Finite Group Databasebeta

← #10720152

Subgroups — #10720152

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
2C2×#4972C211yesmaximal, characteristic
3C2×#4928C211yesmaximal, characteristic
4C2×#4933C212yesmaximal
5#71563C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic
7#71563C211yesmaximal, characteristic
8#71536C212yesmaximal
9C4×C3⋊#95C211yesmaximal, characteristic
10#69417C212yesmaximal
11#69414C212yesmaximal
12#14133—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C3×C4×C4⋊C4—24no—
24C2×C3×#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37#4972—24no—
38#4933—28no—
39#4928C418yescomplemented
40C2×#389—31no—
41C2×#389—31no—
42C2×#387—32no—
43C2×C4×C4⋊C4D311yescharacteristic
44#1443—32no—
45#1442—32no—
46#1436—32no—
47C4⋊#95—31no—
48C4⋊#95—31no—
49C4⋊#95—31no—
50C4×#95—31no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4D414yes—
67C2×C3×C4⋊C4Q814yes—
68C2×C3×C42—22no—
69C2×C3×C42—22no—
70C2×C3×C42—24no—
71C2×C3×C42—24no—
72C2×C3×C42—42no—
73C2×C3×C42C2×C414yes—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3C2×C414yes—
79C2×C4⋊Dic3C2×C414yes—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—42no—
85C2×Dic3⋊C4C2×C418yes—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C3×#95—24no—
93C3⋊#95—24no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—42no—
103C3⋊#95C2×C418yes—
104C3⋊#95D414yes—
105C22×C4⋊C4—32no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4C2×D311yescharacteristic
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C42—31no—
112C22×C42—31no—
113C22×C42C2×D312yes—
114C22×C42C2×D311yescharacteristic
115#389—38no—
116#389—64no—
117#387—68no—
118C4×C4⋊C4—24no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—31no—
122C2×#95—31no—
123C2×#95—31no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95C2×D312yes—
132C24×C3C2411yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C22×C412yes—
156C22×C3×C4C22×C412yes—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×Q814yes—
162C22×C3×C4C2×Q812yes—
163C22×C3×C4#4112yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4112yes—
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C22×C414yes—
205C22×Dic3C22×C414yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×D412yes—
208C22×Dic3#4112yes—
209C22×Dic3#4112yes—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—28no—
212C3×C42—28no—
213C3×C42—48no—
214C4⋊Dic3—28no—
215C4⋊Dic3—28no—
216Dic3⋊C4—216no—
217Dic3⋊C4—48no—
218C4×Dic3—44no—
219C4×Dic3—44no—
220C23×C4—32no—
221C23×C4—32no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4C22×D312yes—
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D312yes—
229C23×C4C22×D311yescharacteristic
230C23×C4C22×D311yescharacteristic
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—38no—
236C2×C4⋊C4—34no—
237C2×C4⋊C4—34no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—122no—
247C2×C4⋊C4Dic614yes—
248C2×C4⋊C4#6714yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—62no—
255C2×C42—62no—
256C2×C42—62no—
257C2×C42—62no—
258C2×C42—62no—
259C2×C42—62no—
260C2×C42C4×D314yes—
261#95—24no—
262#95—38no—
263#95—34no—
264#95—62no—
265#95—62no—
266#95—64no—
267#95—64no—
268#95—64no—
269#95—64no—
270#95—64no—
271#95—62no—
272#95—64no—
273#95—122no—
274C23×C3C23×C411yescharacteristic
275C23×C3C22×D411yescharacteristic
276C23×C3C22×D411yescharacteristic
277C23×C3C22×Q811yescharacteristic
278C23×C3C2×#4111yescharacteristic
279C23×C3C2×#4111yescharacteristic
280C23×C3C2×#4111yescharacteristic
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3H(5,𝔽2)11yescharacteristic
286C23×C32−1+411yescharacteristic
287C23×C32−1+411yescharacteristic
288C23×C32−1+411yescharacteristic
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4#11714yes—
317C2×C3×C4C4×D414yes—
318C2×C3×C4C4×D414yes—
319C2×C3×C4C4×Q814yes—
320C2×C3×C4C22⋊Q814yes—
321C2×C3×C4#12314yes—
322C2×C3×C4#12514yes—
323C2×C3×C4#12614yes—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—41no—
357C2×Dic3—42no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C24C23×D311yescharacteristic
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—34no—
390C22×C4—34no—
391C22×C4—32no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—32no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4C2×Dic614yes—
432C22×C4C2×Dic612yes—
433C22×C4C2×C4×D312yes—
434C22×C4C2×C4×D312yes—
435C22×C4C2×C4×D312yes—
436C22×C4#23512yes—
437C22×C4#23512yes—
438C22×C4#23512yes—
439C22×C4#23512yes—
440C22×C4D3×D412yes—
441C22×C4#23912yes—
442C22×C4C2×#6714yes—
443C22×C4C2×#6712yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—68no—
447C4⋊C4—616no—
448C4⋊C4—68no—
449C4⋊C4—128no—
450C42—28no—
451C42—48no—
452C42—124no—
453C42—124no—
454C22×C3C2×#11711yescharacteristic
455C22×C3C2×C4×D411yescharacteristic
456C22×C3C2×C4×D411yescharacteristic
457C22×C3C2×C4×Q811yescharacteristic
