Finite Group Databasebeta

← #11069436

Subgroups — #11069436

Order 768.

2078 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5729C212yesmaximal, complemented
2C3×#2551C211yesmaximal, characteristic, complemented
3#83256C212yesmaximal, complemented
4#83239C212yesmaximal, complemented
5#83236C212yesmaximal
6#77223C214yesmaximal, complemented
7#77216C214yesmaximal, complemented
8#76251C212yesmaximal, complemented
9#76059C212yesmaximal
10#76029C212yesmaximal, complemented
11#76002C212yesmaximal, complemented
12#74360C211yesmaximal, characteristic
13#74355C211yesmaximal, characteristic, complemented
14#73716C211yesmaximal, characteristic, complemented
15#73638C211yesmaximal, characteristic
16C4×#888C211yesmaximal, characteristic, complemented
17#73617C211yesmaximal, characteristic
18#39370—31nomaximal, Sylow
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2212yes—
21C2×#946C2214yes—
22C2×#944C2214yes—
23#5729—216no—
24C2×#896C2214yes—
25C2×#893C2212yes—
26C2×#892C2212yes—
27C2×D3×C4⋊C4C2212yes—
28C2×#890C2212yes—
29C2×#889C2214yes—
30C2×#888C2211yescharacteristic
31C2×C4×D12C2211yescharacteristic
32C2×#872C2212yes—
33C2×#870C2212yes—
34C2×C4×Dic6C2212yes—
35C3×#395C2214yes—
36C3×#389C2212yes—
37C3×#389C2212yes—
38C3×#384C2211yescharacteristic
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×C4×C4⋊C4C2211yescharacteristic
41#5231C2214yes—
42#5229C2214yes—
43#4994C2214yes—
44#4994C2214yes—
45#4989C2214yes—
46#4989C2214yes—
47#4988C2211yescharacteristic
48#4988C2211yescharacteristic
49#4987C2212yes—
50#4987C2212yes—
51#4984C2212yes—
52#4984C2211yescharacteristic
53#4984C2212yes—
54#4983C2214yes—
55#4983C2214yes—
56#4978C2214yes—
57#4977C2212yes—
58#4975C2212yes—
59#4975C2211yescharacteristic
60#4972C2212yes—
61#4972C2212yes—
62#4972C2211yescharacteristic
63#4972C2212yes—
64#4972C2212yes—
65#4972C2211yescharacteristic
66#4972C2211yescharacteristic
67#4939C2212yes—
68C4×(C2×D3)⋊C4C2212yes—
69#4935C2211yescharacteristic
70C4×C4⋊Dic3C2212yes—
71C4×C4⋊Dic3C2211yescharacteristic
72C4×Dic3⋊C4C2212yes—
73C42×Dic3C2211yescharacteristic
74#4928C2212yes—
75#4676C2214yes—
76#4670C2214yes—
77#4669C2212yes—
78#4669C2212yes—
79#4667C2214yes—
80#4658C2214yes—
81#4657C2214yes—
82#4657C2214yes—
83#4647C2212yes—
84#4646C2212yes—
85C2×#544—32no—
86#2859—31no—
87#2729—34no—
88#2704—34no—
89#2629—32no—
90#2552—31no—
91#2551D311yescharacteristic, complemented
92#2546—32no—
93#2538—32no—
94#2506—32no—
95#2452—32no—
96#2442—32no—
97#2439—32no—
98#2354—31no—
99D4×C4⋊C4—31no—
100C4×C4⋊Q8—31no—
101#2293—31no—
102C22×C3×Q8C2314yes—
103C22×D12C2311yescharacteristic
104C22×C4×D3C2312yes—
105C22×C4×D3C2314yes—
106C22×Dic6C2312yes—
107C3×C4⋊Q8—28no—
108C3×C4⋊Q8—44no—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2314yes—
