Finite Group Databasebeta

← #11069478

Subgroups — #11069478

Order 768.

1954 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×#5731C211yesmaximal, characteristic, complemented
3Dic3×C4⋊Q8C211yesmaximal, characteristic, complemented
4#83102C214yesmaximal
5#83085C214yesmaximal, complemented
6#83065C211yesmaximal, characteristic, complemented
7#83056C211yesmaximal, characteristic, complemented
8#83054C214yesmaximal, complemented
9#83035C214yesmaximal, complemented
10#82777C212yesmaximal
11#82754C212yesmaximal, complemented
12#82494C212yesmaximal, complemented
13#82461C212yesmaximal, complemented
14#82265C211yesmaximal, characteristic, complemented
15#82225C211yesmaximal, characteristic
16#39373—31nomaximal, Sylow
17C3×#541—28no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#126C2214yes—
20C2×C3×#125C2214yes—
21C2×C3×#125C2211yescharacteristic
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C4×Q8C2212yes—
24C2×C3×#117C2211yescharacteristic
25C2×#946C2212yes—
26C2×Q8×Dic3C2212yes—
27C2×Q8×Dic3C2212yes—
28C2×C4×#67C2211yescharacteristic
29#5731—28no—
30C2×#898C2214yes—
31C2×#889C2214yes—
32C2×#872C2211yescharacteristic
33C2×#870C2211yescharacteristic
34#5229C2212yes—
35#5212C2212yes—
36C4×#217C2211yescharacteristic
37#4994C2214yes—
38#4993C2214yes—
39#4990C2212yes—
40#4990C2214yes—
41#4987C2212yes—
42#4984C2214yes—
43#4983C2214yes—
44#4982C2214yes—
45#4982C2214yes—
46#4982C2214yes—
47#4981C2214yes—
48#4978C2214yes—
49#4977C2212yes—
50#4977C2214yes—
51#4976C2212yes—
52#4976C2214yes—
53#4976C2214yes—
54#4975C2212yes—
55Dic3×C4⋊C4C2214yes—
56#4972C2212yes—
57#4972C2214yes—
58#4961C2214yes—
59#4957C2212yes—
60#4949C2214yes—
61#4948C2214yes—
62#4945C2212yes—
63#4945C2214yes—
64#4944C2214yes—
65#4940C2214yes—
66C4×(C2×D3)⋊C4C2211yescharacteristic
67#4935C2211yescharacteristic
68#4933C2214yes—
69C4×Dic3⋊C4C2211yescharacteristic
70C4×Dic3⋊C4C2214yes—
71C42×Dic3C2211yescharacteristic
72#4928C2212yes—
73#4928C2214yes—
74C2×#541—31no—
75C2×#541D311yescharacteristic, complemented
76#2859—31no—
77#2727—34no—
78#2726—34no—
79#2629—31no—
80#2629—31no—
81#2536—32no—
82#2519—34no—
83#2519—34no—
84#2423—32no—
85#2423—32no—
86#2351—32no—
87C4×C4⋊Q8—31no—
88#2293—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2311yescharacteristic
92C22×C4×D3C2311yescharacteristic
93C3×C4⋊Q8D418yescomplemented
94C3×#126—216no—
95C3×#125—216no—
96C3×#125—24no—
97C3×C22⋊Q8—28no—
98C3×C4×Q8—28no—
99C3×#117—24no—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C22⋊C4C2312yes—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2314yes—
110C2×C3×C42C2311yescharacteristic
111C2×#217C2312yes—
112C2×#217C2311yescharacteristic
113C2×#217C2314yes—
114#946—28no—
115Q8×Dic3—28no—
116Q8×Dic3—28no—
117C4×#67—24no—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×(C2×D3)⋊C4C2311yescharacteristic
122C2×(C2×D3)⋊C4C2314yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2314yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2314yes—
127C2×C4⋊Dic3C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2312yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2311yescharacteristic
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2312yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×C4×Dic3C2311yescharacteristic
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2314yes—
140C2×C4×Dic3C2314yes—
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2312yes—
143C2×C4×Dic3C2311yescharacteristic
144#898—216no—
145#889—216no—
146#872—24no—
147#870—24no—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159#541—28no—
160#541—68no—
161C2×C4⋊Q8C2×D311yescharacteristic
162C2×#126—34no—
163C2×#126C2×D314yes—
164C2×#125—34no—
165C2×#125—31no—
166C2×#125C2×D311yescharacteristic
167C2×#125C2×D314yes—
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8C2×D312yes—
