Finite Group Databasebeta

← #10846293

Subgroups — #10846293

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×#466C211yesmaximal, characteristic
2C2×#5155C211yesmaximal, characteristic, complemented
3C2×Dic3×D8C211yesmaximal, characteristic, complemented
4C2×#5116C211yesmaximal, characteristic, complemented
5#82647C212yesmaximal, complemented
6C8×#217C211yesmaximal, characteristic, complemented
7#71685C212yesmaximal, complemented
8#70627C211yesmaximal, characteristic, complemented
9#70611C211yesmaximal, characteristic, complemented
10#70216C212yesmaximal, complemented
11#70171C212yesmaximal, complemented
12#25413—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×#936C2212yes—
17C2×D4×Dic3C2212yes—
18C22×#214C2212yes—
19C2×#924C2212yes—
20C22×Dic12C2211yescharacteristic
21C3×#466—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#102C2212yes—
24#5222—22no—
25#5220—22no—
26C2×#832C2212yes—
27C4×#217C2211yescharacteristic
28C2×#830C2211yescharacteristic
29#5155—24no—
30Dic3×D8—24no—
31#5116—24no—
32C2×#818C2211yescharacteristic
33C2×#816C2212yes—
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4973C2212yes—
37#4972C2212yes—
38C2×#810C2212yes—
39#4952—22no—
40#4942—22no—
41#4550C2211yescharacteristic
42#4550C2211yescharacteristic
43#4529C2212yes—
44C2×#495—31no—
45C2×#467—31no—
46C2×#466D311yescharacteristic
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×D4—22no—
56C22×C3×D4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—22no—
59C22×Dic6C2312yes—
60C2×C3×D8—22no—
61C2×C3×D8—22no—
62C2×C3×D8D414yes—
63C22×C3×C8—22no—
64C22×C3×C8—22no—
65C22×C3×C8—22no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×#121—24no—
69C3×#121—42no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217D411yescharacteristic
77C2×#217D411yescharacteristic
78#936—24no—
79#936—42no—
80D4×Dic3—24no—
81D4×Dic3—42no—
82C2×#214—24no—
83C2×#214—24no—
84C2×#214—42no—
85C2×C4⋊Dic3C2312yes—
86#924—24no—
87#924—42no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C2×Dic12—22no—
95C2×Dic12—22no—
96C2×Dic12—22no—
97C3×#107—22no—
98C3×#102—24no—
99#832—24no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95D411yescharacteristic
104C3⋊#95D411yescharacteristic
105#830—22no—
106#818D414yescomplemented
107#816—24no—
108C8×Dic3—22no—
109C8×Dic3—22no—
110#810—42no—
111C22×Q16—31no—
112C22×SD16—32no—
113C22×D8C2×D311yescharacteristic
114C23×C8C2×D311yescharacteristic
115C2×#124—32no—
116C2×C22⋊Q8—32no—
117C2×#121C2×D312yes—
118C2×C4×D4—32no—
119#495—64no—
120#467—64no—
121#466—24no—
122C4×D8—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#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×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—44no—
150C2×C3×D4—44no—
151C2×C3×D4—82no—
152C2×C3×D4C2×D414yes—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—44no—
165C22×Dic3—44no—
166C22×Dic3—82no—
167C22×Dic3C2×D411yescharacteristic
168C22×Dic3C2×D411yescharacteristic
169C22×Dic3C2×D411yescharacteristic
170C22×Dic3C2×D411yescharacteristic
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C3×D8—28no—
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—22no—
187C3×C22⋊C4—44no—
188#217—22no—
189#217—22no—
190#217—44no—
191#217—44no—
192#217—44no—
193#217—44no—
194#214—48no—
195C4⋊Dic3—22no—
196C4⋊Dic3C2×D414yes—
197Dic3⋊C4—44no—
198Dic3⋊C4—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—
209Dic12—44no—
210C22×Q8—32no—
211C22×D4—22no—
212C22×D4C22×D312yes—
213C23×C4—62no—
214C23×C4C22×D311yescharacteristic
215C2×Q16—62no—
216C2×Q16—62no—
217C2×Q16—62no—
218C2×SD16—64no—
219C2×SD16—64no—
220C2×SD16—122no—
221C2×D8—22no—
222C2×D8—22no—
223C2×D8#6714yes—
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—64no—
234C22⋊Q8—122no—
235#121—24no—
