Finite Group Databasebeta

← #10795138

Subgroups — #10795138

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5546C211yesmaximal, characteristic
2#73867C212yesmaximal, complemented
3C3×#1799C211yesmaximal, characteristic, complemented
4#71619C211yesmaximal, characteristic, complemented
5#70027C212yesmaximal
6Dic3×#99C212yesmaximal, complemented
7#69530C212yesmaximal, complemented
8#68953C212yesmaximal, complemented
9#68952C212yesmaximal, complemented
10#20725—31nomaximal, Sylow
11C22×#237C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×D4×Dic3C2211yescharacteristic
14C2×C4×#67C411yescharacteristic, complemented
15C2×C4×#67C2211yescharacteristic
16C22×Dic3⋊C4C411yescharacteristic, complemented
17C22×Dic3⋊C4C2211yescharacteristic
18#5546—24no—
19#5546—24no—
20C2×D3×C4⋊C4C411yescharacteristic, complemented
21C2×#887C411yescharacteristic, complemented
22C2×#67⋊C4C411yescharacteristic, complemented
23C2×#67⋊C4C2211yescharacteristic
24C2×#877C411yescharacteristic, complemented
25C2×#877C2211yescharacteristic
26C2×#872C411yescharacteristic, complemented
27C2×C4×Dic6C411yescharacteristic, complemented
28C2×C3×#99C2212yes—
29C3×#380C2212yes—
30C2×#834C2212yes—
31#5212C2212yes—
32#4978—24no—
33C2×#806C2214yes—
34#4943C2214yes—
35Dic3×C22⋊C4C2214yes—
36#4655—24no—
37C3×#342C2212yes—
38#4527C2214yes—
39#4526C2212yes—
40C2×#520—31no—
41#2313—32no—
42#1820—32no—
43#1799—32no—
44#1799—31no—
45#1799D311yescharacteristic, complemented
46#1782—32no—
47C4×#99—32no—
48C4×#99—32no—
49C22×C3×D4C2311yescharacteristic
50C23×C3×C4C2×C411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C22×#67C2×C411yescharacteristic
53C22×#67C2311yescharacteristic
54C23×Dic3C2×C411yescharacteristic
55C23×Dic3C2×C411yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C2×#237—22no—
59C2×#237—22no—
60C2×#237—24no—
61C2×#237—24no—
62C22×C4×D3C2×C411yescharacteristic
63C22×C4×D3C2×C411yescharacteristic
64C22×C4×D3C2×C411yescharacteristic
65C22×Dic6C2×C411yescharacteristic
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4C2×C411yescharacteristic
70C2×C3×C22⋊C4C2×C411yescharacteristic
71C2×C3×C22⋊C4C2314yes—
72C2×C3×C22⋊C4C2311yescharacteristic
73C2×C3×C42C2×C411yescharacteristic
74C2×#217C2×C411yescharacteristic
75C2×#217C2314yes—
76C2×#217C2311yescharacteristic
77D4×Dic3—24no—
78C4×#67—22no—
79C4×#67—22no—
80C4×#67—22no—
81C4×#67—22no—
82C2×(C2×D3)⋊C4C2×C411yescharacteristic
83C2×(C2×D3)⋊C4C2×C411yescharacteristic
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3C2×C411yescharacteristic
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4C2×C411yescharacteristic
89C2×Dic3⋊C4C2×C411yescharacteristic
90C2×Dic3⋊C4C2×C411yescharacteristic
91C2×Dic3⋊C4C2×C411yescharacteristic
92C2×Dic3⋊C4C2×C411yescharacteristic
93C2×Dic3⋊C4C2×C411yescharacteristic
94C2×Dic3⋊C4C2×C411yescharacteristic
95C2×Dic3⋊C4C2×C411yescharacteristic
96C2×Dic3⋊C4D414yes—
97C2×Dic3⋊C4D414yes—
98C2×C4×Dic3—24no—
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3C2×C411yescharacteristic
104D3×C4⋊C4—24no—
105#887—24no—
106#67⋊C4—24no—
107#67⋊C4—24no—
108#877—24no—
109#877—24no—
110#872—22no—
111#872—22no—
112C4×Dic6—22no—
113C4×Dic6—22no—
114C3×#99—24no—
115C3×#95—22no—
116#834—24no—
117C3⋊#95—22no—
118C3⋊#95—24no—
119C3⋊#95—28no—
120#806—28no—
121C22×#41—31no—
122#520—64no—
123#520—64no—
124C2×C4×Q8—31no—
125C2×C4×Q8—31no—
126C2×C4×D4—31no—
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×C4×D4—31no—
130C2×C4×D4—31no—
131C2×C4×D4C2×D311yescharacteristic
132C2×#117—31no—
133C2×#117—31no—
134C2×#117—31no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4—31no—
138C2×#99—32no—
139C2×#99—34no—
140C2×#99C2×D312yes—
141#387—64no—
142#387—64no—
143#381—34no—
144#380—32no—
145#380C2×D312yes—
146C4×C22⋊C4—34no—
147#342—32no—
148#342—34no—
149#342C2×D312yes—
150C24×C3C22×C411yescharacteristic
151C24×C3C2411yescharacteristic
152C23×D3C22×C411yescharacteristic
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4#4112yes—
156C2×C3×D4#4112yes—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—24no—
160C22×C3×C4—81no—
161C22×C3×C4—81no—