170C2×C4×Q8—32no—
171C2×C4×Q8—32no—
172C2×C4×Q8C2×D312yes—
173C2×C4×D4—31no—
174C2×#117—31no—
175C2×#117C2×D311yescharacteristic
176#399—34no—
177#399—34no—
178#398—34no—
179#398—34no—
180#398—34no—
181#398—34no—
182#398—34no—
183#396—34no—
184#396—34no—
185#394—34no—
186#391—32no—
187#390—32no—
188#389—34no—
189#389—34no—
190#389—34no—
191#389—32no—
192#389—34no—
193#389—32no—
194#389—32no—
195#389—34no—
196#388—34no—
197#388—32no—
198#388—34no—
199#388—34no—
200#388—32no—
201#388—34no—
202#387—34no—
203#387—34no—
204#384—32no—
205#384—31no—
206#382—32no—
207#382—34no—
208#380—32no—
209#380—32no—
210C4×C4⋊C4—34no—
211C4×C4⋊C4—31no—
212C4×C4⋊C4—34no—
213C4×C22⋊C4—31no—
214C4×C22⋊C4—31no—
215C43—31no—
216C24×C3C2411yescharacteristic
217C23×D3C2411yescharacteristic
218C2×C3×Q8—24no—
219C2×C3×Q8—24no—
220C2×C3×Q8C2×D418yes—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—22no—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2412yes—
226C22×C3×C4C2414yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C2×#67—22no—
231C2×#67—42no—
232C22×Dic3C2414yes—
233C22×Dic3C2411yescharacteristic
234C22×Dic3C2412yes—
235C22×Dic3C2414yes—
236C22×Dic3C2411yescharacteristic
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2412yes—
239C2×C4×D3—22no—
240C2×C4×D3—22no—
241C2×C4×D3—22no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—28no—
245C3×C4⋊C4—28no—
246C3×C4⋊C4—28no—
247C3×C4⋊C4C2×D4116yes—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—28no—
250C3×C42—22no—
251C3×C42—28no—
252C3×C42C2×D414yes—
253#217—22no—
254#217—24no—
255#217—44no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—28no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—44no—
261C4⋊Dic3—24no—
262C4⋊Dic3—28no—
263C4⋊Dic3—24no—
264C4⋊Dic3—28no—
265C4⋊Dic3—44no—
266Dic3⋊C4—28no—
267Dic3⋊C4—24no—
268Dic3⋊C4—28no—
269Dic3⋊C4—22no—
270Dic3⋊C4—28no—
271Dic3⋊C4—44no—
272Dic3⋊C4—42no—
273Dic3⋊C4—44no—
274Dic3⋊C4—44no—
275C4×Dic3—22no—
276C4×Dic3—28no—
277C4×Dic3—22no—
278C4×Dic3—28no—
279C4×Dic3—24no—
280C4×Dic3—42no—
281C4×Dic3#4114yes—
282C22×Q8C22×D312yes—
283C22×D4—31no—
284C23×C4—31no—
285C23×C4C22×D311yescharacteristic
286C4⋊Q8#6718yescomplemented
287#126—216no—
288#126—616no—
289#125—216no—
290#125—24no—
291#125—616no—
292#125—64no—
293C22⋊Q8—28no—
294C22⋊Q8—68no—
295C4×Q8—28no—
296C4×Q8—68no—
297C4×Q8—68no—
298C4×D4—64no—
299#117—24no—
300#117—64no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—34no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4C22×D312yes—
316C2×C4⋊C4C22×D312yes—
317C2×C4⋊C4C22×D314yes—
318C2×C4⋊C4C22×D314yes—
319C2×C4⋊C4C22×D314yes—
320C2×C4⋊C4C22×D314yes—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—32no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—31no—
325C2×C22⋊C4—34no—
326C2×C22⋊C4—32no—
327C2×C22⋊C4—34no—
328C2×C22⋊C4—34no—
329C2×C22⋊C4C22×D312yes—
330C2×C22⋊C4C22×D314yes—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—34no—
334C2×C42—34no—
335C2×C42—31no—
336C2×C42—32no—
337C2×C42—32no—
338C2×C42C22×D311yescharacteristic
339C2×C42C22×D314yes—
340C2×C42C22×D311yescharacteristic
341#95—32no—
342#95—32no—
343#95—32no—
344#95—34no—
345#95—34no—
346#95—34no—
347#95—34no—
348#95—34no—
349#95—34no—
350#95—34no—
351#95—34no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—22no—
357C23×C3C2511yescharacteristic
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—22no—
361C22×D3—21no—
362C22×D3—22no—
363C3×Q8—216no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—24no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—24no—
377C2×C3×C4—44no—
378C2×C3×C4C22×D418yes—
379C2×C3×C4C22×D414yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C42−1+412yes—
383#67—44no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—21no—
391C2×Dic3—24no—
392C2×Dic3—21no—
393C2×Dic3—24no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—24no—
