Finite Group Databasebeta

← #10822711

Subgroups — #10822711

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×#5525C211yesmaximal, characteristic
2C22×#821C211yesmaximal, characteristic, complemented
3C2×#5110C211yesmaximal, characteristic, complemented
4#82270C211yesmaximal, characteristic, complemented
5C2×#5001C211yesmaximal, characteristic, complemented
6C2×#4995C211yesmaximal, characteristic, complemented
7C3×#1735C211yesmaximal, characteristic, complemented
8#68823C214yesmaximal, complemented
9#68822C214yesmaximal, complemented
10#23483—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×#208C2211yescharacteristic
18#5525—24no—
19#5525—24no—
20C2×#895C2212yes—
21C2×#895C2212yes—
22C2×#882—22no—
23C2×#875—21no—
24C2×#874—21no—
25C2×#821—24no—
26C2×#821C2214yes—
27#5110—24no—
28C2×(C2×D3)⋊C8C2214yescomplemented
29#5001—24no—
30#4995—24no—
31#4987C2214yescomplemented
32C2×#809C2214yes—
33C2×#808C2212yescomplemented
34C2×#807C2212yescomplemented
35#4958—22no—
36#4939—21no—
37#4929—21no—
38C3×#340C2212yescomplemented
39C3×#340C2212yescomplemented
40C2×#535—31no—
41C2×#428—31no—
42C2×#418—31no—
43C22×#102—31no—
44C2×#408—31no—
45#2311—31no—
46#2034—34no—
47#2033—34no—
48#1735D311yescharacteristic, complemented
49C24×D3—21no—
50C23×C3×C4C2311yescharacteristic
51C22×#67D412yes—
52C23×Dic3—21no—
53C2×#235—22no—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C22×D12—22no—
60C22×D12—22no—
61C22×D12—22no—
62C22×D12—22no—
63C22×D12—21no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12D411yescharacteristic
68C22×D12D414yes—
69C22×D12C2312yes—
70C22×C4×D3C2312yes—
71C22×Dic6D411yescharacteristic
72C22×C3×C8—22no—
73C22×C3×C8—22no—
74C22×C3×C8—22no—
75C22×C3×C8C2312yes—
76C3×#117—22no—
77C3×#117—22no—
78C3×#117—22no—
79C3×#117D414yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42—21no—
86C2×C3×C42—21no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—24no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2312yes—
94C2×#208—22no—
95C2×#208—22no—
96C2×#208D414yescomplemented
97C22×C3⋊C8C2312yes—
98#895—24no—
99#895—24no—
100#895—24no—
101#895—24no—
102#882—216no—
103#875—28no—
104#874—28no—
105C3⋊#95—24no—
106#821—28no—
107(C2×D3)⋊C8—28no—
108#809—28no—
109#808—24no—
110#807—24no—
111C22×#41—31no—
112C23×D4—31no—
113C22×M16—31no—
114C23×C8C2×D311yescharacteristic
115#535—64no—
116#535—64no—
117C2×#127—61no—
118C2×#124—61no—
119C2×#121—32no—
120C2×#121—32no—
121C2×#120—62no—
122C2×#117C2×D311yescharacteristic
123C2×#117C2×D311yescharacteristic
124C22×C4⋊C4—31no—
125#428—64no—
126C2×#107—32no—
127C2×#106—32no—
128#418—64no—
129C2×#102—34no—
130C2×#102—34no—
131C2×#102—64no—
132#408—64no—
133C2×C22⋊C8—34no—
134#391—61no—
135#390—61no—
136#386—62no—
137#380—34no—
138#340C2×D312yescomplemented
139#340C2×D312yescomplemented
140C24×C3C2×D411yescharacteristic
141C23×D3—22no—
142C23×D3—24no—
143C23×D3—42no—
144C23×D3—42no—
145C23×D3—82no—
146C23×D3—82no—
147C23×D3C2×D412yes—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×C3×C4C2411yescharacteristic
159C2×#67—24no—
160C2×#67—24no—
161C2×#67—24no—
162C22×Dic3—22no—
163C22×Dic3—42no—
164C22×Dic3—82no—
165C22×Dic3C2×D412yes—
166#235—44no—
167#235—44no—
168#235—44no—
169#235—44no—
170C2×D12—28no—
171C2×D12—28no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—24no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—28no—
