Finite Group Databasebeta

← #10828786

Subgroups — #10828786

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5618C211yesmaximal, characteristic
2C22×#817C211yesmaximal, characteristic, complemented
3C2×#5033C211yesmaximal, characteristic, complemented
4#74027C211yesmaximal, characteristic, complemented
5C3×#1893C211yesmaximal, characteristic, complemented
6#71975C211yesmaximal, characteristic, complemented
7#70176C212yesmaximal, complemented
8#70170C212yesmaximal, complemented
9#69502C211yesmaximal, characteristic, complemented
10#68876C212yesmaximal, complemented
11#68828C212yesmaximal, complemented
12#24091—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C2×#939—21no—
16C2×#938C2211yescharacteristic
17C2×D4×Dic3—21no—
18C2×D4×Dic3C2211yescharacteristic
19C2×C4×#67C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22#5618—24no—
23#5618—24no—
24C2×#892C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#886—21no—
27C2×#886—21no—
28C2×#879—21no—
29C2×C3×#102C2212yes—
30C3×#380C2211yescharacteristic
31C2×#832C2212yes—
32C2×#830C2211yescharacteristic
33C2×(C2×D3)⋊C8C2212yes—
34C2×#817—24no—
35C2×#817C2212yes—
36C2×#817C2212yes—
37#5033—24no—
38#4987C2212yes—
39#4984C2212yes—
40C2×#807C2212yes—
41#4943—21no—
42#4674—21no—
43#4661—21no—
44C3×#340C2212yes—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#553—31no—
49C2×#480—31no—
50C22×#106—31no—
51#2486—31no—
52#2059—32no—
53#2034—32no—
54#1893D311yescharacteristic, complemented
55#1869—31no—
56#1864—32no—
57#1854—32no—
58#1824—31no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67D411yescharacteristic
64C23×Dic3—21no—
65C23×Dic3—21no—
66C23×Dic3—21no—
67C22×C4×D3C2311yescharacteristic
68C22×C3×C8—22no—
69C22×C3×C8—24no—
70C22×C3×C8C2312yes—
71C3×#121—22no—
72C3×#121D414yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×#217—21no—
78C2×#217—21no—
79C2×#217D411yescharacteristic
80#939—28no—
81#938—22no—
82#938—22no—
83D4×Dic3—28no—
84D4×Dic3—22no—
85D4×Dic3—22no—
86C4×#67—22no—
87C4×#67—22no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3D414yes—
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2312yes—
107#892—22no—
108#892—22no—
109#890—22no—
110#890—22no—
111#886—28no—
112#886—44no—
113#879—28no—
114C3×#102—24no—
115C3×#95—21no—
116#832—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#830—22no—
120(C2×D3)⋊C8—24no—
121#817—24no—
122#817—24no—
123#807—24no—
124C23×C8C2×D311yescharacteristic
125#553—64no—
126#553—64no—
127C2×#126—61no—
128C2×#125—31no—
129C2×#123—61no—
130C2×#123—61no—
131C2×#121—31no—
132C2×#121—61no—
133C2×#121C2×D311yescharacteristic
134C2×C4×D4—31no—
135C2×C4×D4—31no—
136C2×C4×D4—61no—
137C2×#117—31no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4—31no—
140#480—64no—
141C2×#107—32no—
142C2×#106—32no—
143C2×#106—32no—
144C2×#106—64no—
145C2×#102—32no—
146C2×#102C2×D312yes—
147C2×C22⋊C8—31no—
148C2×C22⋊C8—32no—
149#398—61no—
150#397—61no—
151#389—32no—
152#381—61no—
153#380—32no—
154#380C2×D311yescharacteristic
155#340—32no—
156#340—32no—
157#340C2×D312yes—
158#336—32no—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C23×D3C2×D411yescharacteristic
162C2×C3×D4—24no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4C2×D412yes—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2411yescharacteristic
177C2×#67—22no—
178C2×#67—22no—
179C2×#67—22no—
180C2×#67—24no—
181C2×#67—82no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—22no—
187C22×Dic3—42no—
188C22×Dic3—42no—
189C22×Dic3—42no—
190C22×Dic3—82no—
191C22×Dic3—82no—
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C3×C8—28no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8#4114yes—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4C2×D412yes—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—42no—
209#217—24no—
210#217—24no—
211#217—22no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—22no—
214(C2×D3)⋊C4—44no—
215C4⋊Dic3—24no—
216C4⋊Dic3—24no—
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—44no—
222C4⋊Dic3—42no—
223C4⋊Dic3C2×D412yes—
224C4⋊Dic3C2×D412yes—
225Dic3⋊C4—24no—
226Dic3⋊C4—22no—
227C4×Dic3—24no—
228C4×Dic3—21no—
229C4×Dic3—21no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C22×D4—21no—
234C22×D4—31no—
235C22×D4—61no—
236C22×D4C22×D311yescharacteristic
237C23×C4—31no—
238C23×C4—61no—
239C23×C4—61no—
240C23×C4—61no—
241C23×C4C22×D311yescharacteristic
242C22×C8—22no—
243C22×C8—24no—
244C22×C8—32no—
245C22×C8C22×D312yes—
246#126—68no—
247#125—62no—
248#125—62no—
249#123—68no—
250#123—124no—
251#121—22no—
252#121—62no—
253#121—68no—
254#121—62no—
255#121D1214yes—
256C4×D4—62no—
257C4×D4—68no—
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—62no—
261#117—62no—
