Finite Group Databasebeta

← #10972855

Subgroups — #10972855

Order 768.

2294 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5624C211yesmaximal, characteristic
2C22×#887C211yesmaximal, characteristic, complemented
3C3×#2583C211yesmaximal, characteristic, complemented
4#82785C211yesmaximal, characteristic, complemented
5#82182C211yesmaximal, characteristic
6#77823C212yesmaximal, complemented
7#77784C212yesmaximal, complemented
8#77203C212yesmaximal, complemented
9#77191C212yesmaximal, complemented
10#76097C214yesmaximal, complemented
11#75971C214yesmaximal, complemented
12#74588C212yesmaximal, complemented
13#74588C212yesmaximal, complemented
14#74457C212yesmaximal, complemented
15#74405C212yesmaximal, complemented
16#73550C211yesmaximal, characteristic
17#73437C211yesmaximal, characteristic, complemented
18#35245—31nomaximal, Sylow
19C23×Dic6C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C2×#947C2211yescharacteristic
23C2×Q8×Dic3C2211yescharacteristic
24C2×#930C2211yescharacteristic
25C22×Dic3⋊C4C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27C22×C4×Dic3C2211yescharacteristic
28C22×C4×Dic3C2211yescharacteristic
29C22×C4×Dic3C2211yescharacteristic
30#5624—28no—
31C2×#898C2212yes—
32C2×#896C2214yes—
33C2×#893C2211yescharacteristic
34C2×#892C2212yes—
35C2×#890C2211yescharacteristic
36C2×#887—28no—
37C2×#887C2212yes—
38C2×#887C2212yes—
39C2×#887C2212yes—
40C2×#887C2212yes—
41C2×#886C2212yes—
42C2×#67⋊C4C2212yes—
43C3×#398C2212yes—
44C3×#395C2212yes—
45C3×#389C2214yes—
46C3×#385C2212yes—
47C3×#385C2212yes—
48C3×C4×C22⋊C4C2211yescharacteristic
49#5230C2212yes—
50#5213C2212yes—
51#5213C2212yes—
52C4×#217C2211yescharacteristic
53#4993C2214yes—
54#4991C2212yes—
55#4990C2214yes—
56#4990C2214yes—
57#4990C2214yes—
58#4986C2212yes—
59#4983C2212yes—
60#4980C2214yes—
61#4980C2214yes—
62#4979C2214yes—
63#4976C2214yes—
64#4976C2214yes—
65#4958C2212yes—
66#4958C2212yes—
67#4946C2212yes—
68#4946C2212yes—
69#4943C2212yes—
70#4942C2212yes—
71#4940C2214yes—
72#4932C2212yes—
73C4×Dic3⋊C4C2212yes—
74#4673C2214yes—
75#4672C2212yes—
76#4669C2214yes—
77#4669C2214yes—
78#4667C2214yes—
79#4659C2212yes—
80#4658C2214yes—
81#4655C2214yes—
82#4655C2214yes—
83#4651C2212yes—
84#4650C2212yes—
85#4646C2214yes—
86C2×#529—31no—
87C22×C4×Q8—31no—
88#2853—32no—
89#2853—32no—
90#2634—32no—
91#2632—32no—
92#2604—34no—
93#2591—31no—
94#2591—31no—
95#2583—31no—
96#2583D311yescharacteristic, complemented
97#2581—32no—
98#2536—32no—
99#2487—32no—
100#2409—32no—
101#2323—34no—
102#2283—31no—
103C22×C3×Q8C2311yescharacteristic
104C23×C3×C4C2311yescharacteristic
105C23×C3×C4C2311yescharacteristic
106C23×C3×C4C2311yescharacteristic
107C22×#67C2312yes—
108C23×Dic3C2311yescharacteristic
109C23×Dic3C2311yescharacteristic
110C23×Dic3C2311yescharacteristic
111C23×Dic3C2311yescharacteristic
