C2×Dic3⋊C4C2314yes—
153C2×Dic3⋊C4C2312yes—
154C2×C4×Dic3C2312yes—
155C2×C4×Dic3C2311yescharacteristic
156C2×C4×Dic3C2312yes—
157#897—24no—
158#897—48no—
159D3×C4⋊C4—24no—
160#890—28no—
161#890—24no—
162#888—28no—
163#887—24no—
164#886—44no—
165#883—28no—
166#879—28no—
167#878—216no—
168#872—28no—
169#870—28no—
170#870—28no—
171#869—28no—
172C3×#95C2314yes—
173C3×#95C2312yes—
174C3×#95C2311yescharacteristic
175C3×#95C2314yes—
176C3×#95C2311yescharacteristic
177C3⋊#95C2314yes—
178C3⋊#95C2314yes—
179C3⋊#95C2314yes—
180C3⋊#95C2312yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2312yes—
184C3⋊#95C2312yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2312yes—
188#552—68no—
189#544—68no—
190C2×C4⋊Q8—32no—
191C2×C4⋊Q8—32no—
192C2×#126—32no—
193C2×#125—32no—
194C2×#125—31no—
195C2×#125—32no—
196C2×#125—32no—
197C2×#123—32no—
198C2×#123—32no—
199C2×C22⋊Q8—31no—
200C2×C22⋊Q8—31no—
201C2×C22⋊Q8—34no—
202C2×C22⋊Q8—34no—
203C2×C4×Q8—31no—
204C2×C4×D4—31no—
205C2×#117—31no—
206C2×#117—32no—
207C2×#117C2×D311yescharacteristic
208C22×C4⋊C4—31no—
209C22×C4⋊C4—31no—
210#400—31no—
211#400—32no—
212#400—34no—
213#399—31no—
214#398—34no—
215#398—34no—
216#398—32no—
217#398—32no—
218#398—34no—
219#398—32no—
220#398—32no—
221#398—34no—
222#398—31no—
223#397—34no—
224#397—32no—
225#397—32no—
226#397—31no—
227#397—34no—
228#395—34no—
229#395—34no—
230#394—32no—
231#391—34no—
232#391—32no—
233#391—32no—
234#389—32no—
235#389—32no—
236#389—34no—
237#389C2×D312yes—
238#388—34no—
239#388C2×D314yes—
240#387—34no—
241#385—34no—
242#385—34no—
243#384—34no—
244#384—32no—
245#382—34no—
246#382C2×D312yes—
247#381—32no—
248#380—32no—
249#380C2×D311yescharacteristic
250C4×C4⋊C4—32no—
251C4×C4⋊C4C2×D312yes—
252C4×C22⋊C4—32no—
253C4×C22⋊C4C2×D311yescharacteristic
254#376C2×D312yes—
255C24×C3C2411yescharacteristic
256C23×D3C2411yescharacteristic
257C22×C3×C4—24no—
258C22×C3×C4C2×D414yes—
259C22×C3×C4C2414yes—
260C22×C3×C4C2412yes—
261C22×C3×C4C2411yescharacteristic
262C22×C3×C4C2412yes—
263C22×C3×C4C2412yes—
264C22×C3×C4C2412yes—
265C22×C3×C4C2411yescharacteristic
266C2×#67—22no—
267C2×#67—22no—
268C2×#67—41no—
269C22×Dic3—24no—
270C22×Dic3—42no—
271C22×Dic3—42no—
272C22×Dic3C2412yes—
273C22×Dic3C2411yescharacteristic
274C22×Dic3C2411yescharacteristic
275C22×Dic3C2412yes—
276C22×Dic3C2412yes—
277C22×Dic3C2414yes—
278C22×Dic3C2411yescharacteristic
279C22×Dic3C2412yes—
280C2×C4×D3—24no—
281C2×C4×D3—24no—
282C2×C4×D3—22no—
283C2×C4×D3—24no—
284C2×C4×D3—42no—
285C2×Dic6—22no—
286C2×Dic6—24no—
287C2×Dic6—28no—
288C2×Dic6—44no—
289C2×Dic6—44no—
290C3×C4⋊C4—24no—
291C3×C4⋊C4—44no—
292C3×C4⋊C4C2×D414yes—
293C3×C4⋊C4C2×D414yes—
294C3×C22⋊C4—24no—
295C3×C22⋊C4—42no—
296C3×C22⋊C4C2×D418yes—
297C3×C42—28no—
298C3×C42—24no—
299C3×C42C2×D418yes—
300#217—24no—
301#217—22no—
302#217—24no—
303(C2×D3)⋊C4—28no—
304(C2×D3)⋊C4—24no—
305(C2×D3)⋊C4—22no—
306(C2×D3)⋊C4—24no—
307(C2×D3)⋊C4—24no—
308C4⋊Dic3—22no—
309C4⋊Dic3—24no—
310C4⋊Dic3—28no—
311C4⋊Dic3—22no—
312C4⋊Dic3—24no—
313C4⋊Dic3—24no—
314C4⋊Dic3—24no—
315C4⋊Dic3—22no—
316C4⋊Dic3—24no—
317C4⋊Dic3—44no—
318C4⋊Dic3—44no—
319C4⋊Dic3—44no—
