Finite Group Databasebeta

← #10732322

Subgroups — #10732322

Order 768.

1346 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4973C211yesmaximal, characteristic
2C2×#4973C211yesmaximal, characteristic
3C2×#4929C211yesmaximal, characteristic
4C2×C4×Dic3⋊C4C211yesmaximal, characteristic
5C3×C4⋊#95C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic
7#71563C211yesmaximal, characteristic
8#69421C212yesmaximal
9#69417C212yesmaximal
10#69416C212yesmaximal
11#69414C212yesmaximal
12#15192—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36#4973—24no—
37#4973—24no—
38C4×Dic3⋊C4—24no—
39#4929—24no—
40C2×#391—31no—
41C2×#391—31no—
42C2×#391—31no—
43C2×C4×C4⋊C4—31no—
44#1445—32no—
45#1443—32no—
46#1443—32no—
47#1442—32no—
48C4⋊#95—31no—
49C4⋊#95—31no—
50C4⋊#95D311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6—22no—
68C22×Dic6D414yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4Q814yescomplemented
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3D414yes—
93C3×#95—24no—
94C3×#95—42no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C23×Q8—31no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C42—31no—
114C22×C42—31no—
115C22×C42C2×D311yescharacteristic
116#391—64no—
117#391—64no—
118#391—64no—
119C4×C4⋊C4—64no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95C2×D312yes—
133C2×#95C2×D312yes—
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—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×Q812yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4112yes—
168C22×C3×C4#4112yes—
169C22×C3×C4#4112yes—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
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—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D414yes—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×Q812yes—
209C22×Dic3C2×Q812yes—
210C22×Dic3#4112yes—
211C22×Dic3#4112yes—
212C2×Dic6—28no—
213C2×Dic6—28no—
214C2×Dic6—28no—
215C2×Dic6—44no—
216C2×Dic6—44no—
217C2×Dic6—44no—
218C2×Dic6—44no—
219C3×C4⋊C4—44no—
220C3×C4⋊C4—44no—
221C3×C42—44no—
222C4⋊Dic3—44no—
223Dic3⋊C4—28no—
224Dic3⋊C4—28no—
225C4×Dic3—28no—
226C4×Dic3—44no—
227C22×Q8—34no—
228C22×Q8—62no—
229C22×Q8—62no—
230C22×Q8—62no—
231C22×Q8—62no—
232C22×Q8—62no—
233C22×Q8—62no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4—32no—
240C23×C4C22×D312yes—
241C23×C4C22×D312yes—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C23×C4C22×D311yescharacteristic
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—34no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—62no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—34no—
264C2×C42—62no—
265C2×C42—62no—
266C2×C42—62no—
267C2×C42—62no—
268C2×C42—62no—
269#95—24no—
270#95—42no—
271#95—64no—
272#95—64no—
273#95—62no—
274#95—62no—
275#95—64no—
276#95—64no—
277#95—64no—
278#95—64no—
279#95—64no—
280#95—64no—
281#95—122no—
282#95—122no—
283C23×C3C22×D411yescharacteristic
284C23×C3C22×D411yescharacteristic
285C23×C3C22×Q811yescharacteristic
286C23×C3C2×#4111yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4111yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4111yescharacteristic
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3H(5,𝔽2)11yescharacteristic
295C23×C32−1+411yescharacteristic
296C23×C32−1+411yescharacteristic
297C23×C32−1+411yescharacteristic
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—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4#12114yes—
327C2×C3×C4#12314yes—
328C2×C3×C4#12314yes—
329C2×C3×C4#12314yes—
330C2×C3×C4#12414yes—
331C2×C3×C4#12614yes—
332C2×C3×C4#12614yes—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3C22⋊Q814yes—
375Dic6—416no—
376C24C23×D311yescharacteristic
377C2×Q8—68no—
378C2×Q8—68no—
379C2×Q8—68no—
380C2×Q8—124no—
381C2×Q8—124no—
382C2×Q8—124no—
383C2×Q8—124no—
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—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—34no—
415C22×C4—32no—
416C22×C4—32no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4#23512yes—
453C22×C4#23512yes—
454C22×C4#23512yes—
455C22×C4D3×D412yes—
456C22×C4D3×D412yes—
457C22×C4#23712yes—
458C22×C4#23712yes—
459C22×C4D3×Q812yes—
460C22×C4#23912yes—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—68no—
464C4⋊C4—68no—
465C4⋊C4—124no—
466C42—44no—
467C42—68no—
468C42—124no—
469C22×C3C2×#12111yescharacteristic
470C22×C3C2×C22⋊Q811yescharacteristic
471C22×C3C2×#12311yescharacteristic
472C22×C3C2×#12311yescharacteristic
473C22×C3C2×#12311yescharacteristic
474C22×C3#52511yescharacteristic
475C22×C3C2×#12411yescharacteristic
