Finite Group Databasebeta

← #10729194

Subgroups — #10729194

Order 768.

3030 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C22×#935C212yesmaximal, complemented
3C2×#5220C212yesmaximal, complemented
4C22×C3⋊#95C211yesmaximal, characteristic, complemented
5#72131C212yesmaximal, complemented
6#72127C211yesmaximal, characteristic, complemented
7#71568C212yesmaximal, complemented
8#71547C214yesmaximal, complemented
9#14926—31nomaximal, Sylow
10C23×C3×D4C2212yes—
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C24×Dic3C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C2×C3×#121—28no—
16C2×C3×#121—24no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×#217C2214yes—
20C22×#217C2214yes—
21C22×#217C2212yes—
22C2×#935—28no—
23C2×#935—28no—
24C2×#935—28no—
25C22×Dic3⋊C4C2214yes—
26#5220—28no—
27C2×C3⋊#95—24no—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2214yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C22×#123—32no—
34C22×#121D311yescharacteristic
35C2×#386—32no—
36C22×#95—31no—
37#1444—32no—
38#1439—34no—
39#1438—32no—
40C22⋊#95—31no—
41C25×C3C2312yes—
42C25×C3C2311yescharacteristic
43C22×C3×D4—24no—
44C22×C3×D4—24no—
45C22×C3×D4—24no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4—24no—
49C22×C3×D4—24no—
50C22×C3×D4—22no—
51C22×C3×D4—24no—
52C22×C3×D4—42no—
53C22×C3×D4D418yes—
54C23×C3×C4—24no—
55C23×C3×C4—24no—
56C23×C3×C4—42no—
57C23×C3×C4D414yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—22no—
62C23×Dic3—22no—
63C23×Dic3—24no—
64C23×Dic3—24no—
65C23×Dic3—22no—
66C23×Dic3—24no—
67C23×Dic3—24no—
68C23×Dic3—42no—
69C23×Dic3D414yescomplemented
70C23×Dic3C2312yes—
71C23×Dic3C2314yes—
72C23×Dic3C2312yes—
73C3×#121—232no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—24no—
79C2×#217—24no—
80C2×#217—28no—
81C2×#217—24no—
82C2×#217—28no—
83C2×#217—28no—
84C2×#217—28no—
85C2×#217—28no—
86C2×#217—44no—
87C2×#217D418yescomplemented
88#935—264no—
89C2×Dic3⋊C4—28no—
90C2×Dic3⋊C4—28no—
91C2×Dic3⋊C4—28no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—28no—
96C3⋊#95—42no—
97C23×D4C2×D312yes—
98C23×D4C2×D311yescharacteristic
99C24×C4—31no—
100C24×C4—31no—
101C24×C4C2×D311yescharacteristic
102C2×#123—68no—
103C2×#123—68no—
104C2×#123—68no—
105C2×#121—28no—
106C2×#121—24no—
107C22×C4⋊C4—34no—
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C22⋊C4—32no—
110C22×C22⋊C4—34no—
111C22×C22⋊C4—34no—
112C22×C22⋊C4C2×D312yes—
113#386—68no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—34no—
118C2×#95—32no—
119C2×#95—64no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—24no—
130C24×C3—22no—
131C24×C3—24no—
132C24×C3—24no—
133C24×C3—24no—
134C24×C3C2×D412yes—
135C24×C3C2×D414yes—
136C24×C3C2×D414yes—
137C24×C3C2×D414yes—
138C24×C3C2×D412yes—
139C24×C3#4112yes—
140C24×C3C2411yescharacteristic
141C2×C3×D4—28no—
142C2×C3×D4—216no—
143C2×C3×D4—216no—
144C2×C3×D4—216no—
145C2×C3×D4—416no—
146C2×C3×D4—44no—
147C2×C3×D4—48no—
148C2×C3×D4—48no—
149C2×C3×D4—48no—
150C2×C3×D4—48no—
151C2×C3×D4—44no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—28no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—28no—
165C22×C3×C4—44no—
166C22×C3×C4—416no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4#4114yes—
171C22×Dic3—28no—
172C22×Dic3—216no—
173C22×Dic3—28no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—416no—
189C22×Dic3—44no—
190C22×Dic3—44no—
191C22×Dic3—48no—
192C22×Dic3—44no—
193C22×Dic3—44no—
194C22×Dic3C2×D418yes—
195C22×Dic3C2×D414yes—
196C22×Dic3#4114yes—