262#117—62no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4C22×D312yes—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—31no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—62no—
286C2×C42—31no—
287C2×C42—61no—
288#107—64no—
289#106—64no—
290#106—64no—
291#102—24no—
292#102—64no—
293C22⋊C8—62no—
294C22⋊C8—64no—
295#95—21no—
296#95—62no—
297#95—62no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—42no—
305C23×C3#12012yes—
306C23×C3#13512yes—
307C23×C3AGL(1,ℤ/8)12yes—
308C23×C3C22×D411yescharacteristic
309C22×D3—21no—
310C22×D3—22no—
311C22×D3—21no—
312C22×D3—21no—
313C22×D3—22no—
314C3×D4—24no—
315C3×D4—48no—
316C3×D4—41no—
317C3×D4—41no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4#12012yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4112yes—
332C2×C3×C42−1+412yes—
333#67—44no—
334#67—84no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—22no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—84no—
350C2×Dic3—84no—
351C2×Dic3—84no—
352C4×D3—41no—
353C4×D3—41no—
354C4×D3—42no—
355C3×C8—24no—
356C3⋊C8—42no—
357C24—21no—
358C24—31no—
359C24D3×D411yescharacteristic
360C2×D4—24no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—64no—
370C2×D4—242no—
371C2×D4C2×D1212yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—64no—
382C22×C4—64no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4—122no—
391C22×C4—242no—
392C22×C4—242no—
393C22×C4C2×D1212yes—
394C22×C4D3×D411yescharacteristic
395C22×C4D3×D412yes—
396C22×C4C23×D311yescharacteristic
397C2×C8—28no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—62no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8#23714yes—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—32no—
408C4⋊C4—32no—
409C4⋊C4—64no—
410C4⋊C4—64no—
411C4⋊C4—64no—
412C4⋊C4—62no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—124no—
418C4⋊C4—122no—
419C4⋊C4C2×D1212yes—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—64no—
423C22⋊C4—64no—
424C22⋊C4—64no—
425C22⋊C4—62no—
426C22⋊C4—62no—
427C22⋊C4—124no—
428C42—64no—
429C42—61no—
430C42—61no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3#44914yes—
441C22×C3C2×#12011yescharacteristic
442C22×C3#54712yes—
443C22×C3C2×#13511yescharacteristic
444C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
445C22×C3#57712yes—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—41no—
449C2×D3—41no—
450C2×D3—41no—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4—82no—
454C3×C4#52511yescharacteristic
455C3×C4#53611yescharacteristic
456C3×C4#54712yes—
457Dic3—41no—
458Dic3—41no—
459Dic3—41no—
460Dic3—42no—
461Dic3—42no—
462Dic3—82no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—61no—
471C23—61no—
472C23—61no—
473C23—62no—
474C23—62no—
475C23#88212yes—
476C23#91112yes—
477C23#91612yes—
478C23C2×D3×D411yescharacteristic
479D4—24no—
480D4—48no—
481D4—41no—
482D4—41no—
483D4—124no—
484D4—244no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—61no—
496C2×C4—62no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—121no—
507C2×C4—122no—
508C2×C4—121no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—244no—
513C2×C4—244no—
514C2×C4—244no—
515C2×C4#88212yes—
516C2×C4C22×D1211yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×#23712yes—
519C2×C4#101012yes—
520C8—24no—
521C8—122no—
522C2×C3—24no—
523C2×C3—82no—
524C2×C3#240611yescharacteristic
525C2×C3C2×#44911yescharacteristic
526C2×C3#300911yescharacteristic
527C2×C3#333712yes—
528C2×C3#334512yes—
529D3—81no—
530C22—24no—
531C22—24no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—122no—
540C22—122no—
541C22—121no—
542C22—121no—
543C22—121no—
544C22#478114yes—
545C22C2×#88211yescharacteristic
546C22#561112yes—
547C22C2×#91111yescharacteristic
548C22C2×#91611yescharacteristic
549C22#577012yes—
550C4—22no—
551C4—42no—
552C4—82no—
553C4—121no—
554C4—121no—
555C4—121no—
556C4—122no—
557C4—122no—
558C4—242no—
559C4#555512yes—
560C4#558611yescharacteristic
561C4#564811yescharacteristic
562C3#2409111yescharacteristic, complemented, Sylow
563C2—24no—
564C2—82no—
565C2—241no—
566C2#7468711yescharacteristic
567C2C2×#478111yescharacteristic
568C2#7883511yescharacteristic
569C2#8064912yes—
570C2#8071112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1893C2
socleC23×C3C22×D4
prime radicalC2×#5618C2
hypercenter#1893D3
hypocenterC3#24091
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#817C2semidirect
3. C2×#5033C2semidirect
4. #74027C2semidirect
5. C3×#1893C2semidirect
6. #71975C2semidirect
7. #70176C2semidirect
8. #70170C2semidirect
9. #69502C2semidirect
10. #68876C2semidirect
11. #68828C2semidirect
54. #1893D3semidirect
562. C3#24091semidirect

Maximal subgroups

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

subgroupindex
C2×#56182
C22×#8172
C2×#50332
#740272
C3×#18932
#719752
#701762
#701702
#695022
#688762
#688282
#240913