Finite Group Databasebeta

← #11001998

Subgroups — #11001998

Order 768.

2326 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C2×#5623C211yesmaximal, characteristic, complemented
3C3×#2434C211yesmaximal, characteristic, complemented
4#84065C211yesmaximal, characteristic, complemented
5#83266C214yesmaximal, complemented
6#82819C211yesmaximal, characteristic
7#77261C214yesmaximal, complemented
8#77214C214yesmaximal, complemented
9#76563C214yesmaximal, complemented
10#74655C212yesmaximal
11#74614C212yesmaximal
12#74502C212yesmaximal, complemented
13#74456C212yesmaximal, complemented
14#73605C211yesmaximal, characteristic
15#73445C211yesmaximal, characteristic, complemented
16#36708—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C23×C4×D3C2211yescharacteristic
19C2×C3×C4⋊Q8C2212yescomplemented
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×#946—28no—
22C2×#946C2214yes—
23C2×#946C2214yes—
24C2×#944C2212yescomplemented
25C2×#944C2212yescomplemented
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×(C2×D3)⋊C4C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C2×#924C2211yescharacteristic
30C22×Dic3⋊C4C2211yescharacteristic
31C22×Dic3⋊C4C2211yescharacteristic
32#5623—28no—
33C2×#896C2214yes—
34C2×#889C2212yescomplemented
35C2×#889C2212yescomplemented
36C2×#888C2212yescomplemented
37C2×#878C2212yes—
38C2×#877C2212yes—
39C3×#395C2214yescomplemented
40C3×#391C2212yescomplemented
41C3×#385C2212yes—
42C3×#385C2212yes—
43C3×C4×C22⋊C4C2211yescharacteristic
44#5231C2212yes—
45#5230C2214yescomplemented
46#5229C2212yescomplemented
47#5210C2212yes—
48#5210C2212yes—
49C4×#217C2211yescharacteristic
50#4993C2214yes—
51#4992C2212yes—
52#4990C2214yes—
53#4990C2214yes—
54#4989C2214yes—
55#4983C2214yescomplemented
56#4982C2214yescomplemented
57#4980C2214yescomplemented
58#4979C2214yescomplemented
59#4955C2212yes—
60#4955C2212yes—
61#4943C2212yes—
62#4943C2212yes—
63#4943C2212yes—
64#4942C2212yes—
65#4940C2214yes—
66#4933C2214yescomplemented
67#4675C2214yes—
68#4674C2212yes—
69#4673C2212yes—
70#4669C2214yes—
71#4668C2214yes—
72#4667C2214yes—
73#4660C2214yescomplemented
74#4658C2214yescomplemented
75#4657C2214yescomplemented
76#4655C2214yescomplemented
77#4652C2214yescomplemented
78C2×#544—31no—
79C22×C22⋊Q8—31no—
80#2848—32no—
81#2690—34no—
82#2690—34no—
83#2677—34no—
84#2677—34no—
85#2627—32no—
86#2561—31no—
87#2561—31no—
88#2484—32no—
89#2482—32no—
90#2434—31no—
91#2434D311yescharacteristic, complemented
92#2313—31no—
93C24×D3C2311yescharacteristic
94C22×C3×Q8—24no—
95C22×C3×Q8—44no—
96C22×C3×Q8C2312yes—
97C22×C3×Q8C2312yes—
98C22×C3×Q8C2311yescharacteristic
99C23×C3×C4C2311yescharacteristic
