Finite Group Databasebeta

← #10822707

Subgroups — #10822707

Order 768.

1610 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5805C211yesmaximal, characteristic
2C2×C3×#418C211yesmaximal, characteristic, complemented
3C22×#832C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic, complemented
5C2×#5121C211yesmaximal, characteristic, complemented
6#82270C211yesmaximal, characteristic, complemented
7#71580C211yesmaximal, characteristic, complemented
8#70176C214yesmaximal, complemented
9#70175C214yesmaximal, complemented
10#23483—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#953—22no—
16#5805—24no—
17#5805—24no—
18C2×#940—21no—
19C2×#938C2212yes—
20C2×#938C2212yes—
21C2×#936—21no—
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C3×#418—24no—
26C2×C3×#102C2214yes—
27C2×#832—24no—
28C2×#832C2214yes—
29#5129—24no—
30#5121—24no—
31C2×(C2×D3)⋊C8C2214yescomplemented
32C2×#818C2212yescomplemented
33C2×#817C2212yescomplemented
34#4987C2214yescomplemented
35#4958—22no—
36#4939—21no—
37#4929—21no—
38#4529C2212yescomplemented
39#4529C2212yescomplemented
40C2×#535—31no—
41C2×#428—31no—
42C2×#418D311yescharacteristic, complemented
43C22×#102—31no—
44C2×#408—31no—
45#2311—31no—
46#2034—34no—
47#2033—34no—
48#1735—31no—
49C25×C3—21no—
50C22×C3×D4—22no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4D414yes—
57C22×C3×D4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67—22no—
60C22×#67—22no—
61C22×#67D412yes—
62C23×Dic3—21no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C22×D12D411yescharacteristic
70C22×C4×D3C2312yes—
71C22×Dic6D411yescharacteristic
72C2×C3×M16—22no—
73C2×C3×M16—22no—
74C2×C3×M16D414yescomplemented
75C22×C3×C8C2312yes—
76C3×#117—22no—
77C3×#117—22no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—21no—
81#953—216no—
82C2×#217—21no—
83C2×#217—22no—
84C2×#217—22no—
85#940—28no—
86#938—24no—
87#938—24no—
88#938—24no—
89#938—24no—
90#936—28no—
91C2×(C2×D3)⋊C4—24no—
92#926—22no—
93#926D414yes—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4×Dic3—21no—
101C22×C3⋊C8—22no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8—22no—
104C22×C3⋊C8C2312yes—
105C3×#102—28no—
106#832—28no—
107C3⋊#95—24no—
108(C2×D3)⋊C8—28no—
109#818—24no—
110#817—24no—
111C22×#41—31no—
112C23×D4C2×D311yescharacteristic
113C22×M16C2×D311yescharacteristic
114C23×C8—31no—
115#535—64no—
116#535—64no—
117C2×#127—61no—
118C2×#124—61no—
119C2×#121—32no—
120C2×#121—32no—
121C2×#120—62no—
122C2×#117—31no—
123C2×#117C2×D311yescharacteristic
124C22×C4⋊C4—31no—
125#428—64no—
126C2×#107—32no—
127C2×#106—32no—
128#418—24no—
129C2×#102—34no—
130C2×#102—64no—
131C2×#102C2×D314yes—
132#408—64no—
133C2×C22⋊C8—34no—
134#391—61no—
135#390—61no—
136#386—62no—
137#380—34no—
138#340—32no—
139#340—32no—
140C24×C3—22no—
141C24×C3—24no—
142C24×C3—42no—
143C24×C3—42no—
144C24×C3—82no—
145C24×C3—82no—
146C24×C3C2×D411yescharacteristic
147C23×D3C2×D412yes—
148C2×C3×D4—24no—
149C2×C3×D4—28no—
150C2×C3×D4—28no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—42no—
157C2×C3×D4—44no—
158C2×C3×D4—42no—
159C2×C3×D4—44no—
160C2×C3×D4C2×D414yes—
161C22×C3×C4—22no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4112yes—
168C22×C3×C4#4112yes—
169C22×C3×C4C2411yescharacteristic
170C2×#67—24no—
171C2×#67—24no—
172C2×#67—24no—
173C2×#67—28no—
174C2×#67—28no—
175C2×#67—84no—
