Finite Group Databasebeta

← #10723371

Subgroups — #10723371

Order 768.

1506 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5212C211yesmaximal, characteristic
2C2×#4945C212yesmaximal
3C3×C22⋊#95C211yesmaximal, characteristic
4#72127C211yesmaximal, characteristic
5#71549C212yesmaximal
6#69413C212yesmaximal, complemented
7#69411C212yesmaximal, complemented
8#69407C212yesmaximal, complemented
9#69402C212yesmaximal, complemented
10#14422—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×#217C2212yes—
15C22×#217C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×C4×Dic3C2212yes—
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22#5212C418yescomplemented
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⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33#4945—28no—
34C2×#388—32no—
35C2×#380—31no—
36#1442—32no—
37#1441—32no—
38#1440—32no—
39#1436—32no—
40#1434—32no—
41C22⋊#95—31no—
42C22⋊#95D311yescharacteristic
43C25×C3C2311yescharacteristic
44C23×C3×C4—24no—
45C23×C3×C4—41no—
46C23×C3×C4—41no—
47C23×C3×C4—42no—
48C23×C3×C4C2×C414yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—41no—
59C2×C3×C22⋊C4—41no—
60C2×C3×C22⋊C4—41no—
61C2×C3×C22⋊C4—41no—
62C2×C3×C22⋊C4D414yes—
63C2×#217—24no—
64C2×#217—24no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—42no—
68C2×#217C2×C418yes—
69C2×C4⋊Dic3—41no—
70C2×C4⋊Dic3—42no—
71C2×C4⋊Dic3—41no—
72C2×C4⋊Dic3C2×C414yes—
73C2×C4⋊Dic3C2×C414yes—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—42no—
76C2×Dic3⋊C4—42no—
77C2×C4×Dic3—24no—
78C2×C4×Dic3—24no—
79C2×C4×Dic3—42no—
80C3×#95—24no—
81C3×#95—42no—
82C3⋊#95—24no—
83C3⋊#95—24no—
84C3⋊#95—24no—
85C3⋊#95—24no—
86C3⋊#95—41no—
87C3⋊#95—42no—
88C3⋊#95—42no—
89C3⋊#95—42no—
90C3⋊#95—41no—
91C3⋊#95C2×C418yes—
92C24×C4C2×D311yescharacteristic
93C22×C4⋊C4—31no—
94C22×C4⋊C4—31no—
95C22×C4⋊C4—32no—
96C22×C22⋊C4—32no—
97C22×C22⋊C4—32no—
98C22×C22⋊C4C2×D311yescharacteristic
99C22×C22⋊C4C2×D311yescharacteristic
100C22×C42—32no—
101#388—68no—
102#380—38no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—31no—
109C2×#95—31no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95C2×D312yes—
114C2×#95C2×D312yes—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—21no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—22no—
123C24×C3C22×C412yes—
124C24×C3C2×D412yes—
125C24×C3#4112yes—
126C24×C3C2411yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—28no—
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×C4—22no—
148C22×C3×C4—44no—
149C22×C3×C4—44no—
150C22×C3×C4—48no—
151C22×C3×C4—48no—
152C22×C3×C4C22×C412yes—
153C22×C3×C4C22×C412yes—
154C22×C3×C4C2×D414yes—
155C22×C3×C4#4114yes—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
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×Dic3C22×C414yes—
182C22×Dic3C22×C414yes—
183C22×Dic3#4114yes—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—44no—
186#217—216no—
187#217—48no—
188C4⋊Dic3—44no—
189C4⋊Dic3—44no—
190Dic3⋊C4—48no—
191C4×Dic3—48no—
192C25C22×D311yescharacteristic
193C23×C4—24no—
194C23×C4—32no—
195C23×C4—32no—
196C23×C4—32no—
197C23×C4—32no—
198C23×C4—42no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4C4×D314yes—
202C23×C4C22×D312yes—
203C23×C4C22×D311yescharacteristic
204C23×C4C22×D312yes—
205C23×C4C22×D311yescharacteristic
206C2×C4⋊C4—34no—
207C2×C4⋊C4—34no—
208C2×C4⋊C4—64no—
209C2×C4⋊C4—122no—
210C2×C4⋊C4—122no—
211C2×C4⋊C4—121no—
212C2×C4⋊C4—122no—
213C2×C4⋊C4—121no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—38no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—64no—
221C2×C22⋊C4—64no—
222C2×C22⋊C4—64no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—122no—
225C2×C22⋊C4D1214yes—
226C2×C42—64no—
227C2×C42—64no—
228C2×C42—122no—
229#95—24no—
230#95—38no—
231#95—42no—
232#95—64no—
233#95—64no—
234#95—64no—
235#95—64no—
236#95—121no—
237#95—121no—
238#95—122no—
239#95—122no—
240#95—122no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3C4×D414yes—
264C23×C3C23×C411yescharacteristic
265C23×C3C22×D411yescharacteristic
266C23×C3C2×#4111yescharacteristic
267C23×C3C2×#4112yes—
268C23×C3C2×#4112yes—
269C23×C3H(5,𝔽2)11yescharacteristic
