Finite Group Databasebeta

← #10877201

Subgroups — #10877201

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#121C211yesmaximal, characteristic
2C2×#4975C211yesmaximal, characteristic, complemented
3C3×#1874C211yesmaximal, characteristic
4#71677C211yesmaximal, characteristic
5#71556C211yesmaximal, characteristic, complemented
6#70184C212yesmaximal, complemented
7#70153C212yesmaximal, complemented
8#69928C211yesmaximal, characteristic, complemented
9#69908C211yesmaximal, characteristic, complemented
10#68843C212yesmaximal, complemented
11#68796C212yesmaximal, complemented
12#27678—31nomaximal, Sylow
13C2×C3×#121C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×D4×Dic3—21no—
16C2×D4×Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4—21no—
19C22×Dic3⋊C4—21no—
20C22×C4×Dic3C2211yescharacteristic
21C2×C3×#102C2212yes—
22C2×C3×C22⋊C8C2211yescharacteristic
23#5221C2211yescharacteristic
24#5220—21no—
25C2×#832C2212yes—
26#5212C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#817C2212yes—
29C2×Dic3⋊C8C2212yes—
30#4984C2211yescharacteristic
31#4975—21no—
32#4975—21no—
33#4975C2212yes—
34#4975C2212yes—
35Dic3×C4⋊C4C2211yescharacteristic
36C2×#808C2212yes—
37#4951—21no—
38Dic3×C22⋊C4—21no—
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×#121—31no—
45C2×#384—31no—
46#2100—32no—
47#2094—32no—
48#1948—32no—
49#1874—31no—
50#1874D311yescharacteristic
51#1853—31no—
52#1853—31no—
53#1802—32no—
54#1797—31no—
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—21no—
60C23×Dic3—21no—
61C23×Dic3D411yescharacteristic
62C23×Dic3D411yescharacteristic
63C22×C3×C8C2312yes—
64C3×#121—22no—
65C3×#121Q814yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×#217—21no—
72C2×#217—21no—
73C2×#217—21no—
74C2×#217—21no—
75D4×Dic3—24no—
76D4×Dic3—44no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—42no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—44no—
88C2×Dic3⋊C4—44no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C2×C4×Dic3C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96C3×#102—42no—
97C3×C22⋊C8—22no—
98#832—42no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102#830—22no—
103#817—42no—
104Dic3⋊C8—24no—
105#808—42no—
106C2×#121C2×D311yescharacteristic
107C2×C4×D4—31no—
108C2×C4×D4—61no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—61no—
111C22×C4⋊C4—61no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C42—31no—
114C2×#106—32no—
115C2×#106—32no—
116C2×C4⋊C8—32no—
117C2×#102—32no—
118C2×#102C2×D312yes—
119C2×C22⋊C8—31no—
120C2×C22⋊C8C2×D311yescharacteristic
121#390—31no—
122#389—31no—
123#388—61no—
124#386—61no—
125#384—32no—
126#384—32no—
127#384—61no—
128#384—61no—
129#380—31no—
130C4×C4⋊C4—31no—
131C4×C22⋊C4—61no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340C2×D312yes—
136#336—32no—
137C24×C3—21no—
138C24×C3C2×D411yescharacteristic
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—21no—
142C2×C3×D4—22no—
143C2×C3×D4—42no—
144C2×C3×D4—42no—
145C2×C3×D4C2×Q812yes—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×Q812yes—
155C22×C3×C4#4112yes—
156C22×C3×C4C2411yescharacteristic
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3—42no—
166C22×Dic3—42no—
167C22×Dic3—42no—
168C22×Dic3—42no—
169C22×Dic3—42no—
170C22×Dic3—42no—
171C22×Dic3SD1614yes—
172C22×Dic3C2×D411yescharacteristic
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4C2×Q812yes—
184C3×C22⋊C4—24no—
185#217—42no—
186#217—42no—
187#217—42no—
188#217—42no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195Dic3⋊C4—44no—
196Dic3⋊C4—44no—
197Dic3⋊C4—44no—
198Dic3⋊C4—44no—
199C4×Dic3—22no—
200C4×Dic3—22no—
201C4×Dic3—22no—
202C4×Dic3—42no—
203C4×Dic3#4114yes—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C22×D4—21no—
208C22×D4C22×D311yescharacteristic
209C23×C4—21no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—61no—
213C23×C4—61no—
214C23×C4C22×D311yescharacteristic
215C22×C8—32no—
216C22×C8C22×D312yes—
217#121—22no—
218#121Dic614yes—
219C4×D4—64no—
220C4×D4—124no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—122no—
233C2×C4⋊C4—124no—
234C2×C4⋊C4—124no—
235C2×C4⋊C4C22×D312yes—
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—61no—
242C2×C42—31no—
243C2×C42—31no—
244C2×C42—31no—
245C2×C42—31no—
246C2×C42—64no—
247C2×C42—61no—
248#106—122no—
249#106—122no—
250C4⋊C8—64no—
251#102—42no—
252#102—122no—
253C22⋊C8—22no—
