Finite Group Databasebeta

← #11069327

Subgroups — #11069327

Order 768.

2958 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5553C211yesmaximal, characteristic
2C2×D4×Dic6C211yesmaximal, characteristic, complemented
3C2×#5526C212yesmaximal, complemented
4C3×#2293C211yesmaximal, characteristic, complemented
5#82287C212yesmaximal, complemented
6#76317C214yesmaximal, complemented
7#76011C214yesmaximal, complemented
8#75669C218yesmaximal, complemented
9#73712C211yesmaximal, characteristic, complemented
10#73711C211yesmaximal, characteristic, complemented
11#73491C212yesmaximal, complemented
12#73489C212yesmaximal, complemented
13#73313C211yesmaximal, characteristic, complemented
14C4×#869C211yesmaximal, characteristic, complemented
15#39363—31nomaximal, Sylow
16C22×#235C2212yes—
17C23×Dic6C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117C2212yes—
20C2×#940C2212yes—
21C2×D4×Dic3C2212yes—
22C2×#930C2212yes—
23C2×#924C2212yes—
24#5553—28no—
25D4×Dic6—28no—
26#5526—216no—
27C2×#897C2218yes—
28C2×#893C2212yes—
29C2×#888C2212yes—
30C2×#886C2218yes—
31C2×#884C2214yes—
32C2×#878C2214yes—
33C2×#875C2212yes—
34C2×#875C2212yes—
35C2×#875C2212yes—
36C2×#874C2211yescharacteristic
37C2×#869C2212yes—
38C2×#869C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C2×C4×Dic6C2211yescharacteristic
42C3×#390C2212yes—
43C3×#389C2212yes—
44C3×#388C2214yes—
45C3×C4×C4⋊C4C2211yescharacteristic
46C3×C4×C22⋊C4C2212yes—
47C3×C43C2211yescharacteristic
48#4994C2218yes—
49#4988C2214yes—
50#4984C2214yes—
51#4961C2218yes—
52#4961C2218yes—
53#4958C2218yes—
54#4939C2212yes—
55#4939C2214yes—
56#4939C2214yes—
57#4935C2211yescharacteristic
58C4×C4⋊Dic3C2214yes—
59#4929C2211yescharacteristic
60#4929C2211yescharacteristic
61#4929C2212yes—
62#4928C2214yes—
63#4673C2218yes—
64#4672C2214yes—
65#4668C2218yes—
66#4661C2214yes—
67C2×#550—31no—
68C2×D4×Q8—31no—
69C2×#536—32no—
70#2858—31no—
71#2681—38no—
72#2620—32no—
73#2548—31no—
74#2546—31no—
75#2452—34no—
76#2445—34no—
77#2436—32no—
78#2432—32no—
79C4×C4⋊Q8—31no—
80#2293D311yescharacteristic, complemented
81C22×C3×D4C2311yescharacteristic
82C23×C3×C4C2312yes—
83C22×#67C2314yes—
84C22×#67C2314yes—
85C23×Dic3C2314yes—
86C2×#235—28no—
87C2×#235—28no—
88C2×#235—44no—
89C22×D12C2312yes—
90C22×D12C2312yes—
91C22×D12C2312yes—
92C22×C4×D3C2314yes—
93C22×Dic6—28no—
94C22×Dic6—28no—
95C22×Dic6—44no—
96C22×Dic6C2312yes—
97C22×Dic6C2314yes—
98C22×Dic6C2311yescharacteristic
99C22×Dic6C2311yescharacteristic
100C3×C4×D4D418yes—
101C3×#117—28no—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C22⋊C4C2312yes—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2312yes—
110C2×C3×C42C2312yes—
111C2×C3×C42C2312yes—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×#217C2314yes—
115#940—28no—
116D4×Dic3—28no—
117#930—28no—
118C2×(C2×D3)⋊C4C2318yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2318yes—
121C2×(C2×D3)⋊C4C2318yes—
122C2×(C2×D3)⋊C4C2318yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2318yes—
125C2×C4⋊Dic3C2311yescharacteristic
126C2×C4⋊Dic3C2311yescharacteristic
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2314yes—
