Finite Group Databasebeta

← #10824326

Subgroups — #10824326

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
1Dic3×#127C211yesmaximal, characteristic, complemented
2C2×#4756C212yesmaximal, complemented
3#75961C211yesmaximal, characteristic, complemented
4C3×#2041C211yesmaximal, characteristic, complemented
5#71767C211yesmaximal, characteristic
6#70212C212yesmaximal, complemented
7#70194C212yesmaximal, complemented
8Dic3×#102C212yesmaximal, complemented
9#69607C211yesmaximal, characteristic, complemented
10#68635C211yesmaximal, characteristic, complemented
11#68602C211yesmaximal, characteristic, complemented
12#23676—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×D4×Dic3—21no—
15C2×D4×Dic3—21no—
16C2×D4×Dic3C2212yes—
17C22×#213C2212yes—
18C2×#893C2211yescharacteristic
19C2×#892—21no—
20C2×C3×#102C2212yes—
21C2×C3×#102C2212yes—
22C3×#389C2211yescharacteristic
23#5221—21no—
24#5221—21no—
25#5221C2212yes—
26C2×#832C2212yes—
27C2×#821C2212yes—
28C2×Dic3⋊C8C2212yes—
29C2×C8×Dic3C2212yes—
30#4990—21no—
31#4988C2211yescharacteristic
32#4984C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34C2×#807C2212yes—
35#4939C2211yescharacteristic
36#4935C2211yescharacteristic
37C42×Dic3C2211yescharacteristic
38#4756—28no—
39#4670—21no—
40#4666—21no—
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#2041D311yescharacteristic, complemented
53#2011—31no—
54C4×C4⋊C8—31no—
55C4×#102—32no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C23×Dic3—22no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×C3×C8C2312yes—
63C3×#127—24no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C2×#217—22no—
69C2×#217—22no—
70C2×#217—22no—
71D4×Dic3—24no—
72D4×Dic3—24no—
73D4×Dic3—44no—
74D4×Dic3—44no—
75C2×#213—24no—
76C2×#213—24no—
77C2×#213—24no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4×Dic3—21no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C2×C4×Dic3C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96#893—22no—
97#893D414yescomplemented
98#892—44no—
99C3×#102—24no—
100C3×#102—24no—
101C3×#95—21no—
102#832—24no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105#821—24no—
106Dic3⋊C8—24no—
107C8×Dic3—24no—
108#807—24no—
109C22×D8—32no—
110C2×#127C2×D311yescharacteristic
111C2×C4×D4—32no—
112C2×C4×D4—31no—
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×#102C2×D312yes—
122C2×#102C2×D312yes—
123C2×C4×C8—32no—
124#390—32no—
125#390—31no—
126#390—31no—
127#390—61no—
128#390—61no—
129#389—31no—
130#389C2×D311yescharacteristic
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—22no—
142C23×D3—21no—
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—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4#4114yes—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—44no—
163C22×Dic3—44no—
164C22×Dic3—44no—
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168C22×Dic3C2×D411yescharacteristic
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—21no—
172C2×D12C2×D412yes—
173C2×C4×D3—24no—
174C2×C4×D3—42no—
175C2×C4×D3—42no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×D412yes—
183C3×C42—22no—
184#217—44no—
185#217—44no—
186#217—44no—
187#213—216no—
188(C2×D3)⋊C4—24no—
189(C2×D3)⋊C4—42no—
190(C2×D3)⋊C4—42no—
191(C2×D3)⋊C4—42no—
192(C2×D3)⋊C4—42no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—42no—
197C4⋊Dic3—42no—
198C4⋊Dic3—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#4114yes—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—31no—
213C22×D4C22×D312yes—
214C23×C4—61no—
215C23×C4—62no—
216C2×D8—64no—
217C2×D8—64no—
218C2×D8—64no—
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—21no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
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—24no—
257#102—64no—
258#102—64no—
259C4×C8—64no—
260#95—21no—
261#95—61no—
262#95—61no—
263C23×C3—24no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3C22×D411yescharacteristic
271C22×D3—22no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—41no—
278C3×D4—24no—
279C3×D4—24no—
280C3×D4—48no—
281C3×D4—48no—
282C3×D4—42no—
