Finite Group Databasebeta

← #10876119

Subgroups — #10876119

Order 768.

1128 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5221C211yesmaximal, characteristic
2#73864C211yesmaximal, characteristic, complemented
3C3×#1786C211yesmaximal, characteristic
4#72128C211yesmaximal, characteristic
5#70183C212yesmaximal, complemented
6#70173C212yesmaximal, complemented
7#70165C212yesmaximal, complemented
8#70155C212yesmaximal, complemented
9#69927C211yesmaximal, characteristic, complemented
10#69910C211yesmaximal, characteristic, complemented
11#69504C211yesmaximal, characteristic, complemented
12#27582—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#217—21no—
15C22×#217—21no—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4×Dic3C2211yescharacteristic
18C2×#877—21no—
19C2×C3×#102C2212yes—
20C2×C3×#102C2212yes—
21C2×C3×C22⋊C8C2211yescharacteristic
22C3×#380C2211yescharacteristic
23#5221—21no—
24#5221—21no—
25#5221C2212yes—
26#5221C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5212C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#818C2212yescomplemented
32#4975C2211yescharacteristic, complemented
33#4975C2211yescharacteristic, complemented
34Dic3×C4⋊C4C2211yescharacteristic, complemented
35Dic3×C4⋊C4C2211yescharacteristic, complemented
36C2×#807C2212yescomplemented
37#4942—21no—
38#4650—21no—
39#4649—21no—
40#4551C2212yescomplemented
41#4550C2212yescomplemented
42#4550C2212yescomplemented
43#4529C2212yescomplemented
44C2×#390—31no—
45#2285—31no—
46#2093—32no—
47#2071—32no—
48#1925—32no—
49#1925—32no—
50#1852—31no—
51#1852—31no—
52#1817—31no—
53#1786—31no—
54#1786D311yescharacteristic
55C25×C3—21no—
56C22×C3×D4—24no—
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—42no—
61C22×C3×D4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—21no—
64C23×Dic3D411yescharacteristic
65C23×Dic3D411yescharacteristic
66C22×C3×C8C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—44no—
76C2×#217—44no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—42no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8C2312yes—
95#877—44no—
96C3×#102—24no—
97C3×#102—42no—
98C3×C22⋊C8—22no—
99C3×#95—21no—
100#832—24no—
101#832—42no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105#830—22no—
106#818—42no—
107#807—42no—
108C23×D4C2×D311yescharacteristic
109C2×#117—61no—
110C22×C4⋊C4—31no—
111C22×C22⋊C4—61no—
112C22×C22⋊C4—61no—
113C22×C42—31no—
114C2×#107—32no—
115C2×#107—32no—
116C2×#102—32no—
117C2×#102—32no—
118C2×#102C2×D312yes—
119C2×#102C2×D312yes—
120C2×C22⋊C8—31no—
121C2×C22⋊C8C2×D311yescharacteristic
122#390—32no—
123#390—32no—
124#390—61no—
125#390—61no—
126#384—31no—
127#384—31no—
128#382—61no—
129#380—31no—
130#380—61no—
131#380C2×D311yescharacteristic
132C4×C4⋊C4—31no—
133C4×C4⋊C4—31no—
134C4×C4⋊C4—61no—
135#340—32no—
136#340—32no—
137#336—32no—
138#336—32no—
139C24×C3—24no—
140C24×C3—22no—
141C24×C3—42no—
142C24×C3—42no—
143C24×C3—42no—
144C24×C3—42no—
145C24×C3—42no—
146C24×C3—42no—
147C24×C3C2×D411yescharacteristic
148C2×C3×D4—24no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—48no—
155C2×C3×D4—48no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4#4114yes—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—41no—
175C22×Dic3—42no—
176C22×Dic3—42no—
177C22×Dic3—42no—
178C22×Dic3D814yes—
179C22×Dic3C2×D411yescharacteristic
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C22×Dic3C2×D411yescharacteristic
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C22⋊C4—42no—
189#217—44no—
190#217—42no—
191#217—44no—
192#217—44no—
193#217—44no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—42no—
199Dic3⋊C4—42no—
200Dic3⋊C4—42no—
201Dic3⋊C4—42no—
202Dic3⋊C4—42no—
203C4×Dic3—22no—
204C4×Dic3—22no—
205C4×Dic3—21no—
206C4×Dic3—21no—
207C4×Dic3—21no—
208C4×Dic3—21no—
209C4×Dic3—42no—
210C4×Dic3—42no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C25—21no—
215C22×D4—24no—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—42no—
220C22×D4C22×D312yes—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—61no—
224C23×C4C22×D311yescharacteristic
225C22×C8—32no—
226C22×C8C22×D312yes—
227#117—124no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—122no—
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—124no—
246C2×C22⋊C4—124no—
247C2×C42—31no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—31no—
251C2×C42—62no—
252C2×C42—62no—
253C2×C42—61no—
254C2×C42—61no—
255#107—122no—
256#107—122no—
257#102—24no—
258#102—42no—
259#102—64no—
260#102—122no—
261C22⋊C8—22no—
262C22⋊C8—62no—
263#95—21no—
264#95—61no—
265#95—61no—
266#95—61no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—24no—
272C23×C3—48no—
273C23×C3—48no—
274C23×C3—48no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—84no—
282C23×C3—84no—
283C23×C3—84no—
284C23×C3—84no—
