Finite Group Databasebeta

← #10846287

Subgroups — #10846287

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#467C211yesmaximal, characteristic, complemented
2C2×#5191C211yesmaximal, characteristic, complemented
3C2×#5115C211yesmaximal, characteristic
4#82756C212yesmaximal, complemented
5C2×#4872C211yesmaximal, characteristic, complemented
6#72066C211yesmaximal, characteristic, complemented
7C8×#217C211yesmaximal, characteristic, complemented
8#71699C212yesmaximal, complemented
9#70612C211yesmaximal, characteristic, complemented
10#70276C212yesmaximal, complemented
11#70245C212yesmaximal, complemented
12#25413—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C2×#947C2212yes—
17C22×#215C2212yes—
18C2×#930C2212yes—
19C22×D24C2211yescharacteristic
20C2×#893C2212yes—
21C3×#467—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103C2212yes—
24#5229C2212yes—
25C2×#835C2212yes—
26C4×#217C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5191—24no—
29#5115—24no—
30C2×#821C2212yes—
31C2×#818C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4992—22no—
35#4987—22no—
36#4984C2212yes—
37C2×#809C2212yes—
38#4961—22no—
39#4956—22no—
40#4872—24no—
41#4550C2211yescharacteristic
42#4550C2211yescharacteristic
43#4529C2212yes—
44C2×#495—31no—
45C2×#467D311yescharacteristic, complemented
46C2×#466—31no—
47C2×C4×D8—31no—
48#2427—32no—
49#2042—32no—
50#2000—32no—
51#1969—31no—
52#1942—31no—
53#1877—32no—
54C8×C22⋊C4—31no—
55C22×C3×Q8C2312yes—
56C23×C3×C4C2311yescharacteristic
57C22×#67—22no—
58C22×D12C2312yes—
59C22×C4×D3—22no—
60C2×C3×Q16—22no—
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C22×C3×C8—22no—
64C22×C3×C8—22no—
65C22×C3×C8—22no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C22⋊Q8—24no—
69C3×C22⋊Q8—42no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—22no—
73C2×#217D411yescharacteristic
74C2×#217D411yescharacteristic
75#947—24no—
76#947—42no—
77C2×#215—24no—
78C2×#215—24no—
79C2×#215—42no—
80#930—24no—
81#930—42no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3C2312yes—
88C2×C4×Dic3C2311yescharacteristic
89C2×C4×Dic3C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92C2×D24—22no—
93C2×D24—22no—
94C2×D24D414yes—
95#893—24no—
96#893—42no—
97C3×#107D414yescomplemented
98C3×#103—24no—
99#835—42no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95D411yescharacteristic
104C3⋊#95D411yescharacteristic
105#830—22no—
106#821—24no—
107#818—22no—
108C8×Dic3—22no—
109C8×Dic3—22no—
110#809—24no—
111C22×Q16C2×D311yescharacteristic
112C22×SD16—32no—
113C22×D8—31no—
114C23×C8C2×D311yescharacteristic
115C2×#124—32no—
116C2×C22⋊Q8C2×D312yes—
117C2×#121—32no—
118C2×C4×D4—32no—
119#495—64no—
120#467—24no—
121#466—64no—
122C4×D8—64no—
123C2×#107—31no—
124C2×#107C2×D311yescharacteristic
125C2×#103—32no—
126C2×#103C2×D312yes—
127C2×#102—32no—
128C2×#102—32no—
129C2×C22⋊C8—31no—
130C2×C4×C8—31no—
131C2×C4×C8—31no—
132#397—62no—
133#394—62no—
134#391—32no—
135#389—32no—
136#386—62no—
137#380—62no—
138C4×C22⋊C4—31no—
139#340—32no—
140#336—31no—
141#336—31no—
142C24×C3C2×D411yescharacteristic
143C23×D3—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C2×#67—44no—
161C2×#67—44no—
162C2×#67—82no—
163C22×Dic3—22no—
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12C2×D414yes—
174C2×C4×D3—44no—
175C2×C4×D3—44no—
176C2×C4×D3—82no—
177C3×Q16—44no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—44no—
182C2×C3×C8C2×D412yes—
183C2×C3×C8C2×D412yes—
184C2×C3×C8#4112yes—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—44no—
187C3×C4⋊C4C2×D414yes—
188C3×C22⋊C4—44no—
189#217—22no—
190#217—22no—
191#215—48no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—44no—
195(C2×D3)⋊C4—44no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—82no—
199C4×Dic3—21no—
200C4×Dic3—21no—
201C4×Dic3—21no—
202C4×Dic3—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8#4112yes—
209D24—28no—
210C22×Q8C22×D312yes—
211C22×D4—32no—
212C22×D4—62no—
213C23×C4—62no—
214C23×C4C22×D311yescharacteristic
215C2×Q16—22no—
216C2×Q16—22no—
217C2×Q16—22no—
218C2×SD16—64no—
219C2×SD16—64no—
220C2×SD16—122no—
221C2×D8—34no—
222C2×D8—62no—
223C2×D8—62no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—31no—
228C22×C8—31no—
229C22×C8C22×D311yescharacteristic
230C22×C8C22×D311yescharacteristic
231#124—64no—
232#124—122no—
233C22⋊Q8—24no—
234C22⋊Q8—42no—
235#121—64no—
236#121—122no—
237C4×D4—64no—
238C4×D4—122no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—31no—
245C2×C22⋊C4—31no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—62no—
250C2×C42—31no—
251C2×C42—31no—
252#107—62no—
253#107#6714yescomplemented
254#103—24no—
255#103—122no—
