Finite Group Databasebeta

← #10846282

Subgroups — #10846282

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 544 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×Dic3×Q16C211yesmaximal, characteristic, complemented
4C2×#5115C211yesmaximal, characteristic
5#82755C212yesmaximal, complemented
6C8×#217C211yesmaximal, characteristic, complemented
7#71699C212yesmaximal, complemented
8#70628C211yesmaximal, characteristic, complemented
9#70611C211yesmaximal, characteristic, complemented
10#70279C212yesmaximal, complemented
11#70240C212yesmaximal, complemented
12#25412—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C2×#947C2212yes—
17C2×Q8×Dic3C2212yes—
18C22×#215C2212yes—
19C2×#930C2212yes—
20C22×D24C2211yescharacteristic
21C3×#467—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103C2212yes—
24C2×#835C2212yes—
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#5191—24no—
28Dic3×Q16—24no—
29#5115—24no—
30C2×#821C2212yes—
31C2×#818C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4994—22no—
35#4990—22no—
36#4988C2212yes—
37#4972C2212yes—
38C2×#809C2212yes—
39#4961—22no—
40#4957—22no—
41#4550C2211yescharacteristic
42#4550C2211yescharacteristic
43#4529C2212yes—
44C2×#495—31no—
45C2×#467D311yescharacteristic, complemented
46C2×#466—31no—
47C2×C4×Q16—31no—
48#2426—32no—
49#2042—32no—
50#2000—32no—
51#1970—31no—
52#1942—31no—
53#1880—32no—
54C8×C22⋊C4—31no—
55C22×C3×Q8C2312yes—
56C23×C3×C4C2311yescharacteristic
57C22×#67—22no—
58C22×D12C2312yes—
59C2×C3×Q16—22no—
60C2×C3×Q16—22no—
61C2×C3×Q16D414yescomplemented
62C22×C3×C8—22no—
63C22×C3×C8—22no—
64C22×C3×C8—22no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—42no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—22no—
72C2×#217D411yescharacteristic
73C2×#217D411yescharacteristic
74#947—24no—
75#947—42no—
76Q8×Dic3—24no—
77Q8×Dic3—42no—
78C2×#215—24no—
79C2×#215—24no—
80C2×#215—42no—
81#930—24no—
82#930—42no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96C2×D24—22no—
97C2×D24—22no—
98C2×D24—22no—
99C3×#107—22no—
100C3×#103—24no—
101#835—24no—
102C3⋊#95—22no—
103C3⋊#95D411yescharacteristic
104C3⋊#95D411yescharacteristic
105#830—22no—
106#821—24no—
107#818D414yes—
108C8×Dic3—22no—
109C8×Dic3—22no—
110#809—42no—
111C22×Q16C2×D311yescharacteristic
112C22×SD16—32no—
113C22×D8—31no—
114C23×C8C2×D311yescharacteristic
115C2×#124—32no—
116C2×C22⋊Q8C2×D312yes—
117C2×#121—32no—
118C2×C4×Q8—32no—
119#495—64no—
120#467—24no—
121#466—64no—
122C4×Q16—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#396—62no—
133#394—62no—
134#390—32no—
135#390—62no—
136#389—32no—
137#388—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×Q8C2×D414yes—
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×D12—22no—
174C3×Q16—28no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—44no—
179C2×C3×C8C2×D412yes—
180C2×C3×C8C2×D412yes—
181C2×C3×C8#4112yes—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—44no—
185C3×C22⋊C4—44no—
186#217—22no—
187#217—22no—
188#215—48no—
189(C2×D3)⋊C4—44no—
190(C2×D3)⋊C4—44no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—82no—
194C4⋊Dic3—22no—
195C4⋊Dic3—44no—
196C4⋊Dic3C2×D414yes—
197Dic3⋊C4—82no—
198C4×Dic3—21no—
199C4×Dic3—21no—
200C4×Dic3—21no—
201C4×Dic3—21no—
202C4×Dic3—44no—
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—44no—
210C22×Q8C22×D312yes—
211C22×D4—32no—
212C22×D4—62no—
213C23×C4C22×D311yescharacteristic
214C2×Q16—22no—
215C2×Q16—22no—
216C2×Q16#6714yescomplemented
217C2×SD16—64no—
218C2×SD16—64no—
219C2×SD16—122no—
220C2×D8—62no—
221C2×D8—62no—
222C2×D8—62no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—31no—
227C22×C8—31no—
228C22×C8C22×D311yescharacteristic
229C22×C8C22×D311yescharacteristic
230#124—64no—
231#124—122no—
232C22⋊Q8—24no—
233C22⋊Q8—42no—
234#121—64no—
235#121—122no—
236C4×Q8—64no—
237C4×Q8—122no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—62no—
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×C22⋊C4—62no—
251C2×C42—31no—
252C2×C42—31no—
253C2×C42—62no—
254#107—22no—
255#107—34no—
256#103—24no—
257#103—64no—
258#102—64no—
259#102—122no—
260C22⋊C8—62no—
261C4×C8—62no—
262C4×C8—62no—
263#95—31no—
264#95—31no—
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—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C3×Q8—24no—
282C3×Q8—24no—
