Finite Group Databasebeta

← #11024920

Subgroups — #11024920

Order 768.

2232 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
3#83227C211yesmaximal, characteristic, complemented
4#83096C214yesmaximal, complemented
5#83082C214yesmaximal
6#83072C211yesmaximal, characteristic, complemented
7#83048C214yesmaximal, complemented
8#83043C211yesmaximal, characteristic, complemented
9#83032C214yesmaximal, complemented
10#82758C212yesmaximal, complemented
11#82751C212yesmaximal
12Q8×(C2×D3)⋊C4C212yesmaximal, complemented
13#82468C212yesmaximal, complemented
14#82268C211yesmaximal, characteristic
15#82262C211yesmaximal, characteristic, complemented
16#37695—31nomaximal, Sylow
17C22×D3×Q8C2212yes—
18C3×#541—28no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#126C2214yes—
21C2×C3×#125C2214yes—
22C2×C3×#125C2211yescharacteristic
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×C4×Q8C2212yes—
25C2×C3×#117C2211yescharacteristic
26C2×Q8×Dic3C2212yes—
27C2×#944C2212yes—
28C2×C4×#67C2211yescharacteristic
29C2×#924C2212yes—
30D3×C4⋊Q8—28no—
31C2×D3×C4⋊C4C2214yes—
32C2×#888C2214yes—
33C2×C42×D3C2211yescharacteristic
34C2×#869C2211yescharacteristic
35#5229C2212yes—
36C4×#217C2211yescharacteristic
37#4989C2214yes—
38#4989C2212yes—
39#4989C2214yes—
40#4987C2212yes—
41#4987C2214yes—
42#4986C2214yes—
43#4986C2214yes—
44#4985C2214yes—
45#4983C2214yes—
46#4982C2214yes—
47#4981C2214yes—
48#4980C2214yes—
49#4980C2214yes—
50#4979C2214yes—
51#4978C2212yes—
52#4978C2214yes—
53#4976C2214yes—
54#4973C2212yes—
55#4973C2212yes—
56#4973C2214yes—
57#4972C2212yes—
58#4956C2214yes—
59#4956C2212yes—
60#4953C2214yes—
61#4948C2214yes—
62#4947C2214yes—
63#4945C2214yes—
64#4944C2212yes—
65C4×(C2×D3)⋊C4C2211yescharacteristic
66C4×(C2×D3)⋊C4C2214yes—
67#4936C2211yescharacteristic
68#4933C2214yes—
69#4933C2214yes—
70#4932C2211yescharacteristic
71C4×Dic3⋊C4C2211yescharacteristic
72#4929C2212yes—
73#4929C2214yes—
74C2×#541D311yescharacteristic, complemented
75C22×C4⋊Q8—31no—
76#2851—31no—
77#2716—34no—
78#2713—34no—
79#2642—31no—
80#2621—31no—
81#2481—34no—
82#2473—34no—
83#2446—32no—
84#2429—32no—
85#2409—32no—
86Q8×C22⋊C4—32no—
87#2327—31no—
88#2290—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2311yescharacteristic
92C2×D3×Q8—28no—
93C2×D3×Q8—28no—
94C2×D3×Q8—44no—
95C22×C4×D3C2312yes—
96C22×C4×D3C2314yes—
97C22×C4×D3C2311yescharacteristic
98C22×Dic6C2312yes—
99C22×Dic6C2314yes—
100C3×C4⋊Q8D418yescomplemented
101C3×#126—216no—
102C3×#125—216no—
103C3×#125—24no—
104C3×C22⋊Q8—28no—
105C3×C4×Q8—28no—
106C3×#117—24no—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2314yes—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2314yes—
117C2×C3×C42C2311yescharacteristic
118C2×#217C2311yescharacteristic
119C2×#217C2312yes—
120C2×#217C2314yes—
121Q8×Dic3—44no—
122#944—28no—
123C4×#67—24no—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2311yescharacteristic
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2314yes—
129C2×C4⋊Dic3C2314yes—
130#924—44no—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2311yescharacteristic
136C2×Dic3⋊C4C2314yes—
137C2×C4×Dic3C2311yescharacteristic
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2314yes—
142D3×C4⋊C4—216no—
143#888—216no—
144C42×D3—24no—
145#869—24no—
146C3⋊#95C2312yes—
147C3⋊#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2312yes—
152C3⋊#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162C23×Q8—32no—
163#541—28no—
164C2×C4⋊Q8—32no—
165C2×C4⋊Q8—34no—
166C2×C4⋊Q8—31no—
167C2×C4⋊Q8—68no—
168C2×C4⋊Q8C2×D311yescharacteristic
169C2×#126C2×D314yes—
170C2×#125C2×D311yescharacteristic
171C2×#125C2×D314yes—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8C2×D312yes—
174C2×C4×Q8—32no—
175C2×C4×Q8C2×D312yes—
176C2×C4×D4—31no—
177C2×#117C2×D311yescharacteristic
178C22×C4⋊C4—34no—
179C22×C42—31no—
180#400—34no—
181#400—34no—
182#400—34no—
183#398—34no—
184#398—34no—
185#398—34no—
186#397—34no—
187#396—34no—
