Finite Group Databasebeta

← #29828845

Subgroups — #29828845

Order 15616.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758224C211yesmaximal, characteristic
2C2×#16839969C211yesmaximal, characteristic, complemented
3C2×#16839921C212yesmaximal
4#21454166C211yesmaximal, characteristic, complemented
5#21446157C212yesmaximal, complemented
6#21446155C212yesmaximal, complemented
7#21446128C212yesmaximal, complemented
8#21445825C211yesmaximal, characteristic, complemented
9#21445819C211yesmaximal, characteristic, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C2×C61×#461C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×#11758202C2212yes—
14C2×#14519753C2212yes—
15C4×#11758224—24no—
16C4×#11758224—24no—
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×C4×C22⋊D61—21no—
21C2×#14519737—21no—
22C2×#14519735C2211yescharacteristic
23C2×#14519733—21no—
24C22×C4×Dic61—21no—
25C2×C42×D61C2211yescharacteristic
26#16839969—24no—
27#16839921—28no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519701C2212yes—
31C2×#14519698C2212yes—
32C2×#14519689C2212yes—
33#16839669C2212yes—
34#16839658—21no—
35#16839658—21no—
36#16839632C2211yescharacteristic
37#16839619—21no—
38#16839619—21no—
39#16839525C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C22×SD16×C61C2212yes—
43C2×C4⋊Q8×C61C2211yescharacteristic
44C2×C4×D4×C61C2211yescharacteristic
45C61×#461—24no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×#103×C61C2212yes—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—22no—
51C2×#11758224—22no—
52C2×#11758224—22no—
53C2×#11758224D414yes—
54C22×Dic122C2311yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C23×Dic61—21no—
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C2×#11758202—24no—
62C2×#11758202—24no—
63C2×#11758202—24no—
64C22×#92418C2312yes—
65#14519753—24no—
66#14519753—24no—
67C4×Dic122—22no—
68C4×Dic122D414yes—
69#14519742—22no—
70#14519742—22no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73C4×C22⋊D61—28no—
74#14519737—28no—
75#14519735—22no—
76#14519735—22no—
77#14519733—28no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2312yes—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—22no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—22no—
88C2×C4×Dic61—28no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61—22no—
96C42×D61—22no—
97C4⋊#92418—22no—
98#14519702—24no—
99#14519701—24no—
100#14519698—24no—
101#14519689—24no—
102#14519684—22no—
103#14519684—22no—
104C22×Q8×C61C2312yes—
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61—21no—
107C2×SD16×C61—24no—
108C2×SD16×C61—24no—
109C2×SD16×C61—24no—
110C22×C8×C61C2312yes—
111C4⋊Q8×C61—22no—
112C4⋊Q8×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61D414yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C42×C61C2311yescharacteristic
120C4⋊C8×C61—22no—
121#103×C61—24no—
122C24×C61—21no—
123C23×D61C2×D411yescharacteristic
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×D412yes—
134C2×D4×C61C2×D412yes—
135C2×D4×C61#4112yes—
136C22×C4×C61—21no—
137C22×C4×C61—24no—
138C22×C4×C61—22no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—24no—
145C2×C22⋊D61—24no—
146C2×C22⋊D61—24no—
147C22×Dic61—24no—
148C22×Dic61—24no—
149C22×Dic61—21no—
150C22×Dic61—24no—
151C22×Dic61—24no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159#11758224—28no—
160#11758224—28no—
161#11758224—28no—
162#11758224—28no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—41no—
173C2×C4×D61C2×D412yes—
174C2×C4×D61#4112yes—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C2×D412yes—
179C2×Dic122C2×D412yes—
180C2×Dic122#4112yes—
181SD16×C61—48no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61—42no—
188C4⋊C4×C61C2×D412yes—
189C22⋊C4×C61—24no—
190C42×C61C2×D412yes—
191C42×C61C2×D412yes—
192C22⋊Dic61—24no—
193#11758202—48no—
194#11758198—24no—
195#11758198—44no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61C2×D412yes—
