Finite Group Databasebeta

← #10729041

Subgroups — #10729041

Order 768.

2638 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#930C211yesmaximal, characteristic
2C22×C3×#95C211yesmaximal, characteristic, complemented
3C2×#5213C212yesmaximal, complemented
4C2×#5213C212yesmaximal, complemented
5#72127C211yesmaximal, characteristic, complemented
6#69429C214yesmaximal, complemented
7#69426C212yesmaximal, complemented
8#69415C212yesmaximal, complemented
9#14913—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C23×#67C2212yes—
14C23×D12C2211yescharacteristic
15C22×#217C2212yes—
16C2×#930—24no—
17C2×#930—28no—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2214yes—
20C22×(C2×D3)⋊C4C2214yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C2×C3×#95—24no—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25#5213—28no—
26#5213—28no—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2212yes—
31C22×#121—31no—
32C2×#386—32no—
33C2×#386—32no—
34C22×#95D311yescharacteristic, complemented
35#1442—32no—
36#1440—34no—
37#1438—32no—
38C22⋊#95—31no—
39C25×C3C2311yescharacteristic
40C24×D3C2312yes—
41C23×C3×C4—22no—
42C23×C3×C4—22no—
43C23×C3×C4—24no—
44C23×C3×C4—22no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—42no—
49C23×C3×C4—42no—
50C23×C3×C4—42no—
51C23×C3×C4—42no—
52C23×C3×C4D414yes—
53C23×C3×C4D414yescomplemented
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C22×#67—24no—
59C22×#67—24no—
60C22×#67—24no—
61C22×#67—24no—
62C22×#67—24no—
63C22×#67—24no—
64C22×#67D418yes—
65C23×Dic3C2314yes—
66C23×Dic3C2312yes—
67C22×D12—24no—
68C22×D12—22no—
69C22×D12—24no—
70C22×D12—42no—
71C2×#217—24no—
72C2×#217—24no—
73C2×#217—24no—
74#930—232no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—28no—
79C2×(C2×D3)⋊C4—28no—
80C2×(C2×D3)⋊C4—28no—
81C2×(C2×D3)⋊C4—44no—
82C2×(C2×D3)⋊C4—44no—
83C2×(C2×D3)⋊C4—44no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—24no—
86C3×#95—24no—
87C3×#95—24no—
88C3⋊#95—28no—
89C3⋊#95—24no—
90C3⋊#95—28no—
91C3⋊#95—42no—
92C23×D4—32no—
93C23×D4—31no—
94C24×C4C2×D311yescharacteristic
95C24×C4C2×D311yescharacteristic
96C24×C4C2×D311yescharacteristic
97C2×#121—68no—
98C2×#121—64no—
99C22×C4⋊C4—31no—
100C22×C22⋊C4—32no—
101C22×C22⋊C4—32no—
102C22×C22⋊C4—34no—
103C22×C22⋊C4—34no—
104#386—68no—
105#386—68no—
106C2×#95—24no—
107C2×#95—32no—
108C2×#95—34no—
109C2×#95—34no—
110C2×#95—32no—
111C2×#95C2×D312yes—
112C2×#95C2×D312yes—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3C2×D412yes—
118C24×C3C2×D414yes—
119C24×C3C2×D412yes—
120C24×C3#4112yes—
121C24×C3#4114yes—
122C24×C3C2411yescharacteristic
123C23×D3—22no—
124C23×D3—22no—
125C23×D3—22no—
126C23×D3—24no—
127C23×D3—24no—
128C23×D3—22no—
129C23×D3—22no—
130C23×D3—22no—
131C23×D3—24no—
132C23×D3—24no—
133C23×D3C2×D414yes—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—28no—
137C22×C3×C4—28no—
138C22×C3×C4—28no—
139C22×C3×C4—24no—
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×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—28no—
151C22×C3×C4—44no—
152C22×C3×C4—48no—
153C22×C3×C4—416no—
154C22×C3×C4—416no—
155C22×C3×C4—44no—
156C22×C3×C4—44no—
157C22×C3×C4—44no—
158C22×C3×C4—48no—
159C22×C3×C4—44no—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4#4114yes—
164C22×C3×C4#4114yes—
165C2×#67—216no—
166C2×#67—216no—
167C2×#67—216no—
168C2×#67—48no—
169C2×#67—48no—
170C2×#67—48no—
171C2×#67—48no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3C2×D414yes—
186C2×D12—28no—
187C2×D12—416no—
