Finite Group Databasebeta

← #29689099

Subgroups — #29689099

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840269C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic, complemented
3#380×D61C211yesmaximal, characteristic, complemented
4#21446254C212yesmaximal, complemented
5#21446249C212yesmaximal, complemented
6#21446133C212yesmaximal, complemented
7#21446123C212yesmaximal, complemented
8#21445915C211yesmaximal, characteristic, complemented
9#21445911C211yesmaximal, characteristic, complemented
10#21445759C211yesmaximal, characteristic, complemented
11C61×#1788C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758189C2212yes—
14#16840269—24no—
15#16840269C418yescomplemented
16C2×#14519758—21no—
17C2×#14519753C2211yescharacteristic
18C2×#14519753C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×#14519745—21no—
22C2×C4×C22⋊D61—21no—
23C2×C4×C22⋊D61—21no—
24C2×#14519736C2211yescharacteristic
25C2×C4⋊C4×D61C2212yes—
26C2×C22⋊C4×D61—21no—
27C2×#14519701C2212yescomplemented
28C2×#14519698C2212yescomplemented
29#16839724—24no—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519689C2212yes—
32#16839622C2211yescharacteristic
33#16839621—21no—
34#16839620—21no—
35#16839619C2212yes—
36#95×D61—21no—
37#16839525C2212yescomplemented
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×#41×C61C2211yescharacteristic
42C2×#103×C61C2212yescomplemented
43C2×#102×C61C2212yescomplemented
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#380C2211yescharacteristic
46C24×D61D411yescharacteristic
47C22×C22⋊D61—21no—
48C22×C22⋊D61—21no—
49C23×Dic61D411yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—42no—
52C22×C4×D61—42no—
53C22×C4×D61C2×C414yes—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D414yes—
57C22×C4×D61C2311yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C2×#11758189—24no—
60C2×#11758189—42no—
61C2×#11758189—42no—
62C22×#92418C2312yes—
63#14519758—28no—
64#14519753—22no—
65#14519753—22no—
66#14519753C2×C414yes—
67#14519753C2×C414yescomplemented
68#14519748—22no—
69#14519748—22no—
70#14519748C2×C414yes—
71#14519748C2×C414yescomplemented
72#14519745—28no—
73C4×C22⋊D61—28no—
74C4×C22⋊D61—28no—
75#14519736—22no—
76#14519736C2×C414yescomplemented
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—22no—
83C4⋊C4×D61—42no—
84C4⋊C4×D61—42no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—22no—
92C22⋊C4×D61—44no—
93C2×C4×Dic61—21no—
94#14519701—42no—
95#14519698—42no—
96C22⋊#92418—22no—
97#14519689—24no—
98#14519684—21no—
99#14519684—22no—
100C2×#41×C61—22no—
101C2×#41×C61—42no—
102C2×#41×C61—42no—
103C2×#41×C61C2×C414yescomplemented
104C22×Q8×C61C2311yescharacteristic
105C22×D4×C61—21no—
106C22×D4×C61C2311yescharacteristic
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C22×C8×C61C2312yes—
110C2×C4⋊C4×C61C2312yes—
111C2×C22⋊C4×C61—21no—
112#103×C61—42no—
113#102×C61—42no—
114C22⋊C8×C61—22no—
115#95×C61—21no—
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—42no—
121C23×D61—42no—
122C23×D61C22⋊C414yes—
123C23×D61C2×D411yescharacteristic
124C23×D61C2×D411yescharacteristic
125#41×C61—44no—
126#41×C61—44no—
127#41×C61—48no—
128C2×Q8×C61—21no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61C22×C412yes—
132C2×D4×C61—24no—
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—44no—
137C2×D4×C61C22×C412yes—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—21no—
141C22×C4×C61—24no—
142C22×C4×C61—22no—
143C22×C4×C61—44no—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—24no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156C22×Dic61—42no—
157C22×Dic61C22⋊C414yes—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—44no—
165C2×C4×D61—44no—
166C2×C4×D61—48no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61C22⋊C414yes—
173C2×C4×D61C22⋊C414yes—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C22×C412yes—
176C2×C4×D61C2×D412yes—
177C2×C4×D61C2×D412yes—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C22⋊C4×C61—42no—
184C22⋊Dic61—24no—
185C22⋊Dic61—24no—
186C22⋊Dic61—24no—
187C22⋊Dic61—42no—