197C3×C4⋊C4—28no—
198C3×C22⋊C4—216no—
199#217—232no—
200#217—216no—
201#217—416no—
202Dic3⋊C4—232no—
203C25C22×D312yes—
204C25C22×D311yescharacteristic
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—42no—
215C22×D4#6718yes—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—34no—
219C23×C4—34no—
220C23×C4—32no—
221C23×C4—32no—
222C23×C4—42no—
223C23×C4—64no—
224C23×C4—64no—
225C23×C4—62no—
226C23×C4—64no—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4—64no—
230C23×C4—122no—
231C23×C4#6714yes—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D312yes—
234C23×C4C22×D311yescharacteristic
235#123—664no—
236#121—232no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—68no—
240C2×C4⋊C4—68no—
241C2×C4⋊C4—68no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—38no—
246C2×C22⋊C4—68no—
247C2×C22⋊C4—68no—
248C2×C22⋊C4—64no—
249C2×C22⋊C4—68no—
250C2×C22⋊C4—64no—
251C2×C22⋊C4—68no—
252C2×C22⋊C4—68no—
253C2×C22⋊C4—124no—
254#95—68no—
255#95—64no—
256#95—64no—
257#95—64no—
258#95—122no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—28no—
268C23×C3—28no—
269C23×C3—28no—
270C23×C3—28no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—28no—
275C23×C3—28no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—44no—
282C23×C3—42no—
283C23×C3—44no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—44no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3—44no—
292C23×C3—44no—
293C23×C3—44no—
294C23×C3#12018yes—
295C23×C3#12014yes—
296C23×C3#12114yes—
297C23×C3#12118yes—
298C23×C3#12714yes—
299C23×C3C22×D411yescharacteristic
300C23×C3C22×D412yes—
301C23×C3C22×D412yes—
302C23×C3C22×D411yescharacteristic
303C23×C3C2×#4111yescharacteristic
304C23×C3C2×#4112yes—
305C23×C3C2×#4112yes—
306C23×C3H(5,𝔽2)12yes—
307C23×C32−1+412yes—
308C3×D4—416no—
309C3×D4—432no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—28no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—416no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4#12314yescomplemented
326C2×Dic3—28no—
327C2×Dic3—28no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—28no—
331C2×Dic3—28no—
332C2×Dic3—28no—
333C2×Dic3—416no—
334C2×Dic3—42no—
335C2×Dic3—44no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—44no—
339C2×Dic3—42no—
340C2×Dic3—44no—
341C2×Dic3—44no—
342C2×Dic3—44no—
343C2×Dic3—44no—
344C2×Dic3—416no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24D3×D412yes—
360C24D3×D414yes—
361C24#23712yes—
362C24C2×#6714yes—
363C24C2×#6714yes—
364C24C2×#6712yes—
365C24C23×D311yescharacteristic
366C2×D4—28no—
367C2×D4—216no—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—416no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—44no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—38no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—44no—
394C22×C4—416no—
395C22×C4—68no—
396C22×C4—616no—
397C22×C4—68no—
398C22×C4—64no—
399C22×C4—64no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—62no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—64no—
411C22×C4—64no—
412C22×C4—1216no—
413C22×C4—124no—
414C22×C4—128no—
415C22×C4—124no—
416C22×C4—124no—
417C22×C4—124no—
418C22×C4#23714yes—
419C22×C4C2×#6712yes—
420C22×C4C2×#6714yes—
421C22×C4C2×#6712yes—
422C4⋊C4—28no—
423C4⋊C4—632no—
424C22⋊C4—216no—
425C22⋊C4—632no—
426C22⋊C4—616no—
427C22⋊C4—1216no—
428C22×C3—24no—
429C22×C3—24no—
430C22×C3—24no—
431C22×C3—24no—
432C22×C3—216no—
433C22×C3—216no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—48no—
