Finite Group Databasebeta

← #10824322

Subgroups — #10824322

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×D8C212yesmaximal, complemented
2Dic3×#127C211yesmaximal, characteristic, complemented
3#76289C211yesmaximal, characteristic, complemented
4C3×#2011C211yesmaximal, characteristic
5#72040C212yesmaximal, complemented
6#71762C211yesmaximal, characteristic, complemented
7#70180C212yesmaximal, complemented
8Dic3×#102C212yesmaximal, complemented
9#69640C211yesmaximal, characteristic, complemented
10#68639C211yesmaximal, characteristic, complemented
11#68602C211yesmaximal, characteristic, complemented
12#23676—31nomaximal, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×D4×Dic3—21no—
16C2×D4×Dic3—21no—
17C2×D4×Dic3C2212yes—
18C2×D4×Dic3C2211yescharacteristic
19C2×#877—21no—
20C2×C3×#102C2212yes—
21C3×#390C2211yescharacteristic
22#5221—21no—
23#5221—21no—
24#5221C2211yescharacteristic
25#5221C2212yes—
26C2×#832C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29Dic3×D8—28no—
30C2×#818C2212yes—
31C2×Dic3⋊C8C2212yes—
32C2×C8×Dic3C2212yes—
33Dic3×C4⋊C4C2211yescharacteristic
34#4972C2211yescharacteristic
35#4945—21no—
36#4944—21no—
37Dic3×C22⋊C4—21no—
38#4935C2211yescharacteristic
39C42×Dic3C2211yescharacteristic
40#4928C2211yescharacteristic
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45C2×C4×D8—32no—
46C4×#127—31no—
47#2293—31no—
48#2083—31no—
49#2071—31no—
50#2067—32no—
51#2054—32no—
52#2041—31no—
53#2011D311yescharacteristic
54C4×C4⋊C8—31no—
55C4×#102—32no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C22×C3×D4C2312yes—
60C23×Dic3—21no—
61C23×Dic3—22no—
62C2×C3×D8—24no—
63C2×C3×D8—24no—
64C2×C3×D8—24no—
65C22×C3×C8C2312yes—
66C3×#127—24no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#217—21no—
72C2×#217—21no—
73C2×#217—21no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77D4×Dic3—24no—
78D4×Dic3—22no—
79D4×Dic3—24no—
80D4×Dic3—44no—
81D4×Dic3—44no—
82D4×Dic3D414yescomplemented
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—21no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3D411yescharacteristic
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98#877—44no—
99C3×#102—24no—
100#832—24no—
101#832—24no—
102#832—24no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106#818—24no—
107Dic3⋊C8—24no—
108C8×Dic3—24no—
109C22×D8C2×D312yes—
110C2×#127C2×D311yescharacteristic
111C2×C4×D4—31no—
112C2×C4×D4—32no—
113C2×C4×D4—61no—
114C2×C4×D4—61no—
115C2×#117—61no—
116C4×D8—68no—
117C2×#107—32no—
118C2×C4⋊C8—32no—
119C2×#102—32no—
120C2×#102—32no—
121C2×#102—32no—
122C2×#102C2×D312yes—
123C2×C4×C8—32no—
124#390—32no—
125#390—31no—
126#390—61no—
127#390—61no—
128#390C2×D311yescharacteristic
129#389—31no—
130#389—31no—
131#388—61no—
132#388—61no—
133#384—31no—
134C4×C4⋊C4—31no—
135C4×C22⋊C4—61no—
136C43—31no—
137#340—32no—
138#340—32no—
139#336—31no—
140#336C2×D311yescharacteristic
141C24×C3—21no—
142C24×C3—22no—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—22no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4C2×D412yes—
158C2×C3×D4#4114yes—
159C22×C3×C4—21no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—24no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—42no—
168C22×Dic3—42no—
169C22×Dic3—44no—
170C22×Dic3—44no—
171C22×Dic3—44no—
172C22×Dic3C2×D411yescharacteristic
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C3×D8—216no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8#4114yes—
180C3×C4⋊C4—22no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C42—22no—
184#217—24no—
185#217—44no—
186#217—42no—
187#217—44no—
188#217—42no—
189#217—44no—
190C4⋊Dic3—21no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—42no—
195C4⋊Dic3—42no—
196C4⋊Dic3C2×D412yes—
197Dic3⋊C4—42no—
198Dic3⋊C4—42no—
199C4×Dic3—22no—
200C4×Dic3—22no—
201C4×Dic3—22no—
202C4×Dic3—41no—
203C4×Dic3—42no—
204C4×Dic3D814yescomplemented
205C4×Dic3C2×D412yes—
206C4×Dic3#4112yes—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22×D311yescharacteristic
213C22×D4C22×D312yes—
214C23×C4—61no—
215C23×C4—62no—
216C2×D8—24no—
217C2×D8—24no—
218C2×D8—24no—
219C22×C8—32no—
220C22×C8C22×D312yes—
221#127—24no—
222C4×D4—34no—
223C4×D4—62no—
224C4×D4—64no—
225C4×D4—64no—
226C4×D4—124no—
227C4×D4—124no—
228#117—124no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C42—31no—
246C2×C42—31no—
247C2×C42—31no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—31no—
251C2×C42—61no—
252C2×C42C22×D311yescharacteristic
253#107—64no—
254C4⋊C8—64no—
255#102—24no—
256#102—64no—
257#102—64no—
258#102—64no—
259C4×C8—64no—
260#95—61no—
261#95—61no—
262#95—61no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—42no—
266C23×C3—41no—
