Finite Group Databasebeta

← #10824324

Subgroups — #10824324

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×#4756C212yesmaximal, complemented
2#76289C211yesmaximal, characteristic, complemented
3#74319C211yesmaximal, characteristic, complemented
4C3×#2071C211yesmaximal, characteristic, complemented
5#70212C212yesmaximal, complemented
6#69658C211yesmaximal, characteristic
7#69569C211yesmaximal, characteristic, complemented
8#68901C212yesmaximal, complemented
9#68780C212yesmaximal, complemented
10#68629C211yesmaximal, characteristic, complemented
11#68602C211yesmaximal, characteristic, complemented
12#23676—31nomaximal, Sylow
13C2×D4×Dic3C2211yescharacteristic
14C22×#213C2212yes—
15C2×#893—21no—
16C2×#893—21no—
17C2×#893C2212yes—
18C2×#877—21no—
19C2×#874C2211yescharacteristic
20C2×C3×#102C2212yes—
21C3×#390C2211yescharacteristic
22C3×#384C2211yescharacteristic
23#5221C2211yescharacteristic
24C2×#821C2212yes—
25C2×#821C2212yes—
26C2×Dic3⋊C8C2212yes—
27C2×C8×Dic3C2212yes—
28#4988—21no—
29#4988—21no—
30#4988C2212yes—
31Dic3×C4⋊C4C2211yescharacteristic
32#4972C2211yescharacteristic
33C2×#809C2212yes—
34C2×#807C2212yes—
35#4945—21no—
36#4944—21no—
37Dic3×C22⋊C4—21no—
38C42×Dic3C2211yescharacteristic
39#4928C2211yescharacteristic
40#4756—28no—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43#4529C2212yes—
44#4524C2211yescharacteristic
45C2×C4×D8—32no—
46C4×#127—31no—
47#2293—31no—
48#2083—31no—
49#2071D311yescharacteristic, complemented
50#2067—32no—
51#2054—32no—
52#2041—31no—
53#2011—31no—
54C4×C4⋊C8—31no—
55C4×#102—32no—
56C22×C3×D4C2311yescharacteristic
57C23×Dic3—21no—
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×C3×C8C2312yes—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C42C2311yescharacteristic
70C2×#217—21no—
71C2×#217—21no—
72C2×#217—21no—
73D4×Dic3—22no—
74D4×Dic3D414yescomplemented
75C2×#213—24no—
76C2×#213—24no—
77C2×#213—24no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—21no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3D411yescharacteristic
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#893—24no—
94#893—24no—
95#893—44no—
96#893—44no—
97#877—44no—
98#874—24no—
99C3×#102—24no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103#821—24no—
104#821—24no—
105Dic3⋊C8—24no—
106C8×Dic3—24no—
107#809—24no—
108#807—24no—
109C22×D8—32no—
110C2×#127—31no—
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—31no—
125#390—32no—
126#390—61no—
127#390—61no—
128#390C2×D311yescharacteristic
129#389—31no—
130#389—31no—
131#388—61no—
132#388—61no—
133#384C2×D311yescharacteristic
134C4×C4⋊C4—31no—
135C4×C22⋊C4—61no—
136C43—31no—
137#340—32no—
138#340C2×D312yes—
139#336—31no—
140#336C2×D311yescharacteristic
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—22no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C22×Dic3—21no—
153C22×Dic3—24no—
154C22×Dic3—21no—
155C22×Dic3—42no—
156C22×Dic3—42no—
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C22×Dic3C2×D411yescharacteristic
160C22×Dic3C2×D411yescharacteristic
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—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—42no—
186C3×C42—22no—
187#217—24no—
188#217—42no—
189#217—42no—
190#213—216no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—44no—
194C4⋊Dic3—21no—
195C4⋊Dic3—22no—
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#4114yes—
210C22×D4—32no—
211C22×D4—61no—
212C22×D4—61no—
213C22×D4C22×D311yescharacteristic
214C23×C4—62no—
215C23×C4—61no—
216C2×D8—64no—
217C2×D8—64no—
218C2×D8—64no—
219C22×C8—32no—
220C22×C8C22×D312yes—
221#127—64no—
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—21no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4C22×D312yes—
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
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—64no—
257#102—64no—
258#102—64no—
259C4×C8—64no—
260#95—61no—
261#95—61no—
262#95—61no—
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—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4C2×D812yes—
291C2×C3×C4#13512yes—
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C4C2×#4112yes—
295C2×C3×C4C2×#4111yescharacteristic
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—22no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—48no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3#12112yes—
311C2×Dic3#12112yes—
312C2×Dic3C2×D812yes—
313C2×Dic3C2×D812yes—
314C2×Dic3AGL(1,ℤ/8)12yes—
315D12—24no—
316D12—24no—
317D12—48no—
318D12—48no—
319D12—42no—
320D12—42no—
321C4×D3—88no—
322C3×C8—42no—
323C3⋊C8—24no—
324C24—21no—
325C24—62no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—21no—
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×D4D3×D412yes—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—64no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—124no—
352C22×C4—124no—
353C22×C4—124no—
354C22×C4—122no—
355C22×C4—122no—
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—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—32no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—61no—
373C4⋊C4—62no—
374C4⋊C4—122no—
375C4⋊C4—122no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—64no—
379C22⋊C4—124no—
380C22⋊C4—124no—
381C22⋊C4—122no—
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—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—81no—
417C3×C4—82no—
418C3×C4#52512yes—
419C3×C4#52912yes—
420Dic3—22no—
421Dic3—22no—
422Dic3—41no—
423Dic3—42no—
424Dic3—81no—
425Dic3#45914yes—
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—
440C23C2×D3×D411yescharacteristic
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—32no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—62no—
472C2×C4—128no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—248no—
479C2×C4D3×D812yes—
480C2×C4#91612yes—
481C2×C4C2×D3×D411yescharacteristic
482C2×C4D3×#4112yes—
483C2×C4D3×#4112yes—
484C2×C4D3×#4111yescharacteristic
485C8—42no—
486C8—64no—
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—
510C22D3×#12111yescharacteristic
511C22#565611yescharacteristic
512C22#566811yescharacteristic
513C22C2×D3×D811yescharacteristic
514C22C2×#91611yescharacteristic
515C22D3×AGL(1,ℤ/8)11yescharacteristic
516C22#577711yescharacteristic
517C4—22no—
518C4—22no—
519C4—34no—
520C4—62no—
521C4—62no—
522C4—81no—
523C4—82no—
524C4—121no—
525C4—122no—
526C4—241no—
527C4#565612yes—
528C4#570612yes—
529C3#2367611yescharacteristic, complemented, Sylow
530C2—82no—
531C2—244no—
532C2#7745711yescharacteristic
533C2D3×#45911yescharacteristic
534C2#8042711yescharacteristic
535C2#8126811yescharacteristic
536C2#8130911yescharacteristic
537C2#8155711yescharacteristic
538C2#8166411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radical#69658C2
hypercenter#2071D3
hypocenterC3#23676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
1. C2×#4756C2semidirect
2. #76289C2semidirect
3. #74319C2semidirect
4. C3×#2071C2semidirect
5. #70212C2semidirect
7. #69569C2semidirect
8. #68901C2semidirect
9. #68780C2semidirect
10. #68629C2semidirect
11. #68602C2semidirect
49. #2071D3semidirect
74. 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×#47562
#762892
#743192
C3×#20712
#702122
#696582
#695692
#689012
#687802
#686292
#686022
#236763