202#11758196—24no—
203#11758196—24no—
204#11758196—44no—
205C4×Dic61—24no—
206C4×Dic61—24no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—24no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—22no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61#12112yes—
230Q8×C61—42no—
231Q8×C61—42no—
232Q8×C61—42no—
233Q8×C61—42no—
234D4×C61—22no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61#12114yes—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—48no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61#12012yes—
248C2×C4×C61#12012yes—
249C2×C4×C61#13512yes—
250C2×C4×C61#13712yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C22⋊D61—216no—
258C2×Dic61—216no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—48no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61#12112yes—
277C2×Dic61#13712yes—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—44no—
283C4×D61—44no—
284Dic122—22no—
285Dic122—22no—
286Dic122—22no—
287Dic122#12114yes—
288C8×C61—42no—
289#92418—42no—
290#32907—611nomaximal, Sylow
291C22×C61—24no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61C2×#12011yescharacteristic
299C22×C61C2×#12111yescharacteristic
300C22×C61#52511yescharacteristic
301C22×C61C2×#13511yescharacteristic
302C22×C61C2×#13711yescharacteristic
303C22×C61C2×#13711yescharacteristic
304C22×C61#57511yescharacteristic
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C4×C61—22no—
313C4×C61—42no—
314C4×C61—82no—
315C4×C61#45214yescomplemented
316C4×C61C2×#12111yescharacteristic
317C4×C61C2×#12111yescharacteristic
318C4×C61#52511yescharacteristic
319C4×C61#52511yescharacteristic
320Dic61—24no—
321Dic61—22no—
322Dic61—22no—
323Dic61—41no—
324Dic61—42no—
325Dic61—82no—
326C2×C4×#41—611no—
327C2×#462—611no—
328C2×#461D6111yescharacteristic, complemented
329C2×#452—612no—
330#2857—611no—
331#1878—612no—
332#1877—612no—
333#1873—612no—
334#1803—611no—
335#1802—611no—
336#1797—611no—
337C2×C61—44no—
338C2×C61#284711yescharacteristic
339C2×C61C2×#45211yescharacteristic
340C2×C61#300711yescharacteristic
341C2×C61#303511yescharacteristic
342C2×C61#303911yescharacteristic
343C2×C61#310711yescharacteristic
344C2×C61#311611yescharacteristic
345D61—42no—
346C22×#41—611no—
347C22×Q16—612no—
348C22×SD16C2×D6112yes—
349C2×C4⋊Q8C2×D6111yescharacteristic
350C2×C22⋊Q8—612no—
351C4×#41—1224no—
352C4×#41—1224no—
353C2×C4×Q8—611no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4×D4—1221no—
357C2×C4×D4—1221no—
358C2×C4×D4C2×D6111yescharacteristic
359C2×#117—611no—
360C2×#117—1221no—
361C22×C42—611no—
362C22×C42—1221no—
363#462—1224no—
364#461—24no—
365#452—1228no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8C2×D6111yescharacteristic
368C2×#103—612no—
369C2×#103—612no—
370C2×#103C2×D6112yes—
371C2×#102—612no—
372C2×C22⋊C8—612no—
373#391—612no—
374#389—1221no—
375#389—1221no—
376#384—611no—
377#380—1221no—
378#380—1221no—
379#340—612no—
380#340—612no—
381#336—612no—
382C61#3290711yescharacteristic, complemented, Sylow
383C2×#41—614no—
384C2×#41—1222no—
385C2×#41—1222no—
386C2×#41—1222no—
387C2×#41—1222no—
388C2×#41—1222no—
389C22×Q8—611no—
390C22×Q8C22×D6112yes—
391C22×D4—1221no—
392C22×D4C22×D6111yescharacteristic
393C23×C4—21no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—1221no—
398C23×C4—1221no—
399C2×Q16—1224no—
400C2×Q16—1224no—
401C2×Q16—1224no—
402C2×SD16—24no—
403C2×SD16—24no—
404C2×SD16—24no—
405C22×C8—612no—
406C22×C8C22×D6112yes—
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409C22⋊Q8—1224no—
410C22⋊Q8—1224no—
411C4×Q8—614no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×D4—22no—
416C4×D4—1228no—
417C4×D4—1222no—
418C4×D4—1228no—
419C4×D4—1222no—
420C4×D4C22⋊D6114yes—
421#117—1228no—
422#117—1222no—
423#117—1222no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1222no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1228no—
444C2×C42—1221no—
445C2×C42—1222no—
446C2×C42—1222no—