267C23×C3—42no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—42no—
271C23×C3—41no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—42no—
277C23×C3C22×D411yescharacteristic
278C3×D4—24no—
279C3×D4—24no—
280C3×D4—24no—
281C3×D4—48no—
282C3×D4—48no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—42no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4C2×D812yes—
294C2×C3×C4#13512yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4112yes—
297C2×C3×C4C2×#4112yes—
298C2×C3×C4C2×#4111yescharacteristic
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—22no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—48no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3—88no—
317C2×Dic3#12112yes—
318C2×Dic3#12112yes—
319C2×Dic3C2×D812yes—
320C2×Dic3C2×D812yes—
321C2×Dic3AGL(1,ℤ/8)12yes—
322C3×C8—24no—
323C3⋊C8—42no—
324C24—21no—
325C24—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4D3×D412yes—
341C2×D4#23714yes—
342C22×C4—21no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—64no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—124no—
354C22×C4—124no—
355C22×C4—124no—
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4C23×D311yescharacteristic
359D8—216no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8#23714yes—
366C4⋊C4—22no—
367C4⋊C4—32no—
368C4⋊C4—61no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—62no—
372C4⋊C4—122no—
373C4⋊C4—122no—
374C4⋊C4—122no—
375C4⋊C4—122no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—64no—
379C22⋊C4—122no—
380C22⋊C4—124no—
381C22⋊C4—124no—
382C22⋊C4—124no—
383C22⋊C4—122no—
384C42—22no—
385C42—34no—
386C42—32no—
387C42—32no—
388C42—62no—
389C42—62no—
390C42—62no—
391C42—121no—
392C42—122no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—81no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3C2×#12111yescharacteristic
408C22×C3#52511yescharacteristic
409C22×C3#53211yescharacteristic
410C22×C3C22×D811yescharacteristic
411C22×C3C2×#13511yescharacteristic
412C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C3#57511yescharacteristic
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—81no—
417C3×C4#52512yes—
418C3×C4#52912yes—
419Dic3—22no—
420Dic3—22no—
421Dic3—41no—
422Dic3—42no—
423Dic3—81no—
424Dic3—82no—
425Dic3#45914yes—
426C23—24no—
427C23—22no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—41no—
439C23—42no—
440C23C2×D3×D411yescharacteristic
441D4—24no—
442D4—24no—
443D4—24no—
444D4—48no—
445D4—48no—
446D4—42no—
447D4—42no—
448D4—42no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—32no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—62no—
469C2×C4—128no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—122no—
478C2×C4—248no—
479C2×C4D3×D812yes—
480C2×C4#91212yes—
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×#23712yes—
483C2×C4C2×#23712yes—
484C2×C4D3×#4111yescharacteristic
485C8—24no—
486C8—122no—
487C2×C3—84no—
488C2×C3—82no—
489C2×C3#239211yescharacteristic
490C2×C3C2×#45911yescharacteristic
491C2×C3#303511yescharacteristic
492C2×C3#310011yescharacteristic
493C2×C3#310511yescharacteristic
494C2×C3#316911yescharacteristic
495C2×C3#317111yescharacteristic
496C22—44no—
497C22—44no—
498C22—42no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—81no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22—82no—
507C22—81no—
508C22—82no—
509C22—82no—
510C22#558611yescharacteristic
511C22D3×#12111yescharacteristic
512C22#571611yescharacteristic
513C22C2×D3×D811yescharacteristic
514C22C2×#91211yescharacteristic
515C22D3×AGL(1,ℤ/8)11yescharacteristic
516C22#577311yescharacteristic
517C4—22no—
518C4—22no—
519C4—34no—
520C4—62no—
521C4—62no—
522C4—81no—
523C4—121no—
524C4—122no—
525C4—241no—
526C4—242no—
527C4#558612yes—
528C4#560012yes—
529C3#2367611yescharacteristic, complemented, Sylow
530C2—84no—
531C2—82no—
532C2#7666611yescharacteristic
533C2#7998811yescharacteristic
534C2#7998911yescharacteristic
535C2D3×#45911yescharacteristic
536C2#8033111yescharacteristic
537C2#8177511yescharacteristic
538C2#8183611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2011C2
socleC23×C3C22×D4
hypercenter#2011D3
hypocenterC3#23676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2011

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×D8C2semidirect
2. Dic3×#127C2semidirect
3. #76289C2semidirect
5. #72040C2semidirect
6. #71762C2semidirect
7. #70180C2semidirect
8. Dic3×#102C2semidirect
9. #69640C2semidirect
10. #68639C2semidirect
11. #68602C2semidirect
82. D4×Dic3D4semidirect
204. C4×Dic3D8semidirect
529. C3#23676semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D82
Dic3×#1272
#762892
C3×#20112
#720402
#717622
#701802
Dic3×#1022
#696402
#686392
#686022
#236763