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C42C2312yes—
119C2×C3×C42C2312yes—
120C2×C3×C42C2311yescharacteristic
121#946—216no—
122#944—216no—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×C4⋊Dic3C2314yes—
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2311yescharacteristic
138C2×C4⋊Dic3C2311yescharacteristic
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2312yes—
150C2×C4×Dic3C2314yes—
151C2×C4×Dic3C2311yescharacteristic
152C2×C4×Dic3C2311yescharacteristic
153C2×C4×Dic3C2311yescharacteristic
154C2×C4×Dic3C2312yes—
155C2×C4×Dic3C2311yescharacteristic
156C2×C4×Dic3C2311yescharacteristic
157C2×C4×Dic3C2311yescharacteristic
158#896—216no—
159#893—44no—
160#892—28no—
161D3×C4⋊C4—28no—
162#890—28no—
163#889—216no—
164#888D418yescomplemented
165C4×D12Q818yescomplemented
166#872—28no—
167#870—28no—
168C4×Dic6—44no—
169C3×#95C2312yes—
170C3×#95C2312yes—
171C3×#95C2314yes—
172C3⋊#95C2312yes—
173C3⋊#95C2312yes—
174C3⋊#95C2312yes—
175C3⋊#95C2314yes—
176C3⋊#95C2314yes—
177C3⋊#95C2314yes—
178C3⋊#95C2312yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181C3⋊#95C2312yes—
182#544—616no—
183C2×C4⋊Q8—34no—
184C2×C4⋊Q8—31no—
185C2×C4⋊Q8C2×D312yes—
186C2×C4⋊Q8C2×D312yes—
187C2×#125—32no—
188C2×#125—34no—
189C2×#125—32no—
190C2×C22⋊Q8—34no—
191C2×C22⋊Q8—34no—
192C2×C4×Q8—32no—
193C2×C4×D4—32no—
194C2×C4×D4—31no—
195C2×#117—32no—
196C2×#117—32no—
197C22×C4⋊C4—32no—
198#400—34no—
199#398—34no—
200#398—34no—
201#398—34no—
202#398—34no—
203#396—34no—
204#396—34no—
205#396—34no—
206#395C2×D314yes—
207#393—34no—
208#391—34no—
209#391—32no—
210#390—31no—
211#390—31no—
212#390—32no—
213#389—31no—
214#389—31no—
215#389—32no—
216#389—32no—
217#389—32no—
218#389—32no—
219#389—32no—
220#389—32no—
221#389—31no—
222#389—31no—
223#389—32no—
224#389—32no—
225#389C2×D312yes—
226#389C2×D312yes—
227#388—34no—
228#388—32no—
229#388—32no—
230#387—34no—
231#385—34no—
232#385—34no—
233#384—31no—
234#384—32no—
235#384—31no—
236#384C2×D311yescharacteristic
237#382—32no—
238#380—32no—
239#380—32no—
240#380—34no—
241C4×C4⋊C4—31no—
242C4×C4⋊C4—32no—
243C4×C4⋊C4—32no—
244C4×C4⋊C4C2×D311yescharacteristic
245C4×C4⋊C4C2×D311yescharacteristic
246C4×C22⋊C4—32no—
247C43—31no—
248C23×D3C2412yes—
249C2×C3×Q8—28no—
250C2×C3×Q8—44no—
251C2×C3×Q8—44no—
252C22×C3×C4C2412yes—
253C22×C3×C4C2412yes—
254C22×C3×C4C2414yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2412yes—
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2411yescharacteristic
259C22×C3×C4C2411yescharacteristic
260C22×Dic3C2411yescharacteristic
261C22×Dic3C2411yescharacteristic
262C22×Dic3C2412yes—
263C22×Dic3C2414yes—
264C22×Dic3C2412yes—
265C22×Dic3C2411yescharacteristic
266C22×Dic3C2411yescharacteristic
267C22×Dic3C2412yes—
268C2×D12—22no—
