Finite Group Databasebeta

← #10822298

Subgroups — #10822298

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic
2C23×#216C211yesmaximal, characteristic, complemented
3C22×#909C211yesmaximal, characteristic
4C2×C3×#419C211yesmaximal, characteristic, complemented
5#82230C211yesmaximal, characteristic, complemented
6C2×#5002C211yesmaximal, characteristic
7#71980C211yesmaximal, characteristic, complemented
8#70283C214yesmaximal, complemented
9#70278C214yesmaximal, complemented
10#23425—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16#5816—24no—
17#5816—24no—
18C2×#947C2212yes—
19C2×#946—22no—
20C2×#944C2212yes—
21C22×#216—24no—
22C22×#216—24no—
23C22×#216—24no—
24C22×#216C2214yes—
25C2×#929—22no—
26C2×#926C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28C2×#909—24no—
29C2×#909—24no—
30C2×#909—24no—
31C22×Dic12C2212yes—
32C22×#204C2212yes—
33C3×#419—24no—
34C2×C3×#103C2214yes—
35#5002—24no—
36#4973C2214yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39#4943—22no—
40#4929—22no—
41#4550C2214yes—
42C22×#137—31no—
43C23×Q16—31no—
44C2×#536—31no—
45C2×#419D311yescharacteristic, complemented
46C2×#416—31no—
47#2310—31no—
48#2016—34no—
49#2015—34no—
50#1729—31no—
51C22×C3×Q8—21no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8D414yes—
58C22×C3×Q8C2312yes—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235D414yes—
69C22×D12C2311yescharacteristic
70C22×C4×D3—21no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—21no—
76C22×Dic6—22no—
77C22×Dic6D414yes—
78C22×Dic6C2312yes—
79C22×Dic6C2311yescharacteristic
80C2×C3×M16—22no—
81C2×C3×M16—22no—
82C2×C3×M16D414yes—
83C22×C3×C8C2312yes—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×#217D412yes—
90#947—24no—
91#947—24no—
92#946—216no—
93#944—24no—
94#944—24no—
95C2×#216—232no—
96C2×#216—28no—
97C2×#216—28no—
98C2×#216—28no—
99#929—216no—
100C2×(C2×D3)⋊C4—24no—
101#926—22no—
102#926—22no—
103C2×C4⋊Dic3D412yes—
104C2×Dic3⋊C4—24no—
105C2×C4×Dic3C2312yes—
106C22×C3⋊C8—22no—
107C22×C3⋊C8—22no—
108C22×C3⋊C8D414yes—
109C22×C3⋊C8C2312yes—
110#909—232no—
111C2×Dic12—24no—
112C2×Dic12—24no—
113C2×Dic12—24no—
114C2×#204—24no—
115C2×#204—24no—
116C2×#204—24no—
117C3×#103—28no—
118C3⋊#95—28no—
119#810—24no—
120#809—24no—
121C22×#41—31no—
122C23×Q8—31no—
123C23×Q8C2×D311yescharacteristic
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C22×Q16—34no—
128C22×Q16—32no—
129C22×Q16—64no—
130C22×Q16—64no—
131C22×Q16—64no—
132C22×SD16—32no—
133C22×M16C2×D311yescharacteristic
134C23×C8—31no—
135#536—64no—
136#536—64no—
137C2×C4⋊Q8—32no—
138C2×#124—32no—
139C2×#123—62no—
140C2×C22⋊Q8—62no—
141C2×#117—31no—
142C2×#117C2×D311yescharacteristic
143#419—24no—
144#416—64no—
145C2×#103—32no—
146C2×#103C2×D314yes—
147C2×#102—32no—
148#391—34no—
149#391—62no—
150#381—62no—
151#336—34no—
152C24×C3C2×D411yescharacteristic
153C23×D3—21no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—28no—
157C2×C3×Q8—28no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—44no—
163C2×C3×Q8—42no—
164C2×C3×Q8—44no—
165C2×C3×Q8—42no—
166C2×C3×Q8C2×D414yes—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—24no—
181C2×#67—42no—
182C2×#67—42no—
183C22×Dic3—24no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—42no—
187C22×Dic3—42no—
188C22×Dic3C2×D414yes—
189#235—28no—
190#235—28no—
191#235—44no—
192#235—44no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12C2×D412yes—
199C2×C4×D3—24no—
200C2×C4×D3—42no—
