Finite Group Databasebeta

← #10824323

Subgroups — #10824323

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
1C3×#2293C211yesmaximal, characteristic, complemented
2C2×C4×#213C212yesmaximal, complemented
3C4×#874C211yesmaximal, characteristic, complemented
4#69658C211yesmaximal, characteristic
5#69650C212yesmaximal, complemented
6#69640C211yesmaximal, characteristic, complemented
7#69630C212yesmaximal, complemented
8#69627C211yesmaximal, characteristic, complemented
9#69607C211yesmaximal, characteristic, complemented
10C4×#809C212yesmaximal, complemented
11C4×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
12#23676—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×C3×#117—21no—
15C22×#213C2212yes—
16C2×#874C2211yescharacteristic
17C2×C4×D12—21no—
18C2×C4×D12—21no—
19C2×C4×D12C2212yes—
20C3×#390C2211yescharacteristic
21C3×#390C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#388—21no—
25C3×#388—21no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×C4×C22⋊C4—21no—
28C3×C43C2211yescharacteristic
29C2×#832C2212yes—
30C4×#213—28no—
31C2×#809C2212yes—
32C2×#809C2212yes—
33C2×#809C2212yes—
34C2×#807C2212yes—
35#4939—21no—
36#4939—21no—
37#4939C2212yes—
38#4935C2211yescharacteristic
39C2×C4⋊(C3⋊C8)C2212yes—
40C2×C4×C3⋊C8C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45C2×C4×D8—32no—
46C4×#127—31no—
47#2293D311yescharacteristic, complemented
48#2083—31no—
49#2071—31no—
50#2067—32no—
51#2054—32no—
52#2041—31no—
53#2011—31no—
54C4×C4⋊C8—31no—
55C4×#102—32no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C3×C4×D4—22no—
63C3×C4×D4D414yescomplemented
64C3×#117—44no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×C3×C42D411yescharacteristic
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×#213—24no—
84C2×#213—24no—
85C2×#213—24no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2312yes—
92C22×C3⋊C8C2312yes—
93C22×C3⋊C8C2312yes—
94#874—24no—
95C4×D12—24no—
96C4×D12—24no—
97C4×D12—44no—
98C4×D12—44no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—21no—
102#832—24no—
103#809—24no—
104#809—24no—
105#809—24no—
106#807—24no—
107C4⋊(C3⋊C8)—24no—
108C4×C3⋊C8—24no—
109C22×D8—32no—
110C2×#127—31no—
111C2×C4×D4—32no—
112C2×C4×D4—61no—
113C2×C4×D4—61no—
114C2×C4×D4C2×D311yescharacteristic
115C2×#117—21no—
116C4×D8—68no—
117C2×#107—32no—
118C2×C4⋊C8—32no—
119C2×#102—32no—
120C2×#102—32no—
121C2×#102—32no—
122C2×#102—32no—
123C2×C4×C8—32no—
124#390—32no—
125#390—61no—
126#390—61no—
127#390C2×D311yescharacteristic
128#390C2×D311yescharacteristic
129#389C2×D311yescharacteristic
130#389C2×D311yescharacteristic
131#388—21no—
132#388—21no—
133#384—31no—
134C4×C4⋊C4C2×D311yescharacteristic
135C4×C22⋊C4—21no—
136C43C2×D311yescharacteristic
137#340—32no—
138#340—32no—
139#336—31no—
140#336—31no—
141C24×C3—21no—
142C23×D3—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4C2×D412yes—
147C22×C3×C4—21no—
148C22×C3×C4—24no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—24no—
166C2×D12—24no—
167C2×D12—22no—
168C2×D12—42no—
169C2×D12—42no—
170C2×D12—42no—
171C2×D12—42no—
172C2×D12#4114yes—
173C2×C4×D3—44no—
174C2×C4×D3—44no—
175C2×C4×D3—44no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4C2×D412yes—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—42no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—41no—
192C3×C42—42no—
193C3×C42D814yescomplemented
194C3×C42C2×D412yes—
195C3×C42#4112yes—
196#213—216no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—44no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—42no—
203C4⋊Dic3—42no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8#4114yes—
210C22×D4—32no—
211C22×D4—61no—
212C22×D4—61no—
213C22×D4C22×D311yescharacteristic
214C23×C4—21no—
215C23×C4—62no—
216C2×D8—64no—
217C2×D8—64no—
218C2×D8—64no—
219C22×C8—32no—
220C22×C8—32no—
221#127—64no—
222C4×D4—22no—
223C4×D4—64no—
224C4×D4—64no—
225C4×D4—124no—
226C4×D4—124no—
227C4×D4D1214yescomplemented
228#117—44no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C42—21no—
246C2×C42#6711yescharacteristic
247C2×C42#6711yescharacteristic
248C2×C42#6711yescharacteristic
249C2×C42#6711yescharacteristic
250C2×C42C22×D311yescharacteristic
251C2×C42C22×D311yescharacteristic
252C2×C42C22×D311yescharacteristic
253#107—64no—
254C4⋊C8—64no—
255#102—64no—
256#102—64no—
257#102—64no—
258#102—64no—
259C4×C8—64no—
260#95—21no—
261#95—21no—
262#95—21no—