182C2×D12—28no—
183C2×D12—42no—
184C2×D12—44no—
185C2×D12—42no—
186C2×D12—44no—
187C2×D12—84no—
188C2×D12—84no—
189C2×D12C2×D414yes—
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—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—44no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4C2×D412yes—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—42no—
211C3×C42—24no—
212C3×C42—42no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—84no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—42no—
219#208—28no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8C2×D414yes—
223C25—61no—
224C2×#41—62no—
225C2×#41—62no—
226C2×#41—62no—
227C2×#41—62no—
228C2×#41—62no—
229C2×#41—62no—
230C22×Q8—31no—
231C22×D4—32no—
232C22×D4—34no—
233C22×D4—31no—
234C22×D4—32no—
235C22×D4—62no—
236C22×D4—62no—
237C22×D4—62no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—62no—
241C22×D4—61no—
242C22×D4—62no—
243C23×C4—32no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×M16—34no—
247C2×M16—62no—
248C2×M16—62no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—32no—
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—22no—
264#117D1214yes—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—64no—
277C2×C42—21no—
278C2×C42—21no—
279#107—64no—
280#106—64no—
281#102—68no—
282#102—68no—
283C22⋊C8—68no—
284#95—64no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3#12012yes—
288C23×C3#12112yes—
289C23×C3#12312yes—
290C23×C3C22×D411yescharacteristic
291C22×D3—24no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—24no—
296C22×D3—24no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—44no—
300C22×D3—44no—
301C22×D3—48no—
302C22×D3—84no—
303C22×D3—84no—
304C22×D3—88no—
305C22×D3—88no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4#12012yes—
314C2×C3×C4#12114yes—
315C2×C3×C4#12112yes—
316C2×C3×C4#12114yes—
317C2×C3×C4#12314yes—
318C2×C3×C4#12312yes—
319C2×C3×C4#12414yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326#67—48no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—42no—
331C2×Dic3—44no—
332C2×Dic3—84no—
333D12—24no—
334D12—24no—
335D12—42no—
336D12—42no—
337D12—48no—
338D12—44no—
339D12—88no—
340D12—88no—
341C4×D3—42no—
342C4×D3—42no—
343C4×D3—42no—
344C4×D3—42no—
345Dic6—44no—
346C3×C8—42no—
347C3⋊C8—24no—
348C24—32no—
349C24—62no—
350C24—64no—
351C24—122no—
352C24—122no—
353C24—242no—
354C24—242no—
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—34no—
364C2×D4—68no—
365C2×D4—68no—
366C2×D4—64no—
367C2×D4—64no—
368C2×D4—64no—
369C2×D4—64no—
370C2×D4—62no—
371C2×D4—68no—
372C2×D4—62no—
373C2×D4—68no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—124no—
381C2×D4—122no—
382C2×D4—122no—
383C2×D4—124no—
384C2×D4—244no—
385C2×D4—244no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—32no—
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×C4#23512yes—
402C22×C4D3×D411yescharacteristic
403C22×C4D3×D412yes—
404C22×C4D3×D412yes—
405C22×C4#23912yes—
406C22×C4C23×D311yescharacteristic
407M16—68no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—34no—
412C2×C8—44no—
413C2×C8—62no—
414C2×C8—62no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—62no—
420C4⋊C4—62no—
421C4⋊C4—122no—
422C4⋊C4C2×D1212yes—
423C4⋊C4C2×D1212yes—
424C22⋊C4—24no—
425C22⋊C4—42no—
426C22⋊C4—68no—
427C22⋊C4—68no—
428C22⋊C4—244no—
429C42—24no—
430C42—42no—