100C23×C3×C4C2311yescharacteristic
101C23×C3×C4C2311yescharacteristic
102C23×Dic3C2311yescharacteristic
103C23×Dic3C2311yescharacteristic
104C23×Dic3C2311yescharacteristic
105C22×C4×D3—24no—
106C22×C4×D3—44no—
107C22×C4×D3C2314yes—
108C22×Dic6C2311yescharacteristic
109C22×Dic6C2312yes—
110C3×C4⋊Q8—44no—
111C3×C22⋊Q8—24no—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2314yes—
116C2×C3×C22⋊C4C2312yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C42C2312yes—
119C2×#217C2312yes—
120C2×#217C2312yes—
121#946—216no—
122#946—216no—
123#944—28no—
124#944—44no—
125C2×(C2×D3)⋊C4—24no—
126C2×(C2×D3)⋊C4—24no—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×C4⋊Dic3—24no—
135C2×C4⋊Dic3C2314yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2311yescharacteristic
138#924—24no—
139C2×Dic3⋊C4—24no—
140C2×Dic3⋊C4D418yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2312yes—
151C2×Dic3⋊C4C2312yes—
152C2×C4×Dic3C2312yes—
153C2×C4×Dic3C2312yes—
154C2×C4×Dic3C2312yes—
155#896—48no—
156#889—28no—
157#889—28no—
158#888—28no—
159#878—28no—
160#877—28no—
161C3×#95C2312yes—
162C3×#95C2314yes—
163C3×#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2312yes—
173C3⋊#95C2314yes—
174C23×Q8C2×D311yescharacteristic
175C24×C4—31no—
176#544—68no—
177C2×C4⋊Q8—32no—
178C2×C4⋊Q8—32no—
179C2×C4⋊Q8—32no—
180C2×C4⋊Q8C2×D312yescomplemented
181C2×#125—32no—
182C2×#125—32no—
183C2×C22⋊Q8—34no—
184C2×C22⋊Q8—34no—
185C2×C22⋊Q8—31no—
186C2×C22⋊Q8—34no—
187C2×C22⋊Q8—32no—
188C2×C22⋊Q8—68no—
189C2×C22⋊Q8C2×D311yescharacteristic
190C2×#117—32no—
191C22×C4⋊C4—31no—
192C22×C4⋊C4—31no—
193C22×C4⋊C4—31no—
194C22×C22⋊C4—31no—
195C22×C22⋊C4—31no—
196#400—34no—
197#400—34no—
198#399—34no—
199#398—34no—
200#398—34no—
201#398—34no—
202#397—32no—
203#397—32no—
204#397—34no—
205#395—34no—
206#395—34no—
207#395C2×D314yescomplemented
208#393—32no—
209#393—32no—
210#391—32no—
211#391C2×D312yescomplemented
212#388—34no—
213#388—34no—
214#388—34no—
215#388—34no—
216#387—34no—
217#387—34no—
218#387—34no—
219#387—34no—
220#385—32no—
221#385—32no—
222#385—34no—
223#385—34no—
224#385C2×D312yes—
225#385C2×D312yes—
226#381—32no—
227#381—32no—
228#381—32no—
229#380—32no—
230#380—34no—
231#379—32no—
232#379—32no—
233C4×C22⋊C4—31no—
234C4×C22⋊C4C2×D311yescharacteristic
235C24×C3C2411yescharacteristic
236C23×D3—22no—
237C23×D3—24no—
238C23×D3—24no—
239C23×D3—42no—
240C23×D3C2412yes—
241C2×C3×Q8—24no—
242C2×C3×Q8—24no—
243C2×C3×Q8—22no—
244C2×C3×Q8—24no—
245C2×C3×Q8—24no—
246C2×C3×Q8—42no—
247C2×C3×Q8—42no—
248C2×C3×Q8—42no—
249C2×C3×Q8—416no—
250C22×C3×C4—22no—
251C22×C3×C4—24no—
252C22×C3×C4—22no—
253C22×C3×C4—42no—
254C22×C3×C4—42no—