440C22×C3—48no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—84no—
444C22×C3—84no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3#39218yes—
448C22×C3C2×#12012yes—
449C22×C3C2×#12011yescharacteristic
450C22×C3C2×#12111yescharacteristic
451C22×C3C2×#12112yes—
452C22×C3C2×#12311yescharacteristic
453C22×C3#52511yescharacteristic
454C22×C3#52912yes—
455C22×C3C2×#12711yescharacteristic
456C22×C3#53212yes—
457C22×C3#53411yescharacteristic
458C22×C3#53611yescharacteristic
459C22×C3#53712yes—
460C22×C3#53912yes—
461C22×C3#54614yes—
462C22×C3#54614yes—
463C22×C3#54714yes—
464C22×C3#54714yes—
465C3×C4—24no—
466C3×C4—44no—
467C3×C4—42no—
468Dic3—44no—
469Dic3—48no—
470Dic3—44no—
471C23—22no—
472C23—22no—
473C23—28no—
474C23—28no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23#93718yes—
507C23#93814yes—
508C23#93918yes—
509C23#94014yes—
510C23#95314yes—
511C23C2×D3×D411yescharacteristic
512C23C2×D3×D412yes—
513C23C2×#23712yes—
514C23C2×#23711yescharacteristic
515C23#100412yes—
516C23D3×#4112yes—
517C23#101012yes—
518C23C22×#6711yescharacteristic
519C23C22×#6712yes—
520D4—416no—
521D4—432no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—416no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—68no—
538C2×C4—68no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—68no—
542C2×C4—68no—
543C2×C4—68no—
544C2×C4—1216no—
545C2×C4—1216no—
546C2×C4—124no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—124no—
551C2×C4—124no—
552C2×C4—122no—
553C2×C4—124no—
554C2×C4—124no—
555C2×C4—124no—
556C2×C4#93514yescomplemented
557C2×C3—28no—
558C2×C3—84no—
559C2×C3C2×#39211yescharacteristic
560C2×C3#240511yescharacteristic
561C2×C3#240611yescharacteristic
562C2×C3#242514yes—
563C2×C3#243014yes—
564C2×C3#250111yescharacteristic
565C2×C3#253911yescharacteristic
566C2×C3#261811yescharacteristic
567C2×C3#285411yescharacteristic
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—216no—
573C22—216no—
574C22—24no—
575C22—24no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—48no—
580C22—48no—
581C22—82no—
582C22—82no—
583C22—84no—
584C22—84no—
585C22—82no—
586C22—82no—
587C22#522318yes—
588C22#559612yes—
589C22#559712yes—
590C22#560012yes—
591C22#560212yes—
592C22#561014yes—
593C22#561114yes—
594C22C2×#93511yescharacteristic
595C22C2×#93712yes—
596C22C2×#93811yescharacteristic
597C22C2×#93912yes—
598C22C2×#94011yescharacteristic
599C22#580411yescharacteristic
600C22#580614yes—
601C22#582414yes—
602C22#582511yescharacteristic
603C22#582811yescharacteristic
604C22C2×#95311yescharacteristic
605C4—24no—
606C4—42no—
607C4—44no—
608C4—124no—
609C4—124no—
610C4—128no—
611C3#1492611yescharacteristic, complemented, Sylow
612C2—28no—
613C2—84no—
614C2#8264514yes—
615C2#8265114yes—
616C2#8266411yescharacteristic
617C2#8267511yescharacteristic
618C2#8269911yescharacteristic
619C2#8270811yescharacteristic
620C2#8404711yescharacteristic
621C2C2×#522311yescharacteristic
622C2#8409011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
Fitting, prime radicalC22×C3×#121C2
hypercenterC22×#121D3
hypocenterC3#14926
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
2. C22×#935C2semidirect
3. C2×#5220C2semidirect
4. C22×C3⋊#95C2semidirect
5. #72131C2semidirect
6. #72127C2semidirect
7. #71568C2semidirect
8. #71547C2semidirect
69. C23×Dic3D4semidirect
87. C2×#217D4semidirect
325. C2×C3×C4#123semidirect
556. C2×C4#935semidirect
611. C3#14926semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C22×#9352
C2×#52202
C22×C3⋊#952
#721312
#721272
#715682
#715472
#149263