320C4⋊Dic3—44no—
321C4⋊Dic3—42no—
322Dic3⋊C4—22no—
323Dic3⋊C4—24no—
324Dic3⋊C4—28no—
325Dic3⋊C4—28no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329C4×Dic3—24no—
330C4×Dic3—22no—
331C4×Dic3—24no—
332C22×Q8—31no—
333C22×Q8—34no—
334C22×D4—31no—
335C23×C4—32no—
336C23×C4—31no—
337C23×C4—32no—
338C23×C4C22×D311yescharacteristic
339C4⋊Q8—68no—
340C4⋊Q8—68no—
341#126—68no—
342#125—68no—
343#125—64no—
344#125—68no—
345#125—68no—
346#123—68no—
347#123—124no—
348C22⋊Q8—64no—
349C22⋊Q8—64no—
350C22⋊Q8—616no—
351C22⋊Q8—128no—
352C4×Q8—64no—
353C4×D4—64no—
354#117—64no—
355#117—68no—
356#117D1218yes—
357C2×C4⋊C4—31no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—32no—
361C2×C4⋊C4—32no—
362C2×C4⋊C4—31no—
363C2×C4⋊C4—32no—
364C2×C4⋊C4—31no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—34no—
368C2×C4⋊C4—32no—
369C2×C4⋊C4—34no—
370C2×C4⋊C4—34no—
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—34no—
378C2×C4⋊C4—64no—
379C2×C4⋊C4—64no—
380C2×C4⋊C4C22×D312yes—
381C2×C4⋊C4C22×D314yes—
382C2×C4⋊C4C22×D311yescharacteristic
383C2×C4⋊C4C22×D311yescharacteristic
384C2×C22⋊C4—32no—
385C2×C22⋊C4—31no—
386C2×C22⋊C4—32no—
387C2×C22⋊C4—31no—
388C2×C22⋊C4—32no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4—34no—
391C2×C22⋊C4—32no—
392C2×C22⋊C4C22×D312yes—
393C2×C22⋊C4C22×D312yes—
394C2×C22⋊C4C22×D312yes—
395C2×C42—32no—
396C2×C42—32no—
397C2×C42—31no—
398C2×C42C22×D314yes—
399C2×C42C22×D312yes—
400C2×C42C22×D312yes—
401#95—32no—
402#95—32no—
403#95—32no—
404#95—32no—
405#95—34no—
406#95—34no—
407#95—34no—
408#95—34no—
409#95—34no—
410#95—34no—
411#95—34no—
412#95C22×D311yescharacteristic
413#95C22×D312yes—
414#95C22×D314yes—
415#95C22×D311yescharacteristic
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—
428C2×C3×C4—22no—
429C2×C3×C4—24no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—28no—
435C2×C3×C4—24no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—24no—
439C2×C3×C4—22no—
440C2×C3×C4—24no—
441C2×C3×C4C22×D414yes—
442C2×C3×C4C22×D412yes—
443C2×C3×C4C22×D414yes—
444C2×C3×C4C22×D412yes—
445C2×C3×C42−1+414yes—
446C2×C3×C42−1+414yes—
447#67—44no—
448C2×Dic3—21no—
449C2×Dic3—22no—
450C2×Dic3—21no—
451C2×Dic3—22no—
452C2×Dic3—22no—
453C2×Dic3—21no—
454C2×Dic3—22no—
455C2×Dic3—24no—
456C2×Dic3—22no—
457C2×Dic3—21no—
458C2×Dic3—22no—
459C2×Dic3—24no—
460C2×Dic3—22no—
461C2×Dic3—22no—
462C2×Dic3—22no—
463C2×Dic3—24no—
464C2×Dic3—22no—
465C2×Dic3—22no—
466C2×Dic3—21no—
467C2×Dic3—22no—
468C2×Dic3—24no—
469C2×Dic3—84no—
470C4×D3—48no—
471C4×D3—44no—
472Dic6—44no—
473Dic6—88no—
474C24—31no—
475C24C23×D311yescharacteristic
476C2×Q8—62no—
477C2×Q8—68no—
478C2×Q8—64no—
479C2×Q8—124no—
480C2×Q8—124no—
481C2×D4—62no—
482C2×D4—62no—
483C2×D4—121no—
484C22×C4—24no—
485C22×C4—32no—
486C22×C4—31no—
487C22×C4—31no—
488C22×C4—32no—
489C22×C4—32no—
490C22×C4—34no—
491C22×C4—31no—
492C22×C4—32no—
493C22×C4—64no—
494C22×C4—64no—
495C22×C4—64no—
496C22×C4—62no—
497C22×C4—64no—
498C22×C4—122no—
499C22×C4—122no—
500C22×C4—122no—
501C22×C4C2×D1214yes—
502C22×C4C23×D312yes—
503C22×C4C23×D314yes—
504C22×C4C23×D311yescharacteristic
505C22×C4C23×D312yes—