447C2×C42C22×D6111yescharacteristic
448C4⋊C8—22no—
449C4⋊C8—1222no—
450#103—24no—
451#103—1224no—
452#103—1224no—
453#102—1224no—
454C22⋊C8—1224no—
455#95—1222no—
456#95—1222no—
457C24—21no—
458C24—611no—
459#41—1228no—
460#41—1228no—
461#41—1228no—
462#41—1228no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—612no—
470C2×Q8—612no—
471C2×Q8—612no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—1224no—
479C2×D4—1224no—
480C2×D4—1224no—
481C2×D4D4×D6112yes—
482C2×D4#1175822412yes—
483C2×D4C2×C22⋊D6112yes—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—612no—
492C22×C4—612no—
493C22×C4—611no—
494C22×C4—1222no—
495C22×C4—1224no—
496C22×C4—1224no—
497C22×C4—1224no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1224no—
504C22×C4—1221no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—2441no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4D4×D6111yescharacteristic
514C22×C4D4×D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516Q16—2448no—
517SD16—48no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—1222no—
522C2×C8—1222no—
523C2×C8—1222no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—612no—
528C4⋊C4—1224no—
529C4⋊C4—1221no—
530C4⋊C4—1224no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—2442no—
534C4⋊C4—2444no—
535C4⋊C4—2442no—
536C4⋊C4C2×C22⋊D6112yes—
537C22⋊C4—24no—
538C22⋊C4—1224no—
539C22⋊C4—1224no—
540C22⋊C4—2444no—
541C42—1224no—
542C42—1224no—
543C42—1221no—
544C42—1221no—
545C42—1222no—
546C42—1222no—
547C42—1224no—
548C42D4×D6112yes—
549C42C2×C22⋊D6112yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—612no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23C2×D4×D6111yescharacteristic
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—614no—
570Q8—1222no—
571Q8—1222no—
572Q8—1222no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4—12216no—
577D4#1451974814yes—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—12216no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—2444no—
606C2×C4—2448no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2444no—
613C2×C4#1451974512yes—
614C2×C4#1451974512yes—
615C2×C4#1451979812yes—
616C2×C4#1451980612yes—
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C22×C22⋊D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4C2×#1175822411yescharacteristic
623C8—42no—
624C8—2442no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—2441no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22C2×#1451974511yescharacteristic
640C22#121×D6111yescharacteristic
641C22#1684027211yescharacteristic
642C22C2×#1451979811yescharacteristic
643C22#137×D6111yescharacteristic
644C22C2×#1451980611yescharacteristic
645C22#1684056111yescharacteristic
646C4—22no—
647C4—42no—
648C4—82no—
649C4—1222no—
650C4—1224no—
651C4—1222no—
652C4—4882no—
653C4—2441no—
654C4—2442no—
655C4#1683992114yescomplemented
656C4#121×D6111yescharacteristic
657C4C2×#1451974811yescharacteristic
658C4#1684026911yescharacteristic
659C4#1684027211yescharacteristic
660C2—44no—
661C2—2442no—
662C2#2145409811yescharacteristic
663C2C2×#1683992111yescharacteristic
664C2#2145507511yescharacteristic
665C2#2145527511yescharacteristic
666C2#2145530111yescharacteristic
667C2#2145588711yescharacteristic
668C2#2145597811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenterC2×#461D61
hypocenterC61#32907
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. C2×#16839969C2semidirect
4. #21454166C2semidirect
5. #21446157C2semidirect
6. #21446155C2semidirect
7. #21446128C2semidirect
8. #21445825C2semidirect
9. #21445819C2semidirect
10. #21445796C2semidirect
11. C2×C61×#461C2semidirect
315. C4×C61#452semidirect
328. C2×#461D61semidirect
382. C61#32907semidirect
655. C4#16839921semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582242
C2×#168399692
C2×#168399212
#214541662
#214461572
#214461552
#214461282
#214458252
#214458192
#214457962
C2×C61×#4612
#3290761