283C3×D4—42no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—22no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4C2×D812yes—
295C2×C3×C4#13512yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4112yes—
298C2×C3×C4C2×#4112yes—
299C2×C3×C4C2×#4111yescharacteristic
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—88no—
314C2×Dic3#12112yes—
315C2×Dic3#12112yes—
316C2×Dic3C2×D812yes—
317C2×Dic3C2×D812yes—
318C2×Dic3AGL(1,ℤ/8)12yes—
319D12—24no—
320D12—42no—
321C4×D3—48no—
322C3×C8—42no—
323C3⋊C8—24no—
324C24—22no—
325C24—61no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—32no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—61no—
341C2×D4#23714yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—64no—
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—616no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—34no—
364C2×C8—62no—
365C2×C8—62no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—42no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—62no—
372C4⋊C4—122no—
373C4⋊C4—122no—
374C4⋊C4—122no—
375C4⋊C4D3×D412yes—
376C22⋊C4—64no—
377C22⋊C4—122no—
378C22⋊C4—124no—
379C22⋊C4—124no—
380C22⋊C4—122no—
381C22⋊C4—124no—
382C22⋊C4—122no—
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—44no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3C2×#12111yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#53211yescharacteristic
403C22×C3C22×D811yescharacteristic
404C22×C3C2×#13511yescharacteristic
405C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
406C22×C3#57511yescharacteristic
407C2×D3—42no—
408C2×D3—42no—
409C2×D3—42no—
410C2×D3—81no—
411C2×D3—81no—
412C2×D3—81no—
413C2×D3—81no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—42no—
417C3×C4—81no—
418C3×C4#52512yes—
419C3×C4#52912yes—
420Dic3—22no—
421Dic3—22no—
422Dic3—41no—
423Dic3—81no—
424Dic3—82no—
425Dic3#45914yes—
426C23—24no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—62no—
434C23—121no—
435C23—121no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23C2×D3×D411yescharacteristic
441D4—24no—
442D4—24no—
443D4—48no—
444D4—48no—
445D4—42no—
446D4—42no—
447D4—64no—
448D4—122no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—128no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—122no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—248no—
479C2×C4D3×D812yes—
480C2×C4#91612yes—
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×#23712yes—
483C2×C4D3×#4111yescharacteristic
484C2×C4D3×#4112yes—
485C8—42no—
486C8—64no—
487C2×C3—84no—
488C2×C3#239211yescharacteristic
489C2×C3C2×#45911yescharacteristic
490C2×C3#303511yescharacteristic
491C2×C3#310011yescharacteristic
492C2×C3#310511yescharacteristic
493C2×C3#316911yescharacteristic
494C2×C3#317111yescharacteristic
495D3—82no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—122no—
504C22—122no—
505C22—122no—
506C22—241no—
507C22—241no—
508C22—241no—
509C22—241no—
510C22#558611yescharacteristic
511C22D3×#12111yescharacteristic
512C22#566811yescharacteristic
513C22C2×D3×D811yescharacteristic
514C22C2×#91611yescharacteristic
515C22D3×AGL(1,ℤ/8)11yescharacteristic
516C22#577311yescharacteristic
517C4—22no—
518C4—22no—
519C4—34no—
520C4—42no—
521C4—62no—
522C4—62no—
523C4—81no—
524C4—121no—
525C4—241no—
526C4—242no—
527C4#560612yes—
528C4#564912yes—
529C3#2367611yescharacteristic, complemented, Sylow
530C2—84no—
531C2—242no—
532C2#7689111yescharacteristic
533C2#7998811yescharacteristic
534C2#8013311yescharacteristic
535C2D3×#45911yescharacteristic
536C2#8038211yescharacteristic
537C2#8155711yescharacteristic
538C2#8170411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radical#71767C2
hypercenter#2041D3
hypocenterC3#23676
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
1. Dic3×#127C2semidirect
2. C2×#4756C2semidirect
3. #75961C2semidirect
4. C3×#2041C2semidirect
6. #70212C2semidirect
7. #70194C2semidirect
8. Dic3×#102C2semidirect
9. #69607C2semidirect
10. #68635C2semidirect
11. #68602C2semidirect
52. #2041D3semidirect
97. #893D4semidirect
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
Dic3×#1272
C2×#47562
#759612
C3×#20412
#717672
#702122
#701942
Dic3×#1022
#696072
#686352
#686022
#236763