201C2×C4×D3—42no—
202C2×Dic6—28no—
203C2×Dic6—28no—
204C2×Dic6—21no—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×Dic6—21no—
212C2×Dic6—24no—
213C2×Dic6—21no—
214C2×Dic6—22no—
215C2×Dic6—42no—
216C2×Dic6—44no—
217C2×Dic6—44no—
218C2×Dic6—42no—
219C2×Dic6C2×D414yes—
220C2×Dic6C2×D412yes—
221C3×M16—28no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8C2×D414yes—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—42no—
228C3×C42—42no—
229#217—24no—
230#216—232no—
231(C2×D3)⋊C4—216no—
232C4⋊Dic3—24no—
233Dic3⋊C4—216no—
234C4×Dic3—22no—
235C4×Dic3—22no—
236C2×C3⋊C8—28no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8C2×D414yes—
240Dic12—216no—
241#204—216no—
242C2×#41—34no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—62no—
246C2×#41—62no—
247C2×#41—62no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—34no—
255C22×Q8—31no—
256C22×Q8—32no—
257C22×Q8—62no—
258C22×Q8—61no—
259C22×Q8—62no—
260C22×Q8—62no—
261C22×Q8—62no—
262C22×Q8—62no—
263C22×Q8D1214yes—
264C22×Q8C22×D312yes—
265C22×D4—31no—
266C22×D4—61no—
267C23×C4—21no—
268C23×C4—61no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271#137—632no—
272C2×Q16—64no—
273C2×Q16—64no—
274C2×Q16—68no—
275C2×Q16—68no—
276C2×Q16—632no—
277C2×Q16—68no—
278C2×Q16—64no—
279C2×SD16—64no—
280C2×SD16—64no—
281C2×SD16—64no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16#6714yes—
285C22×C8—34no—
286C22×C8—32no—
287C22×C8—62no—
288C22×C8—62no—
289C22×C8C22×D312yes—
290C4⋊Q8—64no—
291C4⋊Q8—64no—
292#124—64no—
293#124—64no—
294#123—616no—
295C22⋊Q8—616no—
296#117—22no—
297#117—22no—
298#117—62no—
299#117—62no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—64no—
302C2×C4⋊C4C22×D312yes—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—32no—
305C2×C22⋊C4—64no—
306C2×C42—21no—
307C2×C42—32no—
308#103—28no—
309#103—64no—
310#102—64no—
311#95—68no—
312C23×C3—24no—
313C23×C3#12012yes—
314C23×C3#12112yes—
315C23×C3#12712yes—
316C23×C3C22×D411yescharacteristic
317C22×D3—22no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—44no—
321C3×Q8—24no—
322C3×Q8—24no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—48no—
326C2×C3×C4—24no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—48no—
330C2×C3×C4—42no—
331C2×C3×C4—44no—
332C2×C3×C4#12014yes—
333C2×C3×C4#12012yes—
334C2×C3×C4#12014yes—
335C2×C3×C4#12114yes—
336C2×C3×C4#12112yes—
337C2×C3×C4#12114yes—
338C2×C3×C4#12712yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345#67—48no—
346C2×Dic3—24no—
347C2×Dic3—22no—
348C2×Dic3—24no—
349C2×Dic3—28no—
350C2×Dic3—42no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—41no—
354C2×Dic3—48no—
355D12—22no—
356D12—22no—
357D12—41no—
358D12—41no—
359C4×D3—48no—
360Dic6—24no—
361Dic6—22no—
362Dic6—24no—
363Dic6—22no—
364Dic6—41no—
365Dic6—42no—
366Dic6—42no—
367Dic6—41no—
368Dic6—48no—
369C3×C8—24no—
370C3⋊C8—24no—
371C24—61no—
372C24D3×D411yescharacteristic
373#41—68no—
374#41—68no—
375#41—124no—
376#41—124no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—32no—
386C2×Q8—34no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—64no—
392C2×Q8—61no—
393C2×Q8—62no—
394C2×Q8—68no—
395C2×Q8—68no—
396C2×Q8—62no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×Q8—61no—
402C2×Q8—62no—
403C2×Q8—61no—
404C2×Q8—122no—
405C2×Q8—124no—
406C2×Q8—124no—
407C2×Q8—122no—
408C2×Q8C2×D1214yes—
409C2×D4—32no—
410C2×D4—64no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—122no—