399C2×Dic3—24no—
400C2×Dic3C2×#4112yes—
401C2×Dic3C2×#4112yes—
402C2×Dic32−1+414yes—
403C4×D3—44no—
404C24—31no—
405C24C23×D311yescharacteristic
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8C2×#6718yes—
409C2×D4—62no—
410C2×D4—122no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—31no—
415C22×C4—32no—
416C22×C4—31no—
417C22×C4—34no—
418C22×C4—31no—
419C22×C4—32no—
420C22×C4—34no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4C23×D311yescharacteristic
425C22×C4C23×D312yes—
426C22×C4C23×D314yes—
427C22×C4C23×D314yes—
428C22×C4C23×D312yes—
429C22×C4C23×D311yescharacteristic
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—62no—
436C4⋊C4—68no—
437C4⋊C4—68no—
438C4⋊C4—64no—
439C4⋊C4—64no—
440C4⋊C4—64no—
441C4⋊C4—68no—
442C4⋊C4—68no—
443C4⋊C4—68no—
444C4⋊C4—124no—
445C4⋊C4—124no—
446C4⋊C4—122no—
447C4⋊C4—124no—
448C4⋊C4—124no—
449C4⋊C4C2×#67116yes—
450C22⋊C4—24no—
451C22⋊C4—28no—
452C22⋊C4—62no—
453C22⋊C4—64no—
454C22⋊C4—62no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—68no—
458C22⋊C4—124no—
459C22⋊C4—124no—
460C42—22no—
461C42—28no—
462C42—34no—
463C42—68no—
464C42—64no—
465C42—62no—
466C42—62no—
467C42—68no—
468C42—122no—
469C42C2×#6714yes—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—42no—
473C22×C3—41no—
474C22×C3—42no—
475C22×C3C23×D411yescharacteristic
476C22×C3C22×#4111yescharacteristic
477C22×C3C2×2−1+411yescharacteristic
478C22×C3C2×2−1+412yes—
479C22×C3#58512yes—
480C2×D3—41no—
481C2×D3—41no—
482C2×D3—42no—
483C2×D3—41no—
484C2×D3—42no—
485C3×C4—28no—
486C3×C4—24no—
487C3×C4—41no—
488C3×C4—42no—
489C3×C4—44no—
490C3×C4#54714yes—
491Dic3—22no—
492Dic3—22no—
493Dic3—24no—
494Dic3—44no—
495Dic3—41no—
496Dic3—42no—
497Dic3—44no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—61no—
504C23—61no—
505C23—62no—
506C23—62no—
507C23—61no—
508C23C24×D311yescharacteristic
509Q8—216no—
510D4—124no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—32no—
525C2×C4—32no—
526C2×C4—34no—
527C2×C4—44no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—64no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—62no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—64no—
543C2×C4—61no—
544C2×C4—124no—
545C2×C4D3×#4112yes—
546C2×C4#101012yes—
547C2×C4C22×#6718yes—
548C2×C4C22×#6714yes—
549C2×C4C22×#6712yes—
550C2×C3—81no—
551C2×C3#345511yescharacteristic
552C2×C3C2×#54711yescharacteristic
553C2×C3#349712yes—
554C2×C3#350911yescharacteristic
555C2×C3#356312yes—
556D3—81no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—42no—
561C22—42no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22—122no—
566C22—122no—
567C22C2×D3×#4111yescharacteristic
568C22C2×#101011yescharacteristic
569C22D3×2−1+412yes—
570C22#596812yes—
571C22C23×#6711yescharacteristic
572C4—28no—
573C4—24no—
574C4—41no—
575C4—42no—
576C4—44no—
577C4—62no—
578C4—62no—
579C4—64no—
580C4—124no—
581C4—121no—
582C4—122no—
583C4—124no—
584C4#582414yes—
585C3#3937311yescharacteristic, complemented, Sylow
586C2—81no—
587C2—241no—
588C2#8649511yescharacteristic
589C2#8651011yescharacteristic
590C2#8652012yes—
591C2C2×#582411yescharacteristic
592C2#8729112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#541C2
prime radicalC22×#67C23
hypercenterC2×#541D3
hypocenterC3#39373
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×#5731C2semidirect
3. Dic3×C4⋊Q8C2semidirect
5. #83085C2semidirect
6. #83065C2semidirect
7. #83056C2semidirect
8. #83054C2semidirect
9. #83035C2semidirect
11. #82754C2semidirect
12. #82494C2semidirect
13. #82461C2semidirect
14. #82265C2semidirect
75. C2×#541D3semidirect
93. C3×C4⋊Q8D4semidirect
286. C4⋊Q8#67semidirect
585. C3#39373semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#57312
Dic3×C4⋊Q82
#831022
#830852
#830652
#830562
#830542
#830352
#827772
#827542
#824942
#824612
#822652
#822252
#393733