255C22×C3×C4C2×D414yes—
256C22×C3×C4C2414yes—
257C22×C3×C4C2412yes—
258C22×C3×C4C2412yes—
259C22×C3×C4C2412yes—
260C22×C3×C4C2411yescharacteristic
261C22×C3×C4C2411yescharacteristic
262C22×C3×C4C2412yes—
263C22×Dic3—24no—
264C22×Dic3—22no—
265C22×Dic3—24no—
266C22×Dic3—42no—
267C22×Dic3C2×D414yes—
268C22×Dic3C2×D414yes—
269C22×Dic3C2414yes—
270C22×Dic3C2412yes—
271C22×Dic3C2412yes—
272C22×Dic3C2412yes—
273C22×Dic3C2412yes—
274C22×Dic3C2412yes—
275C2×C4×D3—28no—
276C2×C4×D3—28no—
277C2×C4×D3—44no—
278C2×C4×D3—416no—
279C2×Dic6—22no—
280C2×Dic6—24no—
281C2×Dic6—44no—
282C2×Dic6—42no—
283C3×C4⋊C4—24no—
284C3×C4⋊C4—22no—
285C3×C4⋊C4—28no—
286C3×C4⋊C4—44no—
287C3×C22⋊C4—24no—
288C3×C22⋊C4—24no—
289C3×C42—42no—
290#217—24no—
291#217—24no—
292(C2×D3)⋊C4—28no—
293(C2×D3)⋊C4—28no—
294(C2×D3)⋊C4—24no—
295(C2×D3)⋊C4—24no—
296(C2×D3)⋊C4—28no—
297(C2×D3)⋊C4—44no—
298(C2×D3)⋊C4—44no—
299C4⋊Dic3—22no—
300C4⋊Dic3—24no—
301C4⋊Dic3—28no—
302Dic3⋊C4—24no—
303Dic3⋊C4—28no—
304Dic3⋊C4—24no—
305Dic3⋊C4—24no—
306Dic3⋊C4—24no—
307Dic3⋊C4—24no—
308Dic3⋊C4—28no—
309Dic3⋊C4—44no—
310Dic3⋊C4—44no—
311Dic3⋊C4C2×D418yes—
312Dic3⋊C4C2×D418yes—
313C4×Dic3—24no—
314C4×Dic3—24no—
315C4×Dic3—42no—
316C25—31no—
317C22×Q8—24no—
318C22×Q8—32no—
319C22×Q8—31no—
320C22×Q8—44no—
321C22×Q8C22×D312yes—
322C22×Q8C22×D312yes—
323C22×Q8C22×D311yescharacteristic
324C23×C4—31no—
325C23×C4—31no—
326C23×C4—31no—
327C23×C4—34no—
328C23×C4—64no—
329C23×C4—124no—
330C23×C4C22×D311yescharacteristic
331C23×C4C22×D311yescharacteristic
332C23×C4C22×D311yescharacteristic
333C4⋊Q8—44no—
334C4⋊Q8—68no—
335C4⋊Q8—68no—
336C4⋊Q8—124no—
337#125—68no—
338#125—68no—
339C22⋊Q8—24no—
340C22⋊Q8—68no—
341C22⋊Q8—616no—
342C22⋊Q8—616no—
343C22⋊Q8—64no—
344C22⋊Q8—128no—
345#117—68no—
346C2×C4⋊C4—38no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—32no—
354C2×C4⋊C4—31no—
355C2×C4⋊C4—32no—
356C2×C4⋊C4—32no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—34no—
359C2×C4⋊C4—34no—
360C2×C4⋊C4—34no—
361C2×C4⋊C4—64no—
362C2×C4⋊C4—64no—
363C2×C4⋊C4C22×D314yes—
364C2×C4⋊C4C22×D311yescharacteristic
365C2×C4⋊C4C22×D312yes—
366C2×C4⋊C4C22×D314yes—
367C2×C22⋊C4—32no—
368C2×C22⋊C4—32no—
369C2×C22⋊C4—32no—
370C2×C22⋊C4—32no—
371C2×C22⋊C4—34no—
372C2×C22⋊C4—34no—
373C2×C22⋊C4—34no—
374C2×C22⋊C4—34no—
375C2×C22⋊C4—34no—
376C2×C22⋊C4—64no—
377C2×C22⋊C4—64no—
378C2×C22⋊C4C22×D312yes—
379C2×C22⋊C4C22×D312yes—
380C2×C42—32no—
381C2×C42—32no—
382C2×C42—32no—
383C2×C42C22×D312yes—
384#95—32no—
385#95—34no—
386#95—34no—
387#95—34no—
388#95—34no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95—34no—
393#95—34no—
394#95C22×D312yes—
395#95C22×D314yes—
396#95C22×D314yes—
397C23×C3—22no—