263C23×C3—22no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3C22×D411yescharacteristic
271C22×D3—24no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C3×D4—24no—
279C3×D4—42no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—48no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4#12112yes—
302C2×C3×C4#12112yes—
303C2×C3×C4C2×D812yes—
304C2×C3×C4C2×D812yes—
305C2×C3×C4C2×D812yes—
306C2×C3×C4#13512yes—
307C2×C3×C4AGL(1,ℤ/8)12yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4112yes—
311C2×C3×C4C2×#4112yes—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315D12—24no—
316D12—24no—
317D12—48no—
318D12—48no—
319D12—42no—
320D12—42no—
321C4×D3—88no—
322C3⋊C8—24no—
323C3⋊C8—42no—
324C24—21no—
325C24—62no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—34no—
330C2×D4—64no—
331C2×D4—64no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—62no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—122no—
341C2×D4C2×D1212yes—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—62no—
349C22×C4—124no—
350C22×C4—124no—
351C22×C4—124no—
352C22×C4C2×#6711yescharacteristic
353C22×C4C2×#6711yescharacteristic
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C23×D311yescharacteristic
359D8—616no—
360C2×C8—34no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—62no—
372C4⋊C4—62no—
373C4⋊C4—122no—
374C4⋊C4—122no—
375C4⋊C4C2×D1212yes—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—124no—
382C22⋊C4—124no—
383C22⋊C4—124no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—41no—
389C42—42no—
390C42#21314yescomplemented
391C42C2×D1212yes—
392C42#23512yes—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
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—44no—
408C2×D3—44no—
409C2×D3—44no—
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—82no—
413C2×D3—82no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—41no—
419C3×C4—42no—
420C3×C4—81no—
421C3×C4—81no—
422C3×C4#45914yes—
423C3×C4#52512yes—
424C3×C4#52912yes—
425Dic3—82no—
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—64no—
434C23—122no—
435C23—122no—
436C23—122no—
437C23—122no—
438C23—122no—
439C23—122no—
440C23C22×#6711yescharacteristic
441D4—24no—
442D4—42no—
443D4—64no—
444D4—64no—
445D4—128no—
446D4—128no—
447D4—122no—
448D4—122no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—248no—
474C2×C4#93012yes—
475C2×C4#93012yes—
476C2×C4C2×#21312yes—
477C2×C4C2×#21312yes—
478C2×C4C2×#21312yes—
479C2×C4#94912yes—
480C2×C4#95012yes—
481C2×C4C22×D1211yescharacteristic
482C2×C4C2×#23511yescharacteristic
483C2×C4D3×#4112yes—
484C2×C4D3×#4112yes—
485C8—64no—
486C8—122no—
487C2×C3—82no—
488C2×C3#239211yescharacteristic
489C2×C3C2×#45911yescharacteristic
490C2×C3#303511yescharacteristic
491C2×C3#310011yescharacteristic
492C2×C3#310511yescharacteristic
493C2×C3#316911yescharacteristic
494C2×C3#317111yescharacteristic
495D3—84no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—124no—
504C22—124no—
505C22—124no—
506C22—242no—
507C22—242no—
508C22—242no—
509C22—242no—
510C22C2×#93011yescharacteristic
511C22C22×#21311yescharacteristic
512C22C2×#94911yescharacteristic
513C22C2×#95011yescharacteristic
514C22#582211yescharacteristic
515C22#582811yescharacteristic
516C22#583211yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—42no—
522C4—41no—
523C4—81no—
524C4—81no—
525C4—242no—
526C4#501514yes—
527C4#552412yes—
528C4#552812yes—
529C3#2367611yescharacteristic, complemented, Sylow
530C2—82no—
531C2—244no—
532C2#8227311yescharacteristic
533C2C2×#501511yescharacteristic
534C2#8252511yescharacteristic
535C2#8313511yescharacteristic
536C2#8314111yescharacteristic
537C2#8329111yescharacteristic
538C2#8329811yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2293C2
socleC23×C3C22×D4
prime radical#69658C2
hypercenter#2293D3
hypocenterC3#23676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. C3×#2293C2semidirect
2. C2×C4×#213C2semidirect
3. C4×#874C2semidirect
5. #69650C2semidirect
6. #69640C2semidirect
7. #69630C2semidirect
8. #69627C2semidirect
9. #69607C2semidirect
10. C4×#809C2semidirect
11. C4×C4⋊(C3⋊C8)C2semidirect
47. #2293D3semidirect
63. C3×C4×D4D4semidirect
193. C3×C42D8semidirect
227. C4×D4D12semidirect
390. C42#213semidirect
529. C3#23676semidirect

Maximal subgroups

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

subgroupindex
C3×#22932
C2×C4×#2132
C4×#8742
#696582
#696502
#696402
#696302
#696272
#696072
C4×#8092
C4×C4⋊(C3⋊C8)2
#236763