Finite Group Databasebeta

← #10972902

Subgroups — #10972902

Order 768.

2654 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic3C211yesmaximal, characteristic, complemented
2C2×#5596C211yesmaximal, characteristic
3C3×#2436C211yesmaximal, characteristic, complemented
4#84066C211yesmaximal, characteristic, complemented
5#82974C212yesmaximal, complemented
6#82953C212yesmaximal, complemented
7#82664C211yesmaximal, characteristic, complemented
8#76812C214yesmaximal, complemented
9#76802C214yesmaximal, complemented
10#76543C212yesmaximal, complemented
11#76484C212yesmaximal, complemented
12#74872C214yesmaximal, complemented
13#74687C214yesmaximal, complemented
14#73516C211yesmaximal, characteristic, complemented
15#73437C211yesmaximal, characteristic, complemented
16#35249—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C22×#237C2212yes—
19C2×C3×#124C2212yes—
20C2×C3×#121C2211yescharacteristic
21C2×#947C2212yes—
22C2×Q8×Dic3—28no—
23C2×Q8×Dic3C2212yes—
24C2×Q8×Dic3C2212yes—
25C2×Q8×Dic3C2212yes—
26C2×Q8×Dic3C2212yes—
27C2×#940C2211yescharacteristic
28C2×#939C2212yes—
29C2×#935C2214yes—
30C2×D4×Dic3C2211yescharacteristic
31C2×#930C2211yescharacteristic
32C22×C4⋊Dic3C2211yescharacteristic
33C22×C4⋊Dic3C2212yes—
34C22×C4×Dic3C2212yes—
35C22×C4×Dic3C2211yescharacteristic
36#5596—28no—
37C2×#893C2211yescharacteristic
38C2×#888C2211yescharacteristic
39C2×#886C2214yes—
40C2×#885C2212yes—
41C2×#67⋊C4C2212yes—
42C3×#393C2214yes—
43C3×#391C2212yes—
44C3×#386C2214yes—
45C3×C4×C22⋊C4C2211yescharacteristic
46#5231C2214yes—
47#5229C2212yes—
48#5221C2212yes—
49#5213C2214yes—
50C4×#217C2211yescharacteristic
51#4975C2212yes—
52#4961C2218yes—
53#4957C2218yes—
54#4951C2218yes—
55#4950C2218yes—
56#4949C2214yes—
57#4943C2212yes—
58#4942C2212yes—
59#4939C2212yes—
60#4935C2212yes—
61#4932C2212yes—
62C4×Dic3⋊C4C2212yes—
63#4677C2214yes—
64#4669C2218yes—
65#4668C2218yes—
66#4656C2218yes—
67#4651C2212yes—
68#4650C2212yes—
69#4647C2218yes—
70C2×#532—31no—
71C22×C4×Q8—31no—
72#2854—34no—
73#2632—34no—
74#2580—34no—
75#2567—32no—
76#2548—32no—
77#2501—31no—
78#2501—31no—
79#2436—31no—
80#2436D311yescharacteristic, complemented
81#2406—34no—
82#2326—32no—
83#2326—32no—
84#2283—31no—
85C22×C3×Q8—28no—
86C22×C3×Q8—24no—
87C22×C3×Q8C2312yes—
88C22×C3×Q8C2312yes—
89C22×C3×D4C2312yes—
90C22×C3×D4C2311yescharacteristic
91C23×C3×C4C2312yes—
92C23×C3×C4C2311yescharacteristic
93C22×#67C2312yes—
94C22×#67C2312yes—
95C22×#67C2312yes—
96C23×Dic3C2312yes—
97C23×Dic3C2311yescharacteristic
98C23×Dic3C2311yescharacteristic
99C23×Dic3C2312yes—
100C2×#237—28no—
101C2×#237—48no—
102C22×D12C2311yescharacteristic
103C22×C4×D3C2312yes—
104C22×Dic6C2312yes—
105C3×#124—28no—
106C3×#121—24no—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2318yes—
111C2×C3×C42C2312yes—
112C2×#217C2312yes—
113C2×#217C2312yes—
114C2×#217C2318yes—
115#947—28no—
116Q8×Dic3—28no—
117Q8×Dic3—28no—
118Q8×Dic3—28no—
119Q8×Dic3—28no—
120#940—24no—
121#939—28no—
122#935—48no—
123D4×Dic3—24no—
124#930—24no—
125C2×(C2×D3)⋊C4C2318yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2318yes—
129C2×C4⋊Dic3—24no—
130C2×C4⋊Dic3—28no—
