Finite Group Databasebeta

← #10732566

Subgroups — #10732566

Order 768.

1326 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic
3C2×#4928C211yesmaximal, characteristic
4C2×#4936C211yesmaximal, characteristic
5#71568C212yesmaximal
6#71567C212yesmaximal
7#71535C211yesmaximal, characteristic
8#71535C211yesmaximal, characteristic
9#71535C211yesmaximal, characteristic
10#69421C212yesmaximal
11#69414C212yesmaximal
12#15221—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C3×C4×C4⋊C4—24no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37#4936—24no—
38#4928—24no—
39#4928—24no—
40C2×#389—31no—
41C2×#389—31no—
42C2×#382—31no—
43C2×C4×C4⋊C4D311yescharacteristic
44#1445—32no—
45#1444—32no—
46#1442—32no—
47#1442—32no—
48C4⋊#95—31no—
49C4⋊#95—31no—
50C4⋊#95—31no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C42—22no—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C3×C42D414yes—
76C2×C3×C42Q814yes—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3D414yes—
88C2×C4⋊Dic3Q814yes—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—42no—
93C2×Dic3⋊C4—42no—
94C2×Dic3⋊C4—42no—
95C3×#95—22no—
96C3×#95—22no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—42no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C42C2×D311yescharacteristic
116C22×C42C2×D311yescharacteristic
117C22×C42C2×D311yescharacteristic
118#389—64no—
119#389—64no—
120#382—64no—
121C4×C4⋊C4—24no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95C2×D311yescharacteristic
133C2×#95C2×D311yescharacteristic
134C24×C3C2411yescharacteristic
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×Q812yes—
174C22×C3×C4C2×Q812yes—
175C22×C3×C4C2×Q812yes—
176C22×C3×C4C2×Q812yes—
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×C3×C4#4112yes—
181C22×C3×C4#4112yes—
182C22×C3×C4#4112yes—
183C22×C3×C4#4112yes—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×Q814yes—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—28no—
214C3×C42—28no—
215C3×C42—28no—
216C3×C42—44no—
217C4⋊Dic3—28no—
218C4⋊Dic3—28no—
219C4⋊Dic3—44no—
220C4⋊Dic3—44no—
221Dic3⋊C4—48no—
222Dic3⋊C4—48no—
223C23×C4—32no—
224C23×C4—32no—
225C23×C4—32no—
226C23×C4—32no—
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D311yescharacteristic
229C23×C4C22×D311yescharacteristic
230C23×C4C22×D311yescharacteristic
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D311yescharacteristic
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—34no—
241C2×C4⋊C4—34no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—122no—
256C2×C4⋊C4—122no—
257C2×C4⋊C4—122no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42Dic614yes—
266C2×C42#6714yes—
267#95—22no—
268#95—22no—
269#95—64no—
270#95—64no—
271#95—64no—
272#95—64no—
273#95—64no—
274#95—64no—
275#95—64no—
276#95—122no—
277#95—122no—
278#95—122no—
279C23×C3C22×D411yescharacteristic
280C23×C3C22×D411yescharacteristic
281C23×C3C22×Q811yescharacteristic
282C23×C3C22×Q811yescharacteristic
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4111yescharacteristic
286C23×C3C2×#4111yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4111yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3H(5,𝔽2)11yescharacteristic
291C23×C3H(5,𝔽2)11yescharacteristic
292C23×C32−1+411yescharacteristic
293C23×C32−1+411yescharacteristic
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4#12114yes—
336C2×C3×C4C22⋊Q814yes—
337C2×C3×C4C22⋊Q814yes—
338C2×C3×C4#12314yes—
339C2×C3×C4#12514yes—
340C2×C3×C4#12514yes—
341C2×C3×C4#12514yes—
342C2×C3×C4C4⋊Q814yes—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C24C23×D311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—34no—
407C22×C4—34no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4C2×Dic612yes—
435C22×C4C2×Dic612yes—
436C22×C4C2×Dic612yes—
437C22×C4#23512yes—
438C22×C4#23512yes—
439C22×C4#23512yes—
440C22×C4#23512yes—
441C22×C4#23512yes—
442C22×C4D3×D412yes—
443C22×C4#23712yes—
444C22×C4D3×Q812yes—
445C22×C4#23912yes—
446C22×C4C2×#6712yes—
447C22×C4C2×#6712yes—
448C22×C4C2×#6712yes—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—68no—
452C4⋊C4—68no—
453C4⋊C4—128no—
454C4⋊C4—128no—
455C4⋊C4—124no—
456C4⋊C4—124no—
457C42—28no—
458C42—28no—
459C42—44no—
460C22×C3C2×#12111yescharacteristic
461C22×C3C2×C22⋊Q811yescharacteristic
462C22×C3C2×C22⋊Q811yescharacteristic
463C22×C3C2×#12311yescharacteristic
464C22×C3#52511yescharacteristic