188#395—34no—
189#393—34no—
190#391—32no—
191#391—32no—
192#391—34no—
193#391—32no—
194#391—32no—
195#391—34no—
196#389—32no—
197#389—34no—
198#388—32no—
199#388—34no—
200#387—34no—
201#387—34no—
202#387—34no—
203#387—32no—
204#387—34no—
205#387—34no—
206#386—34no—
207#386—32no—
208#385—34no—
209#385—34no—
210#385—32no—
211#382—31no—
212#380—32no—
213#380—34no—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—34no—
218#376—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C2×C3×Q8—24no—
222C2×C3×Q8—24no—
223C2×C3×Q8C2×D418yes—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4C2412yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2412yes—
231C22×C3×C4C2414yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—22no—
234C2×#67—24no—
235C22×Dic3C2411yescharacteristic
236C22×Dic3C2411yescharacteristic
237C22×Dic3C2412yes—
238C22×Dic3C2414yes—
239C22×Dic3C2414yes—
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2412yes—
242D3×Q8—432no—
243C2×C4×D3—24no—
244C2×C4×D3—24no—
245C2×C4×D3—22no—
246C2×C4×D3—22no—
247C2×C4×D3—24no—
248C2×C4×D3—28no—
249C2×C4×D3—28no—
250C2×C4×D3—44no—
251C2×C4×D3#4114yes—
252C2×Dic6—24no—
253C2×Dic6—28no—
254C2×Dic6—24no—
255C2×Dic6—24no—
256C2×Dic6—28no—
257C2×Dic6—44no—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—24no—
260C3×C4⋊C4—28no—
261C3×C4⋊C4—28no—
262C3×C4⋊C4—28no—
263C3×C4⋊C4C2×D4116yes—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—28no—
266C3×C42—22no—
267C3×C42—28no—
268C3×C42C2×D414yes—
269#217—22no—
270#217—24no—
271#217—44no—
272(C2×D3)⋊C4—22no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—28no—
275C4⋊Dic3—28no—
276C4⋊Dic3—24no—
277C4⋊Dic3—44no—
278Dic3⋊C4—28no—
279Dic3⋊C4—28no—
280Dic3⋊C4—22no—
281Dic3⋊C4—24no—
282Dic3⋊C4—28no—
283Dic3⋊C4—44no—
284C4×Dic3—24no—
285C4×Dic3—22no—
286C4×Dic3—22no—
287C4×Dic3—22no—
288C4×Dic3—44no—
289C22×Q8—32no—
290C22×Q8—34no—
291C22×Q8—68no—
292C22×Q8—68no—
293C22×Q8—124no—
294C22×Q8C22×D312yes—
295C22×D4—31no—
296C23×C4—31no—
297C23×C4—32no—
298C23×C4—34no—
299C23×C4C22×D311yescharacteristic
300C4⋊Q8—68no—
301C4⋊Q8—616no—
302C4⋊Q8—64no—
303C4⋊Q8#6718yescomplemented
304#126—216no—
305#125—216no—
306#125—24no—
307C22⋊Q8—28no—
308C22⋊Q8—124no—
309C4×Q8—28no—
310C4×Q8—124no—
311C4×D4—64no—
312#117—24no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—616no—
323C2×C4⋊C4C22×D312yes—
324C2×C4⋊C4C22×D312yes—
325C2×C4⋊C4C22×D314yes—
326C2×C4⋊C4C22×D314yes—
327C2×C4⋊C4C22×D314yes—
328C2×C4⋊C4C22×D314yes—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—32no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—32no—
333C2×C22⋊C4—34no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4C22×D312yes—
336C2×C22⋊C4C22×D314yes—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—32no—
341C2×C42—34no—
342C2×C42—64no—
343C2×C42C22×D311yescharacteristic
344C2×C42C22×D314yes—
345C2×C42C22×D311yescharacteristic
346#95—32no—
347#95—32no—
348#95—32no—
349#95—32no—
350#95—34no—
351#95—34no—
352#95—32no—
353#95—34no—
354#95—34no—
355#95—34no—
356#95—34no—
357#95—34no—
358#95—34no—
359#95—34no—
360#95—34no—
361#95—34no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3C2511yescharacteristic
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—22no—
371C22×D3C2×#4112yes—
372C22×D32−1+414yes—
373C3×Q8—216no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—24no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—24no—
385C2×C3×C4—22no—
386C2×C3×C4—24no—
387C2×C3×C4—44no—
388C2×C3×C4C22×D414yes—
389C2×C3×C4C22×D418yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C2×#4112yes—
392C2×C3×C4H(5,𝔽2)12yes—
393#67—44no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—24no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—24no—
402C2×Dic3—22no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—22no—
406C2×Dic3—24no—
407C2×Dic3—22no—
408C2×Dic3—24no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3C2×#4112yes—
412C2×Dic32−1+414yes—
413C4×D3—216no—
