Finite Group Databasebeta

← #10718871

Subgroups — #10718871

Order 768.

1386 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×Dic3×C4⋊C4C211yesmaximal, characteristic
3C2×C4×C4⋊Dic3C211yesmaximal, characteristic
4#71569C212yesmaximal
5#71564C212yesmaximal
6#71563C211yesmaximal, characteristic
7#71563C211yesmaximal, characteristic
8#71563C211yesmaximal, characteristic
9#71535C211yesmaximal, characteristic
10#69412C212yesmaximal
11#69407C212yesmaximal
12#14041—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24C3×#389C418yescomplemented
25C2×C3×#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37Dic3×C4⋊C4—24no—
38C4×C4⋊Dic3—24no—
39C2×#389D311yescharacteristic
40C2×C4×C4⋊C4—31no—
41C2×C4×C4⋊C4—31no—
42#1445—32no—
43#1442—32no—
44#1436—32no—
45#1436—32no—
46C4⋊#95—31no—
47C4⋊#95—31no—
48C4⋊#95—31no—
49C4⋊#95—31no—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×Dic3C2312yes—
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4C2×C414yes—
68C2×C3×C4⋊C4C2×C414yes—
69C2×C3×C4⋊C4C2×C418yes—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42C2×C414yes—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
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⋊Dic3Q814yes—
84C2×C4⋊Dic3Q814yes—
85C2×C4×Dic3—22no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—24no—
91C2×C4×Dic3—42no—
92C2×C4×Dic3—42no—
93C2×C4×Dic3D414yes—
94C3×#95C2×C418yes—
95C3⋊#95—24no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—42no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95—42no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C4⋊C4C2×D312yes—
113C22×C42—31no—
114C22×C42—31no—
115C22×C42—32no—
116C22×C42C2×D311yescharacteristic
117#389Dic318yescomplemented
118C4×C4⋊C4—64no—
119C4×C4⋊C4—64no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—31no—
124C2×#95—32no—
125C2×#95—31no—
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—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
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—21no—
154C22×C3×C4—22no—
155C22×C3×C4C22×C414yes—
156C22×C3×C4C22×C414yes—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C22×C412yes—
159C22×C3×C4C22×C412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×Q812yes—
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—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
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—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×Q814yes—
206C22×Dic3C2×Q812yes—
207C22×Dic3C2×Q812yes—
208C22×Dic3#4112yes—
209C22×Dic3#4112yes—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—48no—
213C3×C42—28no—
214C4⋊Dic3—28no—
215C4⋊Dic3—28no—
216C4⋊Dic3—28no—
217C4⋊Dic3—44no—
218C4×Dic3—28no—
219C4×Dic3—48no—
220C4×Dic3—44no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4—32no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4C22×D312yes—
228C23×C4C22×D311yescharacteristic
229C23×C4C22×D312yes—
230C23×C4C22×D311yescharacteristic
231C23×C4C22×D311yescharacteristic
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—34no—
237C2×C4⋊C4—34no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4C2×Dic314yes—
251C2×C4⋊C4C2×Dic314yes—
252C2×C4⋊C4C2×Dic318yes—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—34no—
256C2×C42—62no—
257C2×C42—62no—
258C2×C42—64no—
259C2×C42—62no—
260C2×C42—62no—
261C2×C42—62no—
262C2×C42—122no—
263C2×C42—122no—
264C2×C42C2×Dic314yes—
265#95—64no—
266#95—64no—
267#95—62no—
268#95—62no—
269#95—64no—
270#95—64no—
271#95—122no—
272#95—122no—
273#95—122no—
274#95—122no—
275#95—122no—
276#95C2×Dic318yes—
277C23×C3C23×C411yescharacteristic
278C23×C3C22×D411yescharacteristic
279C23×C3C22×Q811yescharacteristic
280C23×C3C22×Q811yescharacteristic
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4111yescharacteristic
286C23×C3C2×#4111yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3H(5,𝔽2)11yescharacteristic
289C23×C3H(5,𝔽2)11yescharacteristic
290C23×C3H(5,𝔽2)11yescharacteristic
291C23×C32−1+411yescharacteristic
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—22no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4#11714yes—
320C2×C3×C4C4×D414yes—
321C2×C3×C4C4×Q814yes—
322C2×C3×C4C4×Q814yes—
323C2×C3×C4C22⋊Q814yes—
324C2×C3×C4#12314yes—
325C2×C3×C4#12514yes—
326C2×C3×C4#12614yes—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
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×Dic3C22⋊Q814yes—
369C24C23×D311yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—32no—
395C22×C4—32no—
396C22×C4—34no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—62no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4#23512yes—
435C22×C4#23512yes—
436C22×C4D3×D412yes—
437C22×C4#23712yes—
438C22×C4D3×Q812yes—
439C22×C4D3×Q812yes—
440C22×C4#23912yes—
441C22×C4#23912yes—
442C22×C4C22×Dic314yes—
443C22×C4C22×Dic314yes—
444C22×C4C22×Dic312yes—
445C22×C4C22×Dic312yes—
446C22×C4C22×Dic312yes—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—48no—
450C4⋊C4—68no—
451C4⋊C4—68no—
452C4⋊C4—68no—
453C4⋊C4—124no—
454C42—28no—
455C42—68no—
