Finite Group Databasebeta

← #10876121

Subgroups — #10876121

Order 768.

1128 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4988C211yesmaximal, characteristic
2#73864C211yesmaximal, characteristic, complemented
3C3×#1852C211yesmaximal, characteristic, complemented
4#71555C211yesmaximal, characteristic, complemented
5#70641C212yesmaximal, complemented
6#70623C212yesmaximal, complemented
7#69959C211yesmaximal, characteristic
8#69910C211yesmaximal, characteristic, complemented
9#69521C211yesmaximal, characteristic
10#68902C212yesmaximal, complemented
11#68791C212yesmaximal, complemented
12#27582—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×(C2×D3)⋊C4—21no—
16C22×(C2×D3)⋊C4—21no—
17C22×C4×Dic3C2211yescharacteristic
18C2×#877—21no—
19C2×C3×#107C2212yescomplemented
20C2×C3×C22⋊C8C2211yescharacteristic
21C3×#380C2211yescharacteristic
22#5212C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#821C2212yes—
25C2×#821C2212yes—
26#4988—21no—
27#4988—21no—
28#4988C2212yes—
29#4988C2212yes—
30#4975C2211yescharacteristic, complemented
31#4975C2211yescharacteristic, complemented
32Dic3×C4⋊C4C2211yescharacteristic, complemented
33Dic3×C4⋊C4C2211yescharacteristic, complemented
34C2×#809C2212yes—
35C2×#809C2212yes—
36C2×#807C2212yescomplemented
37#4942—21no—
38#4650—21no—
39#4649—21no—
40C3×#340C2212yescomplemented
41#4550C2212yescomplemented
42#4550C2212yescomplemented
43#4529C2212yescomplemented
44C2×#390—31no—
45#2285—31no—
46#2093—32no—
47#2071—32no—
48#1925—32no—
49#1925—32no—
50#1852—31no—
51#1852D311yescharacteristic, complemented
52#1817—31no—
53#1786—31no—
54#1786—31no—
55C24×D3—21no—
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3D411yescharacteristic
59C23×Dic3D411yescharacteristic
60C22×D12—21no—
61C22×D12—24no—
62C22×D12—22no—
63C22×D12—22no—
64C22×D12—42no—
65C22×D12C2312yes—
66C22×C3×C8C2312yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×#217—21no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—44no—
79C2×(C2×D3)⋊C4—44no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8C2312yes—
95#877—44no—
96C3×#107—42no—
97C3×C22⋊C8—22no—
98C3×#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102#830—22no—
103#821—24no—
104#821—42no—
105#809—24no—
106#809—42no—
107#807—42no—
108C23×D4—31no—
109C2×#117—61no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C22⋊C4—61no—
112C22×C22⋊C4—61no—
113C22×C42—31no—
114C2×#107—32no—
115C2×#107C2×D312yescomplemented
116C2×#102—32no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—32no—
120C2×C22⋊C8—31no—
121C2×C22⋊C8C2×D311yescharacteristic
122#390—32no—
123#390—32no—
124#390—61no—
125#390—61no—
126#384—31no—
127#384—31no—
128#382—61no—
129#380—31no—
130#380—61no—
131#380C2×D311yescharacteristic
132C4×C4⋊C4—31no—
133C4×C4⋊C4—31no—
134C4×C4⋊C4—61no—
135#340—32no—
136#340C2×D312yescomplemented
137#336—32no—
138#336—32no—
139C24×C3C2×D411yescharacteristic
140C23×D3—24no—
141C23×D3—22no—
142C23×D3—42no—
143C23×D3—42no—
144C23×D3—42no—
145C23×D3—42no—
146C23×D3—42no—
147C23×D3—42no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4#4112yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—41no—
167C22×Dic3D814yes—
168C22×Dic3C2×D411yescharacteristic
169C22×Dic3C2×D411yescharacteristic
170C22×Dic3C2×D411yescharacteristic
171C22×Dic3C2×D411yescharacteristic
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—24no—
177C2×D12—22no—
178C2×D12—48no—
179C2×D12—48no—
180C2×D12—42no—
181C2×D12—42no—
182C2×D12#4114yes—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C22⋊C4—42no—
192#217—42no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—44no—
195(C2×D3)⋊C4—44no—
196(C2×D3)⋊C4—44no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199Dic3⋊C4—42no—
200Dic3⋊C4—42no—
201Dic3⋊C4—42no—
202Dic3⋊C4—42no—
203C4×Dic3—22no—
204C4×Dic3—22no—
205C4×Dic3—21no—
206C4×Dic3—21no—
207C4×Dic3—21no—
208C4×Dic3—21no—
209C4×Dic3—42no—
210C4×Dic3—42no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C25—61no—
215C22×D4—32no—
216C22×D4—64no—
217C22×D4—62no—
218C22×D4—61no—
219C22×D4—62no—
220C22×D4—122no—
221C23×C4—21no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4C22×D311yescharacteristic
225C22×C8—32no—
226C22×C8C22×D312yes—
227#117—124no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4C22×D312yes—
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—124no—
246C2×C22⋊C4—124no—
247C2×C42—31no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—31no—
251C2×C42—62no—
252C2×C42—62no—
253C2×C42—61no—
254C2×C42—61no—
255#107—42no—
256#107—122no—
257#102—64no—
258#102—64no—
259#102—122no—
260#102—122no—
261C22⋊C8—22no—
262C22⋊C8—62no—
263#95—21no—
264#95—61no—
265#95—61no—
266#95—61no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3#12312yes—
272C23×C3C2×D812yes—
273C23×C3#13712yes—
274C23×C3C22×D411yescharacteristic
275C22×D3—24no—
276C22×D3—48no—
277C22×D3—48no—
278C22×D3—48no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—84no—
