Finite Group Databasebeta

← #10732232

Subgroups — #10732232

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×C3×C4×D4C211yesmaximal, characteristic
2C2×#5212C212yesmaximal, complemented
3C2×#5212C212yesmaximal, complemented
4C2×#4928C211yesmaximal, characteristic, complemented
5#72131C212yesmaximal
6#71567C212yesmaximal, complemented
7#71547C214yesmaximal, complemented
8#71535C211yesmaximal, characteristic, complemented
9#15181—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4C2212yes—
12C2×C3×C4×D4—28no—
13C2×C3×C4×D4—24no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×#217C2214yes—
18C22×#217C2214yes—
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23#5212—28no—
24#5212—28no—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30#4928—24no—
31C22×C4×D4D311yescharacteristic
32C2×#389—31no—
33C2×#380—32no—
34C2×#380—32no—
35#1442—32no—
36#1439—34no—
37#1438—32no—
38C4⋊#95—31no—
39C25×C3C2312yes—
40C22×C3×D4—24no—
41C22×C3×D4—24no—
42C22×C3×D4—22no—
43C22×C3×D4—22no—
44C22×C3×D4Q814yes—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4D418yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2314yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C3×C4×D4—232no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C22⋊C4—24no—
63C2×C3×C22⋊C4—24no—
64C2×C3×C22⋊C4—24no—
65C2×C3×C42—24no—
66C2×C3×C42D414yes—
67C2×#217—28no—
68C2×#217—28no—
69C2×#217—28no—
70C2×#217—44no—
71C2×#217—44no—
72C2×#217—44no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3D414yescomplemented
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—44no—
83C3⋊#95—24no—
84C3⋊#95—28no—
85C3⋊#95—24no—
86C3⋊#95—28no—
87C3⋊#95—42no—
88C23×D4C2×D311yescharacteristic
89C24×C4C2×D312yes—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C22×C4⋊C4—31no—
93C22×C4⋊C4—32no—
94C22×C4⋊C4—31no—
95C22×C4⋊C4—32no—
96C22×C4⋊C4C2×D311yescharacteristic
97C22×C22⋊C4—34no—
98C22×C22⋊C4—34no—
99C22×C22⋊C4C2×D312yes—
100C22×C42C2×D311yescharacteristic
101#389—64no—
102#380—68no—
103#380—68no—
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—34no—
107C2×#95—34no—
108C2×#95—32no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3C2×D414yes—
122C24×C3C2×Q814yes—
123C24×C3#4114yes—
124C24×C3C2411yescharacteristic
125C2×C3×D4—28no—
126C2×C3×D4—28no—
127C2×C3×D4—216no—
128C2×C3×D4—216no—
129C2×C3×D4—28no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—216no—
143C22×C3×C4—216no—
144C22×C3×C4—216no—
145C22×C3×C4—48no—
146C22×C3×C4—48no—
147C22×C3×C4—48no—
148C22×C3×C4—48no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D414yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D414yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×Q812yes—
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×C3×C4#4114yes—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—24no—
162C22×Dic3—24no—
163C22×Dic3—22no—
164C22×Dic3—24no—
165C22×Dic3—24no—
166C22×Dic3—24no—
167C22×Dic3—24no—
168C22×Dic3—24no—
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—24no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—22no—
175C22×Dic3C2×D414yes—
176C3×C4⋊C4—28no—
177C3×C22⋊C4—216no—
178C3×C42—28no—
179#217—416no—
180#217—416no—
181C4⋊Dic3—28no—
182C4⋊Dic3—44no—
183C4⋊Dic3—48no—
184Dic3⋊C4—48no—
185C25C22×D312yes—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4Dic614yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—32no—
198C23×C4—32no—
199C23×C4—34no—
200C23×C4#6718yes—
201C23×C4C22×D311yescharacteristic
202C23×C4C22×D312yes—
203C23×C4C22×D311yescharacteristic
204C23×C4C22×D311yescharacteristic
205C4×D4—232no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—34no—
209C2×C4⋊C4—62no—
210C2×C4⋊C4—62no—
211C2×C4⋊C4—64no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—64no—
214C2×C4⋊C4—64no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—62no—
217C2×C4⋊C4—124no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
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—68no—
230#95—64no—
231#95—68no—
232#95—64no—
233#95—122no—
234C23×C3—24no—
235C23×C3—24no—
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—24no—
239C23×C3—24no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—24no—
243C23×C3—24no—
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
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×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3C22⋊Q818yes—
269C23×C3C22×D411yescharacteristic
270C23×C3C22×D412yes—
271C23×C3C22×D411yescharacteristic
272C23×C3C22×Q811yescharacteristic
273C23×C3C2×#4112yes—
274C23×C3C2×#4111yescharacteristic
275C23×C3C2×#4111yescharacteristic
276C23×C3C2×#4112yes—
277C23×C3H(5,𝔽2)12yes—
278C23×C3H(5,𝔽2)11yescharacteristic
279C23×C32−1+411yescharacteristic
280C3×D4—232no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
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—432no—
298C2×C3×C4#12014yes—
299C2×C3×C4#12114yes—
300C2×C3×C4#12118yes—
301C2×C3×C4C22⋊Q814yes—
302C2×C3×C4#12318yes—
303C2×C3×C4#12414yes—
