Finite Group Databasebeta

← #10729037

Subgroups — #10729037

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×C3×#121C211yesmaximal, characteristic
2C2×#5220C212yesmaximal, complemented
3C2×#5220C212yesmaximal, complemented
4C22×C3⋊#95C211yesmaximal, characteristic, complemented
5#72131C212yesmaximal, complemented
6#72127C211yesmaximal, characteristic, complemented
7#71567C212yesmaximal, complemented
8#71549C214yesmaximal, complemented
9#14913—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×#217C2212yes—
20C22×#217C2214yes—
21C22×#217C2214yes—
22#5220—28no—
23#5220—28no—
24C2×C3⋊#95—24no—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C22×#121D311yescharacteristic
32C2×#386—32no—
33C2×#386—32no—
34C22×#95—31no—
35#1442—32no—
36#1440—34no—
37#1438—32no—
38C22⋊#95—31no—
39C25×C3C2312yes—
40C25×C3C2311yescharacteristic
41C22×C3×D4—24no—
42C22×C3×D4—24no—
43C22×C3×D4—24no—
44C22×C3×D4—24no—
45C22×C3×D4—24no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4—22no—
49C22×C3×D4—24no—
50C22×C3×D4—42no—
51C22×C3×D4D418yes—
52C23×C3×C4—24no—
53C23×C3×C4—24no—
54C23×C3×C4—42no—
55C23×C3×C4D414yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—22no—
60C23×Dic3—22no—
61C23×Dic3—22no—
62C23×Dic3—24no—
63C23×Dic3—24no—
64C23×Dic3—42no—
65C23×Dic3—42no—
66C23×Dic3—42no—
67C23×Dic3D414yescomplemented
68C23×Dic3C2314yes—
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C3×#121—232no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×#217—24no—
78C2×#217—28no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—28no—
82C2×#217—28no—
83C2×#217—44no—
84C2×#217—44no—
85C2×#217—44no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—28no—
89C3⋊#95—24no—
90C3⋊#95—28no—
91C3⋊#95—42no—
92C23×D4C2×D312yes—
93C23×D4C2×D311yescharacteristic
94C24×C4—31no—
95C24×C4—31no—
96C24×C4C2×D311yescharacteristic
97C2×#121—28no—
98C2×#121—24no—
99C22×C4⋊C4C2×D311yescharacteristic
100C22×C22⋊C4—32no—
101C22×C22⋊C4—34no—
102C22×C22⋊C4—34no—
103C22×C22⋊C4C2×D312yes—
104#386—68no—
105#386—68no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—34no—
110C2×#95—34no—
111C2×#95—32no—
112C2×#95—64no—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—24no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—24no—
125C24×C3—24no—
126C24×C3—24no—
127C24×C3C2×D412yes—
128C24×C3C2×D414yes—
129C24×C3C2×D414yes—
130C24×C3C2×D412yes—
131C24×C3#4112yes—
132C24×C3#4114yes—
133C24×C3C2411yescharacteristic
134C2×C3×D4—28no—
135C2×C3×D4—216no—
136C2×C3×D4—216no—
137C2×C3×D4—216no—
138C2×C3×D4—416no—
139C2×C3×D4—44no—
140C2×C3×D4—48no—
141C2×C3×D4—48no—
142C2×C3×D4—48no—
143C2×C3×D4—48no—
144C2×C3×D4—44no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—28no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—28no—
158C22×C3×C4—44no—
159C22×C3×C4—416no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D414yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4#4114yes—
164C22×Dic3—28no—
165C22×Dic3—28no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—24no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—24no—
172C22×Dic3—24no—
173C22×Dic3—24no—
174C22×Dic3—24no—
175C22×Dic3—24no—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—48no—
182C22×Dic3—416no—
183C22×Dic3—44no—
184C22×Dic3—44no—
185C22×Dic3—44no—
186C22×Dic3—48no—
187C22×Dic3—44no—
188C22×Dic3C2×D414yes—
189C22×Dic3#4114yes—
190C3×C4⋊C4—28no—
191C3×C22⋊C4—216no—
192#217—416no—
193#217—416no—
194#217—48no—
195C25C22×D312yes—
196C25C22×D311yescharacteristic
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—42no—
207C22×D4#6718yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—32no—
211C23×C4—32no—
212C23×C4—34no—
213C23×C4—34no—
214C23×C4—42no—
215C23×C4—64no—
216C23×C4—62no—
217C23×C4—62no—
218C23×C4—62no—
219C23×C4—64no—
220C23×C4—122no—
221C23×C4—122no—
222C23×C4—122no—
223C23×C4#6714yes—
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D312yes—
226C23×C4C22×D311yescharacteristic
227#121—232no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—68no—
234C2×C22⋊C4—68no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—68no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—124no—
240C2×C22⋊C4—124no—
241C2×C22⋊C4—124no—
242#95—68no—
243#95—64no—
244#95—64no—
245#95—64no—
246#95—68no—
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—24no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—28no—
264C23×C3—28no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—44no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—44no—
279C23×C3—44no—
280C23×C3—44no—
281C23×C3—44no—
282C23×C3—44no—
283C23×C3#12014yes—
284C23×C3#12114yes—
285C23×C3#12118yes—
286C23×C3#12318yes—
287C23×C3#12414yes—
288C23×C3C22×D411yescharacteristic
289C23×C3C22×D411yescharacteristic
290C23×C3C22×D412yes—
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4112yes—
293C23×C3C2×#4112yes—
294C23×C3C2×#4112yes—
295C23×C3H(5,𝔽2)12yes—
296C23×C3H(5,𝔽2)12yes—
297C3×D4—416no—