176C2×#67—84no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—42no—
181C22×Dic3—82no—
182C22×Dic3C2×D412yes—
183#235—44no—
184#235—44no—
185#235—44no—
186#235—44no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C3×M16—28no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8C2×D414yes—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C22⋊C4—42no—
206C3×C42—42no—
207#217—28no—
208#217—28no—
209#217—24no—
210(C2×D3)⋊C4—84no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—42no—
216C4⋊Dic3C2×D412yes—
217C4⋊Dic3C2×D412yes—
218C4×Dic3—24no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—44no—
223C25—21no—
224C2×#41—62no—
225C2×#41—62no—
226C2×#41—62no—
227C2×#41—62no—
228C2×#41—62no—
229C2×#41—62no—
230C22×Q8—31no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—32no—
238C22×D4—31no—
239C22×D4—62no—
240C22×D4—62no—
241C22×D4D1214yes—
242C22×D4C22×D312yes—
243C23×C4—32no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×M16—22no—
247C2×M16—22no—
248C2×M16#6714yescomplemented
249C22×C8—32no—
250C22×C8—62no—
251C22×C8—62no—
252C22×C8—62no—
253C22×C8C22×D312yes—
254#127—68no—
255#124—68no—
256#121—64no—
257#121—64no—
258#121—64no—
259#121—64no—
260#120—616no—
261#117—22no—
262#117—22no—
263#117—34no—
264#117—62no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C22×D312yes—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—64no—
277C2×C42—21no—
278C2×C42—61no—
279#107—64no—
280#106—64no—
281#102—28no—
282#102—68no—
283C22⋊C8—68no—
284#95—64no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—24no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—44no—
291C23×C3—44no—
292C23×C3—48no—
293C23×C3—84no—
294C23×C3—84no—
295C23×C3—88no—
296C23×C3—88no—
297C23×C3#12012yes—
298C23×C3#12112yes—
299C23×C3#12312yes—
300C23×C3C22×D411yescharacteristic
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—24no—
305C22×D3—24no—
306C3×D4—24no—
307C3×D4—24no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—48no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12114yes—
316C2×C3×C4#12112yes—
317C2×C3×C4#12114yes—
318C2×C3×C4#12312yes—
319C2×C3×C4#12314yes—
320C2×C3×C4#12414yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327#67—48no—
328#67—88no—
329#67—88no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—44no—
339C2×Dic3—84no—
340D12—44no—
341C4×D3—42no—
342C4×D3—42no—
343C4×D3—42no—
344C4×D3—42no—
345Dic6—44no—
346C3×C8—24no—
347C3⋊C8—42no—
348C24—22no—
349C24—24no—
350C24—32no—
351C24—42no—
352C24—42no—
353C24—82no—
354C24—82no—
355C24D3×D411yescharacteristic
356#41—124no—
357#41—124no—
358#41—124no—
359#41—124no—
360C2×Q8—62no—
361C2×Q8—62no—
362C2×Q8—62no—
363C2×D4—24no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—64no—
376C2×D4—64no—
377C2×D4—64no—
378C2×D4—68no—
379C2×D4—68no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—244no—
384C2×D4—244no—
385C2×D4C2×D1214yes—
386C22×C4—22no—
387C22×C4—32no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—122no—
398C22×C4—242no—
399C22×C4C2×D1212yes—
400C22×C4#23512yes—
401C22×C4D3×D411yescharacteristic
402C22×C4D3×D412yes—
403C22×C4#23712yes—
404C22×C4#23712yes—
405C22×C4C2×#6712yes—
406C22×C4C23×D311yescharacteristic
407M16—28no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8—124no—
414C2×C8C2×#6714yes—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—32no—
