Finite Group Databasebeta

← #10732230

Subgroups — #10732230

Order 768.

2314 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D12C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic, complemented
3C2×#4987C212yesmaximal, complemented
4C2×#4987C212yesmaximal, complemented
5#71570C214yesmaximal, complemented
6#71535C211yesmaximal, characteristic, complemented
7#69426C212yesmaximal
8#69415C212yesmaximal, complemented
9#15181—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C23×C4×D3C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2214yes—
18C22×(C2×D3)⋊C4C2214yes—
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C2×C4×D12—28no—
22C2×C4×D12—24no—
23C3×#389—24no—
24C2×C3×#95C2212yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29#4987—28no—
30#4987—28no—
31C22×C4×D4—31no—
32C2×#389D311yescharacteristic, complemented
33C2×#380—32no—
34C2×#380—32no—
35#1442—32no—
36#1439—34no—
37#1438—32no—
38C4⋊#95—31no—
39C24×D3C2312yes—
40C23×C3×C4C2312yes—
41C23×C3×C4C2312yes—
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×Dic3C2314yes—
46C23×Dic3C2312yes—
47C22×D12—24no—
48C22×D12—24no—
49C22×D12—22no—
50C22×D12—22no—
51C22×D12Q814yes—
52C22×C4×D3—24no—
53C22×C4×D3—24no—
54C22×C4×D3—24no—
55C22×C4×D3—24no—
56C22×C4×D3—24no—
57C22×C4×D3—24no—
58C22×C4×D3D418yes—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—44no—
65C2×C3×C4⋊C4D414yescomplemented
66C2×C3×C42—24no—
67C2×C3×C42D414yes—
68C2×(C2×D3)⋊C4—24no—
69C2×(C2×D3)⋊C4—24no—
70C2×(C2×D3)⋊C4—24no—
71C2×(C2×D3)⋊C4—28no—
72C2×(C2×D3)⋊C4—28no—
73C2×(C2×D3)⋊C4—28no—
74C2×(C2×D3)⋊C4—44no—
75C2×(C2×D3)⋊C4—44no—
76C2×(C2×D3)⋊C4—44no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3—24no—
81C2×C4⋊Dic3—24no—
82C4×D12—232no—
83C3×#95—24no—
84C3⋊#95—28no—
85C3⋊#95—24no—
86C3⋊#95—28no—
87C3⋊#95—42no—
88C23×D4—31no—
89C24×C4—32no—
90C2×C4×D4—68no—
91C2×C4×D4—64no—
92C22×C4⋊C4—31no—
93C22×C4⋊C4—32no—
94C22×C4⋊C4C2×D311yescharacteristic
95C22×C4⋊C4C2×D311yescharacteristic
96C22×C4⋊C4C2×D312yes—
97C22×C22⋊C4—32no—
98C22×C22⋊C4—34no—
99C22×C22⋊C4—34no—
100C22×C42C2×D311yescharacteristic
101#389—24no—
102#380—68no—
103#380—68no—
104C2×#95—32no—
105C2×#95—34no—
106C2×#95—34no—
107C2×#95—32no—
108C2×#95C2×D312yes—
109C24×C3C2411yescharacteristic
110C23×D3—22no—
111C23×D3—22no—
112C23×D3—22no—
113C23×D3—22no—
114C23×D3—22no—
115C23×D3—22no—
116C23×D3—22no—
117C23×D3—22no—
118C23×D3—22no—
119C23×D3—22no—
120C23×D3—22no—
121C23×D3—22no—
122C23×D3C2×D414yes—
123C23×D3C2×Q814yes—
124C23×D3#4114yes—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—24no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—24no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D414yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D414yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×Q812yes—
147C22×C3×C4#4112yes—
148C22×C3×C4#4112yes—
149C22×C3×C4#4114yes—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—24no—
160C22×Dic3—24no—
161C22×Dic3—24no—
162C22×Dic3—24no—
163C22×Dic3C2×D414yes—
164C2×D12—28no—
165C2×D12—28no—
166C2×D12—216no—
167C2×D12—216no—
168C2×D12—28no—
169C2×C4×D3—216no—
170C2×C4×D3—216no—
171C2×C4×D3—216no—
172C2×C4×D3—48no—
173C2×C4×D3—48no—
174C2×C4×D3—48no—
175C2×C4×D3—48no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—48no—
178C3×C4⋊C4—44no—
179C3×C42—28no—
180(C2×D3)⋊C4—216no—
181(C2×D3)⋊C4—416no—
182(C2×D3)⋊C4—416no—
183C4⋊Dic3—28no—
184C4⋊Dic3—48no—
185C25—32no—
186C22×D4—34no—
187C22×D4—64no—
188C22×D4—64no—
189C22×D4—62no—
190C22×D4—62no—
191C23×C4—32no—