298C3×D4—432no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—28no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—416no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—44no—
314C2×C3×C4#12314yes—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—28no—
318C2×Dic3—28no—
319C2×Dic3—416no—
320C2×Dic3—416no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—44no—
326C2×Dic3—44no—
327C2×Dic3—44no—
328C2×Dic3—44no—
329C2×Dic3—44no—
330C2×Dic3—44no—
331C2×Dic3—44no—
332C2×Dic3—44no—
333C2×Dic3—44no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24D3×D412yes—
349C24#23712yes—
350C24#23714yes—
351C24C2×#6714yes—
352C24C2×#6714yes—
353C24C2×#6712yes—
354C24C23×D311yescharacteristic
355C2×D4—28no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—216no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—44no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—34no—
380C22×C4—34no—
381C22×C4—44no—
382C22×C4—416no—
383C22×C4—68no—
384C22×C4—68no—
385C22×C4—62no—
386C22×C4—64no—
387C22×C4—62no—
388C22×C4—64no—
389C22×C4—64no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—64no—
393C22×C4—64no—
394C22×C4—64no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—64no—
398C22×C4—64no—
399C22×C4—64no—
400C22×C4—128no—
401C22×C4—1216no—
402C22×C4—124no—
403C22×C4—124no—
404C22×C4—124no—
405C22×C4—128no—
406C22×C4—124no—
407C22×C4#23714yes—
408C22×C4C2×#6712yes—
409C22×C4C2×#6714yes—
410C22×C4C2×#6712yes—
411C4⋊C4—28no—
412C22⋊C4—216no—
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—44no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—48no—
428C22×C3—48no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—84no—
433C22×C3—82no—
434C22×C3—84no—
435C22×C3#39418yes—
436C22×C3C2×#12011yescharacteristic
437C22×C3C2×#12111yescharacteristic
438C22×C3C2×#12112yes—
439C22×C3C2×#12311yescharacteristic
440C22×C3C2×#12312yes—
441C22×C3#52511yescharacteristic
442C22×C3C2×#12411yescharacteristic
443C22×C3#52912yes—
444C22×C3#53212yes—
445C22×C3#53411yescharacteristic
446C22×C3#53412yes—
447C22×C3#53511yescharacteristic
448C22×C3#53812yes—
449C22×C3#54614yes—
450C22×C3#54614yes—
451C22×C3#55314yes—
452C22×C3#55314yes—
453C3×C4—24no—
454C3×C4—44no—
455C3×C4—42no—
456Dic3—44no—
457Dic3—44no—
458Dic3—84no—
459C23—22no—
460C23—22no—
461C23—28no—
462C23—22no—
463C23—28no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—24no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23#93518yes—
495C23#93614yes—
496C23#93814yes—
497C23#93918yes—
498C23#95314yes—
499C23C2×D3×D411yescharacteristic
500C23C2×#23711yescharacteristic
501C23C2×#23712yes—
502C23C2×#23712yes—
503C23#100412yes—
504C23D3×#4112yes—
505C23#100912yes—
506C23C22×#6711yescharacteristic
507C23C22×#6712yes—
508D4—416no—
509D4—432no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—416no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—68no—
528C2×C4—68no—
529C2×C4—1216no—
530C2×C4—1216no—
531C2×C4—122no—
532C2×C4—124no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—124no—
537C2×C4—124no—
538C2×C4—124no—
539C2×C4—124no—
540C2×C4—124no—
541C2×C4—124no—
542C2×C4—124no—
543C2×C4—124no—
544C2×C4#93514yes—
545C2×C3—28no—
546C2×C3—84no—
547C2×C3C2×#39411yescharacteristic
548C2×C3#240411yescharacteristic
549C2×C3#240511yescharacteristic
550C2×C3#244414yes—
551C2×C3#250411yescharacteristic
552C2×C3#254314yes—
553C2×C3#259111yescharacteristic
554C2×C3#261711yescharacteristic
555C2×C3#285411yescharacteristic
556C22—24no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—216no—
561C22—24no—
562C22—216no—
563C22—24no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—48no—
568C22—48no—
569C22—82no—
570C22—82no—
571C22—84no—
572C22—82no—
573C22—84no—
574C22—82no—
575C22#522218yes—
576C22#559612yes—
577C22#560012yes—
578C22#560114yes—
579C22#560512yes—
580C22#561712yes—
581C22#561814yes—
582C22C2×#93511yescharacteristic
583C22C2×#93512yes—
584C22C2×#93611yescharacteristic
585C22C2×#93811yescharacteristic
586C22C2×#93912yes—
587C22#580411yescharacteristic
588C22#580614yes—
589C22#582811yescharacteristic
590C22#582914yes—
591C22#583011yescharacteristic
592C22C2×#95311yescharacteristic
593C4—24no—
594C4—44no—
595C4—42no—
596C4—124no—
597C4—124no—
598C4—244no—
599C3#1491311yescharacteristic, complemented, Sylow
600C2—28no—
601C2—84no—
602C2#8265614yes—
603C2#8266511yescharacteristic
604C2#8267014yes—
605C2#8267411yescharacteristic
606C2#8269911yescharacteristic
607C2#8271011yescharacteristic
608C2C2×#522211yescharacteristic
609C2#8404711yescharacteristic
610C2#8409611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
Fitting, prime radicalC22×C3×#121C2
hypercenterC22×#121D3
hypocenterC3#14913
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
2. C2×#5220C2semidirect
3. C2×#5220C2semidirect
4. C22×C3⋊#95C2semidirect
5. #72131C2semidirect
6. #72127C2semidirect
7. #71567C2semidirect
8. #71549C2semidirect
67. C23×Dic3D4semidirect
599. C3#14913semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C2×#52202
C22×C3⋊#952
#721312
#721272
#715672
#715492
#149133