Finite Group Databasebeta

← #29712161

Subgroups — #29712161

Order 15616.

1690 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×#16840412C211yesmaximal, characteristic, complemented
2C22×#14519698C211yesmaximal, characteristic, complemented
3#21449169C211yesmaximal, characteristic, complemented
4#21446953C212yesmaximal, complemented
5#21446814C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446169C212yesmaximal, complemented
8#21446133C212yesmaximal, complemented
9#21446109C211yesmaximal, characteristic, complemented
10#21445906C211yesmaximal, characteristic, complemented
11C2×C61×#447C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C23×#92418C2211yescharacteristic
14#16840412—24no—
15#16840412—24no—
16C2×#14519776C2211yescharacteristic, complemented
17C2×#14519759—21no—
18C2×#14519758—21no—
19C2×#14519754—21no—
20C2×#14519748C2212yes—
21C2×#14519748C2211yescharacteristic, complemented
22C2×#14519746—21no—
23C2×D4×Dic61C2211yescharacteristic, complemented
24C2×C4×C22⋊D61C2211yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic
26C2×C4⋊C4×D61C2211yescharacteristic, complemented
27C2×#14519698—24no—
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2212yescomplemented
31C2×#14519689C2212yescomplemented
32#16839676—21no—
33#16839670—21no—
34#16839665C2212yes—
35#16839622C2211yescharacteristic
36#16839618—21no—
37#16839525C2212yescomplemented
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839516C2212yescomplemented
41C23×D4×C61C2211yescharacteristic
42C22×D8×C61C2212yes—
43C2×#121×C61C2211yescharacteristic
44C61×#447—24no—
45C2×#102×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C2×#11758224—28no—
48C2×#11758224—44no—
49C2×#11758224—44no—
50C22×Dic122—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61—22no—
53C22×C22⋊D61D411yescharacteristic
54C23×Dic61—21no—
55C23×Dic61—21no—
56C22×C4×D61—21no—
57C22×C4×D61C2311yescharacteristic
58C22×#92418—22no—
59C22×#92418—42no—
60C22×#92418C2312yes—
61#14519776—22no—
62#14519776—22no—
63#14519759—28no—
64#14519758—28no—
65#14519754—28no—
66#14519748—22no—
67#14519748—24no—
68#14519748—22no—
69#14519748—42no—
70#14519746—44no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73C4×C22⋊D61—22no—
74C4×C22⋊D61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61D414yescomplemented
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196D411yescharacteristic
84C4⋊C4×D61—22no—
85C4⋊C4×D61—22no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—22no—
90C2×C22⋊Dic61—22no—
91C2×C22⋊Dic61—22no—
92C2×C22⋊Dic61D411yescharacteristic
93C2×#11758198D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519698—24no—
96#14519698—24no—
97#14519698—42no—
98#14519689—42no—
99#14519684—21no—
100C25×C61—21no—
101C22×D4×C61—24no—
102C22×D4×C61—21no—
103C22×D4×C61—22no—
104C22×D4×C61—22no—
105C22×D4×C61—21no—
106C22×D4×C61—42no—
107C22×D4×C61C2311yescharacteristic
108C22×D4×C61C2312yes—
109C23×C4×C61C2311yescharacteristic
110C2×D8×C61—24no—
111C2×D8×C61—24no—
112C2×D8×C61—42no—
113C22×C8×C61C2312yes—
114#121×C61—22no—
115#121×C61D414yescomplemented
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118#102×C61—24no—
119C22⋊C8×C61—22no—
120C24×C61—21no—
121C24×C61—22no—
122C24×C61—44no—
123C24×C61—42no—
124C24×C61—42no—
125C24×C61—42no—
126C24×C61—82no—
127C24×C61C2×D411yescharacteristic
128C23×D61C2×D411yescharacteristic
129C2×D4×C61—28no—
130C2×D4×C61—24no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—48no—
140C2×D4×C61—44no—
141C2×D4×C61—44no—
142C2×D4×C61—42no—
143C2×D4×C61—42no—
144C2×D4×C61—42no—
145C2×D4×C61C2×D412yes—
146C2×D4×C61#4114yescomplemented
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—24no—
157C2×C22⋊D61—42no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61—44no—
160C2×C22⋊D61—44no—
161C2×C22⋊D61—82no—
162C2×C22⋊D61D814yescomplemented
163C22×Dic61—24no—
164C22×Dic61—24no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—22no—
168C22×Dic61—42no—
169C22×Dic61—42no—
170C22×Dic61—42no—
171C22×Dic61—42no—
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175#11758224—432no—
176C2×C4×D61—24no—
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—42no—
182C2×C4×D61—42no—
183C2×Dic122—24no—
184C2×Dic122—42no—
