Finite Group Databasebeta

← #10732201

Subgroups — #10732201

Order 768.

2770 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#897C212yesmaximal, complemented
2C22×C4×D12C211yesmaximal, characteristic
3C2×#4987C212yesmaximal, complemented
4C2×#4929C211yesmaximal, characteristic, complemented
5C3×C4⋊#95C211yesmaximal, characteristic, complemented
6#69427C214yesmaximal, complemented
7#69426C212yesmaximal
8#69419C212yesmaximal, complemented
9#15177—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C23×C4×D3C2212yes—
12C23×Dic6C2211yescharacteristic
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4C2214yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2214yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2214yes—
21C2×#897—28no—
22C2×#897—28no—
23C2×#897—28no—
24C2×C4×D12—28no—
25C2×C4×D12—24no—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31#4987—28no—
32#4929—24no—
33C22×C22⋊Q8—32no—
34C22×C4×D4—31no—
35C2×#391—31no—
36C2×#380—32no—
37#1444—32no—
38#1439—34no—
39#1438—32no—
40C4⋊#95D311yescharacteristic, complemented
41C24×D3C2312yes—
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2314yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×Dic3C2312yes—
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C22×D12—24no—
50C22×D12—24no—
51C22×D12—22no—
52C22×D12—22no—
53C22×D12Q814yescomplemented
54C22×C4×D3—24no—
55C22×C4×D3—24no—
56C22×C4×D3—24no—
57C22×C4×D3—24no—
58C22×C4×D3—24no—
59C22×C4×D3—24no—
60C22×C4×D3D418yes—
61C22×Dic6—22no—
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6—22no—
65C22×Dic6D414yescomplemented
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4D418yescomplemented
69C2×C3×C42—24no—
70C2×C3×C42D414yes—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—28no—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—28no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—28no—
77C2×(C2×D3)⋊C4—28no—
78C2×(C2×D3)⋊C4—28no—
79C2×(C2×D3)⋊C4—44no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—28no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—28no—
84C2×C4⋊Dic3—28no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—24no—
87#897—264no—
88C4×D12—232no—
89C3×#95—24no—
90C3×#95—42no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—28no—
94C23×Q8—31no—
95C23×D4—31no—
96C24×C4—32no—
97C2×C22⋊Q8—68no—
98C2×C22⋊Q8—68no—
99C2×C22⋊Q8—68no—
100C2×C4×D4—68no—
101C2×C4×D4—64no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—34no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4C2×D312yes—
106C22×C22⋊C4—32no—
107C22×C22⋊C4—34no—
108C22×C22⋊C4—34no—
109C22×C42C2×D311yescharacteristic
110#391—64no—
111#380—68no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—34no—
115C2×#95C2×D312yes—
116C2×#95C2×D312yes—
117C24×C3C2411yescharacteristic
118C23×D3—22no—
119C23×D3—22no—
120C23×D3—22no—
121C23×D3—22no—
122C23×D3—22no—
123C23×D3—22no—
124C23×D3—22no—
125C23×D3—22no—
126C23×D3—22no—
127C23×D3—22no—
128C23×D3—22no—
129C23×D3—22no—
130C23×D3C2×D414yes—
131C23×D3C2×Q814yes—
132C23×D3#4114yes—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—21no—
141C22×C3×C4—24no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D418yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D414yes—
150C22×C3×C4C2×D414yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×Q812yes—
153C22×C3×C4#4112yes—
154C22×C3×C4#4112yes—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—24no—
160C22×Dic3—24no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—24no—
167C22×Dic3—24no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3C2×D414yes—
171C22×Dic3C2×D414yes—
172C2×D12—28no—
173C2×D12—28no—
174C2×D12—216no—
175C2×D12—216no—
176C2×D12—28no—
177C2×C4×D3—216no—
178C2×C4×D3—216no—
179C2×C4×D3—216no—
180C2×C4×D3—48no—
181C2×C4×D3—48no—
182C2×C4×D3—48no—
183C2×C4×D3—48no—
184C2×Dic6—216no—
185C2×Dic6—28no—
186C2×Dic6—44no—
187C2×Dic6—48no—
188C2×Dic6—44no—
189C3×C4⋊C4—216no—
190C3×C42—28no—
191(C2×D3)⋊C4—216no—
192(C2×D3)⋊C4—232no—
