Finite Group Databasebeta

← #10731717

Subgroups — #10731717

Order 768.

1238 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×#4978C212yesmaximal
3C2×#4932C211yesmaximal, characteristic
4#71569C211yesmaximal, characteristic
5#71569C211yesmaximal, characteristic
6#71568C211yesmaximal, characteristic
7#71567C211yesmaximal, characteristic
8#71564C212yesmaximal
9#71534C211yesmaximal, characteristic
10#69417C212yesmaximal
11#69413C212yesmaximal
12#15122—31nomaximal, Sylow
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2211yescharacteristic
19C22×C4×Dic3C2211yescharacteristic
20C3×#389—24no—
21C2×C3×#95C2212yes—
22C2×C3⋊#95C2211yescharacteristic
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37#4978—28no—
38#4932—24no—
39C2×#389D311yescharacteristic
40C2×#387—32no—
41C2×#376—31no—
42#1445—31no—
43#1445—31no—
44#1444—31no—
45#1443—32no—
46#1442—32no—
47#1442—31no—
48#1436—32no—
49#1434—31no—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×Dic3C2312yes—
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42D414yes—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—24no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3D414yes—
81C3×#95—24no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C22×C4⋊C4—32no—
98C22×C4⋊C4C2×D312yes—
99C22×C4⋊C4C2×D311yescharacteristic
100C22×C4⋊C4C2×D311yescharacteristic
101C22×C42—31no—
102C22×C42—31no—
103C22×C42C2×D311yescharacteristic
104#389—24no—
105#387—68no—
106#376—64no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—31no—
110C2×#95—31no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—31no—
116C2×#95—32no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—31no—
122C2×#95C2×D312yes—
123C24×C3C2411yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4#4114yes—
152C22×C3×C4#4112yes—
153C22×C3×C4#4112yes—
154C22×C3×C4#4112yes—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3C2×D412yes—
185C22×Dic3C2×D412yes—
186C22×Dic3#4112yes—
187C22×Dic3#4112yes—
188C22×Dic3#4114yes—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—48no—
191C3×C4⋊C4—44no—
192C3×C42—28no—
193Dic3⋊C4—216no—
194C4×Dic3—28no—
195C4×Dic3—28no—
196C23×C4—31no—
197C23×C4—32no—
198C23×C4—31no—
199C23×C4—31no—
200C23×C4—31no—
201C23×C4—32no—
202C23×C4C22×D312yes—
203C23×C4C22×D312yes—
204C23×C4C22×D311yescharacteristic
205C23×C4C22×D311yescharacteristic
206C23×C4C22×D311yescharacteristic
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—64no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—34no—
221C2×C42—62no—
222C2×C42—62no—
223C2×C42—62no—
224C2×C42—62no—
225C2×C42—62no—
226C2×C42#6714yes—
227#95—24no—
228#95—64no—
229#95—64no—
230#95—64no—
231#95—64no—
232#95—64no—
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—62no—
242#95—62no—
243C23×C3C22×D411yescharacteristic
244C23×C3C22×D411yescharacteristic
245C23×C3C2×#4112yes—
246C23×C3C2×#4111yescharacteristic
247C23×C3C2×#4111yescharacteristic
248C23×C3C2×#4111yescharacteristic
249C23×C3C2×#4111yescharacteristic
250C23×C3C2×#4112yes—
251C23×C3C2×#4111yescharacteristic
252C23×C32−1+412yes—
253C23×C32−1+412yes—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4#12114yes—
284C2×C3×C4#12314yes—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—24no—
290C2×Dic3—24no—
291C2×Dic3—24no—
292C2×Dic3—24no—
293C2×Dic3—24no—
294C2×Dic3—24no—
295C2×Dic3—24no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C24C23×D311yescharacteristic
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—32no—