465C22×C3#52511yescharacteristic
466C22×C3C2×#12511yescharacteristic
467C22×C3C2×#12511yescharacteristic
468C22×C3C2×#12511yescharacteristic
469C22×C3#52911yescharacteristic
470C22×C3#52911yescharacteristic
471C22×C3C2×C4⋊Q811yescharacteristic
472C22×C3#53311yescharacteristic
473C22×C3#53511yescharacteristic
474C22×C3#53611yescharacteristic
475C22×C3#54211yescharacteristic
476C22×C3#54211yescharacteristic
477C22×C3#54411yescharacteristic
478C22×C3#54411yescharacteristic
479C22×C3#54611yescharacteristic
480C22×C3#54711yescharacteristic
481C22×C3#54811yescharacteristic
482C22×C3#55011yescharacteristic
483C22×C3#55211yescharacteristic
484C22×C3#55311yescharacteristic
485C22×C3#55411yescharacteristic
486C22×C3#55411yescharacteristic
487C22×C3#55511yescharacteristic
488C22×C3#55611yescharacteristic
489C22×C3#55611yescharacteristic
490C22×C3#55611yescharacteristic
491C22×C3#55711yescharacteristic
492C22×C3#55711yescharacteristic
493C22×C3#55911yescharacteristic
494C22×C3#55911yescharacteristic
495C3×C4—24no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4#39818yes—
502Dic3—44no—
503Dic3—44no—
504Dic3—82no—
505Dic3—82no—
506C23C22×Dic611yescharacteristic
507C23C2×#23511yescharacteristic
508C23C2×#23511yescharacteristic
509C23C2×#23511yescharacteristic
510C23C2×#23511yescharacteristic
511C23C2×#23511yescharacteristic
512C23C2×D3×D411yescharacteristic
513C23C2×#23711yescharacteristic
514C23C2×D3×Q811yescharacteristic
515C23C2×#23911yescharacteristic
516C23#100911yescharacteristic
517C23#100911yescharacteristic
518C23#101011yescharacteristic
519C23#101011yescharacteristic
520C23C22×#6711yescharacteristic
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4#87014yes—
591C2×C4#88814yes—
592C2×C4#88914yes—
593C2×C4#89014yes—
594C2×C4#89414yes—
595C2×C4#92414yes—
596C2×C4#93914yes—
597C2×C4#94614yes—
598C2×C3C2×#39811yescharacteristic
599C2×C3#241411yescharacteristic
600C2×C3#241511yescharacteristic
601C2×C3#244711yescharacteristic
602C2×C3#244811yescharacteristic
603C2×C3#250811yescharacteristic
604C2×C3#250911yescharacteristic
605C2×C3#255011yescharacteristic
606C2×C3#255211yescharacteristic
607C2×C3#257011yescharacteristic
608C2×C3#257111yescharacteristic
609C2×C3#258011yescharacteristic
610C2×C3#258111yescharacteristic
611C2×C3#262811yescharacteristic
612C2×C3#285111yescharacteristic
613C22C2×#87011yescharacteristic
614C22#547011yescharacteristic
615C22#547911yescharacteristic
616C22#547911yescharacteristic
617C22C2×#88811yescharacteristic
618C22C2×#88911yescharacteristic
619C22C2×#89011yescharacteristic
620C22C2×#89411yescharacteristic
621C22#551911yescharacteristic
622C22#551911yescharacteristic
623C22#553411yescharacteristic
624C22#553411yescharacteristic
625C22#554111yescharacteristic
626C22#554211yescharacteristic
627C22#555011yescharacteristic
628C22#555111yescharacteristic
629C22#555211yescharacteristic
630C22#555311yescharacteristic
631C22#556311yescharacteristic
632C22#556511yescharacteristic
633C22#556711yescharacteristic
634C22#556711yescharacteristic
635C22#557911yescharacteristic
636C22#557911yescharacteristic
637C22#558411yescharacteristic
638C22#558411yescharacteristic
639C22#558511yescharacteristic
640C22#558511yescharacteristic
641C22C2×#92411yescharacteristic
642C22#579111yescharacteristic
643C22#579111yescharacteristic
644C22C2×#93911yescharacteristic
645C22C2×#94611yescharacteristic
646C22#582511yescharacteristic
647C22#583011yescharacteristic
648C4—24no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—124no—
655C4—124no—
656C4—242no—
657C4—242no—
658C4#498218yes—
659C3#1522111yescharacteristic, complemented, Sylow
660C2#7367311yescharacteristic
661C2#7368611yescharacteristic
662C2#7368811yescharacteristic
663C2#7371311yescharacteristic
664C2#7371611yescharacteristic
665C2#7372311yescharacteristic
666C2#7372511yescharacteristic
667C2#7378111yescharacteristic
668C2#8207411yescharacteristic
669C2C2×#498211yescharacteristic
670C2#8226411yescharacteristic
671C2#8235011yescharacteristic
672C2#8235111yescharacteristic
673C2#8248011yescharacteristic
674C2#8250111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×C4×C4⋊C4C2
hypercenterC2×C4×C4⋊C4D3
hypocenterC3#15221
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
659. C3#15221semidirect

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×#49282
C2×#49362
#715682
#715672
#715352
#694212
#694142
#152213