Finite Group Databasebeta

← #10900964

Subgroups — #10900964

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×#5003C212yesmaximal
3#82080C211yesmaximal, characteristic
4#71792C211yesmaximal, characteristic, complemented
5#71792C211yesmaximal, characteristic, complemented
6#71783C211yesmaximal, characteristic, complemented
7#71782C211yesmaximal, characteristic, complemented
8#71743C212yesmaximal, complemented
9#71739C212yesmaximal, complemented
10#71575C212yesmaximal, complemented
11#71531C211yesmaximal, characteristic, complemented
12#29644—31nomaximal, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8—21no—
21C2×C3×#117C2212yes—
22C22×C3×C42C2211yescharacteristic
23C2×#930C2211yescharacteristic
24C2×#929—21no—
25#5213—21no—
26#5212C2211yescharacteristic
27C2×#830C2212yes—
28#5003—28no—
29C2×#809C2212yes—
30C2×#809C2212yes—
31C2×#808C2212yes—
32C2×#807C2212yes—
33#4940—21no—
34#4940—21no—
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4928C2211yescharacteristic
38#4928C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#533D311yescharacteristic, complemented
47C2×#482—32no—
48#2854—31no—
49#2059—32no—
50#2058—31no—
51#2057—31no—
52#2025—32no—
53#2023—31no—
54#2023—31no—
55#1825—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8—21no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×D12C2311yescharacteristic
64C3×C4⋊Q8—24no—
65C3×C4⋊Q8—24no—
66C3×#125—24no—
67C3×#125—24no—
68C3×C22⋊Q8—28no—
69C3×C22⋊Q8—28no—
70C3×C4×Q8—28no—
71C3×C4×Q8—28no—
72C3×#117—24no—
73C3×#117D418yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C42—24no—
82C2×C3×C42—22no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88#930—24no—
89#929—44no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2312yes—
102C3⋊#95—21no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105#830—24no—
106#809—24no—
107#809—24no—
108#808—24no—
109#807—24no—
110#533—28no—
111C2×C4⋊Q8C2×D311yescharacteristic
112C2×C4⋊Q8C2×D311yescharacteristic
113C2×#125C2×D312yes—
114C2×#123—61no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×#121—31no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×#117C2×D312yes—
121C22×C42C2×D311yescharacteristic
122#482—68no—
123C2×#107—32no—
124C2×#106—32no—
125C2×#102—32no—
126C2×#102—32no—
127C2×C22⋊C8—32no—
128#390—31no—
129#390—31no—
130#389—31no—
131#389—31no—
132#388—61no—
133#388—61no—
134#386—61no—
135#380—31no—
136#340—32no—
137#340—32no—
138#340—32no—
139#340—32no—
140#336—31no—
141#336—31no—
142#336—32no—
143C24×C3C2×D411yescharacteristic
144C23×D3—21no—
145C2×C3×Q8—24no—
146C2×C3×Q8—24no—
147C2×C3×Q8—42no—
148C2×C3×Q8—42no—
149C2×C3×Q8—42no—
150C2×C3×Q8—42no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—41no—
161C22×C3×C4—41no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×#67—42no—
170C2×#67—42no—
171C2×#67—42no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12—21no—
179C3×C4⋊C4—28no—
180C3×C4⋊C4—28no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4C2×D414yes—
186C3×C4⋊C4C2×D414yes—
187C3×C22⋊C4—28no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—28no—
191C3×C42—22no—
192C3×C42—22no—
193#217—42no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—42no—
196(C2×D3)⋊C4—42no—
197(C2×D3)⋊C4—42no—
198(C2×D3)⋊C4—42no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3—21no—
202C4⋊Dic3—21no—
203Dic3⋊C4—42no—
204Dic3⋊C4—42no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—21no—
212C22×Q8—21no—
213C22×D4—31no—
214C22×D4—61no—
215C23×C4#6711yescharacteristic
216C23×C4#6711yescharacteristic
217C23×C4C22×D311yescharacteristic
218C22×C8—32no—
219C22×C8—32no—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222#125—24no—
223#125—24no—
224#123—124no—
225C22⋊Q8—28no—
226C22⋊Q8—28no—
227#121—64no—
228C4×Q8—28no—
229C4×Q8—28no—
230#117—24no—
231#117#6718yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4C22×D312yes—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—61no—
249C2×C22⋊C4—61no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42#6711yescharacteristic
253C2×C42#6711yescharacteristic
254C2×C42C22×D311yescharacteristic
