Finite Group Databasebeta

← #10846298

Subgroups — #10846298

Order 768.

1238 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#466C211yesmaximal, characteristic, complemented
2C2×#5151C211yesmaximal, characteristic, complemented
3C2×#5115C211yesmaximal, characteristic, complemented
4#82648C212yesmaximal, complemented
5C2×#4872C211yesmaximal, characteristic, complemented
6#72038C211yesmaximal, characteristic, complemented
7C8×#217C211yesmaximal, characteristic, complemented
8#71684C212yesmaximal, complemented
9#70612C211yesmaximal, characteristic, complemented
10#70208C212yesmaximal, complemented
11#70177C212yesmaximal, complemented
12#25414—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×#940C2212yes—
17C2×#938—22no—
18C22×#213C2212yes—
19C2×#930C2212yes—
20C22×D24C2211yescharacteristic
21C2×#893C2212yes—
22C3×#466—24no—
23C2×C3×#107C2211yescharacteristic
24C2×C3×#102C2212yes—
25#5223—22no—
26#5221C2212yes—
27C2×#832C2212yes—
28C4×#217C2211yescharacteristic
29C2×#830C2211yescharacteristic
30#5151—24no—
31#5115—24no—
32C2×#821C2212yes—
33C2×#818C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4984C2212yes—
37C2×#809C2212yes—
38#4961—22no—
39#4956—22no—
40#4872—24no—
41#4550C2211yescharacteristic
42#4550C2211yescharacteristic
43#4529C2212yes—
44C2×#493—31no—
45C2×#466—31no—
46C2×#466D311yescharacteristic, complemented
47C2×C4×D8—31no—
48#2428—32no—
49#2041—32no—
50#1999—32no—
51#1970—31no—
52#1942—31no—
53#1879—32no—
54C8×C22⋊C4—31no—
55C22×C3×D4—22no—
56C22×C3×D4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C22×#67—22no—
59C22×#67—22no—
60C22×#67—22no—
61C22×D12C2312yes—
62C22×C4×D3—22no—
63C2×C3×D8—22no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C22×C3×C8—22no—
67C22×C3×C8—22no—
68C22×C3×C8—22no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×#121—24no—
72C3×#121—42no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—22no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217D411yescharacteristic
78C2×#217D411yescharacteristic
79#940—24no—
80#940—42no—
81#938—48no—
82C2×#213—24no—
83C2×#213—24no—
84C2×#213—42no—
85#930—24no—
86#930—42no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3C2312yes—
91C2×C4×Dic3C2311yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C2×D24—22no—
96C2×D24—22no—
97C2×D24D414yescomplemented
98#893—24no—
99#893—42no—
100C3×#107D414yescomplemented
101C3×#102—24no—
102#832—42no—
103C3⋊#95—22no—
104C3⋊#95D411yescharacteristic
105C3⋊#95D411yescharacteristic
106#830—22no—
107#821—24no—
108#818—22no—
109C8×Dic3—22no—
110C8×Dic3—22no—
111#809—24no—
112C22×D8—32no—
113C22×D8—31no—
114C22×D8C2×D311yescharacteristic
115C23×C8C2×D311yescharacteristic
116C2×#127—32no—
117C2×#121—32no—
118C2×#121—62no—
119C2×#121C2×D312yes—
120C2×C4×D4—32no—
121#493—64no—
122#466—24no—
123#466—64no—
124C4×D8—64no—
125C2×#107—31no—
126C2×#107C2×D311yescharacteristic
127C2×#102—32no—
128C2×#102—32no—
129C2×#102—32no—
130C2×#102C2×D312yes—
131C2×C22⋊C8—31no—
132C2×C4×C8—31no—
133C2×C4×C8—31no—
134#394—62no—
135#392—62no—
136#390—32no—
137#389—32no—
138#386—62no—
139C4×C22⋊C4—31no—
140#340—32no—
141#336—31no—
142#336—31no—
143C24×C3—22no—
144C24×C3C2×D411yescharacteristic
145C23×D3—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—44no—
153C2×C3×D4—44no—
154C2×C3×D4—82no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×#67—44no—
165C2×#67—44no—
166C2×#67—44no—
167C2×#67—44no—
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—82no—
171C2×#67—82no—
172C2×#67—82no—
173C22×Dic3—22no—
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12C2×D414yes—
184C2×C4×D3—44no—
185C2×C4×D3—44no—
186C2×C4×D3—82no—
187C3×D8—44no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—44no—
192C2×C3×C8C2×D412yes—
193C2×C3×C8C2×D412yes—
194C2×C3×C8#4112yes—
195C3×C4⋊C4—22no—
196C3×C4⋊C4C2×D414yes—
197C3×C22⋊C4—44no—
198#217—22no—
199#217—44no—
200#217—82no—
201#217D814yescomplemented
202#213—48no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—44no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—44no—
208C4×Dic3—21no—
209C4×Dic3—21no—
210C4×Dic3—21no—
211C4×Dic3—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8#4112yes—
218D24—28no—
219C22×D4—22no—
220C22×D4—32no—
221C22×D4—62no—
222C22×D4—62no—
223C22×D4—62no—
224C22×D4C22×D312yes—
225C23×C4—62no—
226C23×C4C22×D311yescharacteristic
227C2×D8—22no—
228C2×D8—22no—
229C2×D8—22no—
230C2×D8—34no—
231C2×D8—64no—
232C2×D8—64no—
233C2×D8—62no—
234C2×D8—62no—
235C2×D8—122no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—31no—
240C22×C8—31no—
241C22×C8C22×D311yescharacteristic
242C22×C8C22×D311yescharacteristic
243#127—64no—
244#127—122no—
245#121—24no—
246#121—42no—
247#121—64no—
248#121—128no—
249#121—122no—
250C4×D4—64no—
251C4×D4—122no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4C22×D312yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—31no—
257C2×C22⋊C4—31no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C42—31no—