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—24no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3#13512yes—
320D12—42no—
321D12—42no—
322D12—42no—
323D12—42no—
324C3×C8—41no—
325C3×C8#12114yes—
326C3⋊C8—22no—
327C3⋊C8—41no—
328C24—62no—
329C24D3×D411yescharacteristic
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8C2×#6714yes—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—124no—
343C2×D4—124no—
344C2×D4—242no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—31no—
357C22×C4—62no—
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—28no—
361SD16—128no—
362D8—124no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—32no—
367C2×C8—44no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—61no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8D3×D412yes—
374C2×C8#23712yes—
375C2×C8C2×#6712yes—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—34no—
379C4⋊C4—44no—
380C4⋊C4—62no—
381C4⋊C4—124no—
382C4⋊C4—242no—
383C22⋊C4—44no—
384C22⋊C4—62no—
385C22⋊C4—62no—
386C22⋊C4—124no—
387C22⋊C4—124no—
388C22⋊C4—124no—
389C22⋊C4—124no—
390C22⋊C4—242no—
391C42—61no—
392C42—61no—
393C42—61no—
394C42—61no—
395C42—124no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3#53211yescharacteristic
404C22×C3D4211yescharacteristic
405C22×C3D4211yescharacteristic
406C22×C3C2×#13511yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3#57611yescharacteristic
409C22×C3#57811yescharacteristic
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—82no—
413C2×D3—82no—
414C2×D3—82no—
415C2×D3—82no—
416C2×D3—82no—
417C3×C4—41no—
418C3×C4—44no—
419C3×C4—82no—
420C3×C4C2×#12111yescharacteristic
421C3×C4#54011yescharacteristic
422C3×C4#54611yescharacteristic
423C3×C4#55011yescharacteristic
424Dic3—22no—
425Dic3—41no—
426Dic3—44no—
427Dic3—41no—
428Dic3—41no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—122no—
436C23—122no—
437C23—122no—
438C23—122no—
439C23—122no—
440C23—122no—
441C23—122no—
442C23#91912yes—
443C23C2×D3×D411yescharacteristic
444Q8—24no—
445Q8—24no—
446Q8—42no—
447Q8—42no—
448D4—122no—
449D4—122no—
450D4—122no—
451D4—122no—
452D4—248no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—32no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—61no—
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—64no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×#23711yescharacteristic
483C2×C4#100411yescharacteristic
484C2×C4D3×#4111yescharacteristic
485C2×C4#100911yescharacteristic
486C2×C4C22×#6711yescharacteristic
487C8—41no—
488C8—62no—
489C8—121no—
490C8#93914yes—
491C2×C3—42no—
492C2×C3#242811yescharacteristic
493C2×C3#316811yescharacteristic
494C2×C3#317511yescharacteristic
495C2×C3#328011yescharacteristic
496C2×C3#328511yescharacteristic
497C2×C3#328511yescharacteristic
498C2×C3#328711yescharacteristic
499D3—162no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—242no—
508C22—242no—
509C22—242no—
510C22—242no—
511C22—242no—
512C22—242no—
513C22—242no—
514C22#559611yescharacteristic
515C22#560911yescharacteristic
516C22C2×#91911yescharacteristic
517C22#576611yescharacteristic
518C22D3×#13511yescharacteristic
519C22#576911yescharacteristic
520C22D4×#6711yescharacteristic
521C4—41no—
522C4—44no—
523C4—62no—
524C4—82no—
525C4—121no—
526C4—124no—
527C4—121no—
528C4—121no—
529C4#559911yescharacteristic
530C4#561911yescharacteristic
531C4C2×#93911yescharacteristic
532C4#582911yescharacteristic
533C3#2541211yescharacteristic, complemented, Sylow
534C2—42no—
535C2—482no—
536C2#8074011yescharacteristic
537C2#8074211yescharacteristic
538C2#8077311yescharacteristic
539C2#8077911yescharacteristic
540C2#8264911yescharacteristic
541C2#8388911yescharacteristic
542C2#8392411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#467C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
hypercenterC2×#467D3
hypocenterC3#25412
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
3. C2×Dic3×Q16C2semidirect
5. #82755C2semidirect
6. C8×#217C2semidirect
7. #71699C2semidirect
8. #70628C2semidirect
9. #70611C2semidirect
10. #70279C2semidirect
11. #70240C2semidirect
45. C2×#467D3semidirect
61. C2×C3×Q16D4semidirect
216. C2×Q16#67semidirect
533. C3#25412semidirect

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×Dic3×Q162
C2×#51152
#827552
C8×#2172
#716992
#706282
#706112
#702792
#702402
#254123