255C2×C42C22×D311yescharacteristic
256#107—64no—
257#106—64no—
258#102—64no—
259#102—64no—
260C22⋊C8—64no—
261#95—61no—
262#95—61no—
263#95—61no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3#12012yes—
271C23×C3C22×D411yescharacteristic
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—41no—
279C3×Q8—48no—
280C3×Q8—48no—
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—24no—
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—44no—
298C2×C3×C4—44no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4#12012yes—
304C2×C3×C4#13512yes—
305C2×C3×C4AGL(1,ℤ/8)12yes—
306C2×C3×C4C22×D412yes—
307C2×C3×C42−1+412yes—
308C2×C3×C42−1+411yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310#67—84no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320D12—42no—
321D12—42no—
322C3⋊C8—42no—
323C3⋊C8—42no—
324C24—61no—
325C24C2×#6711yescharacteristic
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×D4—62no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—62no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4—122no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4C2×#6711yescharacteristic
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6714yes—
359C22×C4C23×D311yescharacteristic
360C2×C8—62no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—62no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C4⋊C4—62no—
376C4⋊C4—122no—
377C4⋊C4—122no—
378C4⋊C4C2×#6714yes—
379C4⋊C4C2×#6714yes—
380C22⋊C4—28no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C42—22no—
388C42—22no—
389C42—28no—
390C42—22no—
391C42—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#53611yescharacteristic
401C22×C3#53611yescharacteristic
402C22×C3C2×#13511yescharacteristic
403C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
404C22×C3#57511yescharacteristic
405C22×C3#57511yescharacteristic
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C2×D3—81no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—41no—
417C3×C4—44no—
418C3×C4—41no—
419C3×C4—44no—
420C3×C4#53612yes—
421C3×C4#53612yes—
422Dic3—81no—
423Dic3—81no—
424Dic3—81no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—121no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23—121no—
436C23—121no—
437C23—121no—
438C23#95312yes—
439C23C22×#6711yescharacteristic
440Q8—48no—
441Q8—48no—
442D4—122no—
443D4—122no—
444D4—244no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4#94912yes—
477C2×C4#95012yes—
478C2×C4#95312yes—
479C2×C4#100712yes—
480C2×C4#100711yescharacteristic
481C2×C4#100711yescharacteristic
482C2×C4C22×#6712yes—
483C8—122no—
484C8—122no—
485C2×C3—42no—
486C2×C3#261411yescharacteristic
487C2×C3#300211yescharacteristic
488C2×C3#300611yescharacteristic
489C2×C3#322811yescharacteristic
490C2×C3#323311yescharacteristic
491C2×C3#323511yescharacteristic
492C2×C3#324011yescharacteristic
493D3—161no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—241no—
502C22—241no—
503C22—241no—
504C22—241no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22#581611yescharacteristic
509C22#581611yescharacteristic
510C22C2×#94911yescharacteristic
511C22C2×#95011yescharacteristic
512C22#582211yescharacteristic
513C22#582211yescharacteristic
514C22C2×#95311yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—44no—
520C4—41no—
521C4—44no—
522C4—241no—
523C4—241no—
524C4—241no—
525C4#581612yes—
526C4#581612yes—
527C3#2964411yescharacteristic, complemented, Sylow
528C2—42no—
529C2—481no—
530C2#8335911yescharacteristic
531C2#8336011yescharacteristic
532C2#8336111yescharacteristic
533C2#8336211yescharacteristic
534C2#8406911yescharacteristic
535C2#8408011yescharacteristic
536C2#8408211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#533D3
hypocenterC3#29644
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
4. #71792C2semidirect
5. #71792C2semidirect
6. #71783C2semidirect
7. #71782C2semidirect
8. #71743C2semidirect
9. #71739C2semidirect
10. #71575C2semidirect
11. #71531C2semidirect
46. C2×#533D3semidirect
527. C3#29644semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50032
#820802
#717922
#717832
#717822
#717432
#717392
#715752
#715312
#296443