Finite Group Databasebeta

← #10944077

Subgroups — #10944077

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×#5812C211yesmaximal, characteristic, complemented
2C2×C3×#448C211yesmaximal, characteristic, complemented
3C2×#5227C211yesmaximal, characteristic, complemented
4C2×#5163C212yesmaximal, complemented
5#82676C211yesmaximal, characteristic
6#72048C212yesmaximal, complemented
7#71678C211yesmaximal, characteristic, complemented
8#70240C212yesmaximal, complemented
9#70222C212yesmaximal, complemented
10#69933C211yesmaximal, characteristic, complemented
11#69909C211yesmaximal, characteristic, complemented
12#32912—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#121C2211yescharacteristic
16#5812—28no—
17C2×Q8×Dic3—22no—
18C2×Q8×Dic3C2212yes—
19C2×Q8×Dic3C2212yes—
20C22×#215C2212yes—
21C2×#940C2211yescharacteristic
22C2×#930C2211yescharacteristic
23C2×#926—22no—
24C2×#926C2211yescharacteristic
25C2×#926C2211yescharacteristic
26C3×#448—24no—
27C2×C3×#103C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29#5227—24no—
30C2×#835C2212yes—
31#5223—22no—
32#5221C2211yescharacteristic
33C2×#832C2212yes—
34C2×#830C2211yescharacteristic
35#5163—28no—
36C2×#821C2212yes—
37C2×Dic3⋊C8C2212yes—
38#4988C2211yescharacteristic
39#4977C2211yescharacteristic
40#4961—22no—
41#4551C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44C2×#519—31no—
45C2×#460—32no—
46C2×#448—31no—
47C2×#448D311yescharacteristic, complemented
48#2619—31no—
49#1880—32no—
50#1880—32no—
51#1875—31no—
52#1875—31no—
53#1805—32no—
54#1798—31no—
55C22×C3×Q8—24no—
56C22×C3×Q8—24no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8D414yescomplemented
60C22×C3×Q8C2312yes—
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C22×#67—21no—
66C22×#67—22no—
67C22×D12C2311yescharacteristic
68C2×C3×SD16—24no—
69C2×C3×SD16—24no—
70C2×C3×SD16—24no—
71C22×C3×C8C2312yes—
72C3×#121—22no—
73C3×#121—22no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×#217—21no—
77C2×#217—22no—
78C2×#217D412yes—
79Q8×Dic3—216no—
80Q8×Dic3—24no—
81Q8×Dic3—24no—
82Q8×Dic3—24no—
83Q8×Dic3D418yescomplemented
84C2×#215—24no—
85C2×#215—24no—
86C2×#215—24no—
87#940—24no—
88#930—22no—
89#930—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—22no—
92#926—216no—
93#926—24no—
94#926—24no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3D412yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2312yes—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106C3×#103—24no—
107C3×C22⋊C8—22no—
108#835—24no—
109#832—24no—
110C3⋊#95—22no—
111#830—22no—
112#821—24no—
113Dic3⋊C8—24no—
114C23×Q8C2×D311yescharacteristic
115C22×SD16—32no—
116C22×SD16C2×D312yes—
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—31no—
126C2×#117—62no—
127#460—68no—
128#448—24no—
129#448—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103C2×D312yes—
133C2×#102—32no—
134C2×#102—32no—
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#340—32no—
144#336—32no—
145C24×C3—21no—
146C24×C3C2×D411yescharacteristic
147C23×D3—21no—
148C2×C3×Q8—24no—
149C2×C3×Q8—216no—
150C2×C3×Q8—24no—
151C2×C3×Q8—216no—
152C2×C3×Q8—24no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D414yes—
157C2×C3×Q8C2×D414yes—
158C2×C3×Q8#4114yes—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—22no—
163C2×C3×D4—42no—
164C2×C3×D4—82no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—42no—
177C2×#67—44no—
178C2×#67—82no—
179C2×#67—82no—
180C2×#67—82no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3C2×D412yes—
184C22×Dic3C2×D412yes—
185C2×D12—22no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—22no—
189C3×SD16—48no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C22⋊C4—42no—
196#217—24no—
197#217—24no—
198#217—82no—
199#215—48no—
200(C2×D3)⋊C4—42no—
201(C2×D3)⋊C4—82no—
202C4⋊Dic3—28no—
203C4⋊Dic3—22no—
204C4⋊Dic3—28no—
205C4⋊Dic3—24no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3C2×D414yes—
209C4×Dic3—28no—
210C4×Dic3—24no—
211C4×Dic3—22no—
212C4×Dic3—28no—
213C4×Dic3C2×D414yes—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8#6714yescomplemented
222C22×Q8C22×D312yes—
223C22×D4—21no—
224C22×D4—31no—
225C22×D4—62no—
226C22×D4—61no—
227C22×D4C22×D311yescharacteristic
228C23×C4—21no—
229C23×C4C22×D311yescharacteristic
