Finite Group Databasebeta

← #10822751

Subgroups — #10822751

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5232C211yesmaximal, characteristic, complemented
5C2×#5233C211yesmaximal, characteristic, complemented
6C22×#807C211yesmaximal, characteristic, complemented
7#71695C214yesmaximal, complemented
8#71679C214yesmaximal, complemented
9#71558C211yesmaximal, characteristic, complemented
10#23488—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×C3×#41C2211yescharacteristic
13C3×#537—24no—
14C3×#537—24no—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×#121—21no—
18C2×C3×#121—22no—
19C2×C3×#121C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25#5234—24no—
26#5233—24no—
27#5232—24no—
28#5229—21no—
29C2×#835C2212yes—
30C2×#835C2212yes—
31#5221—21no—
32#5220—22no—
33C2×#832C2212yes—
34C2×#832C2212yes—
35#5212C2214yes—
36C2×#830C2214yes—
37C2×#807—24no—
38C2×#807C2214yes—
39#4529C2214yes—
40C2×#537D311yescharacteristic
41C22×#107—31no—
42C2×#417—31no—
43C2×#416—31no—
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2033—34no—
48#1736—31no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41D414yes—
61C22×C3×Q8D411yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4—22no—
65C22×C3×D4D412yes—
66C22×C3×D4D411yescharacteristic
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×Dic3—21no—
73C3×C22⋊Q8—28no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—24no—
76C3×#121—216no—
77C3×#121—28no—
78C3×#121—24no—
79C3×#121—24no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4D414yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×#217—21no—
88C2×#217—24no—
89#926—22no—
90#926—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2312yes—
96C2×C4×Dic3—21no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208—22no—
100C22×C3⋊C8—22no—
101C22×C3⋊C8—22no—
102C22×C3⋊C8D414yescomplemented
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105#835—24no—
106#835—24no—
107#832—24no—
108#832—24no—
109C3⋊#95—24no—
110#830—28no—
111#807—28no—
112C22×#41C2×D311yescharacteristic
113C22×#41C2×D311yescharacteristic
114C22×M16—31no—
115C23×C8—31no—
116#537—24no—
117#537—24no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2×D312yes—
120C2×#121—21no—
121C2×#121—22no—
122C2×#121C2×D312yes—
123C2×#117—31no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4C2×D311yescharacteristic
126C2×#107—34no—
127C2×#107—64no—
128#417—64no—
129#416—64no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134#408—64no—
135C2×C22⋊C8—34no—
136#391—61no—
137#390—61no—
138#386—62no—
139#380—34no—
140#340—34no—
141C24×C3—21no—
142C24×C3C2×D412yes—
143C24×C3C2×D411yescharacteristic
144C3×#41—28no—
145C3×#41—28no—
146C3×#41—44no—
147C3×#41—44no—
148C3×#41—44no—
149C3×#41—44no—
150C3×#41—44no—
151C3×#41—44no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8C2×D412yes—
161C2×C3×D4—24no—
162C2×C3×D4—24no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—28no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4—84no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4—82no—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4#4112yes—
197C22×C3×C4#4112yes—
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—42no—
203C22×Dic3—82no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4C2×D414yes—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—28no—
210#217—42no—
211#217—84no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—42no—
217C4×Dic3—42no—
218#208—44no—
219C2×C3⋊C8—28no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8C2×D414yes—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41#6714yes—
238C22×Q8#6711yescharacteristic
239C22×Q8C22×D311yescharacteristic
240C22×D4—21no—
241C22×D4—22no—
242C22×D4#6712yes—
243C22×D4#6711yescharacteristic
244C22×D4C22×D311yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—61no—
248C23×C4C22×D312yes—
249C23×C4C22×D311yescharacteristic
250C2×M16—62no—
251C2×M16—62no—
252C2×M16—62no—
253C22×C8—34no—
254C22×C8—32no—
255C22×C8—32no—
256C22×C8—62no—
257C22×C8—62no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261#121—216no—
262#121—28no—
263#121—24no—
264#121—24no—
265#117—62no—
266#117—62no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4#6714yes—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—64no—
281C2×C42—61no—
282#107—68no—
283#103—64no—
284#103—64no—
285#102—64no—
286#102—64no—
287C22⋊C8—68no—