347C22×C4—34no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—62no—
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#23514yes—
378C22×C4D3×D412yes—
379C22×C4#23712yes—
380C22×C4#23712yes—
381C22×C4#23712yes—
382C22×C4C2×#6712yes—
383C22×C4C2×#6712yes—
384C22×C4C2×#6712yes—
385C4⋊C4—28no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—616no—
389C42—28no—
390C42—68no—
391C42—68no—
392C22×C3C2×#12111yescharacteristic
393C22×C3C2×#12311yescharacteristic
394C22×C3#52511yescharacteristic
395C22×C3#52512yes—
396C22×C3#52511yescharacteristic
397C22×C3#52912yes—
398C22×C3#52911yescharacteristic
399C22×C3#52911yescharacteristic
400C22×C3#53211yescharacteristic
401C22×C3#53211yescharacteristic
402C22×C3#53211yescharacteristic
403C22×C3#53612yes—
404C22×C3#53611yescharacteristic
405C22×C3#53611yescharacteristic
406C22×C3#54112yes—
407C22×C3#55212yes—
408C22×C3#55212yes—
409C22×C3#55212yes—
410C22×C3#55212yes—
411C22×C3#55212yes—
412C22×C3#55212yes—
413C22×C3#55612yes—
414C22×C3#55612yes—
415C3×C4—24no—
416C3×C4—24no—
417C3×C4—44no—
418C3×C4—42no—
419C3×C4—82no—
420Dic3—42no—
421Dic3—44no—
422Dic3—42no—
423Dic3—42no—
424Dic3—42no—
425Dic3—82no—
426C23C2×#23512yes—
427C23C2×D3×D411yescharacteristic
428C23C2×#23711yescharacteristic
429C23C2×#23711yescharacteristic
430C23C2×#23711yescharacteristic
431C23#100712yes—
432C23#100712yes—
433C23D3×#4112yes—
434C23D3×#4111yescharacteristic
435C23D3×#4111yescharacteristic
436C23C22×#6711yescharacteristic
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—64no—
467C2×C4—64no—
468C2×C4—64no—
469C2×C4—64no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—64no—
473C2×C4—64no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—122no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4#93514yes—
501C2×C4#93814yes—
502C2×C3#239211yescharacteristic
503C2×C3#248411yescharacteristic
504C2×C3#249911yescharacteristic
505C2×C3#252812yes—
506C2×C3#253312yes—
507C2×C3#259911yescharacteristic
508C2×C3#260412yes—
509C2×C3#261112yes—
510C2×C3#261111yescharacteristic
511C2×C3#262111yescharacteristic
512C2×C3#285411yescharacteristic
513C22#551312yes—
514C22#551312yes—
515C22#553812yes—
516C22#554712yes—
517C22#556812yes—
518C22#558112yes—
519C22#559611yescharacteristic
520C22#559611yescharacteristic
521C22#560011yescharacteristic
522C22#560011yescharacteristic
523C22#563912yes—
524C22#564112yes—
525C22#569312yes—
526C22#569312yes—
527C22#572412yes—
528C22#573112yes—
529C22#579111yescharacteristic
530C22C2×#93511yescharacteristic
531C22C2×#93811yescharacteristic
532C22#581611yescharacteristic
533C22#581611yescharacteristic
534C22#582811yescharacteristic
535C22#583211yescharacteristic
536C4—24no—
537C4—24no—
538C4—42no—
539C4—44no—
540C4—82no—
541C4—122no—
542C4—124no—
543C4—122no—
544C4—122no—
545C4—122no—
546C4—242no—
547C3#1512211yescharacteristic, complemented, Sylow
548C2#7605412yes—
549C2#7605812yes—
550C2#7609712yes—
551C2#7610012yes—
552C2#8217611yescharacteristic
553C2#8240111yescharacteristic
554C2#8250211yescharacteristic
555C2#8269911yescharacteristic
556C2#8277411yescharacteristic
557C2#8306411yescharacteristic
558C2#8326111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#389C2
hypercenterC2×#389D3
hypocenterC3#15122
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
547. C3#15122semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#49782
C2×#49322
#715692
#715682
#715672
#715642
#715342
#694172
#694132
#151223