Finite Group Databasebeta

← #11233340

Subgroups — #11233340

Order 768.

1994 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic, complemented
2C2×#5732C211yesmaximal, characteristic
3#83258C212yesmaximal, complemented
4#83256C212yesmaximal, complemented
5#83249C214yesmaximal
6#83248C214yesmaximal, complemented
7#83234C214yesmaximal, complemented
8#83228C211yesmaximal, characteristic, complemented
9#82499C211yesmaximal, characteristic
10#82496C212yesmaximal, complemented
11#82492C211yesmaximal, characteristic
12#82480C211yesmaximal, characteristic, complemented
13#82473C214yesmaximal, complemented
14#82464C212yesmaximal, complemented
15#82456C211yesmaximal, characteristic, complemented
16#45338—31nomaximal, Sylow
17C3×Q82—28no—
18C2×C3×C4⋊Q8C2214yes—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2212yes—
21C2×C3×C4⋊Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2212yes—
24C2×C3×C4×Q8C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×#947C2212yes—
27C2×#946C2212yes—
28C2×Q8×Dic3C2212yes—
29C2×#944C2211yescharacteristic
30#5732—28no—
31C2×#898C2214yes—
32C2×#897C2212yes—
33C2×#892C2212yes—
34C2×C4×D12C2211yescharacteristic
35C2×C4×Dic6C2211yescharacteristic
36#5231C2212yes—
37#5231C2214yes—
38#5231C2214yes—
39#5230C2212yes—
40#5229C2211yescharacteristic
41#5229C2214yes—
42#5229C2214yes—
43#5229C2214yes—
44#4994C2214yes—
45#4994C2214yes—
46#4992C2212yes—
47#4991C2212yes—
48#4991C2212yes—
49#4990C2214yes—
50#4990C2214yes—
51#4987C2212yes—
52#4986C2212yes—
53#4986C2214yes—
54#4985C2212yes—
55#4984C2212yes—
56#4981C2212yes—
57#4981C2214yes—
58#4980C2214yes—
59#4978C2214yes—
60#4978C2214yes—
61#4978C2214yes—
62#4978C2214yes—
63#4977C2211yescharacteristic
64Dic3×C4⋊C4C2212yes—
65#4973C2211yescharacteristic
66#4940C2214yes—
67#4940C2214yes—
68#4940C2214yes—
69C4×(C2×D3)⋊C4C2212yes—
70C4×(C2×D3)⋊C4C2212yes—
71#4936C2211yescharacteristic
72#4933C2214yes—
73#4932C2211yescharacteristic
74C4×Dic3⋊C4C2214yes—
75#4929C2212yes—
76#4928C2212yes—
77#4928C2214yes—
78C2×Q82D311yescharacteristic, complemented
79C2×#556—31no—
80#2731—34no—
81#2716—34no—
82#2581—31no—
83#2536—31no—
84#2492—32no—
85#2469—34no—
86#2446—34no—
87#2439—32no—
88#2429—32no—
89#2398—31no—
90#2362—32no—
91#2331—31no—
92#2327—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×Q8C2311yescharacteristic
95C22×C3×Q8C2314yes—
96C22×C3×Q8C2312yes—
97C22×D12C2311yescharacteristic
98C22×C4×D3C2312yes—
99C22×Dic6C2311yescharacteristic
100C3×C4⋊Q8—216no—
101C3×C4⋊Q8—28no—
102C3×C4⋊Q8—28no—
103C3×C4⋊Q8D418yes—
104C3×C4×Q8—28no—
105C3×C4×Q8—28no—
106C3×C4×Q8—24no—
107C3×C4×Q8—24no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C42C2312yes—
117C2×C3×C42C2314yes—
118C2×C3×C42C2312yes—
119C2×C3×C42C2311yescharacteristic
120#947—28no—
121#946—28no—
122Q8×Dic3—28no—
123#944—24no—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2314yes—
136C2×C4⋊Dic3C2311yescharacteristic
137C2×C4⋊Dic3C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2312yes—
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2314yes—
147#898—216no—
148#897—28no—
149#892—28no—
150C4×D12—24no—
151C4×Dic6—24no—
152C3⋊#95C2312yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2312yes—
163C3⋊#95C2312yes—
164C3⋊#95C2312yes—
165C3⋊#95C2312yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169Q82—28no—
170#556—68no—
171C2×C4⋊Q8—31no—
172C2×C4⋊Q8C2×D312yes—
173C2×C4⋊Q8C2×D314yes—
174C2×C4⋊Q8C2×D312yes—
175C2×C4⋊Q8C2×D311yescharacteristic
176C2×#126—34no—
177C2×#124—32no—
178C2×C22⋊Q8—32no—
179C2×C22⋊Q8—32no—
180C2×C4×Q8—32no—
181C2×C4×Q8—31no—
182C2×C4×Q8C2×D311yescharacteristic