288#95—64no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3#12012yes—
302C23×C3#12112yes—
303C23×C3#12312yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×Q8—44no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—48no—
313C3×D4—48no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—44no—
317C3×D4—88no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—44no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—48no—
331C2×C3×C4—84no—
332C2×C3×C4#12012yes—
333C2×C3×C4#12114yes—
334C2×C3×C4#12114yes—
335C2×C3×C4#12112yes—
336C2×C3×C4#12314yes—
337C2×C3×C4#12312yes—
338C2×C3×C4#12414yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—44no—
349C2×Dic3—42no—
350C2×Dic3—84no—
351C3⋊C8—24no—
352C3⋊C8—42no—
353C24—21no—
354C24C2×#6712yes—
355C24C2×#6711yescharacteristic
356#41—28no—
357#41—28no—
358#41—44no—
359#41—44no—
360#41—44no—
361#41—44no—
362#41—44no—
363#41—44no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8C2×#6712yes—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—28no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—84no—
389C2×D4C2×#6712yes—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—82no—
405C22×C4—122no—
406C22×C4—242no—
407C22×C4D3×D412yes—
408C22×C4#23712yes—
409C22×C4#23712yes—
410C22×C4#23712yes—
411C22×C4C2×#6712yes—
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6712yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416M16—124no—
417C2×C8—34no—
418C2×C8—68no—
419C2×C8—62no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C4⋊C4—122no—
432C4⋊C4C2×#6714yes—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—122no—
436C22⋊C4—244no—
437C42—122no—
438C22×C3—24no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—24no—
442C22×C3—44no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—44no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3#39414yes—
453C22×C3C2×#12011yescharacteristic
454C22×C3C2×#12111yescharacteristic
455C22×C3C2×#12311yescharacteristic
456C22×C3#57612yes—
457C22×C3#57812yes—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—44no—
461C3×C4#39414yes—
462C3×C4C2×#12111yescharacteristic
463C3×C4C2×#12111yescharacteristic
464C3×C4C2×#12311yescharacteristic
465C3×C4C2×#12411yescharacteristic
466Dic3—82no—
467Dic3—82no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23#93512yes—
481C23#93912yes—
482C23#95312yes—
483C23C22×#6711yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—44no—
487Q8—41no—
488Q8—41no—
489D4—22no—
490D4—22no—
491D4—48no—
492D4—48no—
493D4—44no—
494D4—41no—
495D4—41no—
496D4—88no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—124no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—244no—
517C2×C4#93512yes—
518C2×C4#93514yes—
519C2×C4#93614yes—
520C2×C4#93814yes—
521C2×C4#93912yes—
522C2×C4#93914yes—
523C2×C4#95312yes—
524C2×C4C2×D3×D411yescharacteristic
525C2×C4C2×#23711yescharacteristic
526C2×C4C2×#23711yescharacteristic
527C2×C4C2×#23711yescharacteristic
528C2×C4C22×#6711yescharacteristic
529C2×C4C22×#6711yescharacteristic
530C8—64no—
531C8—122no—
532C2×C3—24no—
533C2×C3—82no—
534C2×C3—82no—
535C2×C3C2×#39411yescharacteristic
536C2×C3#300811yescharacteristic
537C2×C3#301111yescharacteristic
538C2×C3#306112yes—
539C2×C3#309812yes—
540C22—22no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—82no—
553C22—82no—
554C22#522214yes—
555C22C2×#93511yescharacteristic
556C22C2×#93911yescharacteristic
557C22#583512yes—
558C22#583812yes—
559C22C2×#95311yescharacteristic
560C4—42no—
561C4—42no—
562C4—44no—
563C4—242no—
564C4—242no—
565C4#522214yes—
566C4C2×#93511yescharacteristic
567C4C2×#93611yescharacteristic
568C4C2×#93811yescharacteristic
569C4C2×#93911yescharacteristic
570C3#2348811yescharacteristic, complemented, Sylow
571C2—24no—
572C2—82no—
573C2—82no—
574C2#8341812yes—
575C2#8342112yes—
576C2C2×#522211yescharacteristic
577C2#8410111yescharacteristic
578C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#537C2
socleC23×C3C22×D4
hypercenterC2×#537D3
hypocenterC3#23488
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
2. #84087C2semidirect
3. C2×#5234C2semidirect
4. C2×#5232C2semidirect
5. C2×#5233C2semidirect
6. C22×#807C2semidirect
7. #71695C2semidirect
8. #71679C2semidirect
9. #71558C2semidirect
102. C22×C3⋊C8D4semidirect
570. C3#23488semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5372
#840872
C2×#52342
C2×#52322
C2×#52332
C22×#8072
#716952
#716792
#715582
#234883