188C2×D12—44no—
189C2×D12—44no—
190#217—48no—
191(C2×D3)⋊C4—216no—
192(C2×D3)⋊C4—416no—
193(C2×D3)⋊C4—416no—
194C4⋊Dic3—28no—
195C25—32no—
196C25C22×D311yescharacteristic
197C22×D4—38no—
198C22×D4—64no—
199C22×D4—64no—
200C22×D4—64no—
201C22×D4—64no—
202C22×D4—64no—
203C22×D4—64no—
204C22×D4—64no—
205C22×D4—64no—
206C22×D4—62no—
207C22×D4—122no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—32no—
216C23×C4—34no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4D1214yescomplemented
222C23×C4#6714yes—
223C23×C4C22×D312yes—
224C23×C4C22×D312yes—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227#121—632no—
228C2×C4⋊C4—64no—
229C2×C4⋊C4—62no—
230C2×C22⋊C4—64no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—64no—
233C2×C22⋊C4—68no—
234C2×C22⋊C4—64no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—68no—
238C2×C22⋊C4—68no—
239C2×C22⋊C4—124no—
240C2×C22⋊C4—124no—
241C2×C22⋊C4—124no—
242#95—24no—
243#95—24no—
244#95—68no—
245#95—68no—
246#95—64no—
247#95—122no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—24no—
256C23×C3—28no—
257C23×C3—28no—
258C23×C3—28no—
259C23×C3—28no—
260C23×C3#12014yes—
261C23×C3#12118yes—
262C23×C3#12114yes—
263C23×C3#12318yes—
264C23×C3#12414yes—
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D412yes—
267C23×C3C22×D411yescharacteristic
268C23×C3C2×#4111yescharacteristic
269C23×C3C2×#4112yes—
270C23×C3C2×#4112yes—
271C23×C3C2×#4112yes—
272C23×C3H(5,𝔽2)12yes—
273C23×C3H(5,𝔽2)12yes—
274C22×D3—24no—
275C22×D3—24no—
276C22×D3—24no—
277C22×D3—28no—
278C22×D3—28no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—44no—
286C22×D3—44no—
287C22×D3—44no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—44no—
291C22×D3—44no—
292C22×D3—44no—
293C22×D3—44no—
294C22×D3—44no—
295C22×D3—44no—
296C22×D3—44no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—28no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—28no—
305C2×C3×C4—22no—
306C2×C3×C4—28no—
307C2×C3×C4—416no—
308C2×C3×C4—416no—
309C2×C3×C4—416no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—44no—
315C2×C3×C4—44no—
316C2×C3×C4—44no—
317C2×C3×C4#12314yes—
318#67—432no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—44no—
327C2×Dic3—44no—
328C2×Dic3—44no—
329C2×Dic3—44no—
330C2×Dic3—44no—
331C2×Dic3—44no—
332C2×Dic3—44no—
333D12—416no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—34no—
339C24—62no—
340C24—64no—
341C24—64no—
342C24—62no—
343C24—62no—
344C24—64no—
345C24—64no—
346C24—62no—
347C24—62no—
348C24—62no—
349C24C2×D1212yes—
350C24#23512yes—
351C24D3×D414yes—
352C24#23914yes—
353C24C2×#6712yes—
354C24C23×D311yescharacteristic
355C2×D4—68no—
356C2×D4—616no—
357C2×D4—616no—
358C2×D4—616no—
359C2×D4—1216no—
360C2×D4—124no—
361C2×D4—128no—
362C2×D4—128no—
363C2×D4—128no—
364C2×D4—128no—
365C2×D4—124no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—34no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4—416no—
387C22×C4—416no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—64no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4C2×D1214yes—
407C22×C4#23514yes—
408C22×C4#23514yes—
409C22×C4C2×#6712yes—
410C22×C4C2×#6712yes—
411C4⋊C4—68no—
412C22⋊C4—616no—
413C22⋊C4—1216no—
414C22⋊C4—1216no—
415C22⋊C4—128no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—24no—
421C22×C3—216no—
422C22×C3—24no—
423C22×C3—216no—
424C22×C3#39418yes—
425C22×C3C2×#12011yescharacteristic
426C22×C3C2×#12112yes—
427C22×C3C2×#12111yescharacteristic
428C22×C3C2×#12312yes—
429C22×C3C2×#12311yescharacteristic