188#11758198—24no—
189#11758198—24no—
190#11758198—44no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C412yes—
197#11758196—24no—
198#11758196—24no—
199#11758196—44no—
200C4×Dic61—24no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204#11758189—48no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—42no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—42no—
214C23×C61C2×C22⋊C412yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—22no—
222C22×D61—28no—
223C22×D61—44no—
224C22×D61—44no—
225C22×D61—48no—
226C22×D61—48no—
227C22×D61C2×C22⋊C412yes—
228C22×D61C2×C22⋊C412yes—
229C22×D61AGL(1,ℤ/8)12yes—
230C22×D61#13712yes—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—42no—
234D4×C61—22no—
235D4×C61—22no—
236D4×C61—48no—
237D4×C61—42no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—24no—
242C2×C4×C61—22no—
243C2×C4×C61—48no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61H(5,𝔽2)12yes—
252C2×C4×C61H(5,𝔽2)12yes—
253C22⋊D61—48no—
254C22⋊D61—48no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—44no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C22⋊C412yes—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—24no—
273C4×D61—24no—
274C4×D61—44no—
275C4×D61—44no—
276C4×D61—88no—
277C8×C61—42no—
278#92418—42no—
279#21769—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—82no—
289C22×C61—82no—
290C22×C61C22×C22⋊C411yescharacteristic
291C22×C61#53412yes—
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61C2×#13711yescharacteristic
294C22×C61#57512yes—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—24no—
298C2×D61—24no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—416no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61#41714yes—
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—82no—
309C4×C61#51811yescharacteristic
310C4×C61#51811yescharacteristic
311C4×C61#53412yes—
312Dic61—22no—
313Dic61—22no—
314Dic61—42no—
315Dic61—42no—
316Dic61—82no—
317C2×#525—611no—
318C2×#407—611no—
319C2×#380—611no—
320#1895—612no—
321#1894—612no—
322#1874—612no—
323#1872—612no—
324#1825—611no—
325#1825—611no—
326#1788—611no—
327#1788D6111yescharacteristic, complemented
328C2×C61—42no—
329C2×C61—82no—
330C2×C61#230811yescharacteristic
331C2×C61C2×#41711yescharacteristic
332C2×C61#289611yescharacteristic
333C2×C61#319412yes—
334C2×C61#319612yes—
335D61—24no—
336D61—42no—
337C22×#41C2×D6111yescharacteristic
338C24×C4—611no—
339C22×M16—612no—
340#525—618no—
341#525—1224no—
342C2×#123—1221no—
343C2×C22⋊Q8—611no—
344C2×C22⋊Q8—611no—
345C2×#121—611no—
346C2×#121—611no—
347C2×#120—1221no—
348C2×C4×D4—1221no—
349C2×C4×D4—1221no—
350C2×#117—611no—
351C22×C4⋊C4—612no—
352C22×C22⋊C4—1221no—
353C2×#103—612no—
354C2×#103C2×D6112yescomplemented
355C2×#102—612no—
356C2×#102C2×D6112yescomplemented
357#407—1224no—
358C2×C22⋊C8—611no—
359C2×C22⋊C8—612no—
360C2×C22⋊C8C2×D6111yescharacteristic
361#380—611no—
362#380—612no—
363#380—1221no—
364#380—1221no—
365#380C2×D6111yescharacteristic
366C2×#95—1221no—
367#340—612no—
368#340—612no—
369#340—612no—
370#340—612no—
371C61#2176911yescharacteristic, complemented, Sylow
372C25—611no—
373C2×#41—22no—
374C2×#41—42no—
375C2×#41—42no—
376C2×#41C4×D6114yescomplemented
377C22×Q8C22×D6111yescharacteristic
378C22×D4—21no—
379C22×D4—1221no—
380C22×D4—1221no—
381C22×D4C22×D6111yescharacteristic
382C23×C4—21no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—614no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—614no—
390C23×C4—1222no—
391C23×C4—2442no—
392C23×C4—2442no—
393C23×C4C22×D6111yescharacteristic
394C2×M16—1224no—
395C2×M16—2442no—
396C2×M16—2442no—
397C22×C8—612no—
398C22×C8C22×D6112yes—
399#123—1228no—
400C22⋊Q8—614no—
401C22⋊Q8—614no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404#121—614no—
405#121—614no—
406#121—1222no—
407#121—1222no—
408#120—1228no—
409C4×D4—1228no—
410C4×D4—1228no—
411#117—614no—
412#117—1222no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—612no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—2442no—
420C2×C4⋊C4—2442no—
