Finite Group Databasebeta

← #10828457

Subgroups — #10828457

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×#4777C212yesmaximal, complemented
3D3×#389C211yesmaximal, characteristic, complemented
4C3×#2041C211yesmaximal, characteristic, complemented
5#71762C211yesmaximal, characteristic, complemented
6#70200C212yesmaximal, complemented
7#70170C212yesmaximal, complemented
8#70169C212yesmaximal, complemented
9#69605C211yesmaximal, characteristic, complemented
10#68664C211yesmaximal, characteristic, complemented
11#68633C211yesmaximal, characteristic, complemented
12#24061—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×D3×C4⋊C4—21no—
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C3×#389C2211yescharacteristic
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#817C2212yes—
33#4984C2211yescharacteristic
34#4976—21no—
35#4972C2211yescharacteristic
36C2×#808C2212yes—
37#4928C2211yescharacteristic
38#4777—28no—
39D3×#95—21no—
40#4647—21no—
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#481—32no—
48C2×#389—31no—
49#2058—31no—
50#2058—31no—
51#2041—31no—
52#2041D311yescharacteristic, complemented
53#1874—32no—
54#1864—32no—
55#1864—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×D3—21no—
66C22×C4×D3—21no—
67C22×C4×D3D411yescharacteristic
68C22×C4×D3D411yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×Dic6—21no—
71C22×C3×C8C2312yes—
72C3×#127—24no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79#938—28no—
80#938—24no—
81#938D418yes—
82#936—28no—
83D4×Dic3—28no—
84D4×Dic3—24no—
85D4×Dic3—24no—
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×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3D411yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99D3×C4⋊C4—22no—
100D3×C4⋊C4—22no—
101D3×C4⋊C4—24no—
102D3×C4⋊C4—44no—
103C42×D3—24no—
104#869—24no—
105C3×#102—24no—
106C3×#102—24no—
107C3×#95—21no—
108#832—24no—
109#832—24no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113(C2×D3)⋊C8—24no—
114#817—24no—
115#808—24no—
116C22×#41—61no—
117#532—68no—
118C2×C4⋊Q8—31no—
119C2×#127C2×D311yescharacteristic
120C2×#124—61no—
121C2×#121—32no—
122C2×#121—61no—
123C2×C4×D4—32no—
124C2×C4×D4—61no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—61no—
128C22×C42—31no—
129#481—68no—
130C2×#106—32no—
131C2×#106—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102C2×D312yes—
135C2×#102C2×D312yes—
136C2×C22⋊C8—32no—
137#389—31no—
138#389—31no—
139#389—31no—
140#389—61no—
141#389—61no—
142#389C2×D311yescharacteristic
143C2×#95—61no—
144#340—32no—
145#340—32no—
146#336—32no—
147#336—31no—
148#336C2×D311yescharacteristic
149C24×C3—22no—
150C23×D3C2×D411yescharacteristic
151C2×C3×D4—24no—
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4C2×D414yes—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×#67—28no—
169C2×#67—44no—
170C2×#67—44no—
171C22×Dic3—28no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—44no—
176C22×Dic3—44no—
177C22×Dic3C2×D411yescharacteristic
178C22×Dic3C2×D411yescharacteristic
179C22×Dic3C2×D411yescharacteristic
180#237—432no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—22no—
186C2×C4×D3—24no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—41no—
190C2×C4×D3—42no—
191C2×C4×D3—42no—
192C2×C4×D3—42no—
193C2×C4×D3—42no—
194C2×C4×D3—42no—
195C2×C4×D3SD1614yes—
196C2×C4×D3C2×D414yes—
197C2×Dic6—24no—
198C2×Dic6—42no—
199C2×Dic6—42no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4#4112yescomplemented
207C3×C42—22no—
208#217—28no—
209#217—28no—
210C4⋊Dic3—24no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3—42no—
215C4⋊Dic3C2×D414yes—
216C4⋊Dic3#4112yescomplemented
217Dic3⋊C4—24no—
218Dic3⋊C4—42no—
219Dic3⋊C4—42no—
220Dic3⋊C4—42no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C4×Dic3—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×#41—68no—
228C2×#41—124no—
229C2×#41—124no—
230C22×Q8—61no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—62no—
234C22×D4C22×D312yes—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—61no—
239C23×C4—61no—
240C23×C4—62no—
241C22×C8—32no—
242C22×C8C22×D312yes—
243C4⋊Q8—64no—
244#127—24no—
245#124—68no—
246#121—38no—
247#121—64no—
248#121—68no—
249C4×D4—64no—
250C4×D4—68no—
251C4×D4—64no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—124no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—31no—
271C2×C42—31no—
272C2×C42—31no—
273C2×C42—64no—
274C2×C42C22×D311yescharacteristic
275#106—64no—
276#106—64no—
277#102—24no—
278#102—24no—
279#102—64no—
280#102—64no—
281C22⋊C8—64no—
282#95—21no—
283#95—61no—
284#95—61no—
285#95—61no—
286C23×C3—24no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3C22×D411yescharacteristic
294C22×D3—21no—
295C22×D3—21no—
296C22×D3—21no—
297C22×D3—21no—
298C22×D3#12012yes—
299C22×D3C2×SD1612yes—