131C2×C4⋊Dic3C2314yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2318yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2318yes—
140C2×C4×Dic3—28no—
141C2×C4×Dic3D418yes—
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2314yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2311yescharacteristic
146C2×C4×Dic3C2311yescharacteristic
147C2×C4×Dic3C2311yescharacteristic
148C2×C4×Dic3C2311yescharacteristic
149#893—24no—
150#888—24no—
151#886—48no—
152#885—28no—
153#67⋊C4—28no—
154C3×#95C2318yes—
155C3×#95C2312yes—
156C3⋊#95C2318yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C22×#41—32no—
162C23×Q8C2×D311yescharacteristic
163#532—68no—
164C2×C4⋊Q8—31no—
165C2×#127—31no—
166C2×#124—32no—
167C2×#124—32no—
168C2×#124C2×D312yes—
169C2×#123—34no—
170C2×#123—34no—
171C2×#121—32no—
172C2×#121—31no—
173C2×#121C2×D311yescharacteristic
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×Q8—32no—
177C2×C4×Q8—32no—
178C2×C4×Q8—68no—
179C2×C4×D4—31no—
180C2×C4×D4—31no—
181C2×C4×D4—32no—
182C22×C4⋊C4—32no—
183C22×C4⋊C4—31no—
184C22×C42—31no—
185C22×C42—32no—
186#399—34no—
187#399—34no—
188#395—38no—
189#394—38no—
190#393—34no—
191#393C2×D314yes—
192#391—32no—
193#391C2×D312yes—
194#390—32no—
195#390—38no—
196#390—32no—
197#389—38no—
198#388—38no—
199#388—38no—
200#386—34no—
201#386C2×D314yes—
202#385—38no—
203#385—38no—
204#384—32no—
205#384—32no—
206#383—32no—
207#382—32no—
208#381—32no—
209#380—32no—
210C4×C4⋊C4—32no—
211C4×C22⋊C4—31no—
212C4×C22⋊C4C2×D311yescharacteristic
213#376—32no—
214C24×C3C2412yes—
215C24×C3C2411yescharacteristic
216C23×D3C2412yes—
217C2×C3×Q8—232no—
218C2×C3×Q8—24no—
219C2×C3×Q8—24no—
220C2×C3×Q8—28no—
221C2×C3×Q8—24no—
222C2×C3×Q8#4118yescomplemented
223C2×C3×D4—24no—
224C2×C3×D4—28no—
225C2×C3×D4—22no—
226C2×C3×D4—24no—
227C22×C3×C4—24no—
228C22×C3×C4—28no—
229C22×C3×C4—24no—
230C22×C3×C4C2×D414yes—
231C22×C3×C4C2414yes—
232C22×C3×C4C2412yes—
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2414yes—
235C22×C3×C4C2411yescharacteristic
236C2×#67—24no—
237C2×#67—28no—
238C2×#67—24no—
239C2×#67—24no—
240C2×#67—28no—
241C2×#67—44no—
242C22×Dic3—24no—
243C22×Dic3—24no—
244C22×Dic3—24no—
245C22×Dic3—28no—
246C22×Dic3—24no—
247C22×Dic3—44no—
248C22×Dic3C2×D414yes—
249C22×Dic3C2×D414yes—
250C22×Dic3C2414yes—
251C22×Dic3C2412yes—
252C22×Dic3C2414yes—
253C22×Dic3C2411yescharacteristic
254C22×Dic3C2411yescharacteristic
255C22×Dic3C2411yescharacteristic
256C22×Dic3C2411yescharacteristic
257#237—432no—
258C2×D12—22no—
259C2×D12—24no—
260C2×C4×D3—24no—
261C2×C4×D3—44no—
262C2×Dic6—24no—
263C2×Dic6—44no—
264C3×C4⋊C4—22no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—216no—
268C3×C42—24no—
269#217—24no—
270#217—24no—
271#217—48no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—216no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—48no—
276C4⋊Dic3—22no—
277C4⋊Dic3—24no—
278C4⋊Dic3—24no—
279C4⋊Dic3—28no—
280C4⋊Dic3—28no—
281C4⋊Dic3—48no—
282Dic3⋊C4—24no—
283Dic3⋊C4—28no—
284Dic3⋊C4—48no—
285C4×Dic3—24no—
286C4×Dic3—28no—
287C4×Dic3—28no—
288C4×Dic3C2×D414yes—
289C4×Dic3C2×D414yes—
290C4×Dic3C2×D414yes—
291C4×Dic3C2×D414yes—
292C2×#41—68no—
293C2×#41—128no—
294C22×Q8—28no—
295C22×Q8—24no—
296C22×Q8—32no—