286C22×D3—84no—
287C22×D3—84no—
288C22×D3—84no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—84no—
303C2×C3×C4#12312yes—
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4112yes—
307C2×C3×C4H(5,𝔽2)12yes—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—28no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—44no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3C2×D812yes—
326C2×Dic3C2×D812yes—
327C2×Dic3AGL(1,ℤ/8)12yes—
328D12—24no—
329D12—24no—
330D12—48no—
331D12—42no—
332D12—42no—
333C3×C8—42no—
334C3⋊C8—42no—
335C24—64no—
336C24—62no—
337C24—122no—
338C24—122no—
339C24—122no—
340C24—122no—
341C24—122no—
342C24—122no—
343C24D3×D411yescharacteristic
344C2×D4—34no—
345C2×D4—64no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—62no—
349C2×D4—62no—
350C2×D4—62no—
351C2×D4—128no—
352C2×D4—128no—
353C2×D4—122no—
354C2×D4—122no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—34no—
361C22×C4—31no—
362C22×C4—31no—
363C22×C4—31no—
364C22×C4—31no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—121no—
375C22×C4#23512yes—
376C22×C4D3×D411yescharacteristic
377C22×C4#23912yes—
378C22×C4C23×D311yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—122no—
393C4⋊C4—122no—
394C4⋊C4—122no—
395C4⋊C4—122no—
396C22⋊C4—42no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C22⋊C4—124no—
400C22⋊C4—124no—
401C22⋊C4—122no—
402C42—62no—
403C42—62no—
404C42—61no—
405C42—61no—
406C42—61no—
407C42—61no—
408C42—122no—
409C42—122no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3#48014yes—
417C22×C3C2×#12311yescharacteristic
418C22×C3#52511yescharacteristic
419C22×C3#53511yescharacteristic
420C22×C3C22×D811yescharacteristic
421C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
422C22×C3C2×#13711yescharacteristic
423C22×C3#57511yescharacteristic
424C2×D3—44no—
425C2×D3—44no—
426C2×D3—44no—
427C2×D3—816no—
428C2×D3—82no—
429C2×D3—82no—
430C2×D3—82no—
431C2×D3—82no—
432C3×C4—22no—
433C3×C4—81no—
434C3×C4—81no—
435C3×C4—82no—
436C3×C4#55112yes—
437C3×C4#55112yes—
438Dic3—22no—
439Dic3—22no—
440Dic3—22no—
441Dic3—41no—
442Dic3—81no—
443Dic3—81no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—64no—
449C23—128no—
450C23—128no—
451C23—128no—
452C23—122no—
453C23—122no—
454C23—122no—
455C23—122no—
456C23—122no—
457C23—122no—
458C23—244no—
459C23—244no—
460C23—244no—
461C23—244no—
462C23#89412yes—
463C23D3×D812yes—
464C23#91812yes—
465C23C2×D3×D411yescharacteristic
466D4—64no—
467D4—64no—
468D4—128no—
469D4—122no—
470D4—122no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—32no—
476C2×C4—32no—
477C2×C4—32no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—68no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—84no—
498C2×C4—124no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4#89412yes—
506C2×C4C2×#23511yescharacteristic
507C2×C4#100412yes—
508C2×C4C2×#23911yescharacteristic
509C2×C4D3×#4112yes—
510C8—42no—
511C8—122no—
512C2×C3—24no—
513C2×C3#248211yescharacteristic
514C2×C3C2×#48011yescharacteristic
515C2×C3#309411yescharacteristic
516C2×C3#310011yescharacteristic
517C2×C3#311011yescharacteristic
518C2×C3#320511yescharacteristic
519C2×C3#321511yescharacteristic
520D3—84no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—124no—
528C22—124no—
529C22—124no—
530C22—2416no—
531C22—242no—
532C22—242no—
533C22—242no—
534C22—242no—
535C22#488214yes—
536C22C2×#89411yescharacteristic
537C22#559211yescharacteristic
538C22#565611yescharacteristic
539C22C2×D3×D811yescharacteristic
540C22C2×#91811yescharacteristic
541C22D3×AGL(1,ℤ/8)11yescharacteristic
542C22#577711yescharacteristic
543C4—22no—
544C4—62no—
545C4—62no—
546C4—62no—
547C4—81no—
548C4—82no—
549C4—81no—
550C4—121no—
551C4—241no—
552C4—241no—
553C4#555612yes—
554C4#563712yes—
555C3#2758211yescharacteristic, complemented, Sylow
556C2—24no—
557C2—244no—
558C2#7461411yescharacteristic
559C2C2×#488211yescharacteristic
560C2#7921711yescharacteristic
561C2#7999411yescharacteristic
562C2#8020211yescharacteristic
563C2#8126811yescharacteristic
564C2#8145411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1852C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1852D3
hypocenterC3#27582
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. #73864C2semidirect
3. C3×#1852C2semidirect
4. #71555C2semidirect
5. #70641C2semidirect
6. #70623C2semidirect
8. #69910C2semidirect
10. #68902C2semidirect
11. #68791C2semidirect
19. C2×C3×#107C22semidirect
30. #4975C22semidirect
31. #4975C22semidirect
32. Dic3×C4⋊C4C22semidirect
33. Dic3×C4⋊C4C22semidirect
36. C2×#807C22semidirect
40. C3×#340C22semidirect
41. #4550C22semidirect
42. #4550C22semidirect
43. #4529C22semidirect
51. #1852D3semidirect
115. C2×#107C2×D3semidirect
136. #340C2×D3semidirect
555. C3#27582semidirect

Maximal subgroups

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

subgroupindex
C2×#49882
#738642
C3×#18522
#715552
#706412
#706232
#699592
#699102
#695212
#689022
#687912
#275823