300C22×D3AGL(1,ℤ/8)12yes—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—48no—
304C3×D4—48no—
305C3×D4—42no—
306C3×D4—42no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4C2×SD1612yes—
318C2×C3×C4AGL(1,ℤ/8)12yes—
319C2×C3×C4C22×D412yes—
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C42−1+412yes—
323#67—416no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—24no—
330C2×Dic3—22no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—416no—
334C2×Dic3—42no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3#12012yes—
342C2×Dic3C2×SD1612yes—
343C4×D3—28no—
344C4×D3—24no—
345C4×D3—24no—
346C4×D3—44no—
347C4×D3—48no—
348C4×D3—84no—
349Dic6—48no—
350C3×C8—42no—
351C3⋊C8—42no—
352C24—22no—
353C24—31no—
354#41—1232no—
355C2×Q8—64no—
356C2×Q8—122no—
357C2×Q8—122no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—68no—
370C2×D4—124no—
371C2×D4—124no—
372C2×D4D3×D414yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—34no—
378C22×C4—34no—
379C22×C4—31no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—64no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—62no—
387C22×C4—68no—
388C22×C4—61no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—121no—
393C22×C4—124no—
394C22×C4—122no—
395C22×C4—122no—
396C22×C4—122no—
397C22×C4—122no—
398C22×C4—122no—
399C22×C4—124no—
400C22×C4D3×D411yescharacteristic
401C22×C4D3×D411yescharacteristic
402C22×C4C23×D311yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—34no—
412C4⋊C4—32no—
413C4⋊C4—42no—
414C4⋊C4—64no—
415C4⋊C4—64no—
416C4⋊C4—62no—
417C4⋊C4—61no—
418C4⋊C4—62no—
419C4⋊C4—122no—
420C4⋊C4—122no—
421C4⋊C4—122no—
422C4⋊C4—122no—
423C4⋊C4#23712yescomplemented
424C22⋊C4—68no—
425C22⋊C4—68no—
426C42—22no—
427C42—62no—
428C42—62no—
429C42—62no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#52511yescharacteristic
439C22×C3#53611yescharacteristic
440C22×C3C22×SD1611yescharacteristic
441C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
442C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
443C22×C3#57511yescharacteristic
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—22no—
447C2×D3—22no—
448C2×D3—22no—
449C2×D3—22no—
450C2×D3#45014yes—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—81no—
455C3×C4#54712yes—
456C3×C4#54712yes—
457Dic3—22no—
458Dic3—22no—
459Dic3—42no—
460Dic3—44no—
461Dic3—41no—
462Dic3—81no—
463C23—24no—
464C23—32no—
465C23—32no—
466C23—32no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—61no—
474C23—61no—
475C23—61no—
476C23—61no—
477C23C2×D3×D411yescharacteristic
478Q8—128no—
479D4—24no—
480D4—24no—
481D4—48no—
482D4—48no—
483D4—42no—
484D4—42no—
485D4—1216no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—32no—
492C2×C4—32no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—68no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—124no—
511C2×C4—128no—
512C2×C4—1216no—
513C2×C4—121no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—244no—
521C2×C4#91112yes—
522C2×C4D3×SD1612yes—
523C2×C4C2×D3×D412yes—
524C2×C4C2×#23711yescharacteristic
525C2×C4D3×#4111yescharacteristic
526C2×C4#101012yes—
527C8—42no—
528C8—122no—
529C2×C3—84no—
530C2×C3#240611yescharacteristic
531C2×C3C2×#45011yescharacteristic
532C2×C3#300611yescharacteristic
533C2×C3#310311yescharacteristic
534C2×C3#310911yescharacteristic
535C2×C3#322711yescharacteristic
536C2×C3#323411yescharacteristic
537D3—24no—
538C22—34no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—62no—
543C22—62no—
544C22—62no—
545C22—62no—
546C22—62no—
547C22—62no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22D3×#12011yescharacteristic
553C22#560611yescharacteristic
554C22#567511yescharacteristic
555C22C2×#91111yescharacteristic
556C22C2×D3×SD1611yescharacteristic
557C22D3×AGL(1,ℤ/8)11yescharacteristic
558C22#577311yescharacteristic
559C4—22no—
560C4—22no—
561C4—42no—
562C4—62no—
563C4—62no—
564C4—81no—
565C4—124no—
566C4—121no—
567C4—122no—
568C4—241no—
569C4#561112yes—
570C4#565512yes—
571C3#2406111yescharacteristic, complemented, Sylow
572C2—64no—
573C2—84no—
574C2#7710711yescharacteristic
575C2#8000311yescharacteristic
576C2D3×#45011yescharacteristic
577C2#8031411yescharacteristic
578C2#8036011yescharacteristic
579C2#8157211yescharacteristic
580C2#8172111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenter#2041D3
hypocenterC3#24061
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×#4777C2semidirect
3. D3×#389C2semidirect
4. C3×#2041C2semidirect
5. #71762C2semidirect
6. #70200C2semidirect
7. #70170C2semidirect
8. #70169C2semidirect
9. #69605C2semidirect
10. #68664C2semidirect
11. #68633C2semidirect
52. #2041D3semidirect
206. C3×C4⋊C4#41semidirect
216. C4⋊Dic3#41semidirect
423. C4⋊C4#237semidirect
571. C3#24061semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#47772
D3×#3892
C3×#20412
#717622
#702002
#701702
#701692
#696052
#686642
#686332
#240613