183C2×C4×Q8C2×D311yescharacteristic
184C2×C4×Q8C2×D312yes—
185C2×C4×Q8C2×D312yes—
186C2×C4×D4—31no—
187C2×#117—32no—
188#400—32no—
189#400—32no—
190#400—34no—
191#398—34no—
192#398—32no—
193#397—32no—
194#396—34no—
195#396—34no—
196#395—32no—
197#394—32no—
198#394—32no—
199#393—32no—
200#393—34no—
201#393—34no—
202#391—31no—
203#391—34no—
204#391—34no—
205#391—34no—
206#391—31no—
207#391—32no—
208#389—32no—
209#389—32no—
210#389—34no—
211#388—34no—
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#387—34no—
221#387—34no—
222#382—31no—
223#382—31no—
224#380—32no—
225C4×C4⋊C4—32no—
226C4×C4⋊C4—34no—
227C4×C22⋊C4—32no—
228C4×C22⋊C4—32no—
229#376—31no—
230C23×D3C2412yes—
231C2×C3×Q8—22no—
232C2×C3×Q8—22no—
233C2×C3×Q8—28no—
234C2×C3×Q8—24no—
235C2×C3×Q8—22no—
236C2×C3×Q8—44no—
237C2×C3×Q8—44no—
238C2×C3×Q8—44no—
239C2×C3×Q8C2×D418yes—
240C2×C3×Q8#4114yes—
241C22×C3×C4C2412yes—
242C22×C3×C4C2412yes—
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2411yescharacteristic
245C22×C3×C4C2414yes—
246C22×C3×C4C2412yes—
247C22×C3×C4C2412yes—
248C22×C3×C4C2411yescharacteristic
249C22×Dic3C2412yes—
250C22×Dic3C2412yes—
251C22×Dic3C2414yes—
252C22×Dic3C2412yes—
253C22×Dic3C2412yes—
254C22×Dic3C2412yes—
255C2×D12—22no—
256C2×D12—42no—
257C2×C4×D3—24no—
258C2×C4×D3—24no—
259C2×C4×D3—24no—
260C2×Dic6—22no—
261C2×Dic6—22no—
262C2×Dic6—22no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—28no—
266C3×C4⋊C4—28no—
267C3×C4⋊C4—22no—
268C3×C4⋊C4—22no—
269C3×C4⋊C4C2×D418yes—
270C3×C4⋊C4C2×D418yes—
271C3×C42—28no—
272C3×C42—24no—
273C3×C42—24no—
274C3×C42C2×D414yes—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—28no—
277(C2×D3)⋊C4—28no—
278(C2×D3)⋊C4—28no—
279(C2×D3)⋊C4—24no—
280(C2×D3)⋊C4—24no—
281(C2×D3)⋊C4—42no—
282(C2×D3)⋊C4—44no—
283(C2×D3)⋊C4—42no—
284C4⋊Dic3—22no—
285C4⋊Dic3—22no—
286C4⋊Dic3—28no—
287C4⋊Dic3—24no—
288C4⋊Dic3—44no—
289Dic3⋊C4—24no—
290Dic3⋊C4—28no—
291Dic3⋊C4—28no—
292Dic3⋊C4—28no—
293C4×Dic3—24no—
294C4×Dic3—28no—
295C4×Dic3—22no—
296C4×Dic3—22no—
297C4×Dic3—44no—
298C22×Q8—31no—
299C22×Q8C22×D311yescharacteristic
300C22×Q8C22×D311yescharacteristic
301C22×Q8C22×D314yes—
302C22×Q8C22×D312yes—
303C22×D4—31no—
304C23×C4—32no—
305C4⋊Q8—216no—
306C4⋊Q8—28no—
307C4⋊Q8—28no—
308C4⋊Q8—64no—
309C4⋊Q8#6718yes—
310#126—616no—
311#124—68no—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314C4×Q8—28no—
315C4×Q8—28no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—68no—
319C4×Q8—64no—
320C4×D4—64no—
321#117—68no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D312yes—
333C2×C4⋊C4C22×D312yes—
334C2×C4⋊C4C22×D314yes—
335C2×C4⋊C4C22×D314yes—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D312yes—
338C2×C4⋊C4C22×D312yes—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—32no—
346C2×C22⋊C4—32no—
347C2×C22⋊C4—34no—
348C2×C42—34no—
349C2×C42—32no—
350C2×C42—31no—
351C2×C42—31no—
352C2×C42—34no—
353C2×C42C22×D312yes—
354C2×C42C22×D314yes—
355C2×C42C22×D312yes—
356C2×C42C22×D311yescharacteristic
357#95—32no—
358#95—32no—
359#95—32no—
360#95—32no—
361#95—32no—
362#95—32no—
363#95—34no—
364#95—34no—
365#95—34no—
366#95—34no—
367#95—32no—
368#95—32no—
369#95—34no—
370#95—34no—
371#95—34no—
372#95—34no—
373#95—34no—
374C23×C3C2511yescharacteristic
375C22×D3—22no—
376C22×D3—22no—
377C22×D3—22no—
378C22×D3—22no—
379C22×D3—22no—
380C22×D3—22no—
381C22×D3—22no—
382C3×Q8—28no—
383C3×Q8—28no—
384C3×Q8—48no—
385C3×Q8—88no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—24no—
398C2×C3×C4—22no—
399C2×C3×C4C22×D412yes—
400C2×C3×C4C22×D418yes—
401C2×C3×C4C22×D412yes—
402C2×C3×C4C22×D412yes—
403C2×C3×C4C2×#4114yes—
404C2×C3×C4C2×#4112yes—