254C22⋊C8—62no—
255#95—61no—
256#95—61no—
257#95—61no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3C22⋊Q812yes—
270C23×C3C2×SD1612yes—
271C23×C3AGL(1,ℤ/8)12yes—
272C23×C3C22×D411yescharacteristic
273C3×D4—24no—
274C3×D4—48no—
275C3×D4—42no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—84no—
287C2×C3×C4C22⋊Q812yes—
288C2×C3×C4C22×Q811yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4112yes—
291C2×C3×C4H(5,𝔽2)12yes—
292C2×Dic3—28no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—28no—
300C2×Dic3—41no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—84no—
310C2×Dic3—84no—
311C2×Dic3#12312yes—
312C2×Dic3#12312yes—
313C2×Dic3C2×SD1612yes—
314C2×Dic3C2×SD1612yes—
315C2×Dic3AGL(1,ℤ/8)12yes—
316C2×Dic3#13712yes—
317C3×C8—42no—
318C3⋊C8—42no—
319C24—21no—
320C24D3×D411yescharacteristic
321C2×D4—24no—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4C2×Dic612yes—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—31no—
333C22×C4—34no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4—122no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—122no—
354C22×C4C2×Dic612yes—
355C22×C4D3×D411yescharacteristic
356C22×C4#23712yes—
357C22×C4C23×D311yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—62no—
372C4⋊C4—122no—
373C4⋊C4—124no—
374C4⋊C4—122no—
375C4⋊C4—124no—
376C4⋊C4—124no—
377C4⋊C4—124no—
378C4⋊C4C2×Dic612yes—
379C22⋊C4—24no—
380C22⋊C4—122no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C42—34no—
385C42—62no—
386C42—62no—
387C42—62no—
388C42—122no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3#47614yes—
403C22×C3C2×C22⋊Q811yescharacteristic
404C22×C3C2×#12311yescharacteristic
405C22×C3#53511yescharacteristic
406C22×C3C22×SD1611yescharacteristic
407C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
408C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
409C22×C3C2×#13711yescharacteristic
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—82no—
413C3×C4#55312yes—
414C3×C4#55412yes—
415Dic3—22no—
416Dic3—22no—
417Dic3—22no—
418Dic3—81no—
419Dic3—81no—
420Dic3—82no—
421Dic3#48114yes—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23#87812yes—
434C23#91112yes—
435C23D3×SD1612yes—
436C23C2×D3×D411yescharacteristic
437D4—24no—
438D4—48no—
439D4—42no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—32no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—68no—
457C2×C4—68no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—84no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—122no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—122no—
473C2×C4—121no—
474C2×C4—244no—
475C2×C4—244no—
476C2×C4#87812yes—
477C2×C4C22×Dic611yescharacteristic
478C2×C4C2×#23711yescharacteristic
479C2×C4#100412yes—
480C2×C4D3×#4112yes—
481C8—42no—
482C8—122no—
483C2×C3—24no—
484C2×C3—82no—
485C2×C3#248711yescharacteristic
486C2×C3C2×#47611yescharacteristic
487C2×C3#307911yescharacteristic
488C2×C3C2×#48111yescharacteristic
489C2×C3#309211yescharacteristic
490C2×C3#320611yescharacteristic
491C2×C3#321711yescharacteristic
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22#476114yes—
506C22C2×#87811yescharacteristic
507C22#559211yescharacteristic
508C22D3×#12311yescharacteristic
509C22C2×#91111yescharacteristic
510C22C2×D3×SD1611yescharacteristic
511C22D3×AGL(1,ℤ/8)11yescharacteristic
512C22D3×#13711yescharacteristic
513C4—22no—
514C4—34no—
515C4—42no—
516C4—62no—
517C4—62no—
518C4—62no—
519C4—82no—
520C4—242no—
521C4—241no—
522C4—241no—
523C4#553912yes—
524C4#560112yes—
525C3#2767811yescharacteristic, complemented, Sylow
526C2—24no—
527C2—82no—
528C2#7473811yescharacteristic
529C2C2×#476111yescharacteristic
530C2#7887011yescharacteristic
531C2#7999711yescharacteristic
532C2#8003411yescharacteristic
533C2#8128211yescharacteristic
534C2D3×#48111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1874C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#1874D3
hypocenterC3#27678
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#4975C2semidirect
5. #71556C2semidirect
6. #70184C2semidirect
7. #70153C2semidirect
8. #69928C2semidirect
9. #69908C2semidirect
10. #68843C2semidirect
11. #68796C2semidirect
525. C3#27678semidirect

Maximal subgroups

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

subgroupindex
Dic3×#1212
C2×#49752
C3×#18742
#716772
#715562
#701842
#701532
#699282
#699082
#688432
#687962
#276783