129C2×C4⋊Dic3C2314yes—
130#924—28no—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2312yes—
134C2×C4×Dic3C2312yes—
135C2×C4×Dic3C2312yes—
136#897—232no—
137#893—28no—
138#888—28no—
139#886—232no—
140#884—216no—
141#878—216no—
142#875—28no—
143#875—28no—
144#875—44no—
145#874—24no—
146#869—28no—
147#869—24no—
148#869D418yescomplemented
149C4×Dic6—24no—
150C4×Dic6—24no—
151C3×#95C2312yes—
152C3×#95C2314yes—
153C3⋊#95C2318yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C22×#41—32no—
157C23×Q8—32no—
158#550—68no—
159D4×Q8—68no—
160#536—616no—
161C2×C4⋊Q8—32no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—31no—
164C2×C4⋊Q8—32no—
165C2×#127—32no—
166C2×#127—31no—
167C2×#124—32no—
168C2×#124—32no—
169C2×#124—32no—
170C2×#123—38no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—38no—
173C2×C22⋊Q8—34no—
174C2×#121—32no—
175C2×#121—34no—
176C2×C4×Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×D4—32no—
179C2×C4×D4—32no—
180C2×C4×D4C2×D311yescharacteristic
181C2×#117C2×D312yes—
182#398—34no—
183#396—38no—
184#394—38no—
185#394—38no—
186#394—34no—
187#393—38no—
188#391—31no—
189#391—31no—
190#391—32no—
191#390—34no—
192#390—34no—
193#390—34no—
194#390—32no—
195#390C2×D312yes—
196#389—34no—
197#389—34no—
198#389C2×D312yes—
199#388C2×D314yes—
200#386—38no—
201#385—38no—
202#384—31no—
203C4×C4⋊C4—34no—
204C4×C4⋊C4C2×D311yescharacteristic
205C4×C22⋊C4C2×D312yes—
206C43C2×D311yescharacteristic
207C24×C3C2412yes—
208C23×D3C2414yes—
209C2×C3×D4—24no—
210C2×C3×D4C2×D414yes—
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4C2×D418yes—
214C22×C3×C4C2412yes—
215C22×C3×C4C2414yes—
216C22×C3×C4C2411yescharacteristic
217C22×C3×C4C2412yes—
218C22×C3×C4C2412yes—
219C22×C3×C4C2411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C2×#67—28no—
222C2×#67—28no—
223C2×#67—28no—
224C2×#67—44no—
225C2×#67—48no—
226C22×Dic3—28no—
227C22×Dic3—28no—
228C22×Dic3—44no—
229C22×Dic3C2412yes—
230C22×Dic3C2412yes—
231C22×Dic3C2414yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234#235—432no—
235C2×D12—24no—
236C2×D12—24no—
237C2×D12—24no—
238C2×D12—24no—
239C2×D12—28no—
240C2×D12—24no—
241C2×D12—44no—
242C2×C4×D3—28no—
243C2×C4×D3—28no—
244C2×C4×D3—44no—
245C2×Dic6—22no—
246C2×Dic6—22no—
247C2×Dic6—22no—
248C2×Dic6—22no—
249C2×Dic6—24no—
250C2×Dic6—24no—
251C2×Dic6—28no—
252C2×Dic6—24no—
253C2×Dic6—28no—
254C2×Dic6—432no—
255C2×Dic6—44no—
256C2×Dic6C2×D414yes—
257C2×Dic6C2×D414yes—
258C3×C4⋊C4—28no—
259C3×C4⋊C4—24no—
260C3×C4⋊C4C2×D414yes—
261C3×C22⋊C4—28no—
262C3×C22⋊C4—44no—
263C3×C22⋊C4C2×D418yes—
264C3×C42—24no—
265C3×C42—22no—
266C3×C42—24no—
267C3×C42—42no—
268C3×C42C2×D414yes—
269C3×C42C2×D414yes—
270#217—28no—
271(C2×D3)⋊C4—216no—
272(C2×D3)⋊C4—28no—
273(C2×D3)⋊C4—216no—
274(C2×D3)⋊C4—48no—
275(C2×D3)⋊C4—48no—
276C4⋊Dic3—22no—
277C4⋊Dic3—22no—
278C4⋊Dic3—24no—
279C4⋊Dic3—24no—
280C4⋊Dic3—216no—
281C4⋊Dic3—44no—
282C4⋊Dic3C2×D4116yes—
283Dic3⋊C4—24no—
284Dic3⋊C4—24no—
285Dic3⋊C4—28no—
286C4×Dic3—24no—
287C4×Dic3—24no—
288C2×#41—68no—
289C2×#41—68no—
290C2×#41—124no—
291C22×Q8—31no—
292C22×Q8—31no—
293C22×Q8—32no—
294C22×Q8—34no—
