Finite Group Databasebeta

← #10846276

Subgroups — #10846276

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×#466C211yesmaximal, characteristic
2C2×#5155C211yesmaximal, characteristic, complemented
3C2×#5116C211yesmaximal, characteristic, complemented
4#82646C212yesmaximal, complemented
5C2×#4873C211yesmaximal, characteristic, complemented
6#72038C211yesmaximal, characteristic, complemented
7C8×#217C211yesmaximal, characteristic, complemented
8#71685C212yesmaximal, complemented
9#70612C211yesmaximal, characteristic, complemented
10#70213C212yesmaximal, complemented
11#70178C212yesmaximal, complemented
12#25412—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×#936C2212yes—
17C22×#214C2212yes—
18C2×#924C2212yes—
19C22×Dic12C2211yescharacteristic
20C2×#887C2212yes—
21C3×#466—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#102C2212yes—
24#5222—22no—
25#5221—22no—
26#5221C2212yes—
27C2×#832C2212yes—
28C4×#217C2211yescharacteristic
29C2×#830C2211yescharacteristic
30#5155—24no—
31#5116—24no—
32C2×#818C2211yescharacteristic
33C2×#816C2212yes—
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4984C2212yes—
37C2×#810C2212yes—
38#4948—22no—
39#4944—22no—
40#4873—24no—
41#4550C2211yescharacteristic
42#4550C2211yescharacteristic
43#4529C2212yes—
44C2×#495—31no—
45C2×#467—31no—
46C2×#466D311yescharacteristic
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×D4—22no—
56C22×C3×D4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C22×Dic6C2312yes—
59C2×C3×D8—22no—
60C2×C3×D8—22no—
61C2×C3×D8—22no—
62C22×C3×C8—22no—
63C22×C3×C8—22no—
64C22×C3×C8—22no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×#121—24no—
68C3×#121—42no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C22⋊C4—22no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217D411yescharacteristic
77C2×#217D411yescharacteristic
78#936—24no—
79#936—42no—
80C2×#214—24no—
81C2×#214—24no—
82C2×#214—42no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3C2312yes—
85#924—24no—
86#924—42no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C2×Dic12—22no—
95C2×Dic12—22no—
96C2×Dic12D414yescomplemented
97#887—24no—
98#887—42no—
99C3×#107D414yes—
100C3×#102—24no—
101#832—42no—
102C3⋊#95—22no—
103C3⋊#95D411yescharacteristic
104C3⋊#95D411yescharacteristic
105#830—22no—
106#818—22no—
107#816—24no—
108C8×Dic3—22no—
109C8×Dic3—22no—
110#810—24no—
111C22×Q16—31no—
112C22×SD16—32no—
113C22×D8C2×D311yescharacteristic
114C23×C8C2×D311yescharacteristic
115C2×#124—32no—
116C2×C22⋊Q8—32no—
117C2×#121C2×D312yes—
118C2×C4×Q8—32no—
119#495—64no—
120#467—64no—
121#466—24no—
122C4×Q16—64no—
123C2×#107—31no—
124C2×#107C2×D311yescharacteristic
125C2×#103—32no—
126C2×#103—32no—
127C2×#102—32no—
128C2×#102C2×D312yes—
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×C3—22no—
143C24×C3C2×D411yescharacteristic
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4—82no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6C2×D414yes—
174C3×D8—44no—
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⋊C4C2×D414yes—
184C3×C22⋊C4—44no—
185#217—22no—
186#217—22no—
187#217—44no—
188#217—44no—
189#217—44no—
190#217—44no—
191#217—82no—
192#214—48no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—82no—
196Dic3⋊C4—44no—
197Dic3⋊C4—44no—
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—
209Dic12—28no—
210C22×Q8—32no—
211C22×D4—22no—
212C22×D4C22×D312yes—
213C23×C4C22×D311yescharacteristic
214C2×Q16—34no—
215C2×Q16—62no—
216C2×Q16—62no—
217C2×SD16—64no—
218C2×SD16—64no—
219C2×SD16—122no—
220C2×D8—22no—
221C2×D8—22no—
222C2×D8—22no—
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—64no—
233C22⋊Q8—122no—
234#121—24no—
235#121—42no—
236C4×Q8—64no—
237C4×Q8—122no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—62no—