405C2×C3×C42−1+414yes—
406C2×C3×C42−1+412yes—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—24no—
414C2×Dic3—22no—
415C2×Dic3—24no—
416C2×Dic3—24no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—24no—
423C2×Dic3—22no—
424D12—44no—
425C4×D3—48no—
426Dic6—44no—
427C24—32no—
428C2×Q8—24no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—28no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—62no—
437C2×Q8—62no—
438C2×Q8—62no—
439C2×Q8#23914yes—
440C2×Q8C2×#6718yes—
441C2×D4—62no—
442C2×D4—122no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—34no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4C23×D312yes—
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C22×C4C23×D312yes—
456C22×C4C23×D311yescharacteristic
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D314yes—
459C22×C4C23×D311yescharacteristic
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—68no—
467C4⋊C4—64no—
468C4⋊C4—68no—
469C4⋊C4—68no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4—68no—
473C4⋊C4—64no—
474C4⋊C4—124no—
475C4⋊C4C2×#6718yes—
476C4⋊C4C2×#6718yes—
477C22⋊C4—68no—
478C22⋊C4—64no—
479C22⋊C4—64no—
480C22⋊C4—68no—
481C22⋊C4—64no—
482C22⋊C4—68no—
483C22⋊C4—122no—
484C22⋊C4—124no—
485C22⋊C4—122no—
486C42—28no—
487C42—24no—
488C42—24no—
489C42—62no—
490C42—62no—
491C42—68no—
492C42—64no—
493C42—124no—
494C42C2×#6714yes—
495C22×C3C23×D411yescharacteristic
496C22×C3C22×#4111yescharacteristic
497C22×C3C2×2−1+411yescharacteristic
498C22×C3C2×2−1+412yes—
499C22×C3#58512yes—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—42no—
507C3×C4—24no—
508C3×C4—22no—
509C3×C4—28no—
510C3×C4—22no—
511C3×C4—24no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4#54814yes—
515Dic3—42no—
516Dic3—42no—
517Dic3—42no—
518Dic3—42no—
519Dic3—44no—
520Dic3—42no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23C24×D311yescharacteristic
529Q8—28no—
530Q8—28no—
531Q8—48no—
532Q8—88no—
533Q8—124no—
534D4—124no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—62no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—128no—
566C2×C4C2×#23914yes—
567C2×C4C2×#23912yes—
568C2×C4#100714yes—
569C2×C4#100712yes—
570C2×C4C22×#6712yes—
571C2×C4C22×#6718yes—
572C2×C4C22×#6712yes—
573C2×C4C22×#6712yes—
574C2×C3C2×#54811yescharacteristic
575C2×C3#348511yescharacteristic
576C2×C3#348611yescharacteristic
577C2×C3#351812yes—
578C2×C3#355512yes—
579D3—82no—
580C22—122no—
581C22—122no—
582C22—122no—
583C22—122no—
584C22—122no—
585C22—122no—
586C22—122no—
587C22C22×#23911yescharacteristic
588C22C2×#100712yes—
589C22C2×#100711yescharacteristic
590C22#596612yes—
591C22C23×#6711yescharacteristic
592C4—24no—
593C4—24no—
594C4—22no—
595C4—22no—
596C4—28no—
597C4—42no—
598C4—42no—
599C4—122no—
600C4—124no—
601C4—122no—
602C4—122no—
603C4—122no—
604C4—122no—
605C4#581714yes—
606C3#4533811yescharacteristic, complemented, Sylow
607C2—242no—
608C2#8704212yes—
609C2#8704812yes—
610C2C2×#581711yescharacteristic
611C2#8725111yescharacteristic
612C2#8728011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×Q82C2
prime radicalC22×D12C23
hypercenterC2×Q82D3
hypocenterC3#45338
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q82C2semidirect
3. #83258C2semidirect
4. #83256C2semidirect
6. #83248C2semidirect
7. #83234C2semidirect
8. #83228C2semidirect
10. #82496C2semidirect
12. #82480C2semidirect
13. #82473C2semidirect
14. #82464C2semidirect
15. #82456C2semidirect
78. C2×Q82D3semidirect
606. C3#45338semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×#57322
#832582
#832562
#832492
#832482
#832342
#832282
#824992
#824962
#824922
#824802
#824732
#824642
#824562
#453383