418C4⋊C4—32no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—122no—
424C22⋊C4—42no—
425C22⋊C4—68no—
426C22⋊C4—68no—
427C22⋊C4—64no—
428C22⋊C4—244no—
429C42—42no—
430C42—64no—
431C22×C3—24no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—816no—
439C22×C3—84no—
440C22×C3—84no—
441C22×C3#39414yes—
442C22×C3C2×#12011yescharacteristic
443C22×C3C2×#12111yescharacteristic
444C22×C3C2×#12311yescharacteristic
445C22×C3#57612yes—
446C22×C3#57712yes—
447C2×D3—44no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—44no—
452C3×C4—82no—
453C3×C4#39414yes—
454C3×C4C2×#12111yescharacteristic
455C3×C4C2×#12111yescharacteristic
456C3×C4C2×#12311yescharacteristic
457C3×C4C2×#12411yescharacteristic
458Dic3—42no—
459Dic3—42no—
460Dic3—42no—
461Dic3—82no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—48no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—62no—
471C23—62no—
472C23—64no—
473C23—64no—
474C23—62no—
475C23—84no—
476C23—84no—
477C23—88no—
478C23—88no—
479C23#88212yes—
480C23#88312yes—
481C23#93812yes—
482C23C2×D3×D411yescharacteristic
483Q8—124no—
484D4—24no—
485D4—24no—
486D4—48no—
487D4—42no—
488D4—42no—
489D4—128no—
490D4—124no—
491D4—248no—
492D4—248no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—124no—
509C2×C4—244no—
510C2×C4#88212yes—
511C2×C4#88312yes—
512C2×C4#88514yes—
513C2×C4#88614yes—
514C2×C4#93014yes—
515C2×C4#93812yes—
516C2×C4#93914yes—
517C2×C4C22×D1211yescharacteristic
518C2×C4C2×#23511yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C2×C4C2×#23711yescharacteristic
521C2×C4C2×#23711yescharacteristic
522C2×C4C22×#6711yescharacteristic
523C8—24no—
524C8—122no—
525C2×C3—24no—
526C2×C3—84no—
527C2×C3C2×#39411yescharacteristic
528C2×C3#300811yescharacteristic
529C2×C3#300911yescharacteristic
530C2×C3#306212yes—
531C2×C3#309612yes—
532D3—82no—
533C22—24no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—816no—
541C22—84no—
542C22—84no—
543C22—124no—
544C22—122no—
545C22—122no—
546C22—122no—
547C22—124no—
548C22#496114yes—
549C22C2×#88211yescharacteristic
550C22C2×#88311yescharacteristic
551C22#577412yes—
552C22#577512yes—
553C22C2×#93811yescharacteristic
554C4—82no—
555C4—122no—
556C4—122no—
557C4—122no—
558C4—242no—
559C4#496114yes—
560C4C2×#88511yescharacteristic
561C4C2×#88611yescharacteristic
562C4C2×#93011yescharacteristic
563C4C2×#93911yescharacteristic
564C3#2348311yescharacteristic, complemented, Sylow
565C2—24no—
566C2—84no—
567C2—242no—
568C2#7891912yes—
569C2#7892811yescharacteristic
570C2#7892911yescharacteristic
571C2C2×#496111yescharacteristic
572C2#8395112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#418C2
socleC23×C3C22×D4
prime radicalC2×#5805C2
hypercenterC2×#418D3
hypocenterC3#23483
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×C3×#418C2semidirect
3. C22×#832C2semidirect
4. C2×#5129C2semidirect
5. C2×#5121C2semidirect
6. #82270C2semidirect
7. #71580C2semidirect
8. #70176C2semidirect
9. #70175C2semidirect
31. C2×(C2×D3)⋊C8C22semidirect
32. C2×#818C22semidirect
33. C2×#817C22semidirect
34. #4987C22semidirect
38. #4529C22semidirect
39. #4529C22semidirect
42. C2×#418D3semidirect
74. C2×C3×M16D4semidirect
248. C2×M16#67semidirect
564. C3#23483semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
C2×C3×#4182
C22×#8322
C2×#51292
C2×#51212
#822702
#715802
#701762
#701752
#234833