269C2×D12—22no—
270C2×D12C2×Q814yes—
271C2×C4×D3—24no—
272C2×C4×D3—24no—
273C2×C4×D3—28no—
274C2×C4×D3—44no—
275C2×C4×D3—44no—
276C2×C4×D3C2×Q818yes—
277C2×Dic6—42no—
278C2×Dic6—42no—
279C2×Dic6C2×D418yes—
280C3×C4⋊C4—28no—
281C3×C4⋊C4—24no—
282C3×C4⋊C4—24no—
283C3×C4⋊C4—22no—
284C3×C4⋊C4—22no—
285C3×C4⋊C4—22no—
286C3×C4⋊C4—42no—
287C3×C4⋊C4—44no—
288C3×C4⋊C4C2×D414yes—
289C3×C42—24no—
290C3×C42—24no—
291C3×C42C2×Q814yes—
292(C2×D3)⋊C4—28no—
293(C2×D3)⋊C4—24no—
294(C2×D3)⋊C4—24no—
295(C2×D3)⋊C4—44no—
296(C2×D3)⋊C4—42no—
297(C2×D3)⋊C4—42no—
298(C2×D3)⋊C4—44no—
299(C2×D3)⋊C4C2×Q818yes—
300C4⋊Dic3—24no—
301C4⋊Dic3—24no—
302C4⋊Dic3—22no—
303C4⋊Dic3—28no—
304C4⋊Dic3—42no—
305C4⋊Dic3—42no—
306C4⋊Dic3C2×D414yes—
307C4⋊Dic3C2×Q814yes—
308Dic3⋊C4—28no—
309Dic3⋊C4—28no—
310Dic3⋊C4—24no—
311Dic3⋊C4—24no—
312Dic3⋊C4—28no—
313Dic3⋊C4—28no—
314Dic3⋊C4—42no—
315Dic3⋊C4—42no—
316Dic3⋊C4—42no—
317Dic3⋊C4—44no—
318Dic3⋊C4C2×D418yes—
319C4×Dic3—28no—
320C4×Dic3—22no—
321C4×Dic3—24no—
322C4×Dic3—22no—
323C4×Dic3—22no—
324C4×Dic3—41no—
325C4×Dic3—41no—
326C4×Dic3C2×D414yes—
327C22×Q8—32no—
328C22×Q8C22×D314yes—
329C22×D4—31no—
330C23×C4—32no—
331C23×C4—34no—
332C4⋊Q8—28no—
333C4⋊Q8—38no—
334C4⋊Q8—44no—
335C4⋊Q8—616no—
336#125—68no—
337#125—68no—
338#125—616no—
339C22⋊Q8—616no—
340C22⋊Q8—616no—
341C4×Q8—124no—
342C4×D4—38no—
343C4×D4—124no—
344#117—68no—
345#117—68no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—34no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—34no—
356C2×C4⋊C4—34no—
357C2×C4⋊C4—31no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—32no—
361C2×C4⋊C4—32no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4—34no—
364C2×C4⋊C4—32no—
365C2×C4⋊C4—68no—
366C2×C4⋊C4C22×D312yes—
367C2×C4⋊C4C22×D312yes—
368C2×C4⋊C4C22×D314yes—
369C2×C4⋊C4C22×D311yescharacteristic
370C2×C4⋊C4C22×D311yescharacteristic
371C2×C4⋊C4C22×D311yescharacteristic
372C2×C4⋊C4C22×D314yes—
373C2×C4⋊C4C22×D312yes—
374C2×C4⋊C4C22×D311yescharacteristic
375C2×C22⋊C4—32no—
376C2×C22⋊C4—34no—
377C2×C22⋊C4—32no—
378C2×C22⋊C4—32no—
379C2×C22⋊C4—34no—
380C2×C22⋊C4—32no—
381C2×C22⋊C4—32no—
382C2×C22⋊C4—34no—
383C2×C42—31no—
384C2×C42—34no—
385C2×C42—32no—
386C2×C42—31no—
387C2×C42—31no—
388C2×C42—31no—
389C2×C42—31no—
390C2×C42—31no—
391C2×C42C22×D312yes—
392C2×C42C22×D312yes—
393C2×C42C22×D311yescharacteristic
394#95—32no—
395#95—32no—
396#95—34no—
397#95—32no—
398#95—32no—
399#95—32no—
400#95—34no—
401#95—34no—
402#95—34no—
403#95—34no—
404#95C22×D312yes—
405#95C22×D312yes—
406#95C22×D314yes—
407C23×C3C2511yescharacteristic
408C22×D3—22no—
409C22×D3—22no—
410C22×D3—22no—
411C22×D3—22no—
412C22×D3—22no—
413C22×D3—22no—
414C22×D3C22×Q814yes—
415C3×Q8—88no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—24no—
423C2×C3×C4—22no—
