Finite Group Databasebeta

← #29696143

Subgroups — #29696143

Order 15616.

1696 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840408C211yesmaximal, characteristic
2#21448639C212yesmaximal, complemented
3#21446803C212yesmaximal, complemented
4#21446269C214yesmaximal, complemented
5#21445892C212yesmaximal, complemented
6C4×#14519692C211yesmaximal, characteristic, complemented
7#342×D61C212yesmaximal, complemented
8C61×#1799C211yesmaximal, characteristic, complemented
9C22×#11758222C411yescharacteristic, complemented
10C23×D244C2211yescharacteristic
11#16840408—22no—
12#16840408—24no—
13#16840408—22no—
14C2×#14519765C411yescharacteristic, complemented
15C2×#14519762C412yescomplemented
16C2×#14519756C2211yescharacteristic
17C2×#14519751C411yescharacteristic, complemented
18C2×#14519750C412yescomplemented
19C2×#14519745C2212yes—
20C2×C4×D244C411yescharacteristic, complemented
21C2×#14519739—22no—
22C2×C22⋊C4×D61C2214yes—
23C2×C22⋊C4×D61C2212yes—
24C2×#14519692C2212yes—
25#16839663—22no—
26#16839640—24no—
27#16839617C2214yes—
28C4×C22⋊Dic61C2212yes—
29#16839529C2211yescharacteristic
30#16839529C2211yescharacteristic
31#16839528C2214yes—
32#16839528C2214yes—
33C2×C4×D4×C61C2211yescharacteristic
34C2×#99×C61C2212yes—
35C61×#380C2212yes—
36C61×#342C2212yes—
37C24×D61C2312yes—
38C2×#11758222—22no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—44no—
42C22×Dic122C2×C411yescharacteristic, complemented
43C22×D244—44no—
44C22×D244C2×C411yescharacteristic
45C22×D244C2×C412yes—
46C22×D244C2×C411yescharacteristic
47C22×D244D412yes—
48C22×D244D414yes—
49C22×D244D412yes—
50C22×C22⋊D61C2×C412yescomplemented
51C22×C22⋊D61C2×C412yescomplemented
52C22×C4×D61—24no—
53C22×C4×D61C2×C412yescomplemented
54C22×C4×D61C2×C412yescomplemented
55#14519765—22no—
56#14519765—22no—
57#14519762—24no—
58#14519762—24no—
59#14519756—22no—
60#14519756—22no—
61#14519751—22no—
62#14519751—22no—
63#14519750—24no—
64#14519750—24no—
65#14519745—24no—
66#14519745—24no—
67C4×D244—22no—
68C4×D244—21no—
69C4×D244—21no—
70#14519739—48no—
71C2×C4⋊Dic61C2×C411yescharacteristic, complemented
72C2×#11758196C2×C412yescomplemented
73C2×C22⋊Dic61C2312yes—
74C2×C22⋊Dic61C2312yes—
75C2×C22⋊Dic61C2312yes—
76C2×#11758198—24no—
77C2×#11758198—24no—
78C2×#11758198—24no—
79C2×#11758198C2×C412yescomplemented
80C2×#11758198C2×C412yescomplemented
81C2×#11758198C2×C412yescomplemented
82C2×#11758198C2×C412yescomplemented
83C22⋊C4×D61—24no—
84C22⋊C4×D61—24no—
85C22⋊C4×D61—28no—
86C22⋊C4×D61—44no—
87C2×C4×Dic61—22no—
88#14519692—24no—
89#14519684—22no—
90C22×D4×C61C2311yescharacteristic
91C23×C4×C61C2×C411yescharacteristic, complemented
92C23×C4×C61C2311yescharacteristic
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
97C2×C22⋊C4×C61C2×C411yescharacteristic, complemented
98C2×C22⋊C4×C61C2314yes—
99C2×C22⋊C4×C61C2311yescharacteristic
100C2×C42×C61C2×C411yescharacteristic, complemented
101#99×C61—42no—
102#95×C61—22no—
103C24×C61C22×C411yescharacteristic
104C24×C61C2411yescharacteristic
105C23×D61—24no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—44no—
109C23×D61—82no—
110C23×D61C22×C412yes—
111C23×D61C22×C412yes—
112C23×D61C2×D414yes—