112C22×D12C2311yescharacteristic
113C22×C4×D3C2312yes—
114C22×Dic6—24no—
115C22×Dic6—28no—
116C22×Dic6C2312yes—
117C22×Dic6C2312yes—
118C3×C22⋊Q8—24no—
119C2×C3×C4⋊C4—24no—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2314yes—
122C2×C3×C4⋊C4C2311yescharacteristic
123C2×C3×C4⋊C4C2312yes—
124C2×C3×C4⋊C4C2312yes—
125C2×C3×C4⋊C4C2314yes—
126C2×C3×C22⋊C4C2312yes—
127C2×C3×C22⋊C4C2312yes—
128C2×C3×C42C2312yes—
129C2×#217C2312yes—
130#947—24no—
131Q8×Dic3—24no—
132#930—24no—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2312yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2312yes—
138C2×(C2×D3)⋊C4C2312yes—
139C2×(C2×D3)⋊C4C2314yes—
140C2×(C2×D3)⋊C4C2314yes—
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2312yes—
143C2×C4⋊Dic3C2311yescharacteristic
144C2×Dic3⋊C4—24no—
145C2×Dic3⋊C4—24no—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2312yes—
150C2×Dic3⋊C4C2312yes—
151C2×Dic3⋊C4C2312yes—
152C2×Dic3⋊C4C2312yes—
153C2×Dic3⋊C4C2314yes—
154C2×C4×Dic3—24no—
155C2×C4×Dic3—24no—
156C2×C4×Dic3D418yes—
157C2×C4×Dic3C2312yes—
158C2×C4×Dic3C2312yes—
159C2×C4×Dic3C2312yes—
160C2×C4×Dic3C2312yes—
161C2×C4×Dic3C2312yes—
162C2×C4×Dic3C2312yes—
163C2×C4×Dic3C2311yescharacteristic
164C2×C4×Dic3C2311yescharacteristic
165C2×C4×Dic3C2311yescharacteristic
166C2×C4×Dic3C2311yescharacteristic
167#898—28no—
168#896—216no—
169#893—24no—
170#892—28no—
171#890—24no—
172#887—28no—
173#887—28no—
174#887—28no—
175#887—28no—
176#886—28no—
177#67⋊C4—28no—
178C3×#95C2314yes—
179C3×#95C2312yes—
180C3×#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2312yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2314yes—
189C3⋊#95C2314yes—
190C3⋊#95C2314yes—
191C23×Q8—31no—
192#529—68no—
193C2×#126—32no—
194C2×#125—31no—
195C2×#124—31no—
196C2×#123—32no—
197C2×C22⋊Q8—34no—
198C2×C22⋊Q8C2×D311yescharacteristic
199C2×#121—31no—
200C2×C4×Q8—31no—
201C2×C4×Q8—32no—
202C2×C4×Q8—32no—
203C2×C4×Q8—32no—
204C2×C4×Q8—32no—
205C2×C4×Q8—68no—
206C2×C4×D4—31no—
207C2×C4×D4—32no—
208C2×#117—32no—
209C22×C4⋊C4—31no—
210C22×C4⋊C4—31no—
211C22×C4⋊C4C2×D311yescharacteristic
212C22×C42—31no—
213C22×C42—31no—
214C22×C42—31no—
215#400—32no—
216#400—32no—
217#400—34no—
218#399—34no—
219#398—32no—
220#398C2×D312yes—
221#395—32no—
222#395—34no—
223#395C2×D312yes—
224#394—32no—
225#394—32no—
226#393—34no—
227#391—34no—
228#389—34no—
229#389—34no—
230#389C2×D314yes—
231#388—34no—
232#388—34no—
233#388—34no—
234#388—34no—
235#388—34no—
236#388—34no—
237#387—34no—
238#387—34no—
239#387—34no—
240#387—34no—
241#386—32no—
242#386—32no—
243#386—32no—
244#386—32no—
245#385—32no—
246#385—32no—
247#385C2×D312yes—
248#385C2×D312yes—
249#383—32no—
250#382—32no—
251#381—32no—
252#380—32no—
253#380—34no—
254C4×C4⋊C4—32no—
255C4×C22⋊C4—31no—
256C4×C22⋊C4C2×D311yescharacteristic
257#376—32no—
258C24×C3C2411yescharacteristic