304C2×Dic3—24no—
305C2×Dic3—28no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—44no—
309C2×Dic3—42no—
310C2×Dic3—44no—
311C2×Dic3—44no—
312C2×Dic3—42no—
313C2×Dic3—44no—
314C2×Dic3—44no—
315C2×Dic3—42no—
316C2×Dic3—44no—
317C2×Dic3—44no—
318C2×Dic3—44no—
319C2×Dic3—44no—
320C2×Dic3—44no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24C2×Dic614yes—
334C24#23514yes—
335C24C2×#6714yes—
336C24C23×D311yescharacteristic
337C2×D4—28no—
338C2×D4—216no—
339C2×D4—28no—
340C2×D4—216no—
341C2×D4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—34no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—64no—
363C22×C4—62no—
364C22×C4—64no—
365C22×C4—62no—
366C22×C4—64no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4—64no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—62no—
373C22×C4—64no—
374C22×C4—62no—
375C22×C4—64no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4C2×Dic612yes—
379C22×C4#23512yes—
380C22×C4D3×D412yes—
381C22×C4#23712yes—
382C22×C4#23714yes—
383C22×C4C2×#6714yes—
384C22×C4C2×#6712yes—
385C22×C4C2×#6712yes—
386C22×C4C2×#6714yes—
387C22×C4C2×#6712yes—
388C4⋊C4—28no—
389C4⋊C4—68no—
390C4⋊C4—128no—
391C4⋊C4—128no—
392C4⋊C4—124no—
393C22⋊C4—216no—
394C22⋊C4—1216no—
395C22⋊C4—1216no—
396C42—28no—
397C22×C3—28no—
398C22×C3—28no—
399C22×C3—28no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—44no—
403C22×C3—44no—
404C22×C3—44no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—44no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3C2×#12011yescharacteristic
413C22×C3C2×#12111yescharacteristic
414C22×C3C2×#12112yes—
415C22×C3C2×C22⋊Q811yescharacteristic
416C22×C3C2×C22⋊Q812yes—
417C22×C3C2×#12312yes—
418C22×C3#52511yescharacteristic
419C22×C3#52512yes—
420C22×C3C2×#12411yescharacteristic
421C22×C3#53511yescharacteristic
422C22×C3#53512yes—
423C22×C3#53711yescharacteristic
424C22×C3#53712yes—
425C22×C3#54612yes—
426C22×C3D4×Q811yescharacteristic
427C22×C3#55011yescharacteristic
428C22×C3#55212yes—
429C22×C3#55312yes—
430C22×C3#55312yes—
431C22×C3#55412yes—
432C22×C3#55411yescharacteristic
433C22×C3#55512yes—
434C22×C3#55611yescharacteristic
435C3×C4—24no—
436C3×C4—42no—
437C3×C4—44no—
438C3×C4#39418yes—
439Dic3—44no—
440Dic3—82no—
441Dic3—84no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23#92418yes—
477C23C22×Dic611yescharacteristic
478C23C2×#23512yes—
479C23C2×#23511yescharacteristic
480C23C2×D3×D411yescharacteristic
481C23C2×#23711yescharacteristic
482C23C2×#23712yes—
483C23#100912yes—
484C23#100911yescharacteristic
485C23#101011yescharacteristic
486C23C22×#6712yes—
487C23C22×#6711yescharacteristic
488D4—232no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—432no—
506C2×C4—64no—
507C2×C4—68no—
508C2×C4—122no—
509C2×C4—124no—
510C2×C4—122no—
511C2×C4—124no—
512C2×C4—124no—
513C2×C4—124no—
514C2×C4—124no—
515C2×C4—124no—
516C2×C4—124no—
517C2×C4—124no—
518C2×C4—124no—
519C2×C4—122no—
520C2×C4—124no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4#92414yes—
524C2×C4#93518yes—
525C2×C4#93614yes—
526C2×C4#93814yes—
527C2×C4#93918yes—
528C2×C4#95314yes—
529C2×C3—48no—
530C2×C3C2×#39411yescharacteristic
531C2×C3#241412yes—
532C2×C3#241712yes—
533C2×C3#248512yes—
534C2×C3#248712yes—
535C2×C3#253511yescharacteristic
536C2×C3#253811yescharacteristic
537C2×C3#256211yescharacteristic
538C2×C3#256311yescharacteristic
539C2×C3#261511yescharacteristic
540C2×C3#284811yescharacteristic
541C22—28no—
542C22—28no—
543C22—28no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22D4×Dic611yescharacteristic
557C22#554112yes—
558C22#554211yescharacteristic
559C22#554312yes—
560C22#554311yescharacteristic
561C22#555012yes—
562C22#555311yescharacteristic
563C22#556312yes—
564C22#556512yes—
565C22#556512yes—
566C22C2×#92412yes—
567C22C2×#92411yescharacteristic
568C22#579112yes—
569C22#579111yescharacteristic
570C22C2×#93512yes—
571C22C2×#93611yescharacteristic
572C22C2×#93811yescharacteristic
573C22C2×#93912yes—
574C22#583012yes—
575C22#583011yescharacteristic
576C22#583312yes—
577C22#583311yescharacteristic
578C22C2×#95311yescharacteristic
579C4—24no—
580C4—44no—
581C4—42no—
582C4—124no—
583C4—242no—
584C4—244no—
585C4#522218yes—
586C3#1518111yescharacteristic, complemented, Sylow
587C2—48no—
588C2#8235012yes—
589C2#8235312yes—
590C2#8237212yes—
591C2#8237412yes—
592C2#8237611yescharacteristic
593C2#8238311yescharacteristic
594C2#8238411yescharacteristic
595C2#8241511yescharacteristic
596C2#8403111yescharacteristic
597C2C2×#522211yescharacteristic
598C2#8409811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×C4×D4C2
prime radicalC23×C3×D4C22
hypercenterC22×C4×D4D3
hypocenterC3#15181
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5212C2semidirect
3. C2×#5212C2semidirect
4. C2×#4928C2semidirect
6. #71567C2semidirect
7. #71547C2semidirect
8. #71535C2semidirect
80. C2×C4⋊Dic3D4semidirect
586. C3#15181semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×#52122
C2×#49282
#721312
#715672
#715472
#715352
#151813