421C2×C4⋊C4C22×D6112yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—2444no—
431C2×C42—1221no—
432#103—42no—
433#103—2442no—
434#102—42no—
435#102—2442no—
436C22⋊C8—22no—
437C22⋊C8—1224no—
438C22⋊C8—1222no—
439#95—21no—
440#95—1222no—
441#95—1221no—
442C24—21no—
443C24—611no—
444C24—611no—
445C24—614no—
446C24—1222no—
447C24—1222no—
448C24—2442no—
449C24—2442no—
450C24D4×D6111yescharacteristic
451#41—44no—
452#41—44no—
453#41—48no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8C2×C4×D6112yes—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—44no—
463C2×D4—1224no—
464C2×D4—1224no—
465C2×D4—2442no—
466C2×D4—2442no—
467C2×D4—2444no—
468C2×D4C2×C4×D6112yes—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—614no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—614no—
483C22×C4—614no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—2444no—
492C22×C4—2444no—
493C22×C4—2448no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2444no—
497C22×C4—2444no—
498C22×C4—2444no—
499C22×C4—2442no—
500C22×C4C2×C4×D6112yes—
501C22×C4D4×D6112yes—
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504M16—2448no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1222no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—1224no—
516C4⋊C4—1224no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—2444no—
522C22⋊C4—42no—
523C22⋊C4—1224no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—1224no—
528C22⋊C4—2444no—
529C22⋊C4—2442no—
530C42—1224no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—612no—
541C23—612no—
542C23—612no—
543C23—612no—
544C23—1228no—
545C23—1222no—
546C23—1222no—
547C23—1221no—
548C23—1221no—
549C23—1222no—
550C23—1222no—
551C23—2444no—
552C23—2444no—
553C23—2448no—
554C23—2448no—
555C23C22⋊C4×D6112yes—
556C23C2×D4×D6111yescharacteristic
557Q8—22no—
558Q8—22no—
559Q8—42no—
560D4—22no—
561D4—22no—
562D4—48no—
563D4—42no—
564D4—2448no—
565D4—2448no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—4888no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2442no—
598C2×C4C22⋊C4×D6112yes—
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#1451982212yes—
602C2×C4#1451982212yes—
603C8—42no—
604C8—2442no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22—614no—
616C22—1224no—
617C22—1222no—
618C22—1224no—
619C22—1222no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—24416no—
624C22—2442no—
625C22—2442no—
626C22C2×C22⋊C4×D6111yescharacteristic
627C22#1684032212yes—
628C22AGL(1,ℤ/8)×D6111yescharacteristic
629C22#137×D6111yescharacteristic
630C22#1684055912yes—
631C4—22no—
632C4—42no—
633C4—82no—
634C4—1222no—
635C4—1222no—
636C4—4882no—
637C4—2442no—
638C4—2442no—
639C4#1684023511yescharacteristic
640C4#1684023511yescharacteristic
641C4#1684032212yes—
642C2—42no—
643C2—82no—
644C2—1224no—
645C2—2442no—
646C2#2144839511yescharacteristic
647C2#417×D6111yescharacteristic
648C2#2145433411yescharacteristic
649C2#2145663112yes—
650C2#2145664712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1788C2
socleC23×C61C22×D4
prime radicalC2×#16840269C2
cosocleC22×C4×C61C24
hypercenter#1788D61
hypocenterC61#21769
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. C2×#16839724C2semidirect
3. #380×D61C2semidirect
4. #21446254C2semidirect
5. #21446249C2semidirect
6. #21446133C2semidirect
7. #21446123C2semidirect
8. #21445915C2semidirect
9. #21445911C2semidirect
10. #21445759C2semidirect
11. C61×#1788C2semidirect
15. #16840269C4semidirect
27. C2×#14519701C22semidirect
28. C2×#14519698C22semidirect
37. #16839525C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
42. C2×#103×C61C22semidirect
43. C2×#102×C61C22semidirect
67. #14519753C2×C4semidirect
71. #14519748C2×C4semidirect
76. #14519736C2×C4semidirect
103. C2×#41×C61C2×C4semidirect
327. #1788D61semidirect
354. C2×#103C2×D61semidirect
356. C2×#102C2×D61semidirect
371. C61#21769semidirect
376. C2×#41C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402692
C2×#168397242
#380×D612
#214462542
#214462492
#214461332
#214461232
#214459152
#214459112
#214457592
C61×#17882
#2176961