259C23×D3C2412yes—
260C2×C3×Q8—22no—
261C2×C3×Q8—42no—
262C22×C3×C4—22no—
263C22×C3×C4—24no—
264C22×C3×C4—22no—
265C22×C3×C4—42no—
266C22×C3×C4—42no—
267C22×C3×C4C2×D414yes—
268C22×C3×C4C2414yes—
269C22×C3×C4C2412yes—
270C22×C3×C4C2412yes—
271C22×C3×C4C2412yes—
272C22×C3×C4C2411yescharacteristic
273C22×C3×C4C2411yescharacteristic
274C22×C3×C4C2412yes—
275C2×#67—24no—
276C2×#67—28no—
277C22×Dic3—22no—
278C22×Dic3—24no—
279C22×Dic3—24no—
280C22×Dic3—24no—
281C22×Dic3—22no—
282C22×Dic3—42no—
283C22×Dic3C2×D414yes—
284C22×Dic3C2×D414yes—
285C22×Dic3C2414yes—
286C22×Dic3C2412yes—
287C22×Dic3C2412yes—
288C22×Dic3C2411yescharacteristic
289C22×Dic3C2411yescharacteristic
290C22×Dic3C2411yescharacteristic
291C22×Dic3C2412yes—
292C22×Dic3C2411yescharacteristic
293C2×D12—22no—
294C2×D12—42no—
295C2×C4×D3—24no—
296C2×C4×D3—28no—
297C2×Dic6—232no—
298C2×Dic6—24no—
299C2×Dic6—24no—
300C2×Dic6—24no—
301C2×Dic6—24no—
302C2×Dic6—24no—
303C2×Dic6#4118yes—
304C3×C4⋊C4—24no—
305C3×C4⋊C4—24no—
306C3×C4⋊C4—22no—
307C3×C4⋊C4—24no—
308C3×C4⋊C4—28no—
309C3×C4⋊C4—44no—
310C3×C22⋊C4—24no—
311C3×C22⋊C4—24no—
312C3×C42—42no—
313#217—24no—
314(C2×D3)⋊C4—24no—
315(C2×D3)⋊C4—28no—
316(C2×D3)⋊C4—24no—
317(C2×D3)⋊C4—24no—
318(C2×D3)⋊C4—24no—
319(C2×D3)⋊C4—28no—
320(C2×D3)⋊C4—44no—
321(C2×D3)⋊C4—44no—
322C4⋊Dic3—22no—
323C4⋊Dic3—24no—
324C4⋊Dic3—24no—
325Dic3⋊C4—24no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4—24no—
330Dic3⋊C4—28no—
331Dic3⋊C4—28no—
332Dic3⋊C4—44no—
333C4×Dic3—24no—
334C4×Dic3—24no—
335C4×Dic3—24no—
336C4×Dic3—24no—
337C4×Dic3—24no—
338C4×Dic3—42no—
339C4×Dic3C2×D414yes—
340C4×Dic3C2×D414yes—
341C4×Dic3C2×D414yes—
342C4×Dic3C2×D414yes—
343C22×Q8—32no—
344C22×Q8—32no—
345C22×Q8—64no—
346C22×Q8—68no—
347C22×Q8C22×D311yescharacteristic
348C22×D4—32no—
349C22×D4—31no—
350C23×C4—31no—
351C23×C4—31no—
352C23×C4—31no—
353C23×C4—31no—
354C23×C4—32no—
355C23×C4C22×D311yescharacteristic
356C23×C4C22×D311yescharacteristic
357C23×C4C22×D311yescharacteristic
358#126—68no—
359#125—64no—
360#124—64no—
361#123—68no—
362C22⋊Q8—24no—
363C22⋊Q8—616no—
364#121—64no—
365C4×Q8—68no—
366C4×Q8—68no—
367C4×Q8—68no—
368C4×Q8—68no—
369C4×Q8—64no—
370C4×D4—64no—
371C4×D4—68no—
372#117—68no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—32no—
375C2×C4⋊C4—32no—
376C2×C4⋊C4—32no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—32no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—31no—
381C2×C4⋊C4—32no—
382C2×C4⋊C4—34no—
383C2×C4⋊C4—34no—
384C2×C4⋊C4—34no—
385C2×C4⋊C4—64no—
386C2×C4⋊C4—64no—
387C2×C4⋊C4C22×D312yes—
388C2×C4⋊C4C22×D312yes—
389C2×C4⋊C4C22×D312yes—
390C2×C4⋊C4C22×D311yescharacteristic
391C2×C4⋊C4C22×D314yes—
392C2×C4⋊C4C22×D314yes—
393C2×C22⋊C4—32no—
394C2×C22⋊C4—32no—
395C2×C22⋊C4—32no—
396C2×C22⋊C4—32no—
397C2×C22⋊C4—32no—
398C2×C22⋊C4—34no—
399C2×C22⋊C4—34no—
400C2×C22⋊C4—34no—
