Finite Group Databasebeta

← #10822278

Subgroups — #10822278

Order 768.

2130 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic, complemented
3C23×#214C211yesmaximal, characteristic, complemented
4C2×#5233C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6#82230C211yesmaximal, characteristic, complemented
7#71768C214yesmaximal, complemented
8#71729C214yesmaximal, complemented
9#71579C211yesmaximal, characteristic, complemented
10#23423—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C3×#535—24no—
15C3×#535—24no—
16C2×C3×#127C2212yes—
17C2×C3×#124C2212yes—
18C2×C3×#121—22no—
19C2×C3×#120—22no—
20C2×C3×#117C2211yescharacteristic
21C2×#951—24no—
22C2×#951—24no—
23C2×#951—24no—
24C22×#216C2212yescomplemented
25C22×#214—24no—
26C22×#214—24no—
27C22×#214—24no—
28C22×#214C2214yes—
29C22×#214C2212yescomplemented
30C2×#926C2211yescharacteristic
31C22×#208C2211yescharacteristic
32C23×C3⋊C8C2211yescharacteristic
33#5233—24no—
34C2×#835C2212yescomplemented
35#5215—24no—
36C2×#832C2214yes—
37C2×#832C2212yescomplemented
38#4973—22no—
39#4943—22no—
40#4929C2214yescomplemented
41#4524C2214yescomplemented
42C22×#137—31no—
43C23×SD16—31no—
44C2×#535D311yescharacteristic
45C2×#418—31no—
46C2×#416—31no—
47#2310—31no—
48#2015—34no—
49#2014—34no—
50#1729—31no—
51C25×C3—21no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41D414yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—22no—
65C22×C3×D4—24no—
66C22×C3×D4—22no—
67C22×C3×D4D414yes—
68C22×C3×D4C2312yes—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C23×Dic3—21no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—21no—
78C22×Dic6—22no—
79C22×Dic6D414yescomplemented
80C22×Dic6C2312yes—
81C3×#127—24no—
82C3×#127—24no—
83C3×#124—24no—
84C3×#124—24no—
85C3×#121—216no—
86C3×#120—216no—
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42C2312yes—
93C2×#217—21no—
94#951—232no—
95C2×#216—24no—
96C2×#216—24no—
97C2×#216—24no—
98C2×#214—232no—
99C2×#214—28no—
100C2×#214—24no—
101C2×#214—28no—
102C2×#214—24no—
103C2×#214—28no—
104C2×#214—24no—
105#926—22no—
106#926—22no—
107C2×C4⋊Dic3C2312yes—
108C2×C4×Dic3—21no—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208D414yescomplemented
112C22×C3⋊C8—22no—
113C22×C3⋊C8—22no—
114C22×C3⋊C8D414yescomplemented
115C22×C3⋊C8C2312yes—
116C22×C3⋊C8C2312yes—
117#835—24no—
118#832—28no—
119#832—24no—
120C3⋊#95—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×D4C2×D311yescharacteristic
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C22×Q16—32no—
128C22×SD16—34no—
129C22×SD16—32no—
130C22×SD16—64no—
131C22×SD16—64no—
132C22×SD16—64no—
133C22×M16—31no—
134C23×C8—31no—
135#535—24no—
136#535—24no—
137C2×#127C2×D312yes—
138C2×#124C2×D312yes—
139C2×#121—22no—
140C2×#120—22no—
141C2×#117—31no—
142C2×#117C2×D311yescharacteristic
143#418—64no—
144#416—64no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102—34no—
148#391—34no—
149#391—62no—
150#381—62no—
151#336—34no—
152C24×C3—22no—
153C24×C3—21no—
154C24×C3—24no—
155C24×C3—42no—
156C24×C3—42no—
157C24×C3—84no—
158C24×C3C2×D411yescharacteristic
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—44no—
162C3×#41—44no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—24no—
170C2×C3×D4—216no—
171C2×C3×D4—28no—
172C2×C3×D4—24no—
173C2×C3×D4—28no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—44no—
187C2×C3×D4—42no—
188C2×C3×D4—44no—
189C2×C3×D4—42no—
190C2×C3×D4—88no—
191C2×C3×D4C2×D414yes—
192C2×C3×D4C2×D412yes—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D414yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—42no—