193(C2×D3)⋊C4—416no—
194C4⋊Dic3—28no—
195C4⋊Dic3—232no—
196C4⋊Dic3—44no—
197C25—32no—
198C22×Q8—34no—
199C22×Q8—64no—
200C22×Q8—62no—
201C22×Q8—64no—
202C22×Q8—62no—
203C22×D4—34no—
204C22×D4—64no—
205C22×D4—64no—
206C22×D4—62no—
207C22×D4—62no—
208C23×C4—32no—
209C23×C4—32no—
210C23×C4—38no—
211C23×C4—32no—
212C23×C4—64no—
213C23×C4—64no—
214C23×C4—64no—
215C23×C4—64no—
216C23×C4—64no—
217C23×C4—64no—
218C23×C4C22×D314yes—
219C23×C4C22×D311yescharacteristic
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C22⋊Q8—664no—
223C4×D4—632no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—68no—
228C2×C4⋊C4—68no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—68no—
231C2×C4⋊C4—64no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4D1218yescomplemented
234C2×C22⋊C4—64no—
235C2×C22⋊C4—68no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—68no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—68no—
240C2×C22⋊C4—68no—
241C2×C22⋊C4—68no—
242C2×C22⋊C4—124no—
243C2×C42—24no—
244C2×C42D1214yes—
245#95—24no—
246#95—42no—
247#95—68no—
248#95—64no—
249#95—64no—
250C23×C3C22×D411yescharacteristic
251C23×C3C22×D412yes—
252C23×C3C22×D412yes—
253C23×C3C22×D411yescharacteristic
254C23×C3C22×Q811yescharacteristic
255C23×C3C2×#4111yescharacteristic
256C23×C3C2×#4111yescharacteristic
257C23×C3C2×#4112yes—
258C23×C3H(5,𝔽2)11yescharacteristic
259C23×C32−1+411yescharacteristic
260C23×C32−1+412yes—
261C22×D3—24no—
262C22×D3—24no—
263C22×D3—24no—
264C22×D3—24no—
265C22×D3—24no—
266C22×D3—24no—
267C22×D3—24no—
268C22×D3—24no—
269C22×D3—24no—
270C22×D3—24no—
271C22×D3—24no—
272C22×D3—24no—
273C22×D3—24no—
274C22×D3—24no—
275C22×D3—24no—
276C22×D3—24no—
277C22×D3—24no—
278C22×D3—24no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3C22⋊Q818yescomplemented
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—28no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—28no—
305C2×C3×C4—28no—
306C2×C3×C4—44no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—44no—
310C2×C3×C4—44no—
311C2×C3×C4—44no—
312C2×C3×C4#12014yes—
313C2×C3×C4#12018yes—
314C2×C3×C4#12114yes—
315C2×C3×C4#12118yes—
316C2×C3×C4C22⋊Q814yescomplemented
317C2×C3×C4#12714yes—
318C2×Dic3—24no—
319C2×Dic3—28no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—44no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—44no—
334C2×Dic3—44no—
335D12—232no—
336C4×D3—432no—
337Dic6—416no—
338C24—34no—
339C24—34no—
340C24—34no—
341C24—62no—
342C24—62no—
343C24—62no—
344C24—62no—
345C24—62no—
346C24—62no—
347C24—62no—
348C24—62no—
349C24—62no—
350C24—62no—
351C24—62no—
352C24—62no—
353C24C23×D311yescharacteristic
354C2×Q8—616no—
355C2×Q8—68no—
356C2×Q8—124no—
357C2×Q8—124no—
358C2×Q8—128no—
359C2×D4—616no—
360C2×D4—68no—
361C2×D4—68no—
362C2×D4—616no—
363C2×D4—68no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—34no—
377C22×C4—34no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—64no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—64no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—616no—
389C22×C4—616no—
390C22×C4—616no—
391C22×C4—62no—
392C22×C4—64no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—64no—
396C22×C4—128no—
397C22×C4—128no—
398C22×C4—128no—
399C22×C4—128no—
400C22×C4C2×D1212yes—
401C22×C4C2×D1218yes—
402C22×C4C2×D1212yes—
403C22×C4C2×D1212yes—
404C22×C4C2×D1214yes—
405C22×C4D3×D414yes—
406C22×C4D3×D412yes—
407C22×C4#23712yes—
408C22×C4D3×Q812yes—
409C22×C4#23912yes—
410C4⋊C4—216no—
411C4⋊C4—68no—
412C4⋊C4—632no—
413C4⋊C4—124no—
414C22⋊C4—616no—
415C22⋊C4—632no—
416C22⋊C4—1216no—
417C42—28no—
418C22×C3C2×#12011yescharacteristic
419C22×C3C2×#12012yes—
420C22×C3C2×#12112yes—
421C22×C3C2×#12111yescharacteristic
422C22×C3C2×C22⋊Q811yescharacteristic
423C22×C3C2×C22⋊Q812yes—
424C22×C3#52511yescharacteristic
425C22×C3#52512yes—
426C22×C3C2×#12711yescharacteristic
427C22×C3#53611yescharacteristic
428C22×C3#53612yes—
429C22×C3#53711yescharacteristic
430C22×C3#53712yes—