256#102—64no—
257#102—64no—
258C22⋊C8—62no—
259C4×C8—62no—
260C4×C8—62no—
261#95—31no—
262#95—31no—
263#95—62no—
264#95—62no—
265#95—62no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3#13512yes—
273C23×C3C22×D411yescharacteristic
274C22×D3—24no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C3×Q8—42no—
282C3×Q8—42no—
283C3×Q8—42no—
284C3×Q8—42no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—21no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×C3×C4H(5,𝔽2)11yescharacteristic
304#67—88no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3#13512yes—
320D12—24no—
321D12—24no—
322D12—42no—
323D12—42no—
324C4×D3—88no—
325C3×C8—41no—
326C3×C8#12114yes—
327C3⋊C8—22no—
328C3⋊C8—41no—
329C24—62no—
330C24D3×D411yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×D4—34no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—124no—
344C2×D4—124no—
345C2×D4—242no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—62no—
359C22×C4—124no—
360C22×C4—124no—
361C22×C4—242no—
362C22×C4D3×D411yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—44no—
365SD16—128no—
366D8—68no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—32no—
371C2×C8—44no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—61no—
376C2×C8—61no—
377C2×C8D3×D412yes—
378C2×C8#23712yes—
379C2×C8C2×#6712yes—
380C4⋊C4—22no—
381C4⋊C4—44no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—242no—
385C4⋊C4C2×#6714yes—
386C22⋊C4—44no—
387C22⋊C4—62no—
388C22⋊C4—62no—
389C22⋊C4—124no—
390C22⋊C4—124no—
391C22⋊C4—124no—
392C22⋊C4—124no—
393C42—61no—
394C42—61no—
395C42—61no—
396C42—61no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3#53211yescharacteristic
405C22×C3D4211yescharacteristic
406C22×C3D4211yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3C2×#13511yescharacteristic
409C22×C3#57611yescharacteristic
410C22×C3#57811yescharacteristic
411C2×D3—44no—
412C2×D3—44no—
413C2×D3—44no—
414C2×D3—82no—
415C2×D3—82no—
416C2×D3—82no—
417C2×D3—82no—
418C3×C4—41no—
419C3×C4—44no—
420C3×C4—82no—
421C3×C4C2×#12111yescharacteristic
422C3×C4#54011yescharacteristic
423C3×C4#54611yescharacteristic
424C3×C4#55011yescharacteristic
425Dic3—22no—
426Dic3—41no—
427Dic3—41no—
428Dic3—41no—
429Dic3—82no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—64no—
437C23—122no—
438C23—122no—
439C23—122no—
440C23—122no—
441C23—122no—
442C23—122no—
443C23#91912yes—
444C23C2×D3×D411yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449D4—64no—
450D4—64no—
451D4—122no—
452D4—122no—
453D4—248no—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—32no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—248no—
483C2×C4C2×D3×D411yescharacteristic
484C2×C4C2×#23711yescharacteristic
485C2×C4#100411yescharacteristic
486C2×C4D3×#4111yescharacteristic
487C2×C4#100911yescharacteristic
488C2×C4C22×#6711yescharacteristic
489C8—41no—
490C8—62no—
491C8—121no—
492C8#93814yes—
493C2×C3—42no—
494C2×C3#242811yescharacteristic
495C2×C3#316811yescharacteristic
496C2×C3#317511yescharacteristic
497C2×C3#328011yescharacteristic
498C2×C3#328011yescharacteristic
499C2×C3#328311yescharacteristic
500C2×C3#328511yescharacteristic
501D3—84no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—242no—
513C22—242no—
514C22—242no—
515C22—242no—
516C22#559611yescharacteristic
517C22#560911yescharacteristic
518C22C2×#91911yescharacteristic
519C22#576611yescharacteristic
520C22D3×#13511yescharacteristic
521C22#576911yescharacteristic
522C22D4×#6711yescharacteristic
523C4—44no—
524C4—41no—
525C4—62no—
526C4—82no—
527C4—121no—
528C4—121no—
529C4—121no—
530C4—242no—
531C4#561211yescharacteristic
532C4#561911yescharacteristic
533C4C2×#93811yescharacteristic
534C4#583411yescharacteristic
535C3#2541311yescharacteristic, complemented, Sylow
536C2—42no—
537C2—244no—
538C2#8074011yescharacteristic
539C2#8074211yescharacteristic
540C2#8077311yescharacteristic
541C2#8077411yescharacteristic
542C2#8264811yescharacteristic
543C2#8388911yescharacteristic
544C2#8391811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#467C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
hypercenterC2×#467D3
hypocenterC3#25413
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×C3×#467C2semidirect
2. C2×#5191C2semidirect
4. #82756C2semidirect
5. C2×#4872C2semidirect
6. #72066C2semidirect
7. C8×#217C2semidirect
8. #71699C2semidirect
9. #70612C2semidirect
10. #70276C2semidirect
11. #70245C2semidirect
45. C2×#467D3semidirect
97. C3×#107D4semidirect
253. #107#67semidirect
535. C3#25413semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4672
C2×#51912
C2×#51152
#827562
C2×#48722
#720662
C8×#2172
#716992
#706122
#702762
#702452
#254133