458C22×C3C4×#4111yescharacteristic
459C22×C3#51911yescharacteristic
460C22×C3#52011yescharacteristic
461C22×C3C2×C22⋊Q811yescharacteristic
462C22×C3C2×#12311yescharacteristic
463C22×C3C2×#12511yescharacteristic
464C22×C3C2×#12611yescharacteristic
465C22×C3#52911yescharacteristic
466C22×C3#52911yescharacteristic
467C22×C3#53211yescharacteristic
468C22×C3#53211yescharacteristic
469C22×C3#54111yescharacteristic
470C22×C3#54111yescharacteristic
471C22×C3#54211yescharacteristic
472C22×C3#54211yescharacteristic
473C22×C3D4211yescharacteristic
474C22×C3#54711yescharacteristic
475C22×C3#54711yescharacteristic
476C22×C3#54711yescharacteristic
477C22×C3#54811yescharacteristic
478C22×C3#54811yescharacteristic
479C22×C3D4×Q811yescharacteristic
480C22×C3#55211yescharacteristic
481C22×C3#55211yescharacteristic
482C22×C3#55311yescharacteristic
483C22×C3#55311yescharacteristic
484C22×C3#55411yescharacteristic
485C22×C3#55611yescharacteristic
486C22×C3#55611yescharacteristic
487C22×C3#55711yescharacteristic
488C22×C3#55711yescharacteristic
489C3×C4—24no—
490C3×C4—42no—
491C3×C4—44no—
492C3×C4—44no—
493C3×C4#38718yescomplemented
494Dic3—44no—
495Dic3—42no—
496Dic3—42no—
497Dic3—42no—
498Dic3—42no—
499Dic3—44no—
500C23C22×Dic611yescharacteristic
501C23C22×C4×D311yescharacteristic
502C23C2×#23511yescharacteristic
503C23C2×#23511yescharacteristic
504C23C2×#23511yescharacteristic
505C23C2×#23511yescharacteristic
506C23C2×D3×D411yescharacteristic
507C23#100411yescharacteristic
508C23C2×#23911yescharacteristic
509C23#100711yescharacteristic
510C23D3×#4111yescharacteristic
511C23D3×#4111yescharacteristic
512C23#101011yescharacteristic
513C23#101011yescharacteristic
514C23C22×#6711yescharacteristic
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—62no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—122no—
563C2×C4—121no—
564C2×C4—122no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—122no—
576C2×C4—121no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4C4×Dic614yes—
585C2×C4#87014yes—
586C2×C4#87214yes—
587C2×C4#87614yes—
588C2×C4#89314yes—
589C2×C4#92414yes—
590C2×C4C4×#6714yes—
591C2×C4#92914yes—
592C2×C3C2×#38711yescharacteristic
593C2×C3#229311yescharacteristic
594C2×C3#232011yescharacteristic
595C2×C3#236111yescharacteristic
596C2×C3#236211yescharacteristic
597C2×C3#236511yescharacteristic
598C2×C3#237011yescharacteristic
599C2×C3#242111yescharacteristic
600C2×C3#242211yescharacteristic
601C2×C3#245211yescharacteristic
602C2×C3#245311yescharacteristic
603C2×C3#249411yescharacteristic
604C2×C3#249611yescharacteristic
605C2×C3#250611yescharacteristic
606C2×C3#250911yescharacteristic
607C22C2×C4×Dic611yescharacteristic
608C22C2×#87011yescharacteristic
609C22C2×#87211yescharacteristic
610C22C2×#87611yescharacteristic
611C22C2×#89311yescharacteristic
612C22#550411yescharacteristic
613C22#550711yescharacteristic
614C22#551411yescharacteristic
615C22#551611yescharacteristic
616C22#552211yescharacteristic
617C22#552911yescharacteristic
618C22#553311yescharacteristic
619C22D4×Dic611yescharacteristic
620C22#553911yescharacteristic
621C22#554811yescharacteristic
622C22#554811yescharacteristic
623C22#555111yescharacteristic
624C22#555811yescharacteristic
625C22#556111yescharacteristic
626C22#556111yescharacteristic
627C22#556911yescharacteristic
628C22#557011yescharacteristic
629C22#558011yescharacteristic
630C22#558011yescharacteristic
631C22#558111yescharacteristic
632C22#558311yescharacteristic
633C22#558311yescharacteristic
634C22#558411yescharacteristic
635C22C2×#92411yescharacteristic
636C22C2×C4×#6711yescharacteristic
637C22C2×#92911yescharacteristic
638C22D4×#6711yescharacteristic
639C22#581711yescharacteristic
640C22#582411yescharacteristic
641C22#583111yescharacteristic
642C4—24no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—122no—
647C4—122no—
648C4—122no—
649C4—124no—
650C4—122no—
651C4—124no—
652C4#493318yescomplemented
653C3#1413311yescharacteristic, complemented, Sylow
654C2#7361711yescharacteristic
655C2#7362511yescharacteristic
656C2#7365111yescharacteristic
657C2#7365811yescharacteristic
658C2#7368311yescharacteristic
659C2#7368911yescharacteristic
660C2#7379011yescharacteristic
661C2#7379211yescharacteristic
662C2C2×#493311yescharacteristic
663C2#8216311yescharacteristic
664C2#8224311yescharacteristic
665C2#8236211yescharacteristic
666C2#8240611yescharacteristic
667C2#8247111yescharacteristic
668C2#8249311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×C4×C4⋊C4C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×C4×C4⋊C4D3
hypocenterC3#14133
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
39. #4928C4semidirect
493. C3×C4#387semidirect
652. C4#4933semidirect
653. C3#14133semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C4⋊C42
C2×#49722
C2×#49282
C2×#49332
#715632
#715362
C4×C3⋊#952
#694172
#694142
#141333