431C22×C3#54612yes—
432C22×C3#54712yes—
433C22×C3#54812yes—
434C22×C3#54812yes—
435C22×C3D4×Q811yescharacteristic
436C22×C3D4×Q812yes—
437C22×C3#55011yescharacteristic
438C22×C3#55212yes—
439C22×C3#55411yescharacteristic
440C22×C3#55611yescharacteristic
441C2×D3—28no—
442C2×D3—28no—
443C2×D3—28no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—44no—
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—44no—
450C2×D3—44no—
451C2×D3—44no—
452C2×D3—44no—
453C2×D3—44no—
454C2×D3—44no—
455C2×D3—44no—
456C3×C4—24no—
457C3×C4—42no—
458C3×C4—48no—
459C3×C4#39218yes—
460Dic3—44no—
461Dic3—44no—
462Dic3—82no—
463C23—38no—
464C23—64no—
465C23—64no—
466C23—64no—
467C23—64no—
468C23—64no—
469C23—64no—
470C23—64no—
471C23—64no—
472C23—64no—
473C23—64no—
474C23—64no—
475C23—64no—
476C23—64no—
477C23—64no—
478C23—64no—
479C23—64no—
480C23—64no—
481C23—64no—
482C23—122no—
483C23—122no—
484C23—122no—
485C23—122no—
486C23—122no—
487C23—122no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23C22×D1211yescharacteristic
499C23C22×D1212yes—
500C23C2×D3×D411yescharacteristic
501C23C2×D3×D412yes—
502C23C2×#23711yescharacteristic
503C23#100411yescharacteristic
504C23C2×D3×Q811yescharacteristic
505C23C2×#23911yescharacteristic
506C23#100711yescharacteristic
507C23D3×#4112yes—
508C23#101012yes—
509Q8—1216no—
510D4—632no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—64no—
528C2×C4—68no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—124no—
539C2×C4—1232no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—124no—
544C2×C4—124no—
545C2×C4#87414yes—
546C2×C4#88214yes—
547C2×C4#88218yes—
548C2×C4#89518yes—
549C2×C4#89514yes—
550C2×C4#89714yescomplemented
551C2×C3C2×#39211yescharacteristic
552C2×C3#240712yes—
553C2×C3#240912yes—
554C2×C3#241512yes—
555C2×C3#241712yes—
556C2×C3#253611yescharacteristic
557C2×C3#253811yescharacteristic
558C2×C3#254011yescharacteristic
559C2×C3#254211yescharacteristic
560C2×C3#261611yescharacteristic
561C2×C3#284811yescharacteristic
562D3—48no—
563C22—68no—
564C22—68no—
565C22—68no—
566C22—124no—
567C22—124no—
568C22—124no—
569C22—124no—
570C22—124no—
571C22—124no—
572C22—124no—
573C22—124no—
574C22—124no—
575C22—124no—
576C22—124no—
577C22—124no—
578C22C2×#87411yescharacteristic
579C22C2×#88211yescharacteristic
580C22C2×#88212yes—
581C22C2×#89511yescharacteristic
582C22C2×#89512yes—
583C22C2×#89711yescharacteristic
584C22#551011yescharacteristic
585C22#552411yescharacteristic
586C22#552611yescharacteristic
587C22#555412yes—
588C22#555512yes—
589C22Q8×D1212yes—
590C22#557612yes—
591C22#560612yes—
592C22#560812yes—
593C22D3×C22⋊Q812yes—
594C22#562812yes—
595C22#567712yes—
596C22#569412yes—
597C22#571811yescharacteristic
598C22#572711yescharacteristic
599C22#573011yescharacteristic
600C22#573211yescharacteristic
601C4—24no—
602C4—48no—
603C4—42no—
604C4—124no—
605C4—124no—
606C4—242no—
607C4#467118yes—
608C3#1517711yescharacteristic, complemented, Sylow
609C2—128no—
610C2C2×#467111yescharacteristic
611C2#7379811yescharacteristic
612C2#7380011yescharacteristic
613C2#7395011yescharacteristic
614C2#7404911yescharacteristic
615C2#7424011yescharacteristic
616C2#7436911yescharacteristic
617C2#7621012yes—
618C2#7621212yes—
619C2#7643512yes—
620C2#7661912yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C4⋊#95C2
prime radicalC23×D12C22
hypercenterC4⋊#95D3
hypocenterC3#15177
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
1. C22×#897C2semidirect
3. C2×#4987C2semidirect
4. C2×#4929C2semidirect
5. C3×C4⋊#95C2semidirect
6. #69427C2semidirect
8. #69419C2semidirect
40. C4⋊#95D3semidirect
53. C22×D12Q8semidirect
65. C22×Dic6D4semidirect
68. C2×C3×C4⋊C4D4semidirect
233. C2×C4⋊C4D12semidirect
295. C22×D3C22⋊Q8semidirect
316. C2×C3×C4C22⋊Q8semidirect
550. C2×C4#897semidirect
608. C3#15177semidirect

Maximal subgroups

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

subgroupindex
C22×#8972
C22×C4×D122
C2×#49872
C2×#49292
C3×C4⋊#952
#694272
#694262
#694192
#151773