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C22×C411yescharacteristic
164C22×C3×C4C22×C411yescharacteristic
165C22×C3×C4C22×C411yescharacteristic
166C22×C3×C4C22×C411yescharacteristic
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C2×#67—22no—
170C2×#67—22no—
171C2×#67—21no—
172C2×#67—21no—
173C2×#67—22no—
174C2×#67—22no—
175C2×#67#4114yes—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—24no—
184C22×Dic3—81no—
185C22×Dic3—81no—
186C22×Dic3C22×C411yescharacteristic
187C22×Dic3C22×C411yescharacteristic
188C22×Dic3C22×C411yescharacteristic
189C22×Dic3C22×C411yescharacteristic
190C22×Dic3C22×C411yescharacteristic
191C22×Dic3C22×C411yescharacteristic
192C22×Dic3C22×C411yescharacteristic
193C22×Dic3C2×D414yes—
194C22×Dic3C2×D414yes—
195#237—216no—
196#237—216no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—81no—
203C2×C4×D3—81no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—82no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C42—21no—
214C3×C42—21no—
215#217—22no—
216#217—22no—
217#217—24no—
218#217—24no—
219(C2×D3)⋊C4—21no—
220(C2×D3)⋊C4—21no—
221(C2×D3)⋊C4—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—82no—
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226Dic3⋊C4—22no—
227Dic3⋊C4—22no—
228Dic3⋊C4—22no—
229Dic3⋊C4C22⋊C412yes—
230Dic3⋊C4C22⋊C412yes—
231Dic3⋊C4C22⋊C412yes—
232Dic3⋊C4C22⋊C412yes—
233Dic3⋊C4C22⋊C414yes—
234Dic3⋊C4C22⋊C414yes—
235C4×Dic3—22no—
236C4×Dic3—22no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239C4×Dic3—22no—
240C4×Dic3—84no—
241C2×#41—64no—
242C2×#41—64no—
243C2×#41—62no—
244C2×#41—62no—
245C22×Q8—31no—
246C22×D4—31no—
247C22×D4—31no—
248C22×D4C22×D311yescharacteristic
249C23×C4—31no—
250C23×C4—31no—
251C23×C4—31no—
252C23×C4—31no—
253C23×C4—31no—
254C23×C4—31no—
255C23×C4—31no—
256C23×C4C4×D311yescharacteristic
257C23×C4C22×D311yescharacteristic
258C4×Q8—62no—
259C4×Q8—62no—
260C4×Q8—64no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—64no—
264C4×D4—64no—
265C4×D4—62no—
266C4×D4—62no—
267C4×D4—64no—
268C4×D4—62no—
269C4×D4—62no—
270#117—62no—
271#117—62no—
272#117—64no—
273#117—64no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—34no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—64no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4C4×D311yescharacteristic
291C2×C22⋊C4—31no—
292C2×C22⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—34no—
296C2×C22⋊C4C4×D311yescharacteristic
297C2×C22⋊C4C22×D314yes—
298C2×C22⋊C4C22×D311yescharacteristic
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—64no—
305C2×C42C4×D311yescharacteristic
306#99—24no—
307#99—68no—
308#99—64no—
309#95—22no—
310#95—64no—
311#95—68no—
312#95—62no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3#11712yes—
320C23×C3C23×C411yescharacteristic
321C23×C3C22×D412yes—
322C23×C3C2×#4111yescharacteristic
323C23×C3C2×#4111yescharacteristic
324C23×C3C2×#4112yes—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3#11712yes—
330C3×D4—24no—
331C3×D4—24no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—44no—
343C2×C3×C4—44no—
344C2×C3×C4—44no—
345C2×C3×C4—81no—
346C2×C3×C4—81no—
347C2×C3×C4C2×C22⋊C412yes—
348C2×C3×C4C2×C22⋊C411yescharacteristic
349C2×C3×C4C2×C22⋊C411yescharacteristic
350C2×C3×C4#11712yes—
351#67—28no—
352#67—24no—
353#67—24no—
354C2×Dic3—28no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—44no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—44no—
371C2×Dic3—44no—
372C2×Dic3—44no—
373C2×Dic3—82no—
374C2×Dic3C2×C22⋊C412yes—
375C2×Dic3C2×C22⋊C412yes—
376C2×Dic3C2×C22⋊C412yes—
377C2×Dic3C2×C22⋊C412yes—
378C2×Dic3#11712yes—
379C4×D3—28no—
380C4×D3—82no—
381C4×D3—82no—
382Dic6—28no—
383C24—31no—
384C24C2×C4×D311yescharacteristic
385C24C23×D311yescharacteristic
386#41—616no—
387#41—616no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×Q8—62no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—34no—
394C2×D4—62no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4#23712yes—
401C2×D4#23912yes—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—34no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—34no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—64no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—241no—