431C22×C3—24no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3#39414yes—
436C22×C3C2×#12011yescharacteristic
437C22×C3C2×#12111yescharacteristic
438C22×C3C2×#12311yescharacteristic
439C22×C3#57612yes—
440C22×C3#57712yes—
441C2×D3—44no—
442C2×D3—42no—
443C2×D3—44no—
444C2×D3—42no—
445C2×D3—44no—
446C2×D3—42no—
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—816no—
450C2×D3—84no—
451C2×D3—84no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—82no—
455C3×C4#39414yes—
456C3×C4C2×#12111yescharacteristic
457C3×C4C2×#12111yescharacteristic
458C3×C4C2×#12311yescharacteristic
459C3×C4C2×#12411yescharacteristic
460Dic3—42no—
461Dic3—82no—
462C23—22no—
463C23—22no—
464C23—64no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—64no—
469C23—64no—
470C23—128no—
471C23—122no—
472C23—122no—
473C23—124no—
474C23—124no—
475C23—244no—
476C23—244no—
477C23—248no—
478C23—248no—
479C23#88212yes—
480C23#88412yes—
481C23#89412yes—
482C23C2×D3×D411yescharacteristic
483Q8—124no—
484D4—64no—
485D4—64no—
486D4—128no—
487D4—128no—
488D4—124no—
489D4—122no—
490D4—122no—
491D4—248no—
492D4—248no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—124no—
509C2×C4—244no—
510C2×C4#87514yes—
511C2×C4#88212yes—
512C2×C4#88412yes—
513C2×C4#88414yes—
514C2×C4#89412yes—
515C2×C4#89414yes—
516C2×C4#89514yes—
517C2×C4C22×D1211yescharacteristic
518C2×C4C2×#23511yescharacteristic
519C2×C4C2×#23511yescharacteristic
520C2×C4C2×D3×D411yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×#23911yescharacteristic
523C8—42no—
524C8—64no—
525C2×C3—24no—
526C2×C3C2×#39411yescharacteristic
527C2×C3#300811yescharacteristic
528C2×C3#300911yescharacteristic
529C2×C3#306212yes—
530C2×C3#309612yes—
531D3—82no—
532D3—84no—
533C22—24no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—124no—
538C22—122no—
539C22—124no—
540C22—122no—
541C22—124no—
542C22—122no—
543C22—124no—
544C22—124no—
545C22—2416no—
546C22—244no—
547C22—244no—
548C22#467214yes—
549C22C2×#88211yescharacteristic
550C22C2×#88411yescharacteristic
551C22C2×#89411yescharacteristic
552C22#576912yes—
553C22#577012yes—
554C4—42no—
555C4—42no—
556C4—82no—
557C4—122no—
558C4—242no—
559C4#467214yes—
560C4C2×#87511yescharacteristic
561C4C2×#88411yescharacteristic
562C4C2×#89411yescharacteristic
563C4C2×#89511yescharacteristic
564C3#2348311yescharacteristic, complemented, Sylow
565C2—24no—
566C2—242no—
567C2—244no—
568C2C2×#467211yescharacteristic
569C2#7883411yescharacteristic
570C2#7883511yescharacteristic
571C2#7884012yes—
572C2#7917412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1735C2
socleC23×C3C22×D4
prime radicalC2×#5525C2
hypercenter#1735D3
hypocenterC3#23483
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
2. C22×#821C2semidirect
3. C2×#5110C2semidirect
4. #82270C2semidirect
5. C2×#5001C2semidirect
6. C2×#4995C2semidirect
7. C3×#1735C2semidirect
8. #68823C2semidirect
9. #68822C2semidirect
28. C2×(C2×D3)⋊C8C22semidirect
31. #4987C22semidirect
33. C2×#808C22semidirect
34. C2×#807C22semidirect
38. C3×#340C22semidirect
39. C3×#340C22semidirect
48. #1735D3semidirect
96. C2×#208D4semidirect
138. #340C2×D3semidirect
139. #340C2×D3semidirect
564. C3#23483semidirect

Maximal subgroups

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

subgroupindex
C2×#55252
C22×#8212
C2×#51102
#822702
C2×#50012
C2×#49952
C3×#17352
#688232
#688222
#234833