Finite Group Databasebeta

← #10829947

Subgroups — #10829947

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic, complemented
3C2×#4843C211yesmaximal, characteristic
4#74369C211yesmaximal, characteristic, complemented
5C3×#1803C211yesmaximal, characteristic, complemented
6#70472C211yesmaximal, characteristic, complemented
7#70277C212yesmaximal, complemented
8#70245C212yesmaximal, complemented
9#69580C211yesmaximal, characteristic, complemented
10#68897C212yesmaximal, complemented
11#68895C212yesmaximal, complemented
12#24186—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947C2212yes—
15C22×#215C2211yescharacteristic
16C22×D24C2212yes—
17#5585—24no—
18#5585—24no—
19C2×#897—21no—
20C2×#894—22no—
21C2×#893C2212yes—
22C2×#875—21no—
23C2×#874C2211yescharacteristic
24C2×C4×D12—21no—
25C2×C4×D12C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29C3×#384C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2212yes—
32C4×#215—24no—
33#4994—22no—
34#4988C2212yes—
35#4987—21no—
36#4984C2212yes—
37C2×#809C2212yes—
38C2×#809C2211yescharacteristic
39C2×#809C2212yes—
40C2×#808C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4843—24no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×C4×SD16—31no—
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1957—31no—
54#1879—32no—
55#1864—32no—
56#1803D311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×D12C2311yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3—21no—
64C22×Dic6D411yescharacteristic
65C22×C3×C8C2312yes—
66C3×C4×Q8—22no—
67C3×C4×Q8D414yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75#947—24no—
76#947—24no—
77C2×#215—22no—
78C2×#215—22no—
79C2×#215—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×Dic3⋊C4D412yes—
90C2×C4×Dic3C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C2×D24—24no—
94C2×D24—24no—
95C2×D24—42no—
96#897—44no—
97#894—216no—
98#893—24no—
99#893—24no—
100#875—28no—
101#874—22no—
102#874D414yes—
103C4×D12—28no—
104C4×D12—22no—
105C4×D12—22no—
106C4×Dic6—22no—
107C4×Dic6—22no—
108C3×C4⋊C8—22no—
109C3×#103—42no—
110#835—22no—
111C3⋊#95—22no—
112#821—24no—
113#809—22no—
114#809—24no—
115#809—42no—
116#808—22no—
117C4×C3⋊C8—22no—
118C22×SD16—31no—
119C22×D8—32no—
120#559—64no—
121#559—64no—
122C2×#127—31no—
123C2×#124—32no—
124C2×#124—61no—
125C2×#123—62no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—32no—
130C2×C4×D4—31no—
131C2×C4×D4—61no—
132#459—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—31no—
137C2×#103C2×D312yes—
138C2×#102—32no—
139C2×#102—32no—
140C2×#102—31no—
141C2×#102—32no—
142C2×C4×C8—31no—
143#396—62no—
144#390—32no—
145#389—32no—
146#384C2×D311yescharacteristic
147#380—61no—
148#340C2×D312yes—
149#336—32no—
150#336—32no—
151C23×D3—21no—
152C23×D3—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8C2×D412yes—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3C2×D412yes—
167C22×Dic3C2×D412yes—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—21no—
173C2×D12—22no—
174C2×D12—21no—
175C2×D12—22no—
176C2×D12—24no—
177C2×D12—22no—
178C2×D12—21no—
179C2×D12—24no—
180C2×D12—42no—
181C2×D12—42no—
182C2×D12—81no—
183C2×D12—81no—
184C2×D12C2×D414yes—
185C2×D12#4112yes—
186C2×C4×D3—28no—
187C2×C4×D3—42no—
188C2×C4×D3—42no—
189C2×C4×D3—42no—
190C2×C4×D3—82no—
191C2×C4×D3—82no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—81no—
204C3×C4⋊C4C2×D412yes—
205C3×C42—24no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208#215—28no—
209(C2×D3)⋊C4—28no—
210(C2×D3)⋊C4—28no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—44no—
213(C2×D3)⋊C4—42no—
214(C2×D3)⋊C4—82no—
215C4⋊Dic3—22no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—44no—
219Dic3⋊C4—24no—
220C4×Dic3—22no—
221C4×Dic3—22no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8#4112yes—
228D24—48no—
229C22×Q8—31no—
230C22×Q8C22×D311yescharacteristic
231C22×D4—32no—
232C22×D4—31no—
233C22×D4—61no—
234C22×D4—61no—
235C23×C4—62no—
236C23×C4—61no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×SD16—62no—
240C2×D8—64no—
241C2×D8—64no—
242C2×D8—122no—
243C22×C8—31no—
244C22×C8—31no—
245C22×C8C22×D312yes—
246#127—34no—
247#127—62no—
248#124—68no—
249#124—64no—
250#124—64no—
251#123—616no—
252C22⋊Q8—124no—
253C4×Q8—22no—
254C4×Q8—62no—
255C4×Q8—62no—
256C4×Q8D1214yes—
257C4×D4—68no—
258C4×D4—64no—
259C4×D4—64no—
260C4×D4—62no—
261C4×D4—62no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—62no—
277C2×C42—21no—
278C2×C42—32no—
279C2×C42C22×D311yescharacteristic
280#106—62no—
281C4⋊C8—22no—
282#103—42no—
283#103—62no—
284#102—64no—
285#102—64no—
286#102—62no—
287#102—122no—
288C4×C8—62no—
289#95—62no—
290C23×C3C22×D411yescharacteristic
291C22×D3—24no—
292C22×D3—22no—
293C22×D3—41no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—42no—
299C22×D3—41no—
300C22×D3—42no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—41no—
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—41no—
