Finite Group Databasebeta

← #10845913

Subgroups — #10845913

Order 768.

1102 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×C8×D4C211yesmaximal, characteristic, complemented
2C2×#5114C212yesmaximal, complemented
3#82417C211yesmaximal, characteristic
4#82379C211yesmaximal, characteristic, complemented
5C2×#4841C211yesmaximal, characteristic, complemented
6#71972C212yesmaximal, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69962C212yesmaximal, complemented
9#69936C212yesmaximal, complemented
10#69771C211yesmaximal, characteristic, complemented
11C4×#821C211yesmaximal, characteristic, complemented
12#25385—31nomaximal, Sylow
13C23×C3×C8C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#938—21no—
16C2×#930C2212yes—
17C2×#924C2212yes—
18C2×#924C2212yes—
19C22×#204C2212yes—
20C2×C4×D12C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C3×C8×D4—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2212yes—
25C2×C3×C4×C8C2211yescharacteristic
26#5221—21no—
27#5212C2212yes—
28#5114—28no—
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#817C2212yes—
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35#4987—21no—
36#4975—21no—
37#4961—22no—
38#4949—22no—
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4841—24no—
42#4551C2211yescharacteristic
43#4551C2212yes—
44#4551C2211yescharacteristic
45C2×#465—32no—
46C2×#461—31no—
47C2×C8×D4D311yescharacteristic, complemented
48#2542—31no—
49#2537—31no—
50#2035—32no—
51#2000—32no—
52#1956—31no—
53#1948—31no—
54#1892—32no—
55C4×#102—31no—
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×#67—22no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×Dic6C2312yes—
62C22×C3×C8—24no—
63C22×C3×C8—24no—
64C22×C3×C8—42no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2312yes—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C22⋊C4D412yes—
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76#938—44no—
77#930—24no—
78#930—42no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89#924—24no—
90#924—24no—
91#924—42no—
92#924—42no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×C4×Dic3—21no—
96C2×#204—24no—
97C2×#204—24no—
98C2×#204—42no—
99C4×D12—22no—
100C4×D12—22no—
101#869—22no—
102#869—22no—
103C3×C4⋊C8D414yescomplemented
104C3×C22⋊C8—24no—
105C3×C4×C8—22no—
106C3⋊#95—21no—
107C3⋊#95—22no—
108#821—22no—
109#821D414yescomplemented
110#817—24no—
111#816—24no—
112#816—24no—
113#816—42no—
114C22×SD16—32no—
115C23×C8C2×D312yes—
116C2×C4⋊Q8—31no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—32no—
119C2×#121—32no—
120C2×#121—61no—
121C2×C4×D4—31no—
122C2×C4×D4C2×D311yescharacteristic
123#465—68no—
124#461—64no—
125C8×D4—24no—
126C2×#106—32no—
127C2×C4⋊C8C2×D311yescharacteristic
128C2×#103—32no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#102—31no—
132C2×#102—31no—
133C2×C22⋊C8C2×D312yes—
134C2×C4×C8C2×D311yescharacteristic
135#399—62no—
136#394—62no—
137#390—61no—
138#384—31no—
139#384—61no—
140#380—32no—
141#380—61no—
142C4×C4⋊C4—31no—
143#340—31no—
144#340—32no—
145#340—31no—
146C24×C3C2×D412yes—
147C23×D3—21no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4SD1614yescomplemented
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C2×#67—44no—
162C2×#67—44no—
163C2×#67—82no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12C2×D412yes—
175C2×C4×D3—42no—
176C2×C4×D3—42no—
177C2×C4×D3—42no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×C3×C8—22no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—48no—
192C2×C3×C8C2×D412yes—
193C2×C3×C8C2×D414yes—
194C3×C4⋊C4—22no—
195C3×C22⋊C4—24no—
196C3×C42C2×D412yes—
197C3×C42#4112yes—
198#217—44no—
199#217—44no—
200#217—44no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—42no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—22no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—42no—
210C4⋊Dic3—42no—
211C4⋊Dic3—42no—
212C4⋊Dic3—81no—
213C4⋊Dic3C2×D412yes—
214C4⋊Dic3#4112yes—
215Dic3⋊C4—44no—
216Dic3⋊C4—44no—
217C4×Dic3—42no—
218#204—48no—
219C22×Q8—32no—
220C22×D4—31no—
221C22×D4—62no—
222C22×D4D1211yescharacteristic
223C23×C4—61no—
224C23×C4C22×D312yes—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—42no—
231C22×C8C22×D311yescharacteristic
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D312yes—
234C4⋊Q8—62no—
235C4⋊Q8—62no—
236C22⋊Q8—64no—
237C22⋊Q8—64no—
238C22⋊Q8—122no—
239C22⋊Q8—122no—
240#121—64no—
241#121—122no—
242#121—124no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—62no—
246C4×D4—62no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4D1211yescharacteristic
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4D1212yes—
264C2×C42—61no—
265C2×C42C22×D311yescharacteristic
266#106—64no—
267C4⋊C8#6714yescomplemented
268#103—64no—
269#103—64no—
270#103—122no—
271#102—34no—
272#102—62no—
273C22⋊C8—24no—
274C4×C8—22no—
275#95—61no—