113C23×D61C2×D414yes—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—24no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—82no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61C22×C412yes—
138C22×Dic61C22×C412yes—
139#11758222—48no—
140#11758222—44no—
141#11758222—44no—
142C2×D244—24no—
143C2×D244—28no—
144C2×D244—42no—
145C2×D244—44no—
146C2×D244—44no—
147C2×D244—48no—
148C2×D244C22⋊C412yes—
149C2×D244C22⋊C411yescharacteristic
150C2×D244C22⋊C414yes—
151C2×D244C22⋊C412yes—
152C2×D244C22⋊C414yes—
153C2×D244C22⋊C412yes—
154C2×D244C22⋊C411yescharacteristic
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—44no—
160C2×C4×D61—48no—
161C2×C4×D61—48no—
162C2×C4×D61—84no—
163C2×C4×D61—82no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—44no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—24no—
176C22⋊Dic61—22no—
177C22⋊Dic61—22no—
178C22⋊Dic61—22no—
179C22⋊Dic61#4114yescomplemented
180#11758198—24no—
181#11758198—24no—
182#11758198—22no—
183#11758198—22no—
184#11758198—24no—
185#11758198—48no—
186#11758198—48no—
187#11758198—84no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190#11758196—24no—
191C4×Dic61—44no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61C23×C411yescharacteristic
200C23×C61C22×D412yes—
201C23×C61C2×#4112yes—
202C23×C61H(5,𝔽2)11yescharacteristic
203C23×C612−1+411yescharacteristic
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—24no—
211C22×D61—24no—
212C22×D61—28no—
213C22×D61—24no—
214C22×D61—42no—
215C22×D61—44no—
216C22×D61—44no—
217C22×D61—42no—
218C22×D61—44no—
219C22×D61—42no—
220C22×D61—84no—
221C22×D61—88no—
222C22×D61—84no—
223C22×D61C2×C22⋊C414yes—
224C22×D61C2×C22⋊C412yes—
225C22×D61C2×C22⋊C412yes—
226D4×C61—42no—
227D4×C61—42no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—44no—
242C2×C4×C61—44no—
243C2×C4×C61—44no—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C22⋊D61—44no—
248C22⋊D61—44no—
249C22⋊D61—84no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260D244—28no—
261D244—24no—
262D244—24no—
263D244—22no—
264D244—42no—
265D244—44no—
266D244#9914yescomplemented
267C4×D61—44no—
268C4×D61—44no—
269C4×D61—82no—
270C4×D61—82no—
271C4×D61—816no—
272Dic122—42no—
273Dic122—42no—
274#22476—611nomaximal, Sylow
275C22×C61—22no—
276C22×C61—21no—
277C22×C61—22no—
278C22×C61—21no—
279C22×C61—42no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61C4×#4111yescharacteristic
289C22×C61#52011yescharacteristic
290C22×C61#54612yes—
291C22×C61#54712yes—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—24no—
296C2×D61—22no—
297C2×D61—24no—
298C2×D61—22no—
299C2×D61—24no—
300C2×D61—48no—
301C2×D61—48no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—84no—
305C2×D61—84no—
306C2×D61—82no—
307C2×D61—82no—
308C2×D61C2×#9914yes—
309C4×C61—21no—
310C4×C61—22no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—84no—
315C4×C61C2×#9912yes—
316Dic61—44no—
317Dic61—42no—
318Dic61—42no—
319Dic61—82no—
320C2×#546—611no—
321D4×C22⋊C4—612no—
322#1996—612no—
323#1898—614no—
324#1819—612no—
325#1799D6111yescharacteristic, complemented
326C4×#99—611no—
327C2×#342—612no—
328C2×C61—22no—
329C2×C61—22no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—81no—
333C2×C61#233911yescharacteristic
334C2×C61C22×#9911yescharacteristic
335C2×C61#288311yescharacteristic
336D61—24no—
337D61—42no—
338D61—44no—
339C22×#41—611no—
340C23×D4—611no—