192C23×C4—38no—
193C23×C4—34no—
194C23×C4—64no—
195C23×C4—64no—
196C23×C4—64no—
197C23×C4—64no—
198C23×C4—64no—
199C23×C4—64no—
200C23×C4C22×D312yes—
201C23×C4C22×D312yes—
202C23×C4C22×D311yescharacteristic
203C23×C4C22×D311yescharacteristic
204C23×C4C22×D311yescharacteristic
205C4×D4—632no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4—64no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4D1214yescomplemented
218C2×C22⋊C4—64no—
219C2×C22⋊C4—64no—
220C2×C22⋊C4—64no—
221C2×C22⋊C4—68no—
222C2×C22⋊C4—68no—
223C2×C22⋊C4—68no—
224C2×C22⋊C4—124no—
225C2×C22⋊C4—124no—
226C2×C22⋊C4—124no—
227C2×C42—24no—
228C2×C42#6714yes—
229#95—24no—
230#95—68no—
231#95—68no—
232#95—64no—
233#95—122no—
234C23×C3C22×D411yescharacteristic
235C23×C3C22×D412yes—
236C23×C3C22×D411yescharacteristic
237C23×C3C22×Q811yescharacteristic
238C23×C3C2×#4111yescharacteristic
239C23×C3C2×#4112yes—
240C23×C3C2×#4111yescharacteristic
241C23×C3C2×#4112yes—
242C23×C3H(5,𝔽2)12yes—
243C23×C3H(5,𝔽2)11yescharacteristic
244C23×C32−1+411yescharacteristic
245C22×D3—24no—
246C22×D3—24no—
247C22×D3—24no—
248C22×D3—24no—
249C22×D3—24no—
250C22×D3—24no—
251C22×D3—24no—
252C22×D3—24no—
253C22×D3—24no—
254C22×D3—24no—
255C22×D3—24no—
256C22×D3—24no—
257C22×D3—24no—
258C22×D3—24no—
259C22×D3—24no—
260C22×D3—24no—
261C22×D3—24no—
262C22×D3—24no—
263C22×D3—42no—
264C22×D3—42no—
265C22×D3—42no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3C22⋊Q818yes—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—28no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—44no—
294C2×C3×C4—42no—
295C2×C3×C4—44no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—44no—
299C2×C3×C4#12014yes—
300C2×C3×C4#12118yes—
301C2×C3×C4#12114yes—
302C2×C3×C4C22⋊Q814yes—
303C2×C3×C4#12318yes—
304C2×C3×C4#12414yes—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—44no—
313C2×Dic3—44no—
314C2×Dic3—44no—
315C2×Dic3—44no—
316C2×Dic3—44no—
317C2×Dic3—44no—
318C2×Dic3—44no—
319D12—232no—
320C4×D3—432no—
321C24—34no—
322C24—34no—
323C24—34no—
324C24—62no—
325C24—62no—
326C24—62no—
327C24—62no—
328C24—62no—
329C24—62no—
330C24—62no—
331C24—62no—
332C24—62no—
333C24—62no—
334C24—62no—
335C24—62no—
336C24C23×D311yescharacteristic
337C2×D4—616no—
338C2×D4—68no—
339C2×D4—68no—
340C2×D4—616no—
341C2×D4—68no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—34no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—616no—
365C22×C4—616no—
366C22×C4—616no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4—64no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—128no—
375C22×C4—128no—
376C22×C4—128no—
377C22×C4—128no—
378C22×C4C2×D1214yes—
379C22×C4C2×D1212yes—
380C22×C4#23512yes—
381C22×C4D3×D414yes—
382C22×C4D3×Q812yes—
383C22×C4#23912yes—
384C22×C4#23914yes—
385C22×C4C2×#6712yes—
386C22×C4C2×#6712yes—
387C22×C4C2×#6712yes—
388C4⋊C4—28no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—68no—
392C4⋊C4—128no—
393C22⋊C4—616no—
394C22⋊C4—1216no—
395C22⋊C4—1216no—
396C42—28no—
397C22×C3C2×#12011yescharacteristic
398C22×C3C2×#12112yes—
399C22×C3C2×#12111yescharacteristic
400C22×C3C2×C22⋊Q811yescharacteristic
401C22×C3C2×C22⋊Q812yes—
402C22×C3C2×#12312yes—
403C22×C3#52511yescharacteristic
404C22×C3#52512yes—
405C22×C3C2×#12411yescharacteristic
406C22×C3#53511yescharacteristic
407C22×C3#53512yes—
408C22×C3#53711yescharacteristic
409C22×C3#53712yes—
410C22×C3#54612yes—
411C22×C3D4×Q811yescharacteristic
412C22×C3#55011yescharacteristic
413C22×C3#55212yes—
414C22×C3#55312yes—
415C22×C3#55312yes—