476C22×C3C2×#12611yescharacteristic
477C22×C3C2×#12611yescharacteristic
478C22×C3#52911yescharacteristic
479C22×C3#52911yescharacteristic
480C22×C3#53311yescharacteristic
481C22×C3#53311yescharacteristic
482C22×C3#53611yescharacteristic
483C22×C3#53711yescharacteristic
484C22×C3#54111yescharacteristic
485C22×C3#54111yescharacteristic
486C22×C3#54211yescharacteristic
487C22×C3#54211yescharacteristic
488C22×C3#54811yescharacteristic
489C22×C3#54811yescharacteristic
490C22×C3D4×Q811yescharacteristic
491C22×C3#55011yescharacteristic
492C22×C3#55211yescharacteristic
493C22×C3#55211yescharacteristic
494C22×C3#55411yescharacteristic
495C22×C3#55611yescharacteristic
496C22×C3#55611yescharacteristic
497C22×C3#55611yescharacteristic
498C22×C3#55611yescharacteristic
499C22×C3#55611yescharacteristic
500C22×C3#55711yescharacteristic
501C22×C3#55711yescharacteristic
502C22×C3#55911yescharacteristic
503C22×C3#55911yescharacteristic
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—82no—
507C3×C4—82no—
508C3×C4#39718yes—
509Dic3—24no—
510Dic3—42no—
511Dic3—44no—
512Dic3—42no—
513Dic3—42no—
514Dic3—82no—
515C23C2×#23511yescharacteristic
516C23C2×#23511yescharacteristic
517C23C2×#23511yescharacteristic
518C23C2×D3×D411yescharacteristic
519C23C2×D3×D411yescharacteristic
520C23C2×#23711yescharacteristic
521C23C2×#23711yescharacteristic
522C23#100411yescharacteristic
523C23C2×D3×Q811yescharacteristic
524C23C2×#23911yescharacteristic
525C23#100711yescharacteristic
526C23D3×#4111yescharacteristic
527C23D3×#4111yescharacteristic
528C23#101011yescharacteristic
529C23#101011yescharacteristic
530Q8—1216no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—34no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—122no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4#87614yes—
602C2×C4#88314yes—
603C2×C4#88414yes—
604C2×C4#88514yes—
605C2×C4#89414yes—
606C2×C4#89414yes—
607C2×C4#89814yes—
608C2×C3C2×#39711yescharacteristic
609C2×C3#244611yescharacteristic
610C2×C3#244611yescharacteristic
611C2×C3#244811yescharacteristic
612C2×C3#244811yescharacteristic
613C2×C3#251111yescharacteristic
614C2×C3#251111yescharacteristic
615C2×C3#251411yescharacteristic
616C2×C3#251411yescharacteristic
617C2×C3#253611yescharacteristic
618C2×C3#253811yescharacteristic
619C2×C3#256811yescharacteristic
620C2×C3#259211yescharacteristic
621C2×C3#262511yescharacteristic
622C2×C3#285511yescharacteristic
623C22C2×#87611yescharacteristic
624C22C2×#88311yescharacteristic
625C22C2×#88411yescharacteristic
626C22C2×#88511yescharacteristic
627C22C2×#89411yescharacteristic
628C22C2×#89411yescharacteristic
629C22#551111yescharacteristic
630C22C2×#89811yescharacteristic
631C22#551311yescharacteristic
632C22#552911yescharacteristic
633C22#553311yescharacteristic
634C22#554211yescharacteristic
635C22#555711yescharacteristic
636C22#555911yescharacteristic
637C22#556111yescharacteristic
638C22#556711yescharacteristic
639C22#556811yescharacteristic
640C22#557811yescharacteristic
641C22#558311yescharacteristic
642C22#559811yescharacteristic
643C22#559911yescharacteristic
644C22#560611yescharacteristic
645C22#560811yescharacteristic
646C22D3×C22⋊Q811yescharacteristic
647C22#562811yescharacteristic
648C22#563211yescharacteristic
649C22#563311yescharacteristic
650C22#567111yescharacteristic
651C22#567111yescharacteristic
652C22#569111yescharacteristic
653C22#569811yescharacteristic
654C22#571311yescharacteristic
655C22#572111yescharacteristic
656C22#572811yescharacteristic
657C22#573311yescharacteristic
658C4—42no—
659C4—42no—
660C4—64no—
661C4—82no—
662C4—82no—
663C4—122no—
664C4—124no—
665C4—122no—
666C4—122no—
667C4—242no—
668C4#467518yes—
669C3#1519211yescharacteristic, complemented, Sylow
670C2C2×#467511yescharacteristic
671C2#7369111yescharacteristic
672C2#7369711yescharacteristic
673C2#7393411yescharacteristic
674C2#7393611yescharacteristic
675C2#7417311yescharacteristic
676C2#7433211yescharacteristic
677C2#7600511yescharacteristic
678C2#7600711yescharacteristic
679C2#7610211yescharacteristic
680C2#7628711yescharacteristic
681C2#7632011yescharacteristic
682C2#7636511yescharacteristic
683C2#7650211yescharacteristic
684C2#7654511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×C4⋊#95C2
hypercenterC4⋊#95D3
hypocenterC3#15192
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
86. C2×Dic3⋊C4Q8semidirect
669. C3#15192semidirect

Maximal subgroups

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

subgroupindex
C2×#49732
C2×#49292
C2×C4×Dic3⋊C42
C3×C4⋊#952
#715632
#694212
#694172
#694162
#694142
#151923