398C23×C3—22no—
399C23×C3C22×D412yes—
400C23×C32−1+414yes—
401C23×C3C2511yescharacteristic
402C22×D3—28no—
403C22×D3—22no—
404C22×D3—24no—
405C22×D3—24no—
406C22×D3—22no—
407C22×D3—416no—
408C22×D3—44no—
409C22×D3—44no—
410C22×D32−1+414yes—
411C3×Q8—48no—
412C3×Q8—48no—
413C3×Q8—82no—
414C2×C3×C4—28no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—24no—
424C2×C3×C4—24no—
425C2×C3×C4—22no—
426C2×C3×C4—24no—
427C2×C3×C4—24no—
428C2×C3×C4—24no—
429C2×C3×C4—44no—
430C2×C3×C4—44no—
431C2×C3×C4C22×D412yes—
432C2×C3×C4C22×D412yes—
433C2×C3×C4H(5,𝔽2)14yes—
434C2×Dic3—28no—
435C2×Dic3—24no—
436C2×Dic3—24no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—24no—
441C2×Dic3—24no—
442C2×Dic3—24no—
443C2×Dic3—24no—
444C2×Dic3—24no—
445C2×Dic3—24no—
446C2×Dic3—44no—
447C2×Dic3—44no—
448C2×Dic3C22×D414yes—
449C2×Dic3C22×D414yes—
450C4×D3—416no—
451Dic6—84no—
452Dic6—82no—
453C24—32no—
454C24—64no—
455C24—62no—
456C24—64no—
457C24—122no—
458C24C23×D311yescharacteristic
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—22no—
462C2×Q8—24no—
463C2×Q8—24no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—416no—
468C2×Q8—64no—
469C2×Q8—62no—
470C2×Q8—124no—
471C2×Q8—122no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—34no—
476C22×C4—32no—
477C22×C4—32no—
478C22×C4—34no—
479C22×C4—34no—
480C22×C4—32no—
481C22×C4—32no—
482C22×C4—32no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—68no—
486C22×C4—64no—
487C22×C4—62no—
488C22×C4—64no—
489C22×C4—68no—
490C22×C4—124no—
491C22×C4—1216no—
492C22×C4—122no—
493C22×C4D3×D414yes—
494C22×C4C23×D314yes—
495C22×C4C23×D312yes—
496C22×C4C23×D312yes—
497C22×C4C23×D312yes—
498C22×C4C23×D311yescharacteristic
499C22×C4C23×D311yescharacteristic
500C22×C4C23×D312yes—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—28no—
504C4⋊C4—38no—
505C4⋊C4—38no—
506C4⋊C4—44no—
507C4⋊C4—68no—
508C4⋊C4—68no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—64no—
512C4⋊C4—64no—
513C4⋊C4—64no—
514C4⋊C4—62no—
515C4⋊C4—64no—
516C4⋊C4—68no—
517C4⋊C4—124no—
518C4⋊C4—124no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—68no—
522C22⋊C4—68no—
523C22⋊C4—68no—
524C22⋊C4—64no—
525C22⋊C4—64no—
526C22⋊C4—64no—
527C22⋊C4—64no—
528C22⋊C4—124no—
529C22⋊C4—124no—
530C42—42no—
531C42—64no—
532C42—64no—
533C42—122no—
534C22×C3—22no—
535C22×C3—22no—
536C22×C3—28no—
537C22×C3#53614yes—
538C22×C3C23×D411yescharacteristic
539C22×C3C2×H(5,𝔽2)12yes—
540C22×C3C2×2−1+412yes—
541C22×C3C2×2−1+412yes—
542C2×D3—24no—
543C2×D3—28no—
544C2×D3—416no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—44no—
548C3×C4—22no—
549C3×C4—24no—
550C3×C4—22no—
551C3×C4—42no—
552C3×C4—42no—
553C3×C4—42no—
