Finite Group Databasebeta

← #10728996

Subgroups — #10728996

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#376C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic
3C2×#4933C211yesmaximal, characteristic
4C2×#4659C211yesmaximal, characteristic
5#71536C212yesmaximal
6#71536C212yesmaximal
7#69422C211yesmaximal, characteristic
8#69418C212yesmaximal
9#69413C212yesmaximal
10#69412C211yesmaximal, characteristic
11#69410C211yesmaximal, characteristic
12#14908—31nomaximal, Sylow
13C22×C3×C42C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×Dic3⋊C4C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C3×#376—24no—
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37#4933—24no—
38#4928—24no—
39#4659—24no—
40C2×#400—31no—
41C2×#389—31no—
42C2×#387—31no—
43C2×#376D311yescharacteristic
44#1445—31no—
45#1443—32no—
46#1442—31no—
47#1442—32no—
48#1441—31no—
49#1436—32no—
50#1436—32no—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C2×C3×C42—24no—
63C2×C3×C42—22no—
64C2×C3×C42—24no—
65C2×C3×C42—24no—
66C2×C3×C42—22no—
67C2×C3×C42D414yes—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—24no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C3×#95—24no—
83C3×#95—22no—
84C3×#95D414yes—
85C3⋊#95—24no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C22×C4⋊C4—31no—
99C22×C4⋊C4—31no—
100C22×C4⋊C4—32no—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—31no—
103C22×C42C2×D312yes—
104C22×C42C2×D311yescharacteristic
105#400—64no—
106#389—64no—
107#387—64no—
108#376—24no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—31no—
112C2×#95—31no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—31no—
121C2×#95—31no—
122C2×#95C2×D311yescharacteristic
123C2×#95C2×D312yes—
124C2×#95C2×D311yescharacteristic
125C24×C3C2411yescharacteristic
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D414yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4#4114yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4114yes—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3#4112yes—
191C22×Dic3#4112yes—
192C22×Dic3#4114yes—
193C3×C42—216no—
194C3×C42—28no—
195C4⋊Dic3—28no—
196C4⋊Dic3—44no—
197Dic3⋊C4—28no—
198Dic3⋊C4—44no—
199Dic3⋊C4—48no—
200C23×C4—32no—
201C23×C4—31no—
202C23×C4—31no—
203C23×C4—32no—
204C23×C4—31no—
205C23×C4—31no—
206C23×C4C22×D311yescharacteristic
207C23×C4C22×D312yes—
208C23×C4C22×D312yes—
209C23×C4C22×D311yescharacteristic
210C23×C4C22×D311yescharacteristic
211C2×C4⋊C4—62no—
212C2×C4⋊C4—62no—
213C2×C4⋊C4—64no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4—62no—
217C2×C4⋊C4—64no—
218C2×C4⋊C4—62no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—62no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42D1214yes—
231#95—24no—
232#95—22no—
233#95—64no—
234#95—64no—
235#95—64no—
236#95—64no—
237#95—62no—
238#95—62no—
239#95—62no—
240#95—62no—
241#95—64no—
242#95—64no—
243#95—64no—
244#95—64no—
245#95—64no—
246#95#6714yes—
247C23×C3C22×D411yescharacteristic
248C23×C3C22×D411yescharacteristic
249C23×C3C2×#4111yescharacteristic
250C23×C3C2×#4112yes—
251C23×C3C2×#4112yes—
252C23×C3C2×#4111yescharacteristic
253C23×C3C2×#4112yes—
254C23×C3C2×#4111yescharacteristic
255C23×C32−1+412yes—
256C23×C32−1+411yescharacteristic
257C23×C32−1+411yescharacteristic
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4#12114yes—
288C2×C3×C4#12414yes—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—24no—
292C2×Dic3—24no—
293C2×Dic3—24no—
294C2×Dic3—22no—
295C2×Dic3—24no—
296C2×Dic3—22no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C24C23×D311yescharacteristic
