Finite Group Databasebeta

← #10720232

Subgroups — #10720232

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
2C2×#4973C211yesmaximal, characteristic
3C2×#4973C211yesmaximal, characteristic
4C2×#4972C211yesmaximal, characteristic
5C2×#4929C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic
7#71536C212yesmaximal
8#71534C212yesmaximal
9C4×C3⋊#95C211yesmaximal, characteristic
10#69418C212yesmaximal
11#69413C212yesmaximal
12#14142—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2212yes—
17C22×C3×C42C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C3×C4×C4⋊C4—24no—
23C2×C3×#95C2212yes—
24C2×C3⋊#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37#4973—24no—
38#4973—24no—
39#4972—24no—
40#4929C418yescomplemented
41C2×#391—31no—
42C2×#391—31no—
43C2×#391—31no—
44C2×#389—31no—
45C2×C4×C4⋊C4D311yescharacteristic
46#1443—32no—
47#1442—32no—
48#1436—32no—
49C4⋊#95—31no—
50#1434—32no—
51C4×#95—31no—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6C2×C414yes—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4D414yes—
75C2×C3×C4⋊C4D414yes—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42—24no—
79C2×C3×C42—24no—
80C2×C3×C42—42no—
81C2×C3×C42C2×C414yes—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3C2×C414yes—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—42no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C3×#95—24no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—22no—
104C3⋊#95—42no—
105C3⋊#95C2×C418yes—
106C3⋊#95C2×C418yes—
107C3⋊#95Q814yescomplemented
108C23×Q8—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—32no—
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C42—31no—
114C22×C42—31no—
115C22×C42C2×D312yes—
116C22×C42C2×D311yescharacteristic
117#391—38no—
118#391—64no—
119#391—64no—
120#389—64no—
121C4×C4⋊C4—24no—
122C2×#95—31no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95C2×D312yes—
136C24×C3C2411yescharacteristic
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
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—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4C22×C412yes—
160C22×C3×C4C22×C412yes—
161C22×C3×C4C22×C412yes—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×Q812yes—
167C22×C3×C4#4112yes—
168C22×C3×C4#4112yes—
169C22×C3×C4#4112yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
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—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3C22×C414yes—
209C22×Dic3C22×C414yes—
210C22×Dic3C2×Q812yes—
211C22×Dic3C2×Q812yes—
212C22×Dic3#4112yes—
213C22×Dic3#4112yes—
214C2×Dic6—28no—
215C2×Dic6—28no—
216C2×Dic6—28no—
217C2×Dic6—44no—
218C2×Dic6—44no—
219C2×Dic6—44no—
220C2×Dic6—44no—
221C3×C4⋊C4—28no—
222C3×C4⋊C4—28no—
223C3×C42—28no—
224C3×C42—48no—
225C4⋊Dic3—28no—
226Dic3⋊C4—48no—
227C4×Dic3—44no—
228C4×Dic3—44no—
229C22×Q8—34no—
230C22×Q8—62no—
231C22×Q8—62no—
232C22×Q8—62no—
233C22×Q8—62no—
234C22×Q8—62no—
235C22×Q8—62no—
236C23×C4—32no—
237C23×C4—31no—
238C23×C4—32no—
239C23×C4—31no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D312yes—
244C23×C4C22×D312yes—
245C23×C4C22×D311yescharacteristic
246C23×C4C22×D311yescharacteristic
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—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—122no—
257C2×C4⋊C4D1214yes—
258C2×C4⋊C4#6714yes—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—42no—
264C2×C42—62no—
265C2×C42—62no—
266C2×C42—62no—