210C22×Dic3—42no—
211C2×Dic6—28no—
212C2×Dic6—28no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—24no—
218C2×Dic6—22no—
219C2×Dic6—42no—
220C2×Dic6—44no—
221C2×Dic6—44no—
222C2×Dic6—42no—
223C2×Dic6C2×D414yes—
224C3×C4⋊C4—24no—
225C3×C22⋊C4—216no—
226C3×C22⋊C4—24no—
227C3×C42—22no—
228C3×C42—22no—
229#217—42no—
230#216—216no—
231#214—232no—
232#214—216no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235C4×Dic3—42no—
236#208—28no—
237C2×C3⋊C8—28no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8C2×D414yes—
243C2×C3⋊C8C2×D414yes—
244C25—21no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41#6714yes—
251C22×Q8—34no—
252C22×Q8—32no—
253C22×Q8—62no—
254C22×Q8—61no—
255C22×Q8—62no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8C22×D311yescharacteristic
260C22×D4—21no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—22no—
267C22×D4—24no—
268C22×D4#6714yes—
269C22×D4C22×D312yes—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—61no—
273C23×C4C22×D311yescharacteristic
274#137—632no—
275C2×Q16—64no—
276C2×Q16—64no—
277C2×Q16—64no—
278C2×SD16—68no—
279C2×SD16—64no—
280C2×SD16—68no—
281C2×SD16—64no—
282C2×SD16—632no—
283C2×SD16—68no—
284C2×SD16—64no—
285C2×M16—34no—
286C2×M16—62no—
287C2×M16—62no—
288C22×C8—34no—
289C22×C8—32no—
290C22×C8—32no—
291C22×C8—62no—
292C22×C8—62no—
293#127—24no—
294#127—24no—
295#124—24no—
296#124—24no—
297#121—216no—
298#120—216no—
299#117—22no—
300#117—22no—
301#117—62no—
302#117—62no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4#6712yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4#6712yes—
308C2×C42—61no—
309C2×C42C22×D312yes—
310#103—64no—
311#102—64no—
312#102—68no—
313#95—68no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—22no—
317C23×C3—41no—
318C23×C3—42no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—44no—
322C23×C3—48no—
323C23×C3—48no—
324C23×C3—88no—
325C23×C3—88no—
326C23×C3—88no—
327C23×C3#12012yes—
328C23×C3#12112yes—
329C23×C3#12712yes—
330C23×C3C22×D411yescharacteristic
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×D4—24no—
336C3×D4—22no—
337C3×D4—24no—
338C3×D4—22no—
339C3×D4—48no—
340C3×D4—42no—
341C3×D4—41no—
342C3×D4—42no—
343C3×D4—41no—
344C3×D4—48no—
345C3×D4—816no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—28no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—48no—
352C2×C3×C4#12014yes—
353C2×C3×C4#12014yes—
354C2×C3×C4#12012yes—
355C2×C3×C4#12112yes—
356C2×C3×C4#12114yes—
357C2×C3×C4#12114yes—
358C2×C3×C4#12712yes—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×Dic3—24no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—48no—
369C2×Dic3—42no—
370C2×Dic3—44no—
371Dic6—24no—
372Dic6—24no—
373Dic6—42no—
374Dic6—42no—
375Dic6—48no—
376C3⋊C8—24no—
377C3⋊C8—24no—
378C24—22no—
379C24—21no—
380C24—24no—
381C24—42no—
382C24—42no—
383C24—84no—
384C24C2×#6711yescharacteristic
385#41—28no—
386#41—28no—
387#41—44no—
388#41—44no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—34no—
395C2×Q8—64no—
396C2×Q8—62no—
397C2×Q8—68no—
398C2×Q8—68no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—124no—
404C2×Q8—124no—
405C2×Q8—122no—
406C2×Q8—122no—
407C2×Q8C2×#6712yes—
408C2×D4—24no—
409C2×D4—216no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—88no—
430C2×D4C2×#6714yes—
431C2×D4C2×#6712yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—64no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—122no—
440C22×C4—122no—