308C3×Q8—41no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4#12012yes—
321C2×C3×C4#12012yes—
322C2×C3×C4#13512yes—
323C2×C3×C4AGL(1,ℤ/8)12yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329C2×C3×C42−1+411yescharacteristic
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339D12—22no—
340D12—22no—
341D12—24no—
342D12—24no—
343D12—42no—
344D12—41no—
345D12—42no—
346D12—41no—
347D12—48no—
348D12—84no—
349C4×D3—88no—
350C4×D3—84no—
351Dic6—44no—
352C3×C8—42no—
353C3⋊C8—22no—
354C3⋊C8—41no—
355C24—61no—
356C24—62no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—62no—
363C2×Q8—62no—
364C2×Q8—62no—
365C2×Q8C2×D1212yes—
366C2×D4—34no—
367C2×D4—32no—
368C2×D4—64no—
369C2×D4—64no—
370C2×D4—61no—
371C2×D4—62no—
372C2×D4—61no—
373C2×D4—62no—
374C2×D4—61no—
375C2×D4—62no—
376C2×D4—61no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—61no—
380C2×D4—122no—
381C2×D4—122no—
382C2×D4—241no—
383C2×D4—241no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—68no—
390C22×C4—61no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4—242no—
395C22×C4—242no—
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D411yescharacteristic
398C22×C4C23×D311yescharacteristic
399SD16—68no—
400D8—128no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—32no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—62no—
416C4⋊C4—64no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—81no—
420C4⋊C4—124no—
421C4⋊C4C2×D1212yes—
422C22⋊C4—68no—
423C22⋊C4—68no—
424C22⋊C4—68no—
425C22⋊C4—124no—
426C22⋊C4—122no—
427C22⋊C4—242no—
428C42—24no—
429C42—62no—
430C42—62no—
431C42C2×D1212yes—
432C42D3×D412yes—
433C22×C3C2×#12011yescharacteristic
434C22×C3#54811yescharacteristic
435C22×C3#55011yescharacteristic
436C22×C3C2×#13511yescharacteristic
437C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
438C22×C3#57611yescharacteristic
439C22×C3#57711yescharacteristic
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—82no—
447C2×D3—82no—
448C2×D3—81no—
449C2×D3—82no—
450C2×D3—81no—
451C2×D3—82no—
452C2×D3—81no—
453C2×D3—81no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—82no—
458C3×C4#44914yes—
459C3×C4#52511yescharacteristic
460C3×C4#53711yescharacteristic
461C3×C4#54611yescharacteristic
462C3×C4#54711yescharacteristic
463Dic3—42no—
464Dic3—42no—
465Dic3—81no—
466C23—62no—
467C23—64no—
468C23—122no—
469C23—121no—
470C23—121no—
471C23—122no—
472C23—121no—
473C23—122no—
474C23—121no—
475C23—122no—
476C23—122no—
477C23—121no—
478C23—122no—
479C23—121no—
480C23C2×D3×D411yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485Q8—124no—
486D4—62no—
487D4—62no—
488D4—64no—
489D4—64no—
490D4—121no—
491D4—122no—
492D4—122no—
493D4—128no—
494D4—121no—
495D4—244no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—248no—
517C2×C4—244no—
518C2×C4#88212yes—
519C2×C4#88212yes—
520C2×C4#91412yes—
521C2×C4#91912yes—
522C2×C4C22×D1211yescharacteristic
523C2×C4C2×D3×D411yescharacteristic
524C2×C4#100411yescharacteristic
525C2×C4#100711yescharacteristic
526C2×C4D3×#4111yescharacteristic
527C2×C4D3×#4111yescharacteristic
528C8—42no—
529C8—62no—
530C8—121no—
531C2×C3#240711yescharacteristic
532C2×C3C2×#44911yescharacteristic
533C2×C3#301211yescharacteristic
534C2×C3#336111yescharacteristic
535C2×C3#337311yescharacteristic
536C2×C3#339411yescharacteristic
537C2×C3#339911yescharacteristic
538D3—82no—
539D3—84no—
540C22—122no—
541C22—124no—
542C22—124no—
543C22—122no—
544C22—124no—
545C22—122no—
546C22—241no—
547C22—241no—
548C22—242no—
549C22—242no—
550C22—241no—
551C22—242no—
552C22—241no—
553C22—242no—
554C22C2×#88211yescharacteristic
555C22#559911yescharacteristic
556C22#563311yescharacteristic
557C22C2×#91411yescharacteristic
558C22C2×#91911yescharacteristic
559C22#577411yescharacteristic
560C22#577811yescharacteristic
561C4—22no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—122no—
566C4—122no—
567C4—241no—
568C4#481014yes—
569C4#551011yescharacteristic
570C4#552411yescharacteristic
571C4#561211yescharacteristic
572C4#563111yescharacteristic
573C3#2418611yescharacteristic, complemented, Sylow
574C2—244no—
575C2—242no—
576C2#7401511yescharacteristic
577C2C2×#481011yescharacteristic
578C2#7888611yescharacteristic
579C2#8037911yescharacteristic
580C2#8047211yescharacteristic
581C2#8103811yescharacteristic
582C2#8113411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1803C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
hypercenter#1803D3
hypocenterC3#24186
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×#5585C2semidirect
2. C2×C4×#215C2semidirect
4. #74369C2semidirect
5. C3×#1803C2semidirect
6. #70472C2semidirect
7. #70277C2semidirect
8. #70245C2semidirect
9. #69580C2semidirect
10. #68897C2semidirect
11. #68895C2semidirect
56. #1803D3semidirect
573. C3#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C2×C4×#2152
C2×#48432
#743692
C3×#18032
#704722
#702772
#702452
#695802
#688972
#688952
#241863