297C22×Q8C22×D312yes—
298C22×Q8C22×D312yes—
299C22×D4—32no—
300C22×D4—32no—
301C22×D4—32no—
302C22×D4—31no—
303C22×D4C22×D311yescharacteristic
304C22×D4C22×D312yes—
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—32no—
308C23×C4—32no—
309C23×C4—32no—
310C23×C4C22×D312yes—
311C23×C4C22×D311yescharacteristic
312C4⋊Q8—64no—
313#127—64no—
314#124—28no—
315#124—68no—
316#124—68no—
317#123—128no—
318#123—128no—
319#121—24no—
320#121—68no—
321#121—64no—
322C4×Q8—68no—
323C4×Q8—68no—
324C4×Q8—68no—
325C4×Q8—68no—
326C4×D4—68no—
327C4×D4—64no—
328C4×D4—64no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—34no—
336C2×C4⋊C4—38no—
337C2×C4⋊C4—38no—
338C2×C4⋊C4—68no—
339C2×C4⋊C4—64no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C22⋊C4—32no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—38no—
346C2×C22⋊C4—38no—
347C2×C22⋊C4—38no—
348C2×C22⋊C4C22×D312yes—
349C2×C22⋊C4C22×D312yes—
350C2×C22⋊C4C22×D318yes—
351C2×C42—38no—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—34no—
357C2×C42—34no—
358C2×C42—32no—
359C2×C42—68no—
360C2×C42C22×D312yes—
361#95—34no—
362#95—34no—
363#95—32no—
364#95—34no—
365#95—38no—
366#95C22×D312yes—
367#95C22×D318yes—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—24no—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3C22×D412yes—
375C23×C3H(5,𝔽2)14yes—
376C23×C3C2511yescharacteristic
377C22×D3—22no—
378C22×D3—24no—
379C22×D3—24no—
380C22×D3—24no—
381C3×Q8—216no—
382C3×Q8—216no—
383C3×D4—48no—
384C3×D4—44no—
385C2×C3×C4—216no—
386C2×C3×C4—24no—
387C2×C3×C4—22no—
388C2×C3×C4—24no—
389C2×C3×C4—28no—
390C2×C3×C4—22no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—24no—
394C2×C3×C4—24no—
395C2×C3×C4C22×D412yes—
396C2×C3×C4C22×D412yes—
397C2×C3×C4C2×#4118yes—
398C2×C3×C4C2×#4114yes—
399#67—48no—
400#67—48no—
401#67—48no—
402C2×Dic3—28no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—28no—
407C2×Dic3—24no—
408C2×Dic3—22no—
409C2×Dic3—24no—
410C2×Dic3—24no—
411C2×Dic3—28no—
412C2×Dic3—24no—
413C2×Dic3—24no—
414C2×Dic3—48no—
415C2×Dic3—48no—
416C2×Dic3C22×D412yes—
417C2×Dic3C22×D412yes—
418C2×Dic3C22×D412yes—
419C2×Dic3C22×D412yes—
420C2×Dic32−1+414yes—
421D12—44no—
422C4×D3—48no—
423Dic6—48no—
424C24—32no—
425C24C23×D312yes—
426C24C23×D311yescharacteristic
427#41—1232no—
428C2×Q8—232no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—28no—
432C2×Q8—24no—
433C2×Q8—64no—
434C2×Q8—124no—
435C2×Q8#23718yescomplemented
436C2×D4—24no—
437C2×D4—28no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—64no—
441C2×D4—64no—
442C2×D4—68no—
443C2×D4—64no—
444C2×D4—68no—
445C2×D4—62no—
446C2×D4—64no—
447C2×D4—124no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—31no—
452C22×C4—34no—
453C22×C4—34no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—34no—
458C22×C4—32no—
459C22×C4—34no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—68no—
466C22×C4—124no—
467C22×C4—124no—
468C22×C4D3×D414yes—
469C22×C4C23×D314yes—
470C22×C4C23×D312yes—
471C22×C4C23×D311yescharacteristic
472C22×C4C23×D314yes—
473C22×C4C23×D311yescharacteristic
474C4⋊C4—22no—
475C4⋊C4—68no—
476C4⋊C4—68no—
477C4⋊C4—64no—
478C4⋊C4—68no—
479C4⋊C4—64no—
480C4⋊C4—64no—
481C4⋊C4—62no—
482C4⋊C4—128no—