230C2×SD16—24no—
231C2×SD16—24no—
232C2×SD16—24no—
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—64no—
245C4×Q8—64no—
246C4×Q8—616no—
247C4×Q8—64no—
248#117—64no—
249#117—64no—
250#117—616no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C42—31no—
264C2×C42—31no—
265C2×C42—62no—
266C2×C42—62no—
267C2×C42—61no—
268C4⋊C8—64no—
269#103—24no—
270#103—64no—
271#102—64no—
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×Q8—216no—
293C3×Q8—24no—
294C3×Q8—24no—
295C3×Q8—24no—
296C3×Q8#12118yes—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—84no—
300C2×C3×C4—216no—
301C2×C3×C4—24no—
302C2×C3×C4—21no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—21no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4#12112yes—
311C2×C3×C4C22×D412yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4112yes—
314C2×C3×C4C2×#4111yescharacteristic
315#67—88no—
316#67—84no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—24no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3#12014yes—
328D12—42no—
329D12—42no—
330C3×C8—42no—
331C3⋊C8—42no—
332C24—21no—
333C24—61no—
334C24D3×D411yescharacteristic
335C2×Q8—24no—
336C2×Q8—216no—
337C2×Q8—24no—
338C2×Q8—216no—
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8D3×D414yes—
344C2×Q8#23714yes—
345C2×Q8C2×#6714yes—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—22no—
349C2×D4—21no—
350C2×D4—42no—
351C2×D4—62no—
352C2×D4—61no—
353C2×D4—62no—
354C2×D4—61no—
355C2×D4—82no—
356C2×D4—124no—
357C2×D4—122no—
358C2×D4—242no—
359C2×D4—242no—
360C2×D4—242no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4D3×D411yescharacteristic
373C22×C4#23712yes—
374C22×C4C2×#6712yes—
375C22×C4C23×D311yescharacteristic
376SD16—48no—
377SD16—128no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—34no—
387C4⋊C4—64no—
388C4⋊C4—68no—
389C4⋊C4—68no—
390C4⋊C4—62no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C22⋊C4—42no—
394C22⋊C4—64no—
395C22⋊C4—64no—
396C22⋊C4—122no—
397C22⋊C4—242no—
398C22⋊C4—242no—
399C42—34no—
400C42—68no—
401C42—68no—
402C42—64no—
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—28no—
425C3×C4—22no—
426C3×C4—81no—
427C3×C4C2×#12112yes—
428C3×C4#52512yes—
429Dic3—24no—
430Dic3—42no—
431Dic3—44no—
432Dic3—81no—
433C23—22no—
434C23—22no—
435C23—21no—
436C23—21no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—122no—
443C23—122no—
444C23—121no—
445C23—121no—
446C23—121no—
447C23#93912yes—
448C23C2×D3×D411yescharacteristic
449Q8—216no—
450Q8—24no—
451Q8—24no—
452Q8—24no—
453Q8#93918yes—
454D4—42no—
455D4—42no—
456D4—84no—
457D4—122no—
458D4—122no—
459D4—248no—
460D4—244no—
461C2×C4—216no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—34no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4#93912yes—
483C2×C4C2×D3×D412yes—
484C2×C4C2×#23712yes—
485C2×C4C2×#23711yescharacteristic
486C2×C4C22×#6711yescharacteristic
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—82no—
505C22—82no—
506C22—81no—
507C22—241no—
508C22—241no—
509C22—242no—
510C22—242no—
511C22—241no—
512C22#558611yescharacteristic
513C22D3×#12011yescharacteristic
514C22#576212yes—
515C22#577812yes—
516C22C2×#93911yescharacteristic
517C4—28no—
518C4—22no—
519C4—64no—
520C4—81no—
521C4—122no—
522C4—124no—
523C4—241no—
524C4#558612yes—
525C4C2×#93912yes—
526C3#3291211yescharacteristic, complemented, Sylow
527C2—42no—
528C2—161no—
529C2—481no—
530C2#8003312yes—
531C2#8004112yes—
532C2#8260111yescharacteristic
533C2#8384911yescharacteristic
534C2#8396911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#448C2
socleC23×C3C22×D4
prime radical#82676C2
hypercenterC2×#448D3
hypocenterC3#32912
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C2×#5812C2semidirect
2. C2×C3×#448C2semidirect
3. C2×#5227C2semidirect
4. C2×#5163C2semidirect
6. #72048C2semidirect
7. #71678C2semidirect
8. #70240C2semidirect
9. #70222C2semidirect
10. #69933C2semidirect
11. #69909C2semidirect
47. C2×#448D3semidirect
59. C22×C3×Q8D4semidirect
83. Q8×Dic3D4semidirect
221. C22×Q8#67semidirect
526. C3#32912semidirect

Maximal subgroups

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

subgroupindex
C2×#58122
C2×C3×#4482
C2×#52272
C2×#51632
#826762
#720482
#716782
#702402
#702222
#699332
#699092
#329123