285C23×C3#12312yes—
286C23×C3C2×D812yes—
287C23×C3#13712yes—
288C23×C3C22×D411yescharacteristic
289C3×D4—24no—
290C3×D4—24no—
291C3×D4—48no—
292C3×D4—42no—
293C3×D4—42no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4#12312yes—
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C4H(5,𝔽2)12yes—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—28no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—44no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—41no—
329C2×Dic3—84no—
330C2×Dic3C2×D812yes—
331C2×Dic3C2×D812yes—
332C2×Dic3AGL(1,ℤ/8)12yes—
333C3×C8—42no—
334C3⋊C8—42no—
335C24—24no—
336C24—22no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24D3×D411yescharacteristic
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4#23714yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—34no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—31no—
363C22×C4—31no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—121no—
372C22×C4—122no—
373C22×C4—122no—
374C22×C4—122no—
375C22×C4#23512yes—
376C22×C4D3×D411yescharacteristic
377C22×C4#23712yes—
378C22×C4C23×D311yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—62no—
388C4⋊C4—62no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—122no—
392C4⋊C4—122no—
393C4⋊C4—122no—
394C4⋊C4—122no—
395C4⋊C4—122no—
396C22⋊C4—42no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C22⋊C4—124no—
400C22⋊C4—124no—
401C22⋊C4—122no—
402C42—62no—
403C42—62no—
404C42—61no—
405C42—61no—
406C42—61no—
407C42—61no—
408C42—122no—
409C42—122no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—816no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3#48014yes—
425C22×C3C2×#12311yescharacteristic
426C22×C3#52511yescharacteristic
427C22×C3#53511yescharacteristic
428C22×C3C22×D811yescharacteristic
429C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
430C22×C3C2×#13711yescharacteristic
431C22×C3#57511yescharacteristic
432C3×C4—22no—
433C3×C4—81no—
434C3×C4—81no—
435C3×C4#55112yes—
436C3×C4#55112yes—
437Dic3—22no—
438Dic3—22no—
439Dic3—22no—
440Dic3—41no—
441Dic3—81no—
442Dic3—81no—
443Dic3—82no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—24no—
449C23—48no—
450C23—48no—
451C23—48no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—84no—
459C23—84no—
460C23—84no—
461C23—84no—
462C23#88312yes—
463C23D3×D812yes—
464C23#91512yes—
465C23C2×D3×D411yescharacteristic
466D4—24no—
467D4—24no—
468D4—48no—
469D4—42no—
470D4—42no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—32no—
476C2×C4—32no—
477C2×C4—32no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—68no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—124no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—122no—
503C2×C4—121no—
504C2×C4—244no—
505C2×C4#88312yes—
506C2×C4C2×#23511yescharacteristic
507C2×C4C2×#23712yes—
508C2×C4C2×#23711yescharacteristic
509C2×C4#100412yes—
510C8—42no—
511C8—122no—
512C2×C3—24no—
513C2×C3—84no—
514C2×C3#248211yescharacteristic
515C2×C3C2×#48011yescharacteristic
516C2×C3#309411yescharacteristic
517C2×C3#310011yescharacteristic
518C2×C3#311011yescharacteristic
519C2×C3#320511yescharacteristic
520C2×C3#321511yescharacteristic
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—816no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22#477414yes—
536C22C2×#88311yescharacteristic
537C22#558611yescharacteristic
538C22#559211yescharacteristic
539C22C2×D3×D811yescharacteristic
540C22C2×#91511yescharacteristic
541C22D3×AGL(1,ℤ/8)11yescharacteristic
542C22#577311yescharacteristic
543C4—22no—
544C4—62no—
545C4—62no—
546C4—62no—
547C4—81no—
548C4—81no—
549C4—121no—
550C4—241no—
551C4—242no—
552C4—241no—
553C4#549512yes—
554C4#558712yes—
555C3#2758211yescharacteristic, complemented, Sylow
556C2—24no—
557C2—84no—
558C2#7383211yescharacteristic
559C2C2×#477411yescharacteristic
560C2#7893311yescharacteristic
561C2#7998811yescharacteristic
562C2#7999411yescharacteristic
563C2#8013411yescharacteristic
564C2#8014011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1786C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenter#1786D3
hypocenterC3#27582
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. #73864C2semidirect
5. #70183C2semidirect
6. #70173C2semidirect
7. #70165C2semidirect
8. #70155C2semidirect
9. #69927C2semidirect
10. #69910C2semidirect
11. #69504C2semidirect
31. C2×#818C22semidirect
32. #4975C22semidirect
33. #4975C22semidirect
34. Dic3×C4⋊C4C22semidirect
35. Dic3×C4⋊C4C22semidirect
36. C2×#807C22semidirect
40. #4551C22semidirect
41. #4550C22semidirect
42. #4550C22semidirect
43. #4529C22semidirect
555. C3#27582semidirect

Maximal subgroups

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

subgroupindex
C2×#52212
#738642
C3×#17862
#721282
#701832
#701732
#701652
#701552
#699272
#699102
#695042
#275823