Finite Group Databasebeta

← #10822733

Subgroups — #10822733

Order 768.

1610 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5215C211yesmaximal, characteristic, complemented
5C22×#832C211yesmaximal, characteristic, complemented
6C2×#4998C211yesmaximal, characteristic, complemented
7#71680C214yesmaximal, complemented
8#71679C214yesmaximal, complemented
9#71581C211yesmaximal, characteristic, complemented
10#23486—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C3×#535—24no—
14C3×#535—24no—
15C2×C3×#127—21no—
16C2×C3×#124—21no—
17C2×C3×#121C2214yes—
18C2×C3×#120—22no—
19C2×C3×#117C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24#5234—24no—
25#5229—21no—
26#5221—21no—
27#5220—22no—
28#5215—24no—
29C2×#832—24no—
30C2×#832C2212yes—
31C2×#832C2214yes—
32C2×#832C2212yes—
33#5212C2214yescomplemented
34C2×#830C2212yescomplemented
35C2×#830C2212yescomplemented
36#4998—24no—
37C2×#808C2212yescomplemented
38C2×#807C2212yescomplemented
39#4529C2214yescomplemented
40C2×#535D311yescharacteristic
41C2×#427—31no—
42C2×#418—31no—
43C22×#102—31no—
44C2×#408—31no—
45#2311—31no—
46#2034—34no—
47#2033—34no—
48#1736—31no—
49C25×C3—21no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C22×C3×Q8D411yescharacteristic
57C22×C3×D4—24no—
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4D412yes—
65C22×C3×D4D411yescharacteristic
66C22×C3×D4D414yes—
67C22×C3×D4C2312yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×Dic3—21no—
71C3×#127—28no—
72C3×#124—28no—
73C3×#121—28no—
74C3×#121—28no—
75C3×#120—216no—
76C3×#117—22no—
77C3×#117D414yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C42—21no—
82C2×#217—21no—
83C2×#217—24no—
84#926—22no—
85#926—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2312yes—
91C2×C4×Dic3—21no—
92C2×#208—22no—
93C2×#208—22no—
94C2×#208—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8D414yescomplemented
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101#832—28no—
102#832—24no—
103#832—24no—
104C3⋊#95—24no—
105#830—24no—
106#830—24no—
107#808—24no—
108#807—24no—
109C22×#41C2×D311yescharacteristic
110C23×D4C2×D311yescharacteristic
111C22×M16—31no—
112C23×C8—31no—
113#535—24no—
114#535—24no—
115C2×#127—21no—
116C2×#124—21no—
117C2×#121C2×D314yes—
118C2×#120—22no—
119C2×#117—31no—
120C2×#117C2×D311yescharacteristic
121C22×C4⋊C4—31no—
122#427—64no—
123C2×#107—32no—
124C2×#106—32no—
125#418—64no—
126C2×#102—32no—
127C2×#102—32no—
128C2×#102—34no—
129C2×#102—64no—
130#408—64no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133#391—61no—
134#390—61no—
135#386—62no—
136#380—34no—
137#340—34no—
138C24×C3—22no—
139C24×C3—24no—
140C24×C3—42no—
141C24×C3—42no—
142C24×C3—82no—
143C24×C3—82no—
144C24×C3C2×D412yes—
145C24×C3C2×D411yescharacteristic
146C3×#41—44no—
147C3×#41—44no—
148C3×#41—44no—
149C3×#41—44no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—216no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—28no—
160C2×C3×D4—22no—
161C2×C3×D4—28no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—44no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—44no—
172C2×C3×D4—84no—
173C2×C3×D4—84no—
174C2×C3×D4C2×D414yes—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4#4112yes—
188C22×C3×C4#4112yes—
189C22×C3×C4#4112yes—
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—42no—
194C22×Dic3—82no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4C2×D414yes—
197C3×C22⋊C4—216no—
198C3×C22⋊C4—24no—
199C3×C42—24no—
200#217—42no—
201#217—84no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4⋊Dic3—42no—
207C4×Dic3—42no—
208#208—44no—
209C2×C3⋊C8—28no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8C2×D412yes—
218C2×C3⋊C8C2×D412yes—
219C25—21no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C22×Q8#6711yescharacteristic
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4#6712yes—
235C22×D4#6711yescharacteristic
236C22×D4#6714yes—
237C22×D4C22×D312yes—
238C23×C4—61no—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C2×M16—62no—
242C2×M16—62no—
243C2×M16—62no—
244C22×C8—34no—
245C22×C8—31no—
246C22×C8—31no—
247C22×C8—32no—
248C22×C8—62no—
249C22×C8—62no—
250#127—28no—
251#124—28no—
252#121—28no—
253#121—28no—
254#120—216no—
255#117—22no—
256#117—62no—
257#117—62no—
258#117#6714yes—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—64no—
269C2×C42—21no—
270C2×C42—61no—
271#107—64no—
272#106—64no—
273#102—64no—
274#102—64no—
275#102—68no—
276C22⋊C8—64no—
277C22⋊C8—64no—
278#95—64no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3—48no—
292C23×C3—88no—
293C23×C3—88no—
294C23×C3—84no—
295C23×C3—84no—
296C23×C3#12012yes—
297C23×C3#12112yes—