441C22×C4D3×D412yes—
442C22×C4D3×D412yes—
443C22×C4D3×D412yes—
444C22×C4#23712yes—
445C22×C4C2×#6714yes—
446C22×C4C2×#6712yes—
447C22×C4C2×#6712yes—
448C22×C4C2×#6711yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—616no—
451SD16—632no—
452SD16—616no—
453M16—68no—
454C2×C8—34no—
455C2×C8—34no—
456C2×C8—68no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C4⋊C4—24no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C22⋊C4—216no—
465C22⋊C4—24no—
466C22⋊C4—122no—
467C42—22no—
468C42—22no—
469C42—122no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—48no—
477C22×C3—816no—
478C22×C3—82no—
479C22×C3—84no—
480C22×C3—82no—
481C22×C3—84no—
482C22×C3#39214yes—
483C22×C3C2×#12011yescharacteristic
484C22×C3C2×#12111yescharacteristic
485C22×C3C2×#12711yescharacteristic
486C22×C3#57714yes—
487C3×C4—42no—
488C3×C4—44no—
489C3×C4#39214yes—
490C3×C4C2×#12011yescharacteristic
491C3×C4C2×#12011yescharacteristic
492C3×C4C2×#12111yescharacteristic
493C3×C4C2×#12111yescharacteristic
494Dic3—44no—
495Dic3—82no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—48no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—44no—
505C23—48no—
506C23—88no—
507C23—88no—
508C23—88no—
509C23#93812yes—
510C23#94012yes—
511C23#95312yes—
512C23C22×#6711yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517Q8—64no—
518Q8—64no—
519Q8—128no—
520Q8—122no—
521Q8—122no—
522D4—24no—
523D4—22no—
524D4—24no—
525D4—22no—
526D4—48no—
527D4—48no—
528D4—42no—
529D4—41no—
530D4—42no—
531D4—41no—
532D4—816no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—48no—
539C2×C4—64no—
540C2×C4—128no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—124no—
545C2×C4#93714yes—
546C2×C4#93714yes—
547C2×C4#93812yes—
548C2×C4#93914yes—
549C2×C4#93914yes—
550C2×C4#94012yes—
551C2×C4#95312yes—
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×D3×D411yescharacteristic
555C2×C4C2×#23711yescharacteristic
556C2×C4C22×#6711yescharacteristic
557C2×C4C22×#6711yescharacteristic
558C8—64no—
559C8—64no—
560C2×C3—24no—
561C2×C3—82no—
562C2×C3—84no—
563C2×C3C2×#39211yescharacteristic
564C2×C3#300912yes—
565C2×C3#304312yes—
566C2×C3#315412yes—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—48no—
574C22—816no—
575C22—84no—
576C22—82no—
577C22—84no—
578C22—82no—
579C22#522314yes—
580C22C2×#93811yescharacteristic
581C22C2×#94011yescharacteristic
582C22#583614yes—
583C22C2×#95311yescharacteristic
584C4—42no—
585C4—44no—
586C4—124no—
587C4—242no—
588C4#522314yes—
589C4C2×#93711yescharacteristic
590C4C2×#93711yescharacteristic
591C4C2×#93911yescharacteristic
592C4C2×#93911yescharacteristic
593C3#2342311yescharacteristic, complemented, Sylow
594C2—24no—
595C2—84no—
596C2—82no—
597C2#8338112yes—
598C2#8339112yes—
599C2C2×#522311yescharacteristic
600C2#8410512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#535C2
socleC23×C3C22×D4
hypercenterC2×#535D3
hypocenterC3#23423
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. C22×#951C2semidirect
3. C23×#214C2semidirect
4. C2×#5233C2semidirect
5. C2×#5215C2semidirect
6. #82230C2semidirect
7. #71768C2semidirect
8. #71729C2semidirect
9. #71579C2semidirect
24. C22×#216C22semidirect
29. C22×#214C22semidirect
34. C2×#835C22semidirect
37. C2×#832C22semidirect
40. #4929C22semidirect
41. #4524C22semidirect
79. C22×Dic6D4semidirect
111. C2×#208D4semidirect
114. C22×C3⋊C8D4semidirect
593. C3#23423semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C22×#9512
C23×#2142
C2×#52332
C2×#52152
#822302
#717682
#717292
#715792
#234233