401C2×C22⋊C4—34no—
402C2×C22⋊C4C22×D312yes—
403C2×C22⋊C4C22×D312yes—
404C2×C42—38no—
405C2×C42—31no—
406C2×C42—31no—
407C2×C42—31no—
408C2×C42—31no—
409C2×C42—32no—
410C2×C42—32no—
411C2×C42—32no—
412C2×C42—32no—
413C2×C42—32no—
414C2×C42—32no—
415C2×C42—64no—
416C2×C42—64no—
417C2×C42C22×D312yes—
418#95—34no—
419#95—34no—
420#95—32no—
421#95—34no—
422#95—34no—
423#95—34no—
424#95—34no—
425#95—34no—
426#95—34no—
427#95—34no—
428#95C22×D312yes—
429#95C22×D314yes—
430#95C22×D314yes—
431C23×C3—22no—
432C23×C3—22no—
433C23×C3C22×D412yes—
434C23×C32−1+414yes—
435C23×C3C2511yescharacteristic
436C22×D3—22no—
437C22×D3—24no—
438C22×D3—24no—
439C22×D3—24no—
440C3×Q8—44no—
441C2×C3×C4—28no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—22no—
446C2×C3×C4—22no—
447C2×C3×C4—22no—
448C2×C3×C4—22no—
449C2×C3×C4—24no—
450C2×C3×C4—24no—
451C2×C3×C4—24no—
452C2×C3×C4—24no—
453C2×C3×C4—24no—
454C2×C3×C4—24no—
455C2×C3×C4—44no—
456C2×C3×C4—44no—
457C2×C3×C4C22×D412yes—
458C2×C3×C4C22×D412yes—
459C2×C3×C4C2×#4114yes—
460#67—48no—
461C2×Dic3—28no—
462C2×Dic3—28no—
463C2×Dic3—22no—
464C2×Dic3—22no—
465C2×Dic3—22no—
466C2×Dic3—22no—
467C2×Dic3—22no—
468C2×Dic3—28no—
469C2×Dic3—22no—
470C2×Dic3—22no—
471C2×Dic3—24no—
472C2×Dic3—24no—
473C2×Dic3—24no—
474C2×Dic3—24no—
475C2×Dic3—24no—
476C2×Dic3—44no—
477C2×Dic3C22×D412yes—
478C2×Dic3C22×D412yes—
479C2×Dic3C22×D412yes—
480C2×Dic3C22×D412yes—
481C2×Dic3C2×#4114yes—
482C2×Dic3C2×#4114yes—
483D12—44no—
484C4×D3—48no—
485Dic6—216no—
486Dic6—216no—
487C24—32no—
488C24C23×D311yescharacteristic
489C2×Q8—22no—
490C2×Q8—38no—
491C2×Q8—42no—
492C2×Q8—64no—
493C2×Q8—64no—
494C2×Q8—64no—
495C2×Q8—64no—
496C2×Q8—632no—
497C2×Q8—64no—
498C2×D4—64no—
499C2×D4—68no—
500C2×D4—62no—
501C2×D4—122no—
502C22×C4—22no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—31no—
506C22×C4—34no—
507C22×C4—34no—
508C22×C4—31no—
509C22×C4—31no—
510C22×C4—31no—
511C22×C4—32no—
512C22×C4—34no—
513C22×C4—32no—
514C22×C4—32no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—64no—
518C22×C4—68no—
519C22×C4—64no—
520C22×C4—64no—
521C22×C4—62no—
522C22×C4—64no—
523C22×C4—62no—
524C22×C4—122no—
525C22×C4D3×D414yes—
526C22×C4C23×D314yes—
527C22×C4C23×D312yes—
528C22×C4C23×D312yes—
529C22×C4C23×D312yes—
530C22×C4C23×D311yescharacteristic
531C22×C4C23×D311yescharacteristic
532C22×C4C23×D312yes—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—28no—
538C4⋊C4—44no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—64no—
543C4⋊C4—64no—
544C4⋊C4—68no—
545C4⋊C4—68no—
546C4⋊C4—64no—
547C4⋊C4—64no—
548C4⋊C4—62no—
549C4⋊C4—124no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—64no—
553C22⋊C4—64no—
554C22⋊C4—64no—
555C22⋊C4—68no—
556C22⋊C4—64no—
557C22⋊C4—64no—
558C22⋊C4—68no—
559C22⋊C4—124no—
560C22⋊C4—124no—
561C42—34no—
562C42—34no—
563C42—34no—
564C42—34no—
565C42—42no—