417C2×D4—122no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—34no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—64no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4—64no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4C2×D1212yes—
433C22×C4#23512yes—
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D412yes—
436C22×C4D3×D412yes—
437C22×C4D3×D412yes—
438C22×C4C2×#6712yes—
439C22×C4C23×D311yescharacteristic
440Q16—616no—
441Q16—632no—
442SD16—616no—
443M16—28no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—34no—
447C2×C8—68no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8C2×#6714yes—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—616no—
454C4⋊C4—64no—
455C22⋊C4—42no—
456C22⋊C4—616no—
457C22⋊C4—64no—
458C42—42no—
459C42—62no—
460C42—62no—
461C22×C3—24no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3#39214yes—
465C22×C3C2×#12011yescharacteristic
466C22×C3C2×#12111yescharacteristic
467C22×C3C2×#12711yescharacteristic
468C22×C3#57814yes—
469C2×D3—42no—
470C2×D3—44no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—44no—
474C3×C4—82no—
475C3×C4#39214yes—
476C3×C4C2×#12011yescharacteristic
477C3×C4C2×#12011yescharacteristic
478C3×C4C2×#12111yescharacteristic
479C3×C4C2×#12111yescharacteristic
480Dic3—44no—
481Dic3—42no—
482Dic3—44no—
483C23—24no—
484C23—62no—
485C23—121no—
486C23—124no—
487C23—121no—
488C23#88212yes—
489C23#88412yes—
490C23#94012yes—
491C23C2×D3×D411yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—48no—
495Q8—42no—
496Q8—42no—
497Q8—64no—
498Q8—62no—
499Q8—64no—
500Q8—62no—
501Q8—128no—
502Q8—121no—
503Q8—122no—
504Q8—122no—
505Q8—121no—
506D4—62no—
507D4—62no—
508D4—128no—
509D4—121no—
510D4—121no—
511C2×C4—24no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—64no—
518C2×C4—62no—
519C2×C4—64no—
520C2×C4—68no—
521C2×C4—128no—
522C2×C4—122no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—128no—
527C2×C4#88212yes—
528C2×C4#88214yes—
529C2×C4#88412yes—
530C2×C4#88414yes—
531C2×C4#93014yes—
532C2×C4#93714yes—
533C2×C4#94012yes—
534C2×C4C22×D1211yescharacteristic
535C2×C4C2×#23511yescharacteristic
536C2×C4C2×D3×D411yescharacteristic
537C2×C4C2×D3×D411yescharacteristic
538C2×C4C2×D3×D411yescharacteristic
539C2×C4C22×#6711yescharacteristic
540C8—24no—
541C8—64no—
542C2×C3—24no—
543C2×C3C2×#39211yescharacteristic
544C2×C3#301112yes—
545C2×C3#304412yes—
546C2×C3#315612yes—
547D3—82no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—124no—
552C22—122no—
553C22—242no—
554C22—242no—
555C22#496014yes—
556C22C2×#88211yescharacteristic
557C22C2×#88411yescharacteristic
558C22#577914yes—
559C22C2×#94011yescharacteristic
560C4—44no—
561C4—82no—
562C4—124no—
563C4—122no—
564C4—124no—
565C4#496014yes—
566C4C2×#88211yescharacteristic
567C4C2×#88411yescharacteristic
568C4C2×#93011yescharacteristic
569C4C2×#93711yescharacteristic
570C3#2342511yescharacteristic, complemented, Sylow
571C2—24no—
572C2—242no—
573C2#7896112yes—
574C2#7897712yes—
575C2C2×#496011yescharacteristic
576C2#8397512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#419D3
hypocenterC3#23425
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C23×#216C2semidirect
4. C2×C3×#419C2semidirect
5. #82230C2semidirect
7. #71980C2semidirect
8. #70283C2semidirect
9. #70278C2semidirect
45. C2×#419D3semidirect
570. C3#23425semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C23×#2162
C22×#9092
C2×C3×#4192
#822302
C2×#50022
#719802
#702832
#702782
#234253