Finite Group Databasebeta

← #10944079

Subgroups — #10944079

Order 768.

1430 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5516C211yesmaximal, characteristic, complemented
2#82676C211yesmaximal, characteristic
3C2×#5040C211yesmaximal, characteristic, complemented
4C2×#4762C212yesmaximal, complemented
5C2×#4738C211yesmaximal, characteristic, complemented
6C3×#1875C211yesmaximal, characteristic, complemented
7#71586C211yesmaximal, characteristic, complemented
8#70213C212yesmaximal, complemented
9#69909C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68905C212yesmaximal, complemented
12#32912—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#940C2211yescharacteristic
17C22×#214C2212yes—
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×#204C2212yes—
21#5516—28no—
22C2×#887C2212yes—
23C2×#887C2212yes—
24C2×#877—22no—
25C2×C4×Dic6—22no—
26C2×C3×#102C2212yes—
27C2×C3×C22⋊C8C2211yescharacteristic
28#5223—22no—
29#5221C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#816C2212yes—
32C2×Dic3⋊C8C2212yes—
33#5040—24no—
34#4988C2211yescharacteristic
35#4977C2211yescharacteristic
36C2×#810C2212yes—
37C2×#809C2212yes—
38#4961—22no—
39#4762—28no—
40#4738—24no—
41C3×#340C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44C2×#519—31no—
45C2×#460—32no—
46C2×#448—31no—
47C2×#448—31no—
48#2619—31no—
49#1880—32no—
50#1880—32no—
51#1875—31no—
52#1875D311yescharacteristic, complemented
53#1805—32no—
54#1798—31no—
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C22×#67—22no—
60C23×Dic3—21no—
61C22×D12C2311yescharacteristic
62C22×Dic6—21no—
63C22×Dic6—22no—
64C22×Dic6—24no—
65C22×Dic6—24no—
66C22×Dic6D414yescomplemented
67C22×Dic6C2312yes—
68C22×C3×C8C2312yes—
69C3×#121—22no—
70C3×#121—22no—
71C3×#117—24no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42—21no—
77C2×#217—22no—
78C2×#217D412yes—
79#940—24no—
80C2×#214—24no—
81C2×#214—24no—
82C2×#214—24no—
83#930—22no—
84#930—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—22no—
87#926—24no—
88C2×C4⋊Dic3D412yes—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3C2311yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C2×#204—24no—
98C2×#204—24no—
99C2×#204—24no—
100#887—24no—
101#887—24no—
102#887—24no—
103#887D418yescomplemented
104#877—216no—
105C4×Dic6—216no—
106C3×#102—24no—
107C3×C22⋊C8—22no—
108C3⋊#95—22no—
109#830—22no—
110#816—24no—
111Dic3⋊C8—24no—
112#810—24no—
113#809—24no—
114C23×Q8—31no—
115C22×SD16—32no—
116C22×SD16—32no—
117C2×#127—31no—
118C2×#121—31no—
119C2×#121C2×D311yescharacteristic
120#519—68no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—62no—
124C2×#117—31no—
125C2×#117—62no—
126C2×#117C2×D311yescharacteristic
127#460—68no—
128#448—64no—
129#448—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102C2×D312yes—
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#394—62no—
138#392—62no—
139#390—31no—
140#390—31no—
141#382—31no—
142#340—32no—
143#340C2×D312yes—
144#336—32no—
145C24×C3—21no—
146C24×C3C2×D411yescharacteristic
147C23×D3—21no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—42no—
153C2×C3×D4—82no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4#4112yes—
161C22×C3×C4C2411yescharacteristic
162C2×#67—42no—
163C2×#67—44no—
164C2×#67—82no—
165C2×#67—82no—
166C2×#67—82no—
167C22×Dic3—24no—
168C22×Dic3—24no—
169C22×Dic3—24no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3C2×D412yes—
173C22×Dic3C2×D412yes—
174C2×D12—22no—
175C2×D12—21no—
176C2×D12—22no—
177C2×D12—21no—
178C2×Dic6—24no—
179C2×Dic6—24no—
180C2×Dic6—216no—
181C2×Dic6—24no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—216no—
186C2×Dic6C2×D414yes—
187C2×Dic6C2×D414yes—
188C2×Dic6#4114yes—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4C2×D414yes—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—42no—
198C3×C42—24no—
199#217—24no—
200#217—82no—
201#214—48no—
202(C2×D3)⋊C4—42no—
203(C2×D3)⋊C4—82no—
204C4⋊Dic3—24no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209C4×Dic3—28no—
210C4×Dic3—22no—
211C4×Dic3—28no—
212C4×Dic3C2×D414yes—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216#204—48no—
217C22×Q8—34no—
218C22×Q8—32no—
219C22×Q8—64no—
220C22×Q8—64no—
221C22×Q8—62no—
222C22×Q8—61no—
223C22×D4—21no—
224C22×D4—31no—
225C22×D4—62no—
226C22×D4—61no—
227C22×D4C22×D311yescharacteristic
228C23×C4—61no—
229C23×C4C22×D311yescharacteristic
230C2×SD16—64no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—64no—
234C2×SD16—64no—
235C2×SD16—64no—
236C22×C8—32no—
237C22×C8C22×D312yes—
238#127—64no—
239#121—22no—
240#121—22no—
241#121—62no—
242#121—62no—
243C4×Q8—38no—
244C4×Q8—616no—
245C4×Q8—64no—
246C4×Q8—64no—
247C4×Q8—64no—
248#117—24no—
249#117—64no—
250#117—616no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4C22×D311yescharacteristic