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—34no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—61no—
353C22×C4—62no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—61no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4C2×D1212yes—
386C22×C4C2×D1212yes—
387C22×C4C2×D1212yes—
388C22×C4#23514yes—
389C22×C4#23514yes—
390C22×C4#23512yes—
391C22×C4C2×#6714yes—
392C22×C4C2×#6712yes—
393C4⋊C4—68no—
394C4⋊C4—68no—
395C4⋊C4—128no—
396C4⋊C4—124no—
397C4⋊C4—124no—
398C42—216no—
399C42—28no—
400C22×C3C2×#12111yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#52511yescharacteristic
403C22×C3#52512yes—
404C22×C3C2×#12411yescharacteristic
405C22×C3#52912yes—
406C22×C3#52912yes—
407C22×C3#52912yes—
408C22×C3#53211yescharacteristic
409C22×C3#53611yescharacteristic
410C22×C3#53611yescharacteristic
411C22×C3#54112yes—
412C22×C3#54112yes—
413C22×C3#54712yes—
414C22×C3#54711yescharacteristic
415C22×C3#54811yescharacteristic
416C22×C3#55212yes—
417C22×C3#55211yescharacteristic
418C22×C3#55212yes—
419C22×C3#55212yes—
420C22×C3#55212yes—
421C22×C3#55612yes—
422C22×C3#55611yescharacteristic
423C3×C4—24no—
424C3×C4—24no—
425C3×C4—42no—
426C3×C4—44no—
427C3×C4—44no—
428Dic3—42no—
429Dic3—44no—
430Dic3—42no—
431Dic3—82no—
432Dic3—81no—
433Dic3—81no—
434C23C22×D1211yescharacteristic
435C23C2×#23512yes—
436C23C2×#23512yes—
437C23C2×#23511yescharacteristic
438C23D3×#4111yescharacteristic
439C23D3×#4111yescharacteristic
440C23D3×#4112yes—
441C23#101012yes—
442C23#101011yescharacteristic
443C23#101011yescharacteristic
444C23C22×#6711yescharacteristic
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—62no—
475C2×C4—64no—
476C2×C4—62no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—62no—
480C2×C4—64no—
481C2×C4—62no—
482C2×C4—122no—
483C2×C4—121no—
484C2×C4—122no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—121no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4#87514yes—
511C2×C4#93014yes—
512C2×C3#239811yescharacteristic
513C2×C3#241311yescharacteristic
514C2×C3#241511yescharacteristic
515C2×C3#247512yes—
516C2×C3#247812yes—
517C2×C3#252812yes—
518C2×C3#253312yes—
519C2×C3#260411yescharacteristic
520C2×C3#261111yescharacteristic
521C2×C3#264211yescharacteristic
522C2×C3#285011yescharacteristic
523C22C2×#87511yescharacteristic
524C22#547711yescharacteristic
525C22#547912yes—
526C22#551812yes—
527C22#551812yes—
528C22#552411yescharacteristic
529C22#552411yescharacteristic
530C22#552611yescharacteristic
531C22#552611yescharacteristic
532C22#552812yes—
533C22#552812yes—
534C22#552912yes—
535C22#552911yescharacteristic
536C22#553112yes—
537C22#553112yes—
538C22#553112yes—
539C22#553312yes—
540C22#553311yescharacteristic
541C22C2×#93011yescharacteristic
542C22#579112yes—
543C22#582412yes—
544C22#582411yescharacteristic
545C22#583111yescharacteristic
546C4—24no—
547C4—24no—
548C4—44no—
549C4—44no—
550C4—42no—
551C4—122no—
552C4—124no—
553C4—122no—
554C4—241no—
555C4—242no—
556C4—241no—
557C3#1490811yescharacteristic, complemented, Sylow
558C2#7339912yes—
559C2#7340212yes—
560C2#7340311yescharacteristic
561C2#7340711yescharacteristic
562C2#7341911yescharacteristic
563C2#7342711yescharacteristic
564C2#8207711yescharacteristic
565C2#8224611yescharacteristic
566C2#8225112yes—
567C2#8225212yes—
568C2#8227711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#376C2
hypercenterC2×#376D3
hypocenterC3#14908
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
557. C3#14908semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3762
C2×#49282
C2×#49332
C2×#46592
#715362
#694222
#694182
#694132
#694122
#694102
#149083