341#546—1222no—
342#546—1224no—
343#546—1222no—
344C2×#124—611no—
345C2×#123—612no—
346C2×C22⋊Q8—611no—
347C2×#121—611no—
348C2×#121—612no—
349C2×#120—612no—
350C2×C4×D4—611no—
351C2×C4×D4—1222no—
352C2×C4×D4C2×D6111yescharacteristic
353C22×C22⋊C4—612no—
354C22×C22⋊C4—614no—
355C2×#99—612no—
356C2×#99C2×D6112yes—
357#390—1222no—
358#386—1224no—
359#380C2×D6112yes—
360#379—614no—
361C4×C22⋊C4—612no—
362#342—611no—
363#342—614no—
364#342—614no—
365#342—611no—
366#342C2×D6112yes—
367C61#2247611yescharacteristic, complemented, Sylow
368C25—612no—
369C2×#41—1221no—
370C2×#41—1222no—
371C2×#41—1221no—
372C2×#41—2444no—
373C22×Q8—611no—
374C22×D4—612no—
375C22×D4—612no—
376C22×D4—611no—
377C22×D4—612no—
378C22×D4—611no—
379C22×D4—612no—
380C22×D4—614no—
381C22×D4—612no—
382C22×D4—2444no—
383C22×D4C22×D6111yescharacteristic
384C23×C4—612no—
385C23×C4—612no—
386C23×C4—1224no—
387C23×C4C4×D6111yescharacteristic, complemented
388C23×C4C22×D6111yescharacteristic
389#124—1222no—
390#124—1222no—
391#123—1224no—
392#123—1224no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395#121—1224no—
396#121—1224no—
397#121—1222no—
398#121—1222no—
399#120—1224no—
400#120—1224no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—1221no—
404C4×D4—1222no—
405C4×D4—1221no—
406C4×D4—2448no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4C4×D6111yescharacteristic, complemented
411C2×C22⋊C4—612no—
412C2×C22⋊C4—612no—
413C2×C22⋊C4—612no—
414C2×C22⋊C4—612no—
415C2×C22⋊C4—612no—
416C2×C22⋊C4—612no—
417C2×C22⋊C4—612no—
418C2×C22⋊C4—1228no—
419C2×C22⋊C4—1224no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1224no—
422C2×C22⋊C4—1224no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—2444no—
425C2×C22⋊C4C4×D6111yescharacteristic, complemented
426C2×C22⋊C4C22×D6114yes—
427C2×C22⋊C4C22×D6111yescharacteristic
428C2×C42—1222no—
429C2×C42C4×D6111yescharacteristic, complemented
430#99—42no—
431#99—1224no—
432#95—22no—
433#95—1222no—
434C24—612no—
435C24—614no—
436C24—612no—
437C24—614no—
438C24—1224no—
439C24—1222no—
440C24—1222no—
441C24—4882no—
442C24—2444no—
443C24C2×C4×D6111yescharacteristic
444C24C23×D6111yescharacteristic
445#41—2444no—
446#41—2448no—
447#41—2444no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—2442no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—611no—
456C2×D4—612no—
457C2×D4—611no—
458C2×D4—614no—
459C2×D4—612no—
460C2×D4—614no—
461C2×D4—612no—
462C2×D4—1228no—
463C2×D4—1224no—
464C2×D4—1224no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—4882no—
468C2×D4—2444no—
469C2×D4—2442no—
470C2×D4—2444no—
471C2×D4—2448no—
472C2×D4—2444no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—4884no—
487C22×C4—4882no—
488C22×C4—2448no—
489C22×C4—2448no—
490C22×C4—2444no—
491C22×C4C2×C4×D6111yescharacteristic
492C22×C4C2×C4×D6111yescharacteristic
493C22×C4C2×C4×D6111yescharacteristic
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4C2×C4×D6111yescharacteristic
496C22×C4D4×D6112yes—
497C22×C4D4×D6112yes—
498C4⋊C4—22no—
499C4⋊C4—44no—
500C4⋊C4—1224no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—614no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1224no—
513C22⋊C4—1224no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—4884no—
518C22⋊C4—2448no—
519C22⋊C4—2448no—