483C4⋊C4—128no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—216no—
487C22⋊C4—64no—
488C22⋊C4—64no—
489C22⋊C4—64no—
490C22⋊C4—64no—
491C22⋊C4—616no—
492C22⋊C4—128no—
493C22⋊C4—128no—
494C42—24no—
495C42—34no—
496C42—34no—
497C42—34no—
498C42—34no—
499C42—68no—
500C42—68no—
501C42—64no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—28no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—48no—
509C22×C3#53514yes—
510C22×C3C23×D411yescharacteristic
511C22×C3C22×#4112yes—
512C22×C3C2×H(5,𝔽2)12yes—
513C22×C3C2×2−1+412yes—
514C2×D3—42no—
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—48no—
518C3×C4—22no—
519C3×C4—28no—
520C3×C4—42no—
521C3×C4—44no—
522C3×C4#52514yes—
523Dic3—22no—
524Dic3—22no—
525Dic3—22no—
526Dic3—42no—
527Dic3—44no—
528Dic3—44no—
529Dic3#53614yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—62no—
537C23—64no—
538C23—64no—
539C23—64no—
540C23C2×D3×D412yes—
541C23#100914yes—
542C23C24×D311yescharacteristic
543Q8—216no—
544Q8—216no—
545Q8—128no—
546D4—48no—
547D4—44no—
548D4—128no—
549D4—128no—
550D4—128no—
551D4—124no—
552C2×C4—216no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—32no—
563C2×C4—34no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—32no—
567C2×C4—68no—
568C2×C4—68no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—64no—
573C2×C4—62no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—68no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—128no—
580C2×C4—128no—
581C2×C4—128no—
582C2×C4C2×D3×D412yes—
583C2×C4C2×D3×D412yes—
584C2×C4C2×#23718yes—
585C2×C4C2×#23714yes—
586C2×C3—24no—
587C2×C3—82no—
588C2×C3C2×#52511yescharacteristic
589C2×C3C2×#53511yescharacteristic
590C2×C3C2×#53611yescharacteristic
591C2×C3#352214yes—
592D3—82no—
593C22—22no—
594C22—22no—
595C22—28no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—48no—
600C22—122no—
601C22—122no—
602C22—122no—
603C22—128no—
604C22#567414yes—
605C22C22×D3×D411yescharacteristic
606C22C22×#23712yes—
607C22C2×#100912yes—
608C22D3×2−1+412yes—
609C4—22no—
610C4—28no—
611C4—34no—
612C4—42no—
613C4—44no—
614C4—62no—
615C4—62no—
616C4—62no—
617C4—122no—
618C4—124no—
619C4—124no—
620C4#558614yes—
621C3#3524911yescharacteristic, complemented, Sylow
622C2—24no—
623C2—82no—
624C2—242no—
625C2C2×#558611yescharacteristic
626C2C2×#567411yescharacteristic
627C2D3×#53611yescharacteristic
628C2#8672214yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2436C2
prime radicalC2×#5596C2
hypercenter#2436D3
hypocenterC3#35249
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2436

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic3C2semidirect
3. C3×#2436C2semidirect
4. #84066C2semidirect
5. #82974C2semidirect
6. #82953C2semidirect
7. #82664C2semidirect
8. #76812C2semidirect
9. #76802C2semidirect
10. #76543C2semidirect
11. #76484C2semidirect
12. #74872C2semidirect
13. #74687C2semidirect
14. #73516C2semidirect
15. #73437C2semidirect
80. #2436D3semidirect
222. C2×C3×Q8#41semidirect
435. C2×Q8#237semidirect
621. C3#35249semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×#55962
C3×#24362
#840662
#829742
#829532
#826642
#768122
#768022
#765432
#764842
#748722
#746872
#735162
#734372
#352493