566C42—64no—
567C42—64no—
568C42—64no—
569C42—64no—
570C42—64no—
571C42—122no—
572C22×C3—22no—
573C22×C3—22no—
574C22×C3—28no—
575C22×C3#53614yes—
576C22×C3C23×D411yescharacteristic
577C22×C3C22×#4112yes—
578C22×C3C2×2−1+412yes—
579C22×C3#58512yes—
580C2×D3—42no—
581C2×D3—42no—
582C2×D3—42no—
583C2×D3—48no—
584C3×C4—24no—
585C3×C4—22no—
586C3×C4—22no—
587C3×C4—42no—
588C3×C4—42no—
589C3×C4—42no—
590C3×C4—44no—
591Dic3—22no—
592Dic3—24no—
593Dic3—22no—
594Dic3—22no—
595Dic3—42no—
596Dic3—42no—
597Dic3—44no—
598Dic3#52514yes—
599C23—22no—
600C23—22no—
601C23—62no—
602C23—64no—
603C23—64no—
604C23—64no—
605C23C2×D3×D412yes—
606C23#100714yes—
607C23C24×D311yescharacteristic
608Q8—44no—
609Q8—616no—
610Q8—616no—
611D4—124no—
612D4—128no—
613C2×C4—28no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—32no—
628C2×C4—34no—
629C2×C4—32no—
630C2×C4—32no—
631C2×C4—32no—
632C2×C4—34no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—68no—
636C2×C4—68no—
637C2×C4—62no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—68no—
643C2×C4—62no—
644C2×C4—62no—
645C2×C4—64no—
646C2×C4—64no—
647C2×C4—64no—
648C2×C4—64no—
649C2×C4—64no—
650C2×C4—128no—
651C2×C4—124no—
652C2×C4C2×D3×D412yes—
653C2×C4C2×D3×D412yes—
654C2×C4D3×#4114yes—
655C2×C3—24no—
656C2×C3C2×#52511yescharacteristic
657C2×C3C2×#53611yescharacteristic
658C2×C3#344911yescharacteristic
659C2×C3#351614yes—
660D3—82no—
661C22—22no—
662C22—22no—
663C22—28no—
664C22—122no—
665C22—122no—
666C22—128no—
667C22—122no—
668C22#572414yes—
669C22C22×D3×D411yescharacteristic
670C22C2×#100712yes—
671C22C2×D3×#4112yes—
672C22#596612yes—
673C4—24no—
674C4—22no—
675C4—22no—
676C4—34no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—62no—
682C4—64no—
683C4—62no—
684C4—62no—
685C4—122no—
686C4—124no—
687C4—122no—
688C3#3524511yescharacteristic, complemented, Sylow
689C2—24no—
690C2—242no—
691C2D3×#52511yescharacteristic
692C2C2×#572411yescharacteristic
693C2#8629411yescharacteristic
694C2#8693214yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2583C2
prime radicalC2×#930C22
hypercenter#2583D3
hypocenterC3#35245
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2583

Complements

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

NH ≅ G/Ntype
2. C22×#887C2semidirect
3. C3×#2583C2semidirect
4. #82785C2semidirect
6. #77823C2semidirect
7. #77784C2semidirect
8. #77203C2semidirect
9. #77191C2semidirect
10. #76097C2semidirect
11. #75971C2semidirect
12. #74588C2semidirect
13. #74588C2semidirect
14. #74457C2semidirect
15. #74405C2semidirect
17. #73437C2semidirect
96. #2583D3semidirect
688. C3#35245semidirect

Maximal subgroups

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

subgroupindex
C2×#56242
C22×#8872
C3×#25832
#827852
#821822
#778232
#777842
#772032
#771912
#760972
#759712
#745882
#744572
#744052
#735502
#734372
#352453