430C22×C3#52511yescharacteristic
431C22×C3C2×#12411yescharacteristic
432C22×C3#52912yes—
433C22×C3#53212yes—
434C22×C3#53412yes—
435C22×C3#53411yescharacteristic
436C22×C3#53511yescharacteristic
437C22×C3#53812yes—
438C22×C3#54614yes—
439C22×C3#54614yes—
440C22×C3#55314yes—
441C22×C3#55314yes—
442C2×D3—44no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—48no—
446C2×D3—48no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3—84no—
451C2×D3—82no—
452C2×D3—84no—
453C3×C4—24no—
454C3×C4—44no—
455C3×C4—44no—
456C3×C4—42no—
457Dic3—44no—
458Dic3—84no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—68no—
472C23—68no—
473C23—64no—
474C23—64no—
475C23—64no—
476C23—124no—
477C23—124no—
478C23—124no—
479C23—124no—
480C23—124no—
481C23—124no—
482C23—122no—
483C23—122no—
484C23—124no—
485C23—124no—
486C23—124no—
487C23—124no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23#89418yes—
495C23#89518yes—
496C23#93014yes—
497C23#93714yes—
498C23#94714yes—
499C23C22×D1211yescharacteristic
500C23C2×#23512yes—
501C23C2×#23512yes—
502C23C2×#23511yescharacteristic
503C23C2×D3×D412yes—
504C23#100412yes—
505C23#100412yes—
506C23C2×#23912yes—
507C23C22×#6711yescharacteristic
508D4—1216no—
509D4—1232no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—416no—
521C2×C4—416no—
522C2×C4—416no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—124no—
538C2×C4—124no—
539C2×C4—124no—
540C2×C4—124no—
541C2×C4—124no—
542C2×C4—124no—
543C2×C4—124no—
544C2×C4#92914yes—
545C2×C3—28no—
546C2×C3C2×#39411yescharacteristic
547C2×C3#240411yescharacteristic
548C2×C3#240511yescharacteristic
549C2×C3#244414yes—
550C2×C3#250411yescharacteristic
551C2×C3#254314yes—
552C2×C3#259111yescharacteristic
553C2×C3#261711yescharacteristic
554C2×C3#285411yescharacteristic
555D3—84no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—216no—
560C22—216no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—124no—
565C22—128no—
566C22—128no—
567C22—124no—
568C22—124no—
569C22—242no—
570C22—242no—
571C22—242no—
572C22—244no—
573C22—244no—
574C22—242no—
575C22#499118yes—
576C22#547712yes—
577C22#547912yes—
578C22#549014yes—
579C22#549014yes—
580C22#549312yes—
581C22#549412yes—
582C22C2×#89412yes—
583C22C2×#89512yes—
584C22#551414yes—
585C22#551414yes—
586C22C2×#92911yescharacteristic
587C22C2×#93011yescharacteristic
588C22#579111yescharacteristic
589C22C2×#93711yescharacteristic
590C22#580411yescharacteristic
591C22#580511yescharacteristic
592C22C2×#94711yescharacteristic
593C4—24no—
594C4—44no—
595C4—42no—
596C4—44no—
597C4—124no—
598C4—244no—
599C3#1491311yescharacteristic, complemented, Sylow
600C2—28no—
601C2—244no—
602C2#7337714yes—
603C2#7337814yes—
604C2#8208011yescharacteristic
605C2#8212511yescharacteristic
606C2#8212611yescharacteristic
607C2#8213011yescharacteristic
608C2#8218011yescharacteristic
609C2#8218211yescharacteristic
610C2C2×#499111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#95C2
prime radicalC22×#930C2
hypercenterC22×#95D3
hypocenterC3#14913
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
2. C22×C3×#95C2semidirect
3. C2×#5213C2semidirect
4. C2×#5213C2semidirect
5. #72127C2semidirect
6. #69429C2semidirect
7. #69426C2semidirect
8. #69415C2semidirect
34. C22×#95D3semidirect
53. C23×C3×C4D4semidirect
221. C23×C4D12semidirect
599. C3#14913semidirect

Maximal subgroups

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

subgroupindex
C22×#9302
C22×C3×#952
C2×#52132
#721272
#694292
#694262
#694152
#149133