424C2×C3×C4—24no—
425C2×C3×C4—22no—
426C2×C3×C4—22no—
427C2×C3×C4—22no—
428C2×C3×C4—22no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—24no—
432C2×C3×C4C22×D414yes—
433C2×C3×C4C22×D412yes—
434C2×C3×C4C22×Q812yes—
435C2×C3×C4C22×Q812yes—
436C2×C3×C4C22×Q812yes—
437C2×C3×C4H(5,𝔽2)12yes—
438C2×C3×C42−1+412yes—
439C2×C3×C42−1+412yes—
440C2×Dic3—22no—
441C2×Dic3—21no—
442C2×Dic3—22no—
443C2×Dic3—21no—
444C2×Dic3—22no—
445C2×Dic3—21no—
446C2×Dic3—21no—
447C2×Dic3—22no—
448C2×Dic3—24no—
449C2×Dic3—21no—
450C2×Dic3—22no—
451C2×Dic3—21no—
452C2×Dic3—21no—
453C2×Dic3—21no—
454C2×Dic3—24no—
455C2×Dic3—22no—
456C2×Dic3—21no—
457C2×Dic3—21no—
458C2×Dic3—22no—
459C2×Dic3—24no—
460C2×Dic3C22×D412yes—
461C2×Dic3C22×D412yes—
462C2×Dic3C22×D414yes—
463C2×Dic3C22×Q814yes—
464D12—28no—
465C4×D3—216no—
466C4×D3—88no—
467Dic6—48no—
468C24—32no—
469C2×Q8—28no—
470C2×Q8—38no—
471C2×Q8—44no—
472C2×Q8—44no—
473C2×Q8—122no—
474C2×Q8—122no—
475C2×D4—34no—
476C2×D4—62no—
477C2×D4—62no—
478C22×C4—32no—
479C22×C4—32no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—38no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—32no—
486C22×C4—34no—
487C22×C4—68no—
488C22×C4—64no—
489C22×C4—64no—
490C22×C4—124no—
491C22×C4—124no—
492C22×C4C23×D312yes—
493C22×C4C23×D312yes—
494C22×C4C23×D314yes—
495C22×C4C23×D312yes—
496C22×C4C23×D312yes—
497C22×C4C23×D311yescharacteristic
498C22×C4C23×D311yescharacteristic
499C22×C4C23×D311yescharacteristic
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—34no—
507C4⋊C4—38no—
508C4⋊C4—34no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—68no—
512C4⋊C4—64no—
513C4⋊C4—64no—
514C4⋊C4—68no—
515C4⋊C4—68no—
516C4⋊C4—68no—
517C4⋊C4—68no—
518C4⋊C4—64no—
519C4⋊C4—62no—
520C4⋊C4—64no—
521C4⋊C4—122no—
522C4⋊C4—122no—
523C4⋊C4—122no—
524C4⋊C4—122no—
525C4⋊C4—124no—
526C4⋊C4—122no—
527C4⋊C4D3×D414yes—
528C22⋊C4—38no—
529C22⋊C4—64no—
530C22⋊C4—64no—
531C22⋊C4—68no—
532C22⋊C4—124no—
533C22⋊C4—122no—
534C22⋊C4—124no—
535C22⋊C4—122no—
536C42—24no—
537C42—24no—
538C42—34no—
539C42—68no—
540C42—64no—
541C42—62no—
542C42—62no—
543C42—62no—
544C42—121no—
545C42—121no—
546C42D3×Q814yes—
547C22×C3C23×D411yescharacteristic
548C22×C3C23×Q811yescharacteristic
549C22×C3C2×H(5,𝔽2)11yescharacteristic
550C22×C3C2×2−1+411yescharacteristic
551C22×C3C2×2−1+411yescharacteristic
552C22×C3#58511yescharacteristic
553C22×C3#58511yescharacteristic
554C2×D3—24no—
555C2×D3—24no—
556C2×D3—24no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C3×C4—24no—
562C3×C4—22no—
563C3×C4—44no—
564C3×C4—42no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4#54414yescomplemented
568C3×C4D4×Q814yes—
569Dic3—24no—
570Dic3—22no—
571Dic3—24no—
572Dic3—22no—
573Dic3—44no—
574Dic3—41no—
575Dic3—42no—
576Dic3—41no—
577C23—34no—