236#121—42no—
237C4×D4—64no—
238C4×D4—122no—
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×C42—31no—
251C2×C42—31no—
252#107—22no—
253#107—34no—
254#103—64no—
255#103—122no—
256#102—24no—
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—24no—
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—24no—
282C3×D4—24no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—88no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
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—88no—
320C2×Dic3#13512yes—
321Dic6—42no—
322Dic6—42no—
323Dic6—42no—
324Dic6—42no—
325C3×C8—41no—
326C3×C8#12114yes—
327C3⋊C8—22no—
328C3⋊C8—41no—
329C24—22no—
330C24D3×D411yescharacteristic
331C2×Q8—62no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—82no—
345C2×D4C2×#6714yes—
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×C4—62no—
359C22×C4—124no—
360C22×C4—124no—
361C22×C4—242no—
362C22×C4D3×D411yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—124no—
365SD16—128no—
366D8—28no—
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—22no—
382C4⋊C4—34no—
383C4⋊C4—62no—
384C4⋊C4—124no—
385C4⋊C4—242no—
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—44no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—82no—
411C22×C3#53211yescharacteristic
412C22×C3D4211yescharacteristic
413C22×C3D4211yescharacteristic
414C22×C3C2×#13511yescharacteristic
415C22×C3C2×#13511yescharacteristic
416C22×C3#57611yescharacteristic
417C22×C3#57811yescharacteristic
418C3×C4—41no—
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—
429Dic3—82no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—24no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23#91212yes—
444C23C2×D3×D411yescharacteristic
445Q8—122no—
446Q8—122no—
447Q8—122no—
448Q8—122no—
449D4—24no—
450D4—24no—
451D4—42no—
452D4—42no—
453D4—88no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—32no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
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—61no—
476C2×C4—64no—
477C2×C4—122no—
478C2×C4—122no—
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#93914yes—
493C2×C3—42no—
494C2×C3—84no—
495C2×C3#242811yescharacteristic
496C2×C3#316811yescharacteristic
497C2×C3#317511yescharacteristic
498C2×C3#328011yescharacteristic
499C2×C3#328011yescharacteristic
500C2×C3#328311yescharacteristic
501C2×C3#328511yescharacteristic
502C22—22no—
503C22—22no—
504C22—22no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22#559611yescharacteristic
517C22#560911yescharacteristic
518C22C2×#91211yescharacteristic
519C22#575711yescharacteristic
520C22D3×#13511yescharacteristic
521C22#577111yescharacteristic
522C22D4×#6711yescharacteristic
523C4—41no—
524C4—62no—
525C4—82no—
526C4—124no—
527C4—121no—
528C4—121no—
529C4—121no—
530C4—242no—
531C4#559911yescharacteristic
532C4#561911yescharacteristic
533C4C2×#93911yescharacteristic
534C4#582911yescharacteristic
535C3#2541311yescharacteristic, complemented, Sylow
536C2—42no—
537C2—84no—
538C2#8069611yescharacteristic
539C2#8069811yescharacteristic
540C2#8073011yescharacteristic
541C2#8073411yescharacteristic
542C2#8264911yescharacteristic
543C2#8383911yescharacteristic
544C2#8392611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#466C2
socleC23×C3C22×D4
hypercenterC2×#466D3
hypocenterC3#25413
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×Dic3×D8C2semidirect
4. C2×#5116C2semidirect
5. #82647C2semidirect
6. C8×#217C2semidirect
7. #71685C2semidirect
8. #70627C2semidirect
9. #70611C2semidirect
10. #70216C2semidirect
11. #70171C2semidirect
106. #818D4semidirect
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×#4662
C2×#51552
C2×Dic3×D82
C2×#51162
#826472
C8×#2172
#716852
#706272
#706112
#702162
#701712
#254133