295C22×Q8—68no—
296C22×Q8—68no—
297C22×Q8—124no—
298C22×D4—34no—
299C22×D4—34no—
300C22×D4—32no—
301C22×D4—32no—
302C22×D4—32no—
303C22×D4C22×D311yescharacteristic
304C23×C4—34no—
305C23×C4—34no—
306C23×C4C22×D312yes—
307C4⋊Q8—38no—
308C4⋊Q8—68no—
309C4⋊Q8—68no—
310C4⋊Q8—64no—
311#127—68no—
312#127—64no—
313#124—68no—
314#124—68no—
315#124—124no—
316#123—632no—
317C22⋊Q8—632no—
318C22⋊Q8—68no—
319C22⋊Q8—616no—
320#121—68no—
321#121—616no—
322C4×Q8—64no—
323C4×Q8—64no—
324C4×D4—68no—
325C4×D4—68no—
326C4×D4D1218yes—
327#117—28no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—38no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4C22×D314yes—
339C2×C4⋊C4C22×D312yes—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—38no—
344C2×C22⋊C4—38no—
345C2×C22⋊C4—38no—
346C2×C22⋊C4—38no—
347C2×C22⋊C4C22×D314yes—
348C2×C22⋊C4C22×D314yes—
349C2×C22⋊C4C22×D312yes—
350C2×C42—32no—
351C2×C42—32no—
352C2×C42C22×D312yes—
353C2×C42C22×D311yescharacteristic
354C2×C42C22×D312yes—
355C2×C42C22×D312yes—
356C2×C42C22×D311yescharacteristic
357C2×C42C22×D311yescharacteristic
358#95—34no—
359#95—34no—
360#95—38no—
361#95C22×D314yes—
362#95C22×D312yes—
363C23×C3—22no—
364C23×C3—22no—
365C23×C3—24no—
366C23×C3—24no—
367C23×C3C22×D414yes—
368C23×C3C2511yescharacteristic
369C22×D3—24no—
370C22×D3—24no—
371C22×D3—24no—
372C22×D3—28no—
373C22×D3—28no—
374C3×D4—28no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—216no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4—22no—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D414yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D414yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4H(5,𝔽2)12yes—
394C2×C3×C42−1+414yes—
395#67—416no—
396#67—88no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—24no—
407C2×Dic3—24no—
408C2×Dic3—416no—
409C2×Dic3C22×D414yes—
410C2×Dic3C22×D414yes—
411D12—48no—
412D12—48no—
413D12—84no—
414C4×D3—416no—
415Dic6—28no—
416Dic6—28no—
417Dic6—416no—
418Dic6—48no—
419C24—34no—
420C24C23×D312yes—
421#41—1232no—
422C2×Q8—34no—
423C2×Q8—34no—
424C2×Q8—68no—
425C2×Q8—62no—
426C2×Q8—62no—
427C2×Q8—68no—
428C2×Q8—64no—
429C2×Q8—64no—
430C2×Q8—62no—
431C2×Q8—62no—
432C2×Q8—64no—
433C2×Q8—124no—
434C2×Q8—1232no—
435C2×D4—24no—
436C2×D4—68no—
437C2×D4—68no—
438C2×D4—68no—
439C2×D4—64no—
440C2×D4—64no—
441C2×D4—64no—
442C2×D4—64no—
443C2×D4—64no—
444C2×D4—68no—
445C2×D4—124no—
446C2×D4—124no—
447C2×D4—128no—
448C2×D4C2×D1214yes—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—32no—
452C22×C4—32no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—34no—
456C22×C4—68no—
457C22×C4—68no—
458C22×C4—68no—
459C22×C4—68no—
460C22×C4—124no—
461C22×C4—124no—
462C22×C4C2×D1218yes—
463C22×C4C23×D312yes—
464C22×C4C23×D314yes—
465C22×C4C23×D311yescharacteristic
466C22×C4C23×D312yes—
467C22×C4C23×D312yes—
468C22×C4C23×D311yescharacteristic
469C22×C4C23×D311yescharacteristic
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—316no—
473C4⋊C4—616no—
474C4⋊C4—64no—
475C4⋊C4—64no—
476C4⋊C4—62no—
477C4⋊C4—64no—
478C4⋊C4—62no—
479C4⋊C4—68no—
480C4⋊C4—64no—
481C4⋊C4—124no—
482C4⋊C4C2×D1214yes—
483C22⋊C4—28no—