456C42—128no—
457C42—124no—
458C22×C3C2×#11711yescharacteristic
459C22×C3C2×C4×D411yescharacteristic
460C22×C3C2×C4×Q811yescharacteristic
461C22×C3C2×C4×Q811yescharacteristic
462C22×C3C4×#4111yescharacteristic
463C22×C3#51811yescharacteristic
464C22×C3#52011yescharacteristic
465C22×C3C2×C22⋊Q811yescharacteristic
466C22×C3C2×C22⋊Q811yescharacteristic
467C22×C3C2×#12311yescharacteristic
468C22×C3#52511yescharacteristic
469C22×C3C2×#12511yescharacteristic
470C22×C3C2×#12611yescharacteristic
471C22×C3#52911yescharacteristic
472C22×C3#53311yescharacteristic
473C22×C3#53511yescharacteristic
474C22×C3#53711yescharacteristic
475C22×C3#54011yescharacteristic
476C22×C3#54211yescharacteristic
477C22×C3#55211yescharacteristic
478C22×C3#55311yescharacteristic
479C22×C3#55311yescharacteristic
480C22×C3#55311yescharacteristic
481C22×C3#55411yescharacteristic
482C22×C3#55411yescharacteristic
483C22×C3#55411yescharacteristic
484C22×C3#55411yescharacteristic
485C22×C3#55511yescharacteristic
486C22×C3#55611yescharacteristic
487C22×C3#55711yescharacteristic
488C22×C3#55711yescharacteristic
489C22×C3Q8211yescharacteristic
490C22×C3#55911yescharacteristic
491C22×C3#55911yescharacteristic
492C22×C3#55911yescharacteristic
493C3×C4—24no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4—44no—
497C3×C4#38718yes—
498Dic3—24no—
499Dic3—44no—
500Dic3—42no—
501Dic3—42no—
502Dic3—42no—
503Dic3—82no—
504C23C2×#23511yescharacteristic
505C23C2×#23511yescharacteristic
506C23C2×D3×D411yescharacteristic
507C23C2×#23711yescharacteristic
508C23#100411yescharacteristic
509C23#100411yescharacteristic
510C23C2×D3×Q811yescharacteristic
511C23C2×D3×Q811yescharacteristic
512C23C2×#23911yescharacteristic
513C23C2×#23911yescharacteristic
514C23D3×#4111yescharacteristic
515C23D3×#4111yescharacteristic
516C23#100911yescharacteristic
517C23#101011yescharacteristic
518C23C23×Dic311yescharacteristic
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—34no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—64no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—122no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—121no—
579C2×C4—121no—
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#88914yes—
589C2×C4#89414yes—
590C2×C4#89614yes—
591C2×C4#89814yes—
592C2×C4#92614yes—
593C2×C4D4×Dic314yes—
594C2×C4Q8×Dic314yes—
595C2×C4Q8×Dic314yes—
596C2×C3C2×#38711yescharacteristic
597C2×C3C4×C22⋊Q811yescharacteristic
598C2×C3#232411yescharacteristic
599C2×C3#235511yescharacteristic
600C2×C3#235711yescharacteristic
601C2×C3#236911yescharacteristic
602C2×C3#237011yescharacteristic
603C2×C3#248511yescharacteristic
604C2×C3#248711yescharacteristic
605C2×C3#250711yescharacteristic
606C2×C3#250911yescharacteristic
607C2×C3#251111yescharacteristic
608C2×C3#251311yescharacteristic
609C2×C3#258011yescharacteristic
610C2×C3#258211yescharacteristic
611C22C2×#88911yescharacteristic
612C22C2×#89411yescharacteristic
613C22C2×#89611yescharacteristic
614C22C2×#89811yescharacteristic
615C22#555811yescharacteristic
616C22#555811yescharacteristic
617C22#555911yescharacteristic
618C22#555911yescharacteristic
619C22#556711yescharacteristic
620C22#557911yescharacteristic
621C22#558411yescharacteristic
622C22#558511yescharacteristic
623C22#560311yescharacteristic
624C22#560711yescharacteristic
625C22#560811yescharacteristic
626C22#561911yescharacteristic
627C22#562211yescharacteristic
628C22#562411yescharacteristic
629C22D3×C22⋊Q811yescharacteristic
630C22#562711yescharacteristic
631C22#568511yescharacteristic
632C22#569011yescharacteristic
633C22#569411yescharacteristic
634C22#569511yescharacteristic
635C22#572211yescharacteristic
636C22#573011yescharacteristic
637C22#573011yescharacteristic
638C22#573411yescharacteristic
639C22C2×#92611yescharacteristic
640C22C2×D4×Dic311yescharacteristic
641C22#579811yescharacteristic
642C22C2×Q8×Dic311yescharacteristic
643C22C2×Q8×Dic311yescharacteristic
644C22Dic3×#4111yescharacteristic
645C22#582711yescharacteristic
646C4—24no—
647C4—44no—
648C4—42no—
649C4—44no—
650C4—64no—
651C4—124no—
652C4—122no—
653C4—122no—
654C4—122no—
655C4—242no—
656C4#498018yes—
657C3#1404111yescharacteristic, complemented, Sylow
658C2#7603411yescharacteristic
659C2#7604311yescharacteristic
660C2#7604611yescharacteristic
661C2#7604811yescharacteristic
662C2#7607511yescharacteristic
663C2#7607711yescharacteristic
664C2#7623011yescharacteristic
665C2#7623211yescharacteristic
666C2C2×#498011yescharacteristic
667C2#8233811yescharacteristic
668C2#8246211yescharacteristic
669C2#8263311yescharacteristic
670C2Dic3×C22⋊Q811yescharacteristic
671C2#8302511yescharacteristic
672C2#8323011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#389C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#389D3
hypocenterC3#14041
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
24. C3×#389C4semidirect
117. #389Dic3semidirect
657. C3#14041semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×Dic3×C4⋊C42
C2×C4×C4⋊Dic32
#715692
#715642
#715632
#715352
#694122
#694072
#140413