Finite Group Databasebeta

← #10828465

Subgroups — #10828465

Order 768.

1352 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×#4784C212yesmaximal, complemented
3#75963C211yesmaximal, characteristic, complemented
4C3×#2041C211yesmaximal, characteristic, complemented
5#71762C211yesmaximal, characteristic, complemented
6#70201C212yesmaximal, complemented
7#70169C212yesmaximal, complemented
8#70168C212yesmaximal, complemented
9#69605C211yesmaximal, characteristic, complemented
10#68664C211yesmaximal, characteristic, complemented
11#68632C211yesmaximal, characteristic, complemented
12#24062—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#127C2211yescharacteristic
15C2×#938—21no—
16C2×#938C2212yes—
17C2×#936—21no—
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2212yes—
20#5716—28no—
21C2×#892C2211yescharacteristic
22C2×#892C2211yescharacteristic
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C3×#389C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×#818C2212yes—
32#4989—21no—
33#4984C2211yescharacteristic
34Dic3×C4⋊C4—21no—
35#4972C2211yescharacteristic
36C2×#808C2212yes—
37#4928C2211yescharacteristic
38#4784—28no—
39#4668—21no—
40#4649—21no—
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#482—32no—
48#2296—31no—
49#2058—31no—
50#2057—31no—
51#2041—31no—
52#2041D311yescharacteristic, complemented
53#1875—32no—
54#1864—32no—
55#1862—32no—
56#1817—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×#67—22no—
61C23×Dic3—22no—
62C2×#237—28no—
63C2×#237—44no—
64C2×#237—44no—
65C22×C4×D3D411yescharacteristic
66C22×C4×D3D411yescharacteristic
67C22×C4×D3C2311yescharacteristic
68C22×Dic6—21no—
69C22×C3×C8C2312yes—
70C3×#127—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×#217—22no—
77#938—28no—
78#938—24no—
79#938D418yes—
80#936—28no—
81D4×Dic3—28no—
82D4×Dic3—24no—
83D4×Dic3—24no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3D411yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99#892—22no—
100#892—22no—
101#892—24no—
102C42×D3—24no—
103#869—24no—
104C3×#102—24no—
105C3×#102—24no—
106C3×#95—21no—
107#832—24no—
108#832—24no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112(C2×D3)⋊C8—24no—
113#818—24no—
114#808—24no—
115C22×#41—61no—
116#532—68no—
117C2×C4⋊Q8—31no—
118C2×#127C2×D311yescharacteristic
119C2×#124—61no—
120C2×#121—32no—
121C2×#121—61no—
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124C2×#117—31no—
125C2×#117—31no—
126C22×C42—31no—
127#482—68no—
128C2×#107—32no—
129C2×#106—32no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102C2×D312yes—
133C2×#102C2×D312yes—
134C2×C22⋊C8—32no—
135#389—31no—
136#389—31no—
137#389—31no—
138#389C2×D311yescharacteristic
139#385—61no—
140#385—61no—
141C4×C4⋊C4—61no—
142C4×C4⋊C4—61no—
143#340—32no—
144#340—32no—
145#336—32no—
146#336—31no—
147#336C2×D311yescharacteristic
148C24×C3—22no—
149C23×D3C2×D411yescharacteristic
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4C2×D414yes—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C2×#67—28no—
168C2×#67—44no—
169C2×#67—44no—
170C22×Dic3—28no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—44no—
175C22×Dic3—44no—
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C22×Dic3C2×D411yescharacteristic
179#237—432no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—41no—
189C2×C4×D3C2×D414yes—
190C2×Dic6—24no—
191C2×Dic6—42no—
192C2×Dic6—42no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4#4112yescomplemented
200C3×C42—22no—
201#217—28no—
202#217—28no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—42no—
205C4⋊Dic3—24no—
206C4⋊Dic3—22no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—42no—
210C4⋊Dic3C2×D414yes—
211C4⋊Dic3#4112yescomplemented
212Dic3⋊C4—42no—
213Dic3⋊C4—42no—
214C4×Dic3—24no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—42no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×#41—68no—
223C2×#41—124no—
224C2×#41—124no—
225C22×Q8—61no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—62no—
229C22×D4C22×D312yes—
230C23×C4—31no—
231C23×C4—31no—
232C23×C4—31no—
233C23×C4—62no—
234C22×C8—32no—
235C22×C8C22×D312yes—
236C4⋊Q8—64no—
237#127—24no—
238#124—68no—
239#121—38no—
240#121—64no—
241#121—68no—
242C4×D4—64no—
243C4×D4—68no—
244C4×D4—64no—
245#117—62no—
246#117—62no—
247#117—64no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—61no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C42—31no—
263C2×C42—31no—
264C2×C42—31no—
265C2×C42—64no—
266C2×C42—61no—
267C2×C42—61no—
268C2×C42C22×D311yescharacteristic
269#107—64no—
270#106—64no—
271#102—24no—
272#102—24no—
273#102—64no—
274#102—64no—
275C22⋊C8—64no—
276#95—21no—
277#95—61no—
278#95—61no—
279#95—61no—