484C22⋊C4—44no—
485C22⋊C4—616no—
486C22⋊C4—68no—
487C22⋊C4—68no—
488C22⋊C4—616no—
489C22⋊C4—128no—
490C22⋊C4—128no—
491C22⋊C4C2×D1218yes—
492C42—24no—
493C42—22no—
494C42—24no—
495C42—42no—
496C42—64no—
497C42—64no—
498C42C2×D1214yes—
499C42D3×D414yes—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—24no—
503C22×C3—44no—
504C22×C3—44no—
505C22×C3C23×D411yescharacteristic
506C22×C3C23×D411yescharacteristic
507C22×C3C2×H(5,𝔽2)11yescharacteristic
508C22×C3C2×2−1+412yes—
509C22×C3#58512yes—
510C2×D3—44no—
511C2×D3—44no—
512C2×D3—48no—
513C2×D3—44no—
514C2×D3—48no—
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—22no—
518C3×C4—44no—
519C3×C4—42no—
520C3×C4#53614yescomplemented
521C3×C4D4214yes—
522Dic3—24no—
523Dic3—24no—
524Dic3—44no—
525Dic3—42no—
526Dic3—42no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—64no—
532C23—64no—
533C23—64no—
534C23—68no—
535C23—68no—
536C23C22×D1214yes—
537C23C24×D311yescharacteristic
538Q8—68no—
539Q8—68no—
540Q8—1216no—
541Q8—128no—
542D4—28no—
543D4—1216no—
544D4—128no—
545D4—128no—
546D4—244no—
547D4—248no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—216no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—34no—
561C2×C4—34no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—64no—
573C2×C4—1216no—
574C2×C4—1216no—
575C2×C4C22×D1212yes—
576C2×C4C22×D1214yes—
577C2×C4C22×D1212yes—
578C2×C4C22×D1212yes—
579C2×C4C2×D3×D414yes—
580C2×C4C2×D3×D412yes—
581C2×C4#100912yes—
582C2×C4#101014yes—
583C2×C3—44no—
584C2×C3C2×#53611yescharacteristic
585C2×C3C2×D4211yescharacteristic
586C2×C3#349712yes—
587C2×C3#350011yescharacteristic
588C2×C3#355412yes—
589D3—84no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—44no—
594C22—44no—
595C22—124no—
596C22—124no—
597C22—128no—
598C22—128no—
599C22—124no—
600C22C23×D1211yescharacteristic
601C22C22×D3×D411yescharacteristic
602C22C2×#100911yescharacteristic
603C22C2×#101012yes—
604C22#596412yes—
605C4—24no—
606C4—22no—
607C4—24no—
608C4—42no—
609C4—44no—
610C4—64no—
611C4—64no—
612C4—124no—
613C4—122no—
614C4—122no—
615C4#552614yescomplemented
616C4D4×D1214yes—
617C3#3936311yescharacteristic, complemented, Sylow
618C2—44no—
619C2—244no—
620C2C2×#552611yescharacteristic
621C2C2×D4×D1211yescharacteristic
622C2#8584711yescharacteristic
623C2#8613712yes—
624C2#8622212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2293C2
prime radicalC2×#5553C2
hypercenter#2293D3
hypocenterC3#39363
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic6C2semidirect
3. C2×#5526C2semidirect
4. C3×#2293C2semidirect
5. #82287C2semidirect
6. #76317C2semidirect
7. #76011C2semidirect
8. #75669C2semidirect
9. #73712C2semidirect
10. #73711C2semidirect
11. #73491C2semidirect
12. #73489C2semidirect
13. #73313C2semidirect
14. C4×#869C2semidirect
80. #2293D3semidirect
148. #869D4semidirect
520. C3×C4#536semidirect
615. C4#5526semidirect
617. C3#39363semidirect

Maximal subgroups

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

subgroupindex
C2×#55532
C2×D4×Dic62
C2×#55262
C3×#22932
#822872
#763172
#760112
#756692
#737122
#737112
#734912
#734892
#733132
C4×#8692
#393633