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—62no—
255#107#6714yes—
256#103—64no—
257#103—64no—
258#102—24no—
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—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3#13512yes—
280C23×C3C22×D411yescharacteristic
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—88no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4H(5,𝔽2)11yescharacteristic
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—24no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3#13512yes—
320Dic6—24no—
321Dic6—24no—
322Dic6—42no—
323Dic6—42no—
324C3×C8—41no—
325C3×C8#12114yes—
326C3⋊C8—22no—
327C3⋊C8—41no—
328C24—22no—
329C24D3×D411yescharacteristic
330C2×Q8—34no—
331C2×Q8—62no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—82no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—68no—
361SD16—128no—
362D8—44no—
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—62no—
378C4⋊C4—62no—
379C4⋊C4—124no—
380C4⋊C4—124no—
381C4⋊C4—242no—
382C4⋊C4C2×#6714yes—
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—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3#53211yescharacteristic
411C22×C3D4211yescharacteristic
412C22×C3D4211yescharacteristic
413C22×C3C2×#13511yescharacteristic
414C22×C3C2×#13511yescharacteristic
415C22×C3#57611yescharacteristic
416C22×C3#57811yescharacteristic
417C3×C4—41no—
418C3×C4—44no—
419C3×C4C2×#12111yescharacteristic
420C3×C4#54011yescharacteristic
421C3×C4#54611yescharacteristic
422C3×C4#55011yescharacteristic
423Dic3—22no—
424Dic3—44no—
425Dic3—41no—
426Dic3—41no—
427Dic3—41no—
428Dic3—82no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23#91212yes—
443C23C2×D3×D411yescharacteristic
444Q8—64no—
445Q8—64no—
446Q8—122no—
447Q8—122no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—88no—
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—61no—
465C2×C4—61no—
466C2×C4—61no—
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—64no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
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#93814yes—
491C2×C3—42no—
492C2×C3—162no—
493C2×C3#242811yescharacteristic
494C2×C3#316811yescharacteristic
495C2×C3#317511yescharacteristic
496C2×C3#328011yescharacteristic
497C2×C3#328511yescharacteristic
498C2×C3#328511yescharacteristic
499C2×C3#328711yescharacteristic
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22#559611yescharacteristic
515C22#560911yescharacteristic
516C22C2×#91211yescharacteristic
517C22#575711yescharacteristic
518C22D3×#13511yescharacteristic
519C22#577111yescharacteristic
520C22D4×#6711yescharacteristic
521C4—44no—
522C4—41no—
523C4—62no—
524C4—121no—
525C4—121no—
526C4—121no—
527C4—124no—
528C4—242no—
529C4#561211yescharacteristic
530C4#561911yescharacteristic
531C4C2×#93811yescharacteristic
532C4#583411yescharacteristic
533C3#2541211yescharacteristic, complemented, Sylow
534C2—42no—
535C2—162no—
536C2#8069611yescharacteristic
537C2#8069811yescharacteristic
538C2#8073411yescharacteristic
539C2#8073511yescharacteristic
540C2#8264811yescharacteristic
541C2#8383911yescharacteristic
542C2#8392011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#466C2
socleC23×C3C22×D4
hypercenterC2×#466D3
hypocenterC3#25412
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
2. C2×#5155C2semidirect
3. C2×#5116C2semidirect
4. #82646C2semidirect
5. C2×#4873C2semidirect
6. #72038C2semidirect
7. C8×#217C2semidirect
8. #71685C2semidirect
9. #70612C2semidirect
10. #70213C2semidirect
11. #70178C2semidirect
96. C2×Dic12D4semidirect
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×#4662
C2×#51552
C2×#51162
#826462
C2×#48732
#720382
C8×#2172
#716852
#706122
#702132
#701782
#254123