280C23×C3—24no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3C22×D411yescharacteristic
288C22×D3—21no—
289C22×D3—21no—
290C22×D3—21no—
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—21no—
294C22×D3#12012yes—
295C3×D4—24no—
296C3×D4—24no—
297C3×D4—48no—
298C3×D4—48no—
299C3×D4—42no—
300C3×D4—42no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4#13512yes—
312C2×C3×C4AGL(1,ℤ/8)12yes—
313C2×C3×C4C22×D412yes—
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C42−1+412yes—
317#67—416no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—24no—
324C2×Dic3—22no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—416no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3#12012yes—
336C2×Dic3AGL(1,ℤ/8)12yes—
337C4×D3—28no—
338C4×D3—24no—
339C4×D3—24no—
340C4×D3—44no—
341Dic6—48no—
342C3×C8—42no—
343C3⋊C8—42no—
344C24—22no—
345C24—31no—
346#41—1232no—
347C2×Q8—64no—
348C2×Q8—122no—
349C2×Q8—122no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—68no—
362C2×D4—124no—
363C2×D4—124no—
364C2×D4D3×D414yes—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—34no—
370C22×C4—31no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4—68no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4—121no—
384C22×C4—124no—
385C22×C4—124no—
386C22×C4D3×D411yescharacteristic
387C22×C4D3×D411yescharacteristic
388C22×C4C23×D311yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—34no—
398C4⋊C4—32no—
399C4⋊C4—42no—
400C4⋊C4—64no—
401C4⋊C4—62no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—122no—
405C4⋊C4—122no—
406C4⋊C4—122no—
407C4⋊C4#23712yescomplemented
408C22⋊C4—64no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—122no—
412C42—22no—
413C42—62no—
414C42—62no—
415C42—64no—
416C42—62no—
417C42—122no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#52511yescharacteristic
427C22×C3#53611yescharacteristic
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3#57511yescharacteristic
432C2×D3—22no—
433C2×D3—22no—
434C2×D3—22no—
435C2×D3—41no—
436C2×D3—41no—
437C2×D3—41no—
438C2×D3—41no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—42no—
442C3×C4—81no—
443C3×C4#54712yes—
444C3×C4#54712yes—
445Dic3—22no—
446Dic3—22no—
447Dic3—42no—
448Dic3—41no—
449Dic3—44no—
450Dic3—81no—
451C23—24no—
452C23—32no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—61no—
460C23—61no—
461C23—61no—
462C23—61no—
463C23—61no—
464C23—61no—
465C23C2×D3×D411yescharacteristic
466Q8—128no—
467D4—24no—
468D4—24no—
469D4—48no—
470D4—48no—
471D4—42no—
472D4—42no—
473D4—1216no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—32no—
480C2×C4—32no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—68no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—64no—
497C2×C4—62no—
498C2×C4—124no—
499C2×C4—1216no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4#91212yes—
508C2×C4#91412yes—
509C2×C4C2×D3×D412yes—
510C2×C4C2×#23711yescharacteristic
511C2×C4D3×#4111yescharacteristic
512C2×C4#101012yes—
513C8—42no—
514C8—122no—
515C2×C3—84no—
516C2×C3#240611yescharacteristic
517C2×C3#300211yescharacteristic
518C2×C3#300311yescharacteristic
519C2×C3#310611yescharacteristic
520C2×C3#311511yescharacteristic
521C2×C3#322811yescharacteristic
522C2×C3#323511yescharacteristic
523D3—42no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—62no—
528C22—62no—
529C22—62no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—121no—
535C22—121no—
536C22—121no—
537C22—121no—
538C22D3×#12011yescharacteristic
539C22#560611yescharacteristic
540C22#567511yescharacteristic
541C22C2×#91211yescharacteristic
542C22C2×#91411yescharacteristic
543C22D3×AGL(1,ℤ/8)11yescharacteristic
544C22#577311yescharacteristic
545C4—22no—
546C4—22no—
547C4—42no—
548C4—62no—
549C4—62no—
550C4—81no—
551C4—124no—
552C4—121no—
553C4—122no—
554C4—241no—
555C4#561112yes—
556C4#565512yes—
557C3#2406211yescharacteristic, complemented, Sylow
558C2—84no—
559C2—122no—
560C2#7710711yescharacteristic
561C2#8000111yescharacteristic
562C2#8014611yescharacteristic
563C2#8031311yescharacteristic
564C2#8036311yescharacteristic
565C2#8157311yescharacteristic
566C2#8172211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenter#2041D3
hypocenterC3#24062
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
2. C2×#4784C2semidirect
3. #75963C2semidirect
4. C3×#2041C2semidirect
5. #71762C2semidirect
6. #70201C2semidirect
7. #70169C2semidirect
8. #70168C2semidirect
9. #69605C2semidirect
10. #68664C2semidirect
11. #68632C2semidirect
52. #2041D3semidirect
199. C3×C4⋊C4#41semidirect
211. C4⋊Dic3#41semidirect
407. C4⋊C4#237semidirect
557. C3#24062semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#47842
#759632
C3×#20412
#717622
#702012
#701692
#701682
#696052
#686642
#686322
#240623