267C2×C42—62no—
268C2×C42—62no—
269C2×C42—62no—
270C2×C42C4×D314yes—
271#95—24no—
272#95—38no—
273#95—38no—
274#95—34no—
275#95—62no—
276#95—64no—
277#95—62no—
278#95—64no—
279#95—64no—
280#95—64no—
281#95—62no—
282#95—64no—
283#95—64no—
284#95—122no—
285C23×C3C23×C411yescharacteristic
286C23×C3C22×D411yescharacteristic
287C23×C3C22×D411yescharacteristic
288C23×C3C22×Q811yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4111yescharacteristic
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3H(5,𝔽2)11yescharacteristic
297C23×C32−1+411yescharacteristic
298C23×C32−1+411yescharacteristic
299C23×C32−1+411yescharacteristic
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—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4#11714yes—
328C2×C3×C4C4×D414yes—
329C2×C3×C4C4×D414yes—
330C2×C3×C4C4×Q814yes—
331C2×C3×C4#12114yes—
332C2×C3×C4#12314yes—
333C2×C3×C4#12414yes—
334C2×C3×C4#12614yes—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—42no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377Dic6—416no—
378C24C23×D311yescharacteristic
379C2×Q8—68no—
380C2×Q8—68no—
381C2×Q8—68no—
382C2×Q8—124no—
383C2×Q8—124no—
384C2×Q8—124no—
385C2×Q8—124no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—34no—
409C22×C4—34no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—62no—
450C22×C4C2×C4×D312yes—
451C22×C4C2×C4×D312yes—
452C22×C4C2×C4×D312yes—
453C22×C4C2×D1214yes—
454C22×C4C2×D1212yes—
455C22×C4#23512yes—
456C22×C4#23512yes—
457C22×C4#23512yes—
458C22×C4#23512yes—
459C22×C4#23712yes—
460C22×C4D3×Q812yes—
461C22×C4C2×#6714yes—
462C22×C4C2×#6712yes—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—68no—
466C4⋊C4—128no—
467C42—28no—
468C42—48no—
469C42—124no—
470C42—124no—
471C22×C3C2×#11711yescharacteristic
472C22×C3C2×C4×D411yescharacteristic
473C22×C3C2×C4×D411yescharacteristic
474C22×C3C2×C4×Q811yescharacteristic
475C22×C3C4×#4111yescharacteristic
476C22×C3#51911yescharacteristic
477C22×C3#52011yescharacteristic
478C22×C3C2×#12111yescharacteristic
479C22×C3C2×#12311yescharacteristic
480C22×C3C2×#12411yescharacteristic
481C22×C3C2×#12611yescharacteristic
482C22×C3#52911yescharacteristic
483C22×C3#52911yescharacteristic
484C22×C3#53311yescharacteristic
485C22×C3#53311yescharacteristic
486C22×C3#54111yescharacteristic
487C22×C3#54111yescharacteristic
488C22×C3#54211yescharacteristic
489C22×C3#54211yescharacteristic
490C22×C3#54611yescharacteristic
491C22×C3#54611yescharacteristic
492C22×C3#54711yescharacteristic
493C22×C3#54711yescharacteristic
494C22×C3#54811yescharacteristic
495C22×C3#54811yescharacteristic
496C22×C3D4×Q811yescharacteristic
497C22×C3D4×Q811yescharacteristic
498C22×C3#55211yescharacteristic
499C22×C3#55211yescharacteristic
500C22×C3#55411yescharacteristic
501C22×C3#55611yescharacteristic
502C22×C3#55611yescharacteristic
503C22×C3#55611yescharacteristic
504C22×C3#55711yescharacteristic
505C22×C3#55911yescharacteristic
506C3×C4—24no—
507C3×C4—42no—
508C3×C4—44no—
509C3×C4—44no—
510C3×C4#38818yes—
511Dic3—44no—
512Dic3—42no—
513Dic3—42no—
514Dic3—42no—
515Dic3—42no—
516Dic3—44no—
517C23C22×C4×D311yescharacteristic
518C23C22×D1211yescharacteristic
519C23C2×#23511yescharacteristic
520C23C2×#23511yescharacteristic
521C23C2×#23511yescharacteristic
522C23C2×#23511yescharacteristic
523C23C2×#23711yescharacteristic
524C23#100411yescharacteristic
525C23C2×D3×Q811yescharacteristic
526C23#100711yescharacteristic