506C22×C4C23×D312yes—
507C22×C4C23×D312yes—
508C22×C4C23×D311yescharacteristic
509C4⋊C4—24no—
510C4⋊C4—44no—
511C4⋊C4—62no—
512C4⋊C4—64no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—64no—
516C4⋊C4—62no—
517C4⋊C4—64no—
518C4⋊C4—62no—
519C4⋊C4—62no—
520C4⋊C4—64no—
521C4⋊C4—68no—
522C4⋊C4—64no—
523C4⋊C4—68no—
524C4⋊C4—68no—
525C4⋊C4—64no—
526C4⋊C4—64no—
527C4⋊C4—124no—
528C4⋊C4—124no—
529C4⋊C4—124no—
530C4⋊C4—124no—
531C4⋊C4—122no—
532C4⋊C4C2×D1214yes—
533C4⋊C4C2×D1214yes—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—64no—
537C22⋊C4—68no—
538C22⋊C4—64no—
539C22⋊C4—62no—
540C22⋊C4—64no—
541C22⋊C4—62no—
542C22⋊C4—64no—
543C22⋊C4—64no—
544C22⋊C4C2×D1218yes—
545C42—28no—
546C42—24no—
547C42—64no—
548C42—64no—
549C42—62no—
550C42C2×D1218yes—
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—24no—
569C3×C4—24no—
570C3×C4—44no—
571C3×C4—42no—
572C3×C4#53614yes—
573Dic3—42no—
574Dic3—41no—
575Dic3—42no—
576Dic3—44no—
577Dic3—42no—
578Dic3—41no—
579Dic3—41no—
580Dic3—42no—
581C23—22no—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—32no—
586C23—61no—
587C23—62no—
588C23—62no—
589C23—61no—
590C23C22×D1212yes—
591C23C24×D311yescharacteristic
592Q8—124no—
593Q8—248no—
594D4—124no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—61no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—61no—
615C2×C4—64no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—61no—
625C2×C4—62no—
626C2×C4—61no—
627C2×C4—64no—
628C2×C4—62no—
629C2×C4—128no—
630C2×C4—124no—
631C2×C4—244no—
632C2×C4C22×D1214yes—
633C2×C4C22×D1212yes—
634C2×C4C22×D1214yes—
635C2×C4C22×D1212yes—
636C2×C4#101014yes—
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—121no—
653C22—122no—
654C22—121no—
655C22C23×D1211yescharacteristic
656C22C2×#101012yes—
657C22C2×#101012yes—
658C22D3×H(5,𝔽2)11yescharacteristic
659C22#596811yescharacteristic
660C4—24no—
661C4—24no—
662C4—24no—
663C4—22no—
664C4—42no—
665C4—44no—
666C4—121no—
667C4—122no—
668C4—122no—
669C4—124no—
670C4—121no—
671C4—122no—
672C4—121no—
673C4—122no—
674C4#552614yes—
675C3#4296411yescharacteristic, complemented, Sylow
676C2—42no—
677C2—122no—
678C2C2×#552611yescharacteristic
679C2#8582711yescharacteristic
680C2#8590011yescharacteristic
681C2#8685212yes—
682C2#8697512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2346C2
prime radicalC22×#67C23
hypercenter#2346D3
hypocenterC3#42964
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2346

Complements

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

NH ≅ G/Ntype
2. C2×#5519C2semidirect
3. C3×#2346C2semidirect
4. #76009C2semidirect
5. #75999C2semidirect
6. #75678C2semidirect
8. #74217C2semidirect
9. #74193C2semidirect
11. #73951C2semidirect
12. #73786C2semidirect
13. #73717C2semidirect
15. #73517C2semidirect
16. #73427C2semidirect
17. #73410C2semidirect
102. #2346D3semidirect
675. C3#42964semidirect

Maximal subgroups

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

subgroupindex
C2×#55292
C2×#55192
C3×#23462
#760092
#759992
#756782
#756682
#742172
#741932
#740272
#739512
#737862
#737172
#735832
#735172
#734272
#734102
#429643