270C23×C3H(5,𝔽2)11yescharacteristic
271C23×C3H(5,𝔽2)12yes—
272C23×C3H(5,𝔽2)11yescharacteristic
273C23×C32−1+411yescharacteristic
274C23×C32−1+412yes—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—416no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4#11714yes—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
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—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24C2×C4×D312yes—
337C24C2×D1212yes—
338C24#23512yes—
339C24C23×D311yescharacteristic
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—34no—
362C22×C4—34no—
363C22×C4—34no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4C2×C4×D312yes—
394C22×C4C2×C4×D312yes—
395C22×C4C2×D1214yes—
396C22×C4#23714yes—
397C4⋊C4—124no—
398C4⋊C4—124no—
399C4⋊C4—128no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—616no—
403C22⋊C4—128no—
404C42—128no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—24no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3C2×#11711yescharacteristic
417C22×C3C2×C4×D411yescharacteristic
418C22×C3C4×#4111yescharacteristic
419C22×C3#51811yescharacteristic
420C22×C3#51811yescharacteristic
421C22×C3#51911yescharacteristic
422C22×C3#52011yescharacteristic
423C22×C3#53411yescharacteristic
424C22×C3#53711yescharacteristic
425C22×C3#53811yescharacteristic
426C22×C3#53812yes—
427C22×C3#53812yes—
428C22×C3#53912yes—
429C22×C3#53911yescharacteristic
430C22×C3#54212yes—
431C22×C3#54612yes—
432C22×C3#54712yes—
433C22×C3#55212yes—
434C22×C3#55212yes—
435C22×C3#55312yes—
436C22×C3#55512yes—
437C22×C3#56212yes—
438C22×C3#56312yes—
439C3×C4—24no—
440C3×C4—42no—
441C3×C4—44no—
442C3×C4—82no—
443Dic3—44no—
444Dic3—44no—
445Dic3—44no—
446Dic3—82no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23C4×D1214yes—
470C23C22×C4×D311yescharacteristic
471C23C22×D1211yescharacteristic
472C23C2×#23511yescharacteristic
473C23C2×#23712yes—
474C23#100412yes—
475C23#100411yescharacteristic
476C23#100411yescharacteristic
477C23#100411yescharacteristic
478C23#100711yescharacteristic
479C23D3×#4112yes—
480C23#101012yes—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—416no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4#87714yes—
534C2×C3—44no—
535C2×C3#231211yescharacteristic
536C2×C3#232511yescharacteristic
537C2×C3#233211yescharacteristic
538C2×C3#234411yescharacteristic
539C2×C3#234611yescharacteristic
540C2×C3#236411yescharacteristic
541C2×C3#237811yescharacteristic
542C2×C3#268612yes—
543C2×C3#269512yes—
544C2×C3#277012yes—
545C2×C3#278112yes—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22C2×C4×D1211yescharacteristic
558C22C2×#87711yescharacteristic
559C22#548611yescharacteristic
560C22#548611yescharacteristic
561C22#549311yescharacteristic
562C22#549411yescharacteristic
563C22#550411yescharacteristic
564C22#551011yescharacteristic
565C22#551511yescharacteristic
566C22#551611yescharacteristic
567C22#552211yescharacteristic
568C22#554012yes—
569C22#554112yes—
570C22#555412yes—
571C22#555512yes—
572C22#559312yes—
573C22#559312yes—
574C22#561612yes—
575C22#562112yes—
576C22#563912yes—
577C22#564612yes—
578C22#566012yes—
579C22#566212yes—
580C4—24no—
581C4—44no—
582C4—42no—
583C4—82no—
584C4—124no—
585C4—124no—
586C4—124no—
587C4—242no—
588C3#1442211yescharacteristic, complemented, Sylow
589C2—44no—
590C2#7333311yescharacteristic
591C2#7345611yescharacteristic
592C2#7347111yescharacteristic
593C2#7382011yescharacteristic
594C2#7388111yescharacteristic
595C2#7388611yescharacteristic
596C2#7405711yescharacteristic
597C2#7452012yes—
598C2#7456112yes—
599C2#7479612yes—
600C2#7485912yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C22⋊#95C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC22⋊#95D3
hypocenterC3#14422
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #69413C2semidirect
7. #69411C2semidirect
8. #69407C2semidirect
9. #69402C2semidirect
22. #5212C4semidirect
588. C3#14422semidirect

Maximal subgroups

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

subgroupindex
C2×#52122
C2×#49452
C3×C22⋊#952
#721272
#715492
#694132
#694112
#694072
#694022
#144223