554C3×C4—44no—
555Dic3—24no—
556Dic3—24no—
557Dic3—42no—
558Dic3—42no—
559Dic3—42no—
560Dic3—44no—
561C23—22no—
562C23—22no—
563C23—34no—
564C23—68no—
565C23—62no—
566C23—64no—
567C23—64no—
568C23—62no—
569C23—1216no—
570C23—124no—
571C23—124no—
572C23C2×D3×D412yes—
573C23#100714yes—
574C23C24×D311yescharacteristic
575Q8—48no—
576Q8—48no—
577Q8—82no—
578Q8—242no—
579Q8—244no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—34no—
596C2×C4—34no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—68no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—1216no—
612C2×C4—124no—
613C2×C4—124no—
614C2×C4C2×D3×D412yes—
615C2×C4C2×D3×D412yes—
616C2×C4#100414yes—
617C2×C3—24no—
618C2×C3C2×#53611yescharacteristic
619C2×C3#348411yescharacteristic
620C2×C3#348611yescharacteristic
621C2×C3#353814yes—
622D3—44no—
623C22—22no—
624C22—22no—
625C22—28no—
626C22—64no—
627C22—68no—
628C22—1216no—
629C22—122no—
630C22—122no—
631C22—124no—
632C22#572414yes—
633C22C22×D3×D411yescharacteristic
634C22C2×#100412yes—
635C22C2×#100712yes—
636C22D3×2−1+412yes—
637C4—24no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—64no—
645C4—64no—
646C4—122no—
647C4—122no—
648C4—122no—
649C4—124no—
650C3#3670811yescharacteristic, complemented, Sylow
651C2—24no—
652C2—124no—
653C2#8591911yescharacteristic
654C2C2×#572411yescharacteristic
655C2#8635511yescharacteristic
656C2#8676214yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2434C2
prime radicalC24×D3C23
hypercenter#2434D3
hypocenterC3#36708
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2434

Complements

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

NH ≅ G/Ntype
2. C2×#5623C2semidirect
3. C3×#2434C2semidirect
4. #84065C2semidirect
5. #83266C2semidirect
7. #77261C2semidirect
8. #77214C2semidirect
9. #76563C2semidirect
12. #74502C2semidirect
13. #74456C2semidirect
15. #73445C2semidirect
19. C2×C3×C4⋊Q8C22semidirect
24. C2×#944C22semidirect
25. C2×#944C22semidirect
34. C2×#889C22semidirect
35. C2×#889C22semidirect
36. C2×#888C22semidirect
39. C3×#395C22semidirect
40. C3×#391C22semidirect
45. #5230C22semidirect
46. #5229C22semidirect
55. #4983C22semidirect
56. #4982C22semidirect
57. #4980C22semidirect
58. #4979C22semidirect
66. #4933C22semidirect
73. #4660C22semidirect
74. #4658C22semidirect
75. #4657C22semidirect
76. #4655C22semidirect
77. #4652C22semidirect
91. #2434D3semidirect
180. C2×C4⋊Q8C2×D3semidirect
207. #395C2×D3semidirect
211. #391C2×D3semidirect
650. C3#36708semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
C2×#56232
C3×#24342
#840652
#832662
#828192
#772612
#772142
#765632
#746552
#746142
#745022
#744562
#736052
#734452
#367083