256C2×C4⋊C4C22×D312yes—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C42—21no—
264C2×C42—31no—
265C2×C42—31no—
266C2×C42—62no—
267C2×C42—62no—
268C4⋊C8—64no—
269#103—64no—
270#103—64no—
271#102—24no—
272#102—64no—
273C22⋊C8—22no—
274C22⋊C8—62no—
275#95—62no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3#12112yes—
286C23×C3C22×D411yescharacteristic
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—42no—
291C22×D3—42no—
292C3×D4—42no—
293C3×D4—42no—
294C3×D4—84no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4#12112yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D412yes—
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4112yes—
309#67—88no—
310#67—84no—
311C2×Dic3—216no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3#12014yes—
323D12—42no—
324D12—42no—
325Dic6—216no—
326Dic6—24no—
327Dic6—24no—
328Dic6—24no—
329Dic6#12118yes—
330C3×C8—42no—
331C3⋊C8—42no—
332C24—21no—
333C24—61no—
334C24D3×D411yescharacteristic
335C2×Q8—34no—
336C2×Q8—34no—
337C2×Q8—34no—
338C2×Q8—616no—
339C2×Q8—64no—
340C2×Q8—64no—
341C2×Q8—616no—
342C2×Q8—64no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—42no—
351C2×D4—61no—
352C2×D4—62no—
353C2×D4—61no—
354C2×D4—62no—
355C2×D4—82no—
356C2×D4—124no—
357C2×D4—122no—
358C2×D4—242no—
359C2×D4—242no—
360C2×D4—242no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—32no—
366C22×C4—32no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4—64no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4#23512yes—
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D412yes—
375C22×C4C23×D311yescharacteristic
376SD16—128no—
377SD16—128no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—64no—
388C4⋊C4—68no—
389C4⋊C4—68no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4D3×D414yes—
393C22⋊C4—24no—
394C22⋊C4—42no—
395C22⋊C4—64no—
396C22⋊C4—122no—
397C22⋊C4—242no—
398C22⋊C4—242no—
399C42—24no—
400C42—34no—
401C42—68no—
402C42—68no—
403C42—62no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—81no—
410C22×C3—82no—
411C22×C3—82no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3C2×#12011yescharacteristic
415C22×C3C2×#12111yescharacteristic
416C22×C3#52511yescharacteristic
417C22×C3#57712yes—
418C22×C3#57712yes—
419C2×D3—81no—
420C2×D3—82no—
421C2×D3—82no—
422C2×D3—81no—
423C2×D3—81no—
424C3×C4—22no—
425C3×C4—44no—
426C3×C4—81no—
427C3×C4C2×#12112yes—
428C3×C4#52512yes—
429Dic3—28no—
430Dic3—24no—
431Dic3—42no—
432Dic3—81no—
433C23—21no—
434C23—22no—
435C23—22no—
436C23—21no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—121no—
443C23—121no—
444C23—122no—
445C23—122no—
446C23—121no—
447C23#88412yes—
448C23C2×D3×D411yescharacteristic
449Q8—38no—
450Q8—616no—
451Q8—64no—
452Q8—64no—
453Q8—64no—
454D4—42no—
455D4—42no—
456D4—84no—
457D4—122no—
458D4—122no—
459D4—248no—
460D4—244no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—34no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—616no—
472C2×C4—64no—
473C2×C4—64no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—121no—
481C2×C4—122no—
482C2×C4#88412yes—
483C2×C4C2×#23511yescharacteristic
484C2×C4C2×D3×D412yes—
485C2×C4C2×D3×D411yescharacteristic
486C2×C4D3×#4112yes—
487C8—42no—
488C8—122no—
489C2×C3—42no—
490C2×C3—161no—
491C2×C3#284711yescharacteristic
492C2×C3#301012yes—
493C2×C3#304311yescharacteristic
494C2×C3#304311yescharacteristic
495C2×C3#312612yes—
496D3—161no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—42no—
501C22—42no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—82no—
506C22—82no—
507C22—241no—
508C22—241no—
509C22—241no—
510C22—242no—
511C22—242no—
512C22C2×#88411yescharacteristic
513C22D3×#12011yescharacteristic
514C22#565611yescharacteristic
515C22#577012yes—
516C22#577512yes—
517C4—22no—
518C4—44no—
519C4—68no—
520C4—64no—
521C4—81no—
522C4—122no—
523C4—241no—
524C4#554712yes—
525C4D3×#12112yes—
526C3#3291211yescharacteristic, complemented, Sylow
527C2—42no—
528C2—161no—
529C2—481no—
530C2#7493511yescharacteristic
531C2#7886411yescharacteristic
532C2#7894511yescharacteristic
533C2#8008612yes—
534C2#8135512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1875C2
socleC23×C3C22×D4
prime radical#82676C2
hypercenter#1875D3
hypocenterC3#32912
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
1. C2×#5516C2semidirect
3. C2×#5040C2semidirect
4. C2×#4762C2semidirect
5. C2×#4738C2semidirect
6. C3×#1875C2semidirect
7. #71586C2semidirect
8. #70213C2semidirect
9. #69909C2semidirect
10. #68909C2semidirect
11. #68905C2semidirect
52. #1875D3semidirect
66. C22×Dic6D4semidirect
103. #887D4semidirect
526. C3#32912semidirect

Maximal subgroups

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

subgroupindex
C2×#55162
#826762
C2×#50402
C2×#47622
C2×#47382
C3×#18752
#715862
#702132
#699092
#689092
#689052
#329123