430C22×C4—241no—
431C22×C4—241no—
432C22×C4—241no—
433C22×C4C2×C4×D311yescharacteristic
434C22×C4C2×C4×D311yescharacteristic
435C22×C4C2×C4×D311yescharacteristic
436C22×C4C2×C4×D311yescharacteristic
437C22×C4C2×C4×D311yescharacteristic
438C22×C4D3×D412yes—
439C22×C4D3×D412yes—
440C4⋊C4—22no—
441C4⋊C4—32no—
442C4⋊C4—32no—
443C4⋊C4—34no—
444C4⋊C4—32no—
445C4⋊C4—32no—
446C4⋊C4—34no—
447C4⋊C4—62no—
448C4⋊C4—62no—
449C4⋊C4—62no—
450C4⋊C4—62no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—82no—
454C4⋊C4—242no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—62no—
460C22⋊C4—62no—
461C22⋊C4—62no—
462C22⋊C4—61no—
463C22⋊C4—61no—
464C22⋊C4—64no—
465C22⋊C4—64no—
466C42—21no—
467C42—21no—
468C42—62no—
469C42—62no—
470C42—62no—
471C42—62no—
472C42—62no—
473C42—244no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—22no—
477C22×C3—21no—
478C22×C3—22no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C22×C22⋊C411yescharacteristic
487C22×C3C2×#11711yescharacteristic
488C22×C3C2×#11711yescharacteristic
489C22×C3#52512yes—
490C22×C3#52512yes—
491C2×D3—22no—
492C2×D3—42no—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—42no—
496C3×C4—22no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—22no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—84no—
503Dic3—22no—
504Dic3—22no—
505Dic3—22no—
506Dic3—22no—
507Dic3—22no—
508Dic3—41no—
509Dic3—41no—
510Dic3—84no—
511C23—21no—
512C23—21no—
513C23—32no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—61no—
519C23—61no—
520C23—121no—
521C23—121no—
522C23#89212yes—
523C23C22×C4×D311yescharacteristic
524C23C2×D3×D412yes—
525C23C2×#23711yescharacteristic
526C23C2×#23911yescharacteristic
527C23D3×#4112yes—
528Q8—68no—
529D4—24no—
530D4—24no—
531D4—68no—
532D4—64no—
533D4—64no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—32no—
539C2×C4—32no—
540C2×C4—32no—
541C2×C4—32no—
542C2×C4—32no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—68no—
553C2×C4—68no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—124no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—124no—
572C2×C4—124no—
573C2×C4—124no—
574C2×C4—242no—
575C2×C4—242no—
576C2×C4—242no—
577C2×C4D3×C22⋊C411yescharacteristic
578C2×C4D3×C22⋊C411yescharacteristic
579C2×C4D3×C22⋊C412yes—
580C2×C4#89212yes—
581C2×C3—22no—
582C2×C3—22no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—42no—
586C2×C3#228011yescharacteristic
587C2×C3#288311yescharacteristic
588C2×C3#288411yescharacteristic
589D3—42no—
590C22—22no—
591C22—21no—
592C22—22no—
593C22—21no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—62no—
603C22—122no—
604C22—122no—
605C22—121no—
606C22—121no—
607C22C2×D3×C22⋊C411yescharacteristic
608C22C2×#89211yescharacteristic
609C22D3×#11711yescharacteristic
610C22#560612yes—
611C22#562712yes—
612C4—21no—
613C4—21no—
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—62no—
619C4—62no—
620C4—62no—
621C4—62no—
622C4—62no—
623C4—84no—
624C4—121no—
625C4—121no—
626C4—244no—
627C3#2072511yescharacteristic, complemented, Sylow
628C2—22no—
629C2—22no—
630C2—21no—
631C2—21no—
632C2—42no—
633C2—122no—
634C2#7386211yescharacteristic
635C2#7868811yescharacteristic
636C2#7868911yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1799C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#237C22
cosocleC24×C3C24
hypercenter#1799D3
hypocenterC3#20725
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #73867C2semidirect
3. C3×#1799C2semidirect
4. #71619C2semidirect
6. Dic3×#99C2semidirect
7. #69530C2semidirect
8. #68953C2semidirect
9. #68952C2semidirect
14. C2×C4×#67C4semidirect
16. C22×Dic3⋊C4C4semidirect
20. C2×D3×C4⋊C4C4semidirect
21. C2×#887C4semidirect
22. C2×#67⋊C4C4semidirect
24. C2×#877C4semidirect
26. C2×#872C4semidirect
27. C2×C4×Dic6C4semidirect
45. #1799D3semidirect
627. C3#20725semidirect

Maximal subgroups

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

subgroupindex
C2×#55462
#738672
C3×#17992
#716192
#700272
Dic3×#992
#695302
#689532
#689522
#207253