298C23×C3#12312yes—
299C23×C3C22×D411yescharacteristic
300C3×Q8—44no—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—48no—
304C3×D4—42no—
305C3×D4—42no—
306C3×D4—48no—
307C3×D4—44no—
308C3×D4—816no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4#12012yes—
320C2×C3×C4#12114yes—
321C2×C3×C4#12114yes—
322C2×C3×C4#12112yes—
323C2×C3×C4#12314yes—
324C2×C3×C4#12312yes—
325C2×C3×C4#12414yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—44no—
336C2×Dic3—42no—
337C2×Dic3—84no—
338C3⋊C8—22no—
339C3⋊C8—22no—
340C3⋊C8—42no—
341C24—22no—
342C24—24no—
343C24—42no—
344C24—42no—
345C24—82no—
346C24—82no—
347C24C2×#6712yes—
348C24C2×#6711yescharacteristic
349#41—44no—
350#41—44no—
351#41—44no—
352#41—44no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—216no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—28no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4C2×#6714yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—122no—
388C22×C4—242no—
389C22×C4D3×D412yes—
390C22×C4#23712yes—
391C22×C4#23712yes—
392C22×C4#23712yes—
393C22×C4C2×#6712yes—
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6712yes—
396C22×C4C2×#6712yes—
397C22×C4C23×D311yescharacteristic
398M16—124no—
399C2×C8—32no—
400C2×C8—32no—
401C2×C8—68no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C4⋊C4—22no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—122no—
415C4⋊C4C2×#6714yes—
416C22⋊C4—216no—
417C22⋊C4—24no—
418C22⋊C4—122no—
419C22⋊C4—244no—
420C42—24no—
421C42—122no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—44no—
427C22×C3—42no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—48no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—816no—
434C22×C3—84no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3#39414yes—
438C22×C3C2×#12011yescharacteristic
439C22×C3C2×#12111yescharacteristic
440C22×C3C2×#12311yescharacteristic
441C22×C3#57612yes—
442C22×C3#57712yes—
443C3×C4—42no—
444C3×C4—44no—
445C3×C4#39414yes—
446C3×C4C2×#12111yescharacteristic
447C3×C4C2×#12111yescharacteristic
448C3×C4C2×#12311yescharacteristic
449C3×C4C2×#12411yescharacteristic
450Dic3—82no—
451Dic3—82no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—48no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—88no—
466C23—88no—
467C23—84no—
468C23—84no—
469C23#93512yes—
470C23#93912yes—
471C23#95312yes—
472C23C22×#6711yescharacteristic
473Q8—44no—
474D4—24no—
475D4—24no—
476D4—48no—
477D4—48no—
478D4—44no—
479D4—42no—
480D4—42no—
481D4—816no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—124no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—244no—
498C2×C4#93512yes—
499C2×C4#93514yes—
500C2×C4#93614yes—
501C2×C4#93814yes—
502C2×C4#93912yes—
503C2×C4#93914yes—
504C2×C4#95312yes—
505C2×C4C2×D3×D411yescharacteristic
506C2×C4C2×#23711yescharacteristic
507C2×C4C2×#23711yescharacteristic
508C2×C4C2×#23711yescharacteristic
509C2×C4C22×#6711yescharacteristic
510C2×C4C22×#6711yescharacteristic
511C8—62no—
512C8—62no—
513C8—122no—
514C2×C3—24no—
515C2×C3—82no—
516C2×C3—84no—
517C2×C3C2×#39411yescharacteristic
518C2×C3#301212yes—
519C2×C3#305711yescharacteristic
520C2×C3#305811yescharacteristic
521C2×C3#309612yes—
522C22—24no—
523C22—24no—
524C22—22no—
525C22—22no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—48no—
530C22—44no—
531C22—42no—
532C22—42no—
533C22—816no—
534C22—84no—
535C22—82no—
536C22—82no—
537C22#522214yes—
538C22C2×#93511yescharacteristic
539C22C2×#93911yescharacteristic
540C22#583512yes—
541C22#583612yes—
542C22C2×#95311yescharacteristic
543C4—42no—
544C4—44no—
545C4—242no—
546C4—242no—
547C4#522214yes—
548C4C2×#93511yescharacteristic
549C4C2×#93611yescharacteristic
550C4C2×#93811yescharacteristic
551C4C2×#93911yescharacteristic
552C3#2348611yescharacteristic, complemented, Sylow
553C2—24no—
554C2—82no—
555C2—84no—
556C2#8337612yes—
557C2#8337711yescharacteristic
558C2#8337811yescharacteristic
559C2C2×#522211yescharacteristic
560C2#8410212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#535C2
socleC23×C3C22×D4
hypercenterC2×#535D3
hypocenterC3#23486
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. #84087C2semidirect
3. C2×#5234C2semidirect
4. C2×#5215C2semidirect
5. C22×#832C2semidirect
6. C2×#4998C2semidirect
7. #71680C2semidirect
8. #71679C2semidirect
9. #71581C2semidirect
33. #5212C22semidirect
34. C2×#830C22semidirect
35. C2×#830C22semidirect
37. C2×#808C22semidirect
38. C2×#807C22semidirect
39. #4529C22semidirect
97. C22×C3⋊C8D4semidirect
552. C3#23486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
#840872
C2×#52342
C2×#52152
C22×#8322
C2×#49982
#716802
#716792
#715812
#234863