578C23—62no—
579C23—62no—
580C23—62no—
581C23—62no—
582C23—62no—
583C23—62no—
584C23C24×D311yescharacteristic
585Q8—88no—
586Q8—128no—
587D4—68no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—32no—
605C2×C4—32no—
606C2×C4—34no—
607C2×C4—34no—
608C2×C4—62no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—62no—
613C2×C4—61no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—616no—
617C2×C4—64no—
618C2×C4—62no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—64no—
624C2×C4—62no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—64no—
628C2×C4—62no—
629C2×C4—248no—
630C2×C4C2×D3×D414yes—
631C2×C4C2×D3×D412yes—
632C2×C4#100412yes—
633C2×C4C2×D3×Q812yes—
634C2×C4C2×D3×Q812yes—
635C2×C4C2×D3×Q812yes—
636C2×C4#100712yes—
637C2×C4#100712yes—
638C2×C3C2×#54411yescharacteristic
639C2×C3C2×D4×Q811yescharacteristic
640C2×C3#349611yescharacteristic
641C2×C3#349711yescharacteristic
642C2×C3#350411yescharacteristic
643C2×C3#354711yescharacteristic
644C2×C3#355511yescharacteristic
645D3—44no—
646C22—64no—
647C22—64no—
648C22—64no—
649C22—122no—
650C22—122no—
651C22—122no—
652C22—122no—
653C22C22×D3×D411yescharacteristic
654C22C2×#100411yescharacteristic
655C22C22×D3×Q811yescharacteristic
656C22C2×#100711yescharacteristic
657C22C2×#100711yescharacteristic
658C22#596611yescharacteristic
659C22#596811yescharacteristic
660C4—22no—
661C4—24no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—64no—
667C4—62no—
668C4—64no—
669C4—62no—
670C4—121no—
671C4—124no—
672C4—122no—
673C4—121no—
674C4#572714yes—
675C4#572914yescomplemented
676C3#3937011yescharacteristic, complemented, Sylow
677C2—124no—
678C2C2×#572911yescharacteristic
679C2C2×#572711yescharacteristic
680C2#8658711yescharacteristic
681C2#8662111yescharacteristic
682C2#8676011yescharacteristic
683C2#8676311yescharacteristic
684C2#8704811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2551C2
prime radicalC22×D12C23
hypercenter#2551D3
hypocenterC3#39370
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2551

Complements

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

NH ≅ G/Ntype
1. C2×#5729C2semidirect
2. C3×#2551C2semidirect
3. #83256C2semidirect
4. #83239C2semidirect
6. #77223C2semidirect
7. #77216C2semidirect
8. #76251C2semidirect
10. #76029C2semidirect
11. #76002C2semidirect
13. #74355C2semidirect
14. #73716C2semidirect
16. C4×#888C2semidirect
91. #2551D3semidirect
164. #888D4semidirect
165. C4×D12Q8semidirect
567. C3×C4#544semidirect
675. C4#5729semidirect
676. C3#39370semidirect

Maximal subgroups

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

subgroupindex
C2×#57292
C3×#25512
#832562
#832392
#832362
#772232
#772162
#762512
#760592
#760292
#760022
#743602
#743552
#737162
#736382
C4×#8882
#736172
#393703