263C2×C42—31no—
264#107—62no—
265#107#6714yescomplemented
266#102—24no—
267#102—64no—
268#102—64no—
269#102—122no—
270C22⋊C8—62no—
271C4×C8—62no—
272C4×C8—62no—
273#95—31no—
274#95—31no—
275#95—62no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3C2×D812yes—
290C23×C3C22×D411yescharacteristic
291C22×D3—24no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—42no—
301C3×D4—42no—
302C3×D4—88no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319#67—88no—
320#67—88no—
321#67—88no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3C2×D812yes—
335C2×Dic3C2×D812yes—
336C2×Dic3#13512yes—
337D12—24no—
338D12—24no—
339D12—42no—
340D12—42no—
341C4×D3—88no—
342C3×C8—41no—
343C3×C8#12114yescomplemented
344C3⋊C8—22no—
345C3⋊C8—41no—
346C24—22no—
347C24—62no—
348C24D3×D411yescharacteristic
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—34no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—82no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—124no—
368C2×D4—124no—
369C2×D4—124no—
370C2×D4—242no—
371C2×D4—242no—
372C2×D4—242no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—62no—
385C22×C4—124no—
386C22×C4—124no—
387C22×C4—242no—
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390D8—44no—
391D8—68no—
392D8—128no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—32no—
397C2×C8—44no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8D3×D412yes—
404C2×C8#23712yes—
405C2×C8C2×#6712yes—
406C4⋊C4—22no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—124no—
410C4⋊C4C2×#6714yes—
411C22⋊C4—34no—
412C22⋊C4—44no—
413C22⋊C4—62no—
414C22⋊C4—124no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—242no—
418C42—61no—
419C42—61no—
420C42—61no—
421C42—61no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3#53211yescharacteristic
437C22×C3D4211yescharacteristic
438C22×C3D4211yescharacteristic
439C22×C3C22×D811yescharacteristic
440C22×C3C2×#13511yescharacteristic
441C22×C3#57611yescharacteristic
442C22×C3#57611yescharacteristic
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—82no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C3×C4—41no—
451C3×C4—44no—
452C3×C4C2×#12111yescharacteristic
453C3×C4#54011yescharacteristic
454C3×C4#54611yescharacteristic
455C3×C4#55011yescharacteristic
456Dic3—22no—
457Dic3—22no—
458Dic3—22no—
459Dic3—41no—
460Dic3—82no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—64no—
475C23—122no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23D3×D812yes—
482C23C2×D3×D411yescharacteristic
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—64no—
488D4—64no—
489D4—88no—
490D4—122no—
491D4—122no—
492D4—248no—
493D4—248no—
494D4—248no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—32no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—248no—
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×#23711yescharacteristic
523C2×C4#100411yescharacteristic
524C2×C4D3×#4111yescharacteristic
525C2×C4#100911yescharacteristic
526C2×C4C22×#6711yescharacteristic
527C8—41no—
528C8—62no—
529C8—121no—
530C8#93814yescomplemented
531C2×C3—42no—
532C2×C3—162no—
533C2×C3#242811yescharacteristic
534C2×C3#316911yescharacteristic
535C2×C3#317611yescharacteristic
536C2×C3D4×D811yescharacteristic
537C2×C3D4×D811yescharacteristic
538C2×C3#328011yescharacteristic
539C2×C3#328711yescharacteristic
540D3—84no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—124no—
556C22—124no—
557C22—124no—
558C22—242no—
559C22—242no—
560C22—242no—
561C22—242no—
562C22#559611yescharacteristic
563C22#560911yescharacteristic
564C22C2×D3×D811yescharacteristic
565C22#575711yescharacteristic
566C22D3×#13511yescharacteristic
567C22#576911yescharacteristic
568C22D4×#6711yescharacteristic
569C4—44no—
570C4—41no—
571C4—62no—
572C4—62no—
573C4—62no—
574C4—121no—
575C4—242no—
576C4#561211yescharacteristic
577C4#561911yescharacteristic
578C4C2×#93811yescharacteristic
579C4#583411yescharacteristic
580C3#2541411yescharacteristic, complemented, Sylow
581C2—42no—
582C2—162no—
583C2—244no—
584C2#8069711yescharacteristic
585C2#8069911yescharacteristic
586C2#8072911yescharacteristic
587C2#8073011yescharacteristic
588C2#8264811yescharacteristic
589C2D8×#6711yescharacteristic
590C2#8392411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#466C2
socleC23×C3C22×D4
hypercenterC2×#466D3
hypocenterC3#25414
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×C3×#466C2semidirect
2. C2×#5151C2semidirect
3. C2×#5115C2semidirect
4. #82648C2semidirect
5. C2×#4872C2semidirect
6. #72038C2semidirect
7. C8×#217C2semidirect
8. #71684C2semidirect
9. #70612C2semidirect
10. #70208C2semidirect
11. #70177C2semidirect
46. C2×#466D3semidirect
97. C2×D24D4semidirect
100. C3×#107D4semidirect
201. #217D8semidirect
265. #107#67semidirect
343. C3×C8#121semidirect
530. C8#938semidirect
580. C3#25414semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4662
C2×#51512
C2×#51152
#826482
C2×#48722
#720382
C8×#2172
#716842
#706122
#702082
#701772
#254143