Finite Group Databasebeta

← #10845935

Subgroups — #10845935

Order 768.

1154 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×C3×C8×D4C211yesmaximal, characteristic, complemented
2C2×#5114C212yesmaximal, complemented
3#82378C211yesmaximal, characteristic
4#82376C211yesmaximal, characteristic, complemented
5C2×#4848C211yesmaximal, characteristic, complemented
6#71973C212yesmaximal, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69962C212yesmaximal, complemented
9#69935C212yesmaximal, complemented
10#69772C211yesmaximal, characteristic, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#25387—31nomaximal, Sylow
13C23×C3×C8C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#940—21no—
16C2×#930C2212yes—
17C2×#930C2212yes—
18C2×#924C2212yes—
19C22×#204C2212yes—
20C2×#874C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
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×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#817C2212yes—
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#4988—21no—
36#4972—21no—
37#4960—22no—
38#4948—22no—
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4848—24no—
42#4551C2211yescharacteristic
43#4551C2212yes—
44#4551C2211yescharacteristic
45C2×#465—32no—
46C2×#460—31no—
47C2×C8×D4D311yescharacteristic, complemented
48#2541—31no—
49#2538—31no—
50#2034—32no—
51#2000—32no—
52#1957—31no—
53#1948—31no—
54#1893—32no—
55C4×#103—31no—
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×#67—22no—
59C22×#67—22no—
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2312yes—
63C22×Dic6C2311yescharacteristic
64C22×C3×C8—24no—
65C22×C3×C8—24no—
66C22×C3×C8—42no—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2312yes—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C22⋊C4D412yes—
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×#217—22no—
77#940—44no—
78#930—24no—
79#930—24no—
80#930—42no—
81#930—42no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90#924—24no—
91#924—42no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3—21no—
96C2×#204—24no—
97C2×#204—24no—
98C2×#204—42no—
99#874—22no—
100#874—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×C4⋊C8D414yescomplemented
104C3×C22⋊C8—24no—
105C3×C4×C8—22no—
106C3⋊#95—21no—
107C3⋊#95—22no—
108#821—24no—
109#821—24no—
110#821—42no—
111#817—24no—
112#816—22no—
113#816D414yescomplemented
114C22×SD16—32no—
115C23×C8C2×D312yes—
116C2×#127—31no—
117C2×#127—61no—
118C2×C22⋊Q8—32no—
119C2×#121—32no—
120C2×#121—32no—
121C2×C4×Q8—31no—
122C2×C4×D4C2×D311yescharacteristic
123#465—68no—
124#460—64no—
125C8×D4—24no—
126C2×#106—32no—
127C2×C4⋊C8C2×D311yescharacteristic
128C2×#103—31no—
129C2×#103—31no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×C22⋊C8C2×D312yes—
134C2×C4×C8C2×D311yescharacteristic
135#396—62no—
136#392—62no—
137#390—61no—
138#390—61no—
139#389—61no—
140#384—31no—
141#380—32no—
142C4×C4⋊C4—31no—
143#340—31no—
144#340—32no—
145#340—31no—
146C24×C3C2×D412yes—
147C23×D3—22no—
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—44no—
164C2×#67—44no—
165C2×#67—82no—
166C2×#67—82no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—42no—
179C2×D12—42no—
180C2×D12—42no—
181C2×D12—42no—
182C2×D12—81no—
183C2×D12—81no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6C2×D412yes—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—48no—
198C2×C3×C8C2×D412yes—
199C2×C3×C8C2×D414yes—
200C3×C4⋊C4—22no—
201C3×C22⋊C4—24no—
202C3×C42C2×D412yes—
203C3×C42#4112yes—
204#217—44no—
205#217—82no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—44no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3—42no—
215C4⋊Dic3—81no—
216C4⋊Dic3C2×D412yes—
217C4⋊Dic3#4112yes—
218Dic3⋊C4—44no—
219Dic3⋊C4—42no—
220C4×Dic3—42no—
221C4×Dic3—42no—
222#204—48no—
223C22×Q8—31no—
224C22×D4—32no—
225C22×D4—62no—
226C22×D4—62no—
227C22×D4—61no—
228C22×D4—61no—
229C22×D4D1211yescharacteristic
230C23×C4C22×D312yes—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C22×C8—24no—
235C22×C8—24no—
236C22×C8—42no—
237C22×C8C22×D311yescharacteristic
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D312yes—
240#127—62no—
241#127—62no—
242#127—124no—
243C22⋊Q8—64no—
244C22⋊Q8—122no—
245#121—64no—
246#121—64no—
247#121—122no—
248#121—122no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—22no—
252C4×D4—22no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4D1211yescharacteristic
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4D1212yes—
267C2×C42—61no—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270#106—64no—
271C4⋊C8#6714yescomplemented
272#103—34no—
273#103—62no—
274#102—64no—
275#102—64no—
276#102—122no—
277C22⋊C8—24no—
278C4×C8—22no—
279#95—61no—
280#95—62no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3C2×SD1614yes—