520C42—21no—
521C42—21no—
522C42—2444no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—612no—
531C23—614no—
532C23—612no—
533C23—1228no—
534C23—1224no—
535C23—1222no—
536C23—1222no—
537C23—1224no—
538C23—1224no—
539C23—1222no—
540C23—1222no—
541C23—1224no—
542C23—1224no—
543C23—4888no—
544C23—4884no—
545C23—4884no—
546C23—2444no—
547C23—2442no—
548C23—2444no—
549C23—2442no—
550C23—2444no—
551C23—2442no—
552C23C22×C4×D6111yescharacteristic
553C23C2×D4×D6112yes—
554C23#41×D6112yes—
555C23#1451982211yescharacteristic
556C23#1451982511yescharacteristic
557Q8—2442no—
558Q8—2442no—
559D4—42no—
560D4—42no—
561D4—614no—
562D4—1228no—
563D4—1224no—
564D4—1224no—
565D4—1222no—
566D4—4884no—
567D4—2442no—
568D4—2444no—
569D4—2444no—
570D4—2444no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—4882no—
591C2×C4—4882no—
592C2×C4—48816no—
593C2×C4—2444no—
594C2×C4—2444no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C22⋊C4×D6112yes—
604C2×C4C22⋊C4×D6111yescharacteristic
605C22—22no—
606C22—21no—
607C22—22no—
608C22—21no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—81no—
616C22—81no—
617C22—614no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—1224no—
622C22—1222no—
623C22—1224no—
624C22—1224no—
625C22—1222no—
626C22—4884no—
627C22—4884no—
628C22—4882no—
629C22—4882no—
630C22—2448no—
631C22—2448no—
632C22—2444no—
633C22—2444no—
634C22C2×C22⋊C4×D6111yescharacteristic
635C22#1684023011yescharacteristic
636C22#1684024311yescharacteristic
637C22#1684040912yes—
638C22#1684041412yes—
639C4—21no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—84no—
645C4—4882no—
646C4—2444no—
647C4—2442no—
648C4—2442no—
649C4#99×D6112yes—
650C2—22no—
651C2—22no—
652C2—21no—
653C2—21no—
654C2—81no—
655C2—1224no—
656C2—2442no—
657C2—2444no—
658C2#2144864711yescharacteristic
659C2C2×#99×D6111yescharacteristic
660C2#2145427811yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1799C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840408C2
cosocleC24×C61C24
hypercenter#1799D61
hypocenterC61#22476
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21448639C2semidirect
3. #21446803C2semidirect
4. #21446269C2semidirect
5. #21445892C2semidirect
6. C4×#14519692C2semidirect
7. #342×D61C2semidirect
8. C61×#1799C2semidirect
9. C22×#11758222C4semidirect
14. C2×#14519765C4semidirect
15. C2×#14519762C4semidirect
17. C2×#14519751C4semidirect
18. C2×#14519750C4semidirect
20. C2×C4×D244C4semidirect
42. C22×Dic122C2×C4semidirect
50. C22×C22⋊D61C2×C4semidirect
51. C22×C22⋊D61C2×C4semidirect
53. C22×C4×D61C2×C4semidirect
54. C22×C4×D61C2×C4semidirect
71. C2×C4⋊Dic61C2×C4semidirect
72. C2×#11758196C2×C4semidirect
79. C2×#11758198C2×C4semidirect
80. C2×#11758198C2×C4semidirect
81. C2×#11758198C2×C4semidirect
82. C2×#11758198C2×C4semidirect
91. C23×C4×C61C2×C4semidirect
96. C2×C4⋊C4×C61C2×C4semidirect
97. C2×C22⋊C4×C61C2×C4semidirect
100. C2×C42×C61C2×C4semidirect
179. C22⋊Dic61#41semidirect
266. D244#99semidirect
325. #1799D61semidirect
367. C61#22476semidirect
387. C23×C4C4×D61semidirect
410. C2×C4⋊C4C4×D61semidirect
425. C2×C22⋊C4C4×D61semidirect
429. C2×C42C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404082
#214486392
#214468032
#214462692
#214458922
C4×#145196922
#342×D612
C61×#17992
#2247661