527C23D3×#4111yescharacteristic
528C23D3×#4111yescharacteristic
529C23#101011yescharacteristic
530C23#101011yescharacteristic
531C23C22×#6711yescharacteristic
532Q8—1216no—
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—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—62no—
567C2×C4—64no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
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—122no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—122no—
588C2×C4—121no—
589C2×C4—122no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—122no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4#87214yes—
603C2×C4C4×D1214yes—
604C2×C4#87514yes—
605C2×C4#87614yes—
606C2×C4#88714yes—
607C2×C4C4×#6714yes—
608C2×C4#92914yes—
609C2×C4#93014yes—
610C2×C3C2×#38811yescharacteristic
611C2×C3#229511yescharacteristic
612C2×C3#232011yescharacteristic
613C2×C3#236011yescharacteristic
614C2×C3#236211yescharacteristic
615C2×C3#236511yescharacteristic
616C2×C3#236711yescharacteristic
617C2×C3#242711yescharacteristic
618C2×C3#242911yescharacteristic
619C2×C3#244611yescharacteristic
620C2×C3#244711yescharacteristic
621C2×C3#248911yescharacteristic
622C2×C3#249211yescharacteristic
623C2×C3#251111yescharacteristic
624C2×C3#251411yescharacteristic
625C22C2×#87211yescharacteristic
626C22C2×C4×D1211yescharacteristic
627C22C2×#87511yescharacteristic
628C22C2×#87611yescharacteristic
629C22C2×#88711yescharacteristic
630C22#550411yescharacteristic
631C22#551111yescharacteristic
632C22#551311yescharacteristic
633C22#551611yescharacteristic
634C22#552211yescharacteristic
635C22#552911yescharacteristic
636C22#553311yescharacteristic
637C22#554211yescharacteristic
638C22#555411yescharacteristic
639C22#555511yescharacteristic
640C22#555711yescharacteristic
641C22#555711yescharacteristic
642C22#555911yescharacteristic
643C22#556111yescharacteristic
644C22#556111yescharacteristic
645C22#556711yescharacteristic
646C22#556811yescharacteristic
647C22Q8×D1211yescharacteristic
648C22#557611yescharacteristic
649C22#557811yescharacteristic
650C22#557811yescharacteristic
651C22#558311yescharacteristic
652C22#558311yescharacteristic
653C22C2×C4×#6711yescharacteristic
654C22C2×#92911yescharacteristic
655C22C2×#93011yescharacteristic
656C22#580611yescharacteristic
657C22Q8×#6711yescharacteristic
658C22#582411yescharacteristic
659C22#583111yescharacteristic
660C4—24no—
661C4—42no—
662C4—44no—
663C4—44no—
664C4—122no—
665C4—122no—
666C4—122no—
667C4—124no—
668C4—124no—
669C4—122no—
670C4#494018yes—
671C3#1414211yescharacteristic, complemented, Sylow
672C2#7361811yescharacteristic
673C2#7362511yescharacteristic
674C2#7365011yescharacteristic
675C2#7365411yescharacteristic
676C2#7369111yescharacteristic
677C2#7369711yescharacteristic
678C2#7378611yescharacteristic
679C2#7378911yescharacteristic
680C2C2×#494011yescharacteristic
681C2#8216211yescharacteristic
682C2#8224311yescharacteristic
683C2#8235911yescharacteristic
684C2#8240811yescharacteristic
685C2#8246811yescharacteristic
686C2#8249611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×C4×C4⋊C4C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×C4×C4⋊C4D3
hypocenterC3#14142
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
40. #4929C4semidirect
107. C3⋊#95Q8semidirect
671. C3#14142semidirect

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×#49732
C2×#49722
C2×#49292
#715632
#715362
#715342
C4×C3⋊#952
#694182
#694132
#141423