416C22×C3#55412yes—
417C22×C3#55411yescharacteristic
418C22×C3#55512yes—
419C22×C3#55611yescharacteristic
420C2×D3—28no—
421C2×D3—28no—
422C2×D3—28no—
423C2×D3—44no—
424C2×D3—44no—
425C2×D3—44no—
426C2×D3—44no—
427C2×D3—44no—
428C2×D3—44no—
429C2×D3—44no—
430C2×D3—44no—
431C2×D3—44no—
432C2×D3—44no—
433C2×D3—44no—
434C2×D3—44no—
435C3×C4—24no—
436C3×C4—42no—
437C3×C4—44no—
438C3×C4—82no—
439C3×C4#39418yes—
440Dic3—44no—
441Dic3—84no—
442C23—38no—
443C23—64no—
444C23—64no—
445C23—64no—
446C23—64no—
447C23—64no—
448C23—64no—
449C23—64no—
450C23—64no—
451C23—64no—
452C23—64no—
453C23—64no—
454C23—64no—
455C23—64no—
456C23—64no—
457C23—64no—
458C23—64no—
459C23—64no—
460C23—64no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23—122no—
471C23—122no—
472C23—122no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23—122no—
477C23C22×D1211yescharacteristic
478C23C2×#23511yescharacteristic
479C23C2×D3×D412yes—
480C23#100412yes—
481C23C2×D3×Q811yescharacteristic
482C23C2×#23912yes—
483C23C2×#23911yescharacteristic
484C23D3×#4112yes—
485C23#100911yescharacteristic
486C23#101011yescharacteristic
487C23C22×#6711yescharacteristic
488D4—632no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—1232no—
516C2×C4—124no—
517C2×C4—124no—
518C2×C4—124no—
519C2×C4—124no—
520C2×C4—124no—
521C2×C4—124no—
522C2×C4—124no—
523C2×C4#89418yes—
524C2×C4#89518yes—
525C2×C4#93014yes—
526C2×C4#93714yes—
527C2×C4#94614yes—
528C2×C4#94714yes—
529C2×C3C2×#39411yescharacteristic
530C2×C3#241412yes—
531C2×C3#241712yes—
532C2×C3#248512yes—
533C2×C3#248712yes—
534C2×C3#253511yescharacteristic
535C2×C3#253811yescharacteristic
536C2×C3#256211yescharacteristic
537C2×C3#256311yescharacteristic
538C2×C3#261511yescharacteristic
539C2×C3#284811yescharacteristic
540D3—48no—
541C22—68no—
542C22—68no—
543C22—68no—
544C22—124no—
545C22—124no—
546C22—124no—
547C22—124no—
548C22—124no—
549C22—124no—
550C22—124no—
551C22—124no—
552C22—124no—
553C22—124no—
554C22—124no—
555C22—124no—
556C22C2×#89412yes—
557C22C2×#89512yes—
558C22#555412yes—
559C22#555812yes—
560C22Q8×D1211yescharacteristic
561C22#557711yescharacteristic
562C22#557911yescharacteristic
563C22#558411yescharacteristic
564C22#560712yes—
565C22#560812yes—
566C22D3×C22⋊Q812yes—
567C22#562712yes—
568C22#569412yes—
569C22#569512yes—
570C22#573012yes—
571C22#573412yes—
572C22C2×#93011yescharacteristic
573C22C2×#93711yescharacteristic
574C22#580511yescharacteristic
575C22C2×#94611yescharacteristic
576C22C2×#94711yescharacteristic
577C22#582811yescharacteristic
578C22#583311yescharacteristic
579C4—24no—
580C4—42no—
581C4—44no—
582C4—82no—
583C4—124no—
584C4—244no—
585C4#499118yes—
586C3#1518111yescharacteristic, complemented, Sylow
587C2—128no—
588C2#7620912yes—
589C2#7621212yes—
590C2#7623012yes—
591C2#7623212yes—
592C2C2×#499111yescharacteristic
593C2#8241611yescharacteristic
594C2#8250011yescharacteristic
595C2#8270911yescharacteristic
596C2#8283411yescharacteristic
597C2#8306311yescharacteristic
598C2#8326011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#389C2
prime radicalC23×D12C22
hypercenterC2×#389D3
hypocenterC3#15181
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×C3×#389C2semidirect
3. C2×#4987C2semidirect
4. C2×#4987C2semidirect
5. #71570C2semidirect
6. #71535C2semidirect
8. #69415C2semidirect
32. C2×#389D3semidirect
65. C2×C3×C4⋊C4D4semidirect
217. C2×C4⋊C4D12semidirect
586. C3#15181semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
C2×C3×#3892
C2×#49872
#715702
#715352
#694262
#694152
#151813