414C4×D3—28no—
415C4×D3—416no—
416Dic6—416no—
417Dic6—48no—
418C24—31no—
419C24C23×D311yescharacteristic
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—68no—
423C2×Q8—64no—
424C2×Q8—64no—
425C2×Q8—68no—
426C2×Q8—64no—
427C2×Q8—124no—
428C2×Q8—1232no—
429C2×Q8C2×#6718yes—
430C2×D4—62no—
431C2×D4—64no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—31no—
436C22×C4—32no—
437C22×C4—34no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—32no—
441C22×C4—34no—
442C22×C4—34no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—68no—
446C22×C4—62no—
447C22×C4—64no—
448C22×C4—62no—
449C22×C4—68no—
450C22×C4—124no—
451C22×C4C23×D312yes—
452C22×C4C23×D314yes—
453C22×C4C23×D311yescharacteristic
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D311yescharacteristic
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—62no—
463C4⋊C4—64no—
464C4⋊C4—68no—
465C4⋊C4—64no—
466C4⋊C4—68no—
467C4⋊C4—68no—
468C4⋊C4—68no—
469C4⋊C4—124no—
470C4⋊C4—124no—
471C4⋊C4C2×#67116yes—
472C22⋊C4—24no—
473C22⋊C4—28no—
474C22⋊C4—62no—
475C22⋊C4—64no—
476C22⋊C4—68no—
477C22⋊C4—62no—
478C22⋊C4—64no—
479C22⋊C4—124no—
480C42—22no—
481C42—28no—
482C42—62no—
483C42—62no—
484C42—62no—
485C42—64no—
486C42—124no—
487C42C2×#6714yes—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—42no—
491C22×C3—41no—
492C22×C3—42no—
493C22×C3C23×D411yescharacteristic
494C22×C3C22×#4111yescharacteristic
495C22×C3C2×H(5,𝔽2)11yescharacteristic
496C22×C3C2×2−1+412yes—
497C22×C3C2×2−1+412yes—
498C2×D3—22no—
499C2×D3—24no—
500C2×D3—22no—
501C2×D3—24no—
502C2×D3#54114yescomplemented
503C3×C4—28no—
504C3×C4—24no—
505C3×C4—41no—
506C3×C4—42no—
507C3×C4—44no—
508C3×C4#54614yes—
509Dic3—22no—
510Dic3—24no—
511Dic3—41no—
512Dic3—44no—
513Dic3—41no—
514Dic3—42no—
515Dic3—44no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—21no—
521C23—32no—
522C23—34no—
523C23—61no—
524C23—62no—
525C23—61no—
526C23C24×D311yescharacteristic
527Q8—216no—
528Q8—1216no—
529Q8—128no—
530D4—124no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—32no—
545C2×C4—34no—
546C2×C4—44no—
547C2×C4—616no—
548C2×C4—61no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—68no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—62no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—1216no—
567C2×C4D3×#4112yes—
568C2×C4#100912yes—
569C2×C4C22×#6718yes—
570C2×C4C22×#6714yes—
571C2×C4C22×#6712yes—
572C2×C3—81no—
573C2×C3C2×#54111yescharacteristic
574C2×C3C2×#54611yescharacteristic
575C2×C3#348612yes—
576C2×C3#350911yescharacteristic
577C2×C3#355212yes—
578D3—24no—
579C22—34no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—42no—
585C22—62no—
586C22—62no—
587C22—64no—
588C22—64no—
589C22C2×D3×#4111yescharacteristic
590C22C2×#100911yescharacteristic
591C22D3×2−1+412yes—
592C22D3×2−1+412yes—
593C22C23×#6711yescharacteristic
594C4—28no—
595C4—24no—
596C4—41no—
597C4—42no—
598C4—44no—
599C4—62no—
600C4—64no—
601C4—124no—
602C4—121no—
603C4—121no—
604C4—122no—
605C4—124no—
606C4#582914yes—
607C3#3769511yescharacteristic, complemented, Sylow
608C2—64no—
609C2—81no—
610C2D3×#54111yescharacteristic
611C2#8651011yescharacteristic
612C2#8651712yes—
613C2C2×#582911yescharacteristic
614C2#8729412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#541C2
prime radicalC22×#67C23
hypercenterC2×#541D3
hypocenterC3#37695
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×C3×#541C2semidirect
2. C2×D3×C4⋊Q8C2semidirect
3. #83227C2semidirect
4. #83096C2semidirect
6. #83072C2semidirect
7. #83048C2semidirect
8. #83043C2semidirect
9. #83032C2semidirect
10. #82758C2semidirect
12. Q8×(C2×D3)⋊C4C2semidirect
13. #82468C2semidirect
15. #82262C2semidirect
74. C2×#541D3semidirect
100. C3×C4⋊Q8D4semidirect
303. C4⋊Q8#67semidirect
502. C2×D3#541semidirect
607. C3#37695semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×D3×C4⋊Q82
#832272
#830962
#830822
#830722
#830482
#830432
#830322
#827582
#827512
Q8×(C2×D3)⋊C42
#824682
#822682
#822622
#376953