288C23×C3C22×D411yescharacteristic
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C3×D4—28no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4#12112yes—
307C2×C3×C4#12112yes—
308C2×C3×C4C2×SD1612yes—
309C2×C3×C4#13512yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316#67—88no—
317#67—88no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333D12—42no—
334D12—42no—
335D12—42no—
336D12—42no—
337D12—84no—
338D12—84no—
339Dic6—22no—
340Dic6—22no—
341Dic6—41no—
342Dic6—41no—
343C3×C8—24no—
344C3×C8—22no—
345C3×C8—41no—
346C24—62no—
347C24C2×D1212yes—
348C2×Q8—32no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—124no—
363C2×D4—124no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—241no—
371C2×D4—241no—
372C2×D4—242no—
373C2×D4—242no—
374C2×D4#20414yescomplemented
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4C2×D1212yes—
387C22×C4C2×D1211yescharacteristic
388C22×C4C2×D1211yescharacteristic
389C22×C4C23×D311yescharacteristic
390SD16—128no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—48no—
399C2×C8D3×D412yes—
400C2×C8C2×#6714yes—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—32no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—61no—
410C4⋊C4—124no—
411C4⋊C4—122no—
412C4⋊C4—122no—
413C4⋊C4—241no—
414C22⋊C4—24no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—124no—
418C22⋊C4—242no—
419C42—122no—
420C42—122no—
421C42#23512yes—
422C42C2×#6712yes—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3C2×#12111yescharacteristic
431C22×C3D4211yescharacteristic
432C22×C3#54611yescharacteristic
433C22×C3C22×SD1611yescharacteristic
434C22×C3C2×#13511yescharacteristic
435C22×C3#57611yescharacteristic
436C22×C3#57711yescharacteristic
437C2×D3—82no—
438C2×D3—82no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C2×D3—82no—
443C2×D3—82no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4#46514yescomplemented
447C3×C4#53211yescharacteristic
448C3×C4#54011yescharacteristic
449C3×C4D4211yescharacteristic
450C3×C4#55011yescharacteristic
451Dic3—42no—
452Dic3—42no—
453Dic3—42no—
454Dic3—82no—
455Dic3—81no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23C2×#20414yes—
470C23C22×D1211yescharacteristic
471Q8—62no—
472Q8—62no—
473Q8—121no—
474Q8—121no—
475D4—28no—
476D4—122no—
477D4—122no—
478D4—122no—
479D4—122no—
480D4—244no—
481D4—248no—
482D4—248no—
483D4—244no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—122no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—122no—
506C2×C4—121no—
507C2×C4—122no—
508C2×C4C2×#20412yes—
509C2×C4#90412yes—
510C2×C4#93012yes—
511C2×C4#93012yes—
512C2×C4C2×#23511yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4#100411yescharacteristic
515C2×C4D3×#4111yescharacteristic
516C2×C4#100911yescharacteristic
517C2×C4C22×#6711yescharacteristic
518C8—22no—
519C8—24no—
520C8—41no—
521C2×C3—44no—
522C2×C3#242811yescharacteristic
523C2×C3C2×#46511yescharacteristic
524C2×C3#306211yescharacteristic
525C2×C3D4×SD1611yescharacteristic
526C2×C3#328711yescharacteristic
527C2×C3#329511yescharacteristic
528C2×C3#330411yescharacteristic
529D3—162no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—242no—
538C22—242no—
539C22—242no—
540C22—242no—
541C22—242no—
542C22—242no—
543C22—242no—
544C22D4×D1211yescharacteristic
545C22#555411yescharacteristic
546C22C22×#20411yescharacteristic
547C22C2×#90411yescharacteristic
548C22#575111yescharacteristic
549C22#575211yescharacteristic
550C22C2×#93011yescharacteristic
551C4—22no—
552C4—42no—
553C4—122no—
554C4—122no—
555C4—122no—
556C4—242no—
557C4—241no—
558C4#511414yescomplemented
559C4#554811yescharacteristic
560C4#556211yescharacteristic
561C4D4×#6711yescharacteristic
562C4#583411yescharacteristic
563C3#2538711yescharacteristic, complemented, Sylow
564C2—44no—
565C2—482no—
566C2D4×#20411yescharacteristic
567C2#7943911yescharacteristic
568C2#7944611yescharacteristic
569C2#7945111yescharacteristic
570C2#8240911yescharacteristic
571C2C2×#511411yescharacteristic
572C2#8380211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C8×D4C2
socleC23×C3C22×D4
prime radical#82378C2
hypercenterC2×C8×D4D3
hypocenterC3#25387
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. #82376C2semidirect
5. C2×#4848C2semidirect
6. #71973C2semidirect
7. #70454C2semidirect
8. #69962C2semidirect
9. #69935C2semidirect
10. #69772C2semidirect
11. C4×#816C2semidirect
47. C2×C8×D4D3semidirect
103. C3×C4⋊C8D4semidirect
113. #816D4semidirect
150. C2×C3×D4SD16semidirect
271. C4⋊C8#67semidirect
374. C2×D4#204semidirect
446. C3×C4#465semidirect
558. C4#5114semidirect
563. C3#25387semidirect

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
#823782
#823762
C2×#48482
#719732
#704542
#699622
#699352
#697722
C4×#8162
#253873