276#95—62no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3C2×SD1614yes—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C3×D4—28no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12112yes—
303C2×C3×C4#12112yes—
304C2×C3×C4C2×SD1612yes—
305C2×C3×C4#13512yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C2×C3×C4H(5,𝔽2)11yescharacteristic
312#67—88no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328D12—22no—
329D12—22no—
330D12—41no—
331D12—41no—
332C4×D3—84no—
333Dic6—42no—
334Dic6—42no—
335Dic6—42no—
336Dic6—42no—
337C3×C8—24no—
338C3×C8—22no—
339C3×C8—41no—
340C24—61no—
341C24C2×D1212yes—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—32no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—124no—
357C2×D4—124no—
358C2×D4—242no—
359C2×D4#20414yescomplemented
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—122no—
372C22×C4—122no—
373C22×C4—122no—
374C22×C4C2×D1212yes—
375C22×C4C2×D1211yescharacteristic
376C22×C4C2×D1211yescharacteristic
377C22×C4C23×D311yescharacteristic
378SD16—128no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—48no—
387C2×C8D3×D412yes—
388C2×C8C2×#6714yes—
389C4⋊C4—22no—
390C4⋊C4—32no—
391C4⋊C4—32no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—61no—
398C4⋊C4—124no—
399C4⋊C4—124no—
400C4⋊C4—122no—
401C4⋊C4—122no—
402C4⋊C4—122no—
403C4⋊C4—241no—
404C22⋊C4—24no—
405C22⋊C4—124no—
406C22⋊C4—124no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C22⋊C4—122no—
410C42—122no—
411C42#23512yes—
412C42C2×#6712yes—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3C2×#12111yescharacteristic
421C22×C3D4211yescharacteristic
422C22×C3#54611yescharacteristic
423C22×C3C22×SD1611yescharacteristic
424C22×C3C2×#13511yescharacteristic
425C22×C3#57711yescharacteristic
426C22×C3#57811yescharacteristic
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—81no—
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—81no—
434C3×C4—22no—
435C3×C4—42no—
436C3×C4#46514yescomplemented
437C3×C4#53211yescharacteristic
438C3×C4#54011yescharacteristic
439C3×C4D4211yescharacteristic
440C3×C4#55011yescharacteristic
441Dic3—42no—
442Dic3—42no—
443Dic3—82no—
444Dic3—82no—
445Dic3—81no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—62no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23C2×#20414yes—
460C23C22×D1211yescharacteristic
461Q8—122no—
462Q8—122no—
463Q8—122no—
464Q8—122no—
465D4—28no—
466D4—62no—
467D4—62no—
468D4—121no—
469D4—121no—
470D4—248no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4—122no—
490C2×C4—121no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—244no—
496C2×C4C2×#20412yes—
497C2×C4#90412yes—
498C2×C4#93012yes—
499C2×C4#93012yes—
500C2×C4C2×#23511yescharacteristic
501C2×C4C2×D3×D411yescharacteristic
502C2×C4#100411yescharacteristic
503C2×C4D3×#4111yescharacteristic
504C2×C4#100911yescharacteristic
505C2×C4C22×#6711yescharacteristic
506C8—22no—
507C8—24no—
508C8—41no—
509C2×C3—44no—
510C2×C3#242811yescharacteristic
511C2×C3C2×#46511yescharacteristic
512C2×C3#306311yescharacteristic
513C2×C3D4×SD1611yescharacteristic
514C2×C3#328311yescharacteristic
515C2×C3#329811yescharacteristic
516C2×C3#330311yescharacteristic
517D3—82no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—122no—
526C22—122no—
527C22—122no—
528C22—241no—
529C22—241no—
530C22—241no—
531C22—241no—
532C22D4×D1211yescharacteristic
533C22#555411yescharacteristic
534C22C22×#20411yescharacteristic
535C22C2×#90411yescharacteristic
536C22#575111yescharacteristic
537C22#575311yescharacteristic
538C22C2×#93011yescharacteristic
539C4—22no—
540C4—42no—
541C4—122no—
542C4—122no—
543C4—241no—
544C4—242no—
545C4—242no—
546C4#511414yescomplemented
547C4#554811yescharacteristic
548C4#556211yescharacteristic
549C4D4×#6711yescharacteristic
550C4#583411yescharacteristic
551C3#2538511yescharacteristic, complemented, Sylow
552C2—44no—
553C2—242no—
554C2D4×#20411yescharacteristic
555C2#7943611yescharacteristic
556C2#7944311yescharacteristic
557C2#7944711yescharacteristic
558C2#8240911yescharacteristic
559C2C2×#511411yescharacteristic
560C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C8×D4C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC2×C8×D4D3
hypocenterC3#25385
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C8×D4C2semidirect
2. C2×#5114C2semidirect
4. #82379C2semidirect
5. C2×#4841C2semidirect
6. #71972C2semidirect
7. #70454C2semidirect
8. #69962C2semidirect
9. #69936C2semidirect
10. #69771C2semidirect
11. C4×#821C2semidirect
47. C2×C8×D4D3semidirect
103. C3×C4⋊C8D4semidirect
109. #821D4semidirect
150. C2×C3×D4SD16semidirect
267. C4⋊C8#67semidirect
359. C2×D4#204semidirect
436. C3×C4#465semidirect
546. C4#5114semidirect
551. C3#25385semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×D42
C2×#51142
#824172
#823792
C2×#48412
#719722
#704542
#699622
#699362
#697712
C4×#8212
#253853