185C2×Dic122—42no—
186D8×C61—48no—
187C2×C8×C61—22no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C2×D412yes—
192C22⋊C4×C61—24no—
193C22⋊Dic61—24no—
194C22⋊Dic61—24no—
195C22⋊Dic61—22no—
196C22⋊Dic61—44no—
197C22⋊Dic61—42no—
198C22⋊Dic61—44no—
199C22⋊Dic61—82no—
200#11758198—22no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61C2×D412yes—
207C4⋊Dic61C2×D412yes—
208#11758196—24no—
209#11758196—24no—
210#11758196—22no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C2×#92418—22no—
214C2×#92418—22no—
215C2×#92418—22no—
216C2×#92418—44no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—24no—
221C23×C61—48no—
222C23×C61—42no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61—42no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61—42no—
231C23×C61—42no—
232C23×C61—88no—
233C23×C61—88no—
234C23×C61—84no—
235C23×C61—84no—
236C23×C61#12012yes—
237C23×C61C2×D812yes—
238C23×C61AGL(1,ℤ/8)12yes—
239C23×C61C22×D411yescharacteristic
240C22×D61—21no—
241C22×D61—22no—
242C22×D61—21no—
243C22×D61—22no—
244C22×D61C2×D812yes—
245D4×C61—24no—
246D4×C61—24no—
247D4×C61—24no—
248D4×C61—48no—
249D4×C61—48no—
250D4×C61—41no—
251D4×C61—41no—
252D4×C61—42no—
253D4×C61—42no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—24no—
258C2×C4×C61—22no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61#12012yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4112yes—
265C2×C4×C61H(5,𝔽2)12yes—
266C22⋊D61—28no—
267C22⋊D61—48no—
268C22⋊D61—88no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—48no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—48no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61C2×D812yes—
287C4×D61—48no—
288C4×D61—41no—
289C4×D61—41no—
290C4×D61—42no—
291Dic122—48no—
292C8×C61—42no—
293#92418—42no—
294#23960—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—24no—
298C22×C61—24no—
299C22×C61—44no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—44no—
304C22×C61—44no—
305C22×C61—816no—
306C22×C61—82no—
307C22×C61—82no—
308C22×C61—82no—
309C22×C61—82no—
310C22×C61—82no—
311C22×C61—82no—
312C22×C61#44714yes—
313C22×C61C2×#12011yescharacteristic
314C22×C61#54612yes—
315C22×C61C22×D811yescharacteristic
316C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
317C22×C61#57712yes—
318C2×D61—22no—
319C2×D61—24no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—42no—
323C4×C61—22no—
324C4×C61—42no—
325C4×C61#52511yescharacteristic
326C4×C61#53511yescharacteristic
327C4×C61#54612yes—
328Dic61—22no—
329Dic61—41no—
330Dic61—42no—
331Dic61—42no—
332Dic61—41no—
333Dic61—82no—
334C2×#547—611no—
335C2×#447D6111yescharacteristic, complemented
336C22×#102—611no—
337#2408—611no—
338#2023—612no—
339#1999—612no—
340#1893—611no—
341#1879—612no—
342#1874—612no—
343#1869—611no—
344#1823—611no—
345C2×C61—24no—
346C2×C61—82no—
347C2×C61—84no—
348C2×C61#240411yescharacteristic
349C2×C61C2×#44711yescharacteristic
350C2×C61#301011yescharacteristic
351C2×C61#329412yes—
352C2×C61#330612yes—
353D61—42no—
354C22×#41—1221no—
355C23×D4C2×D6111yescharacteristic
356C22×D8C2×D6112yes—
357C23×C8—611no—
358#547—1224no—
359#547—1224no—
360C2×C4⋊Q8—611no—
361C2×#123—1221no—
362C2×#123—1221no—
363C2×C22⋊Q8—1221no—
364C2×#121—611no—
365C2×#121—612no—
366C2×#121C2×D6111yescharacteristic
367C2×#120—1221no—
368C2×C4×D4—611no—
369C2×C4×D4—611no—
370C22×C4⋊C4—611no—
371C22×C4⋊C4—611no—
372#447—24no—
373C2×#102—612no—
374C2×#102—612no—
375C2×#102—612no—
376C2×#102—1224no—
377C2×#102C2×D6112yes—
378C2×C22⋊C8—612no—
379C2×C22⋊C8C2×D6111yescharacteristic
380#394—1221no—
381#392—1221no—
382#390—612no—
383#380—611no—
384#379—1221no—
385#340—612no—
386#340—612no—
387#340—612no—
388#336—612no—
389C61#2396011yescharacteristic, complemented, Sylow
390C25—21no—
391C2×#41—1228no—
392C2×#41—2444no—
393C2×#41—2444no—
394C22×Q8—1221no—
395C22×D4—24no—
396C22×D4—21no—
397C22×D4—22no—
398C22×D4—21no—
399C22×D4—22no—
400C22×D4—42no—
401C22×D4—611no—
402C22×D4—1221no—
403C22×D4—1222no—
404C22×D4C22×D6112yes—
405C22×D4C22×D6111yescharacteristic
406C23×C4—611no—
407C23×C4—1221no—
408C23×C4—1221no—
409C23×C4—1221no—
410C23×C4C22×D6111yescharacteristic
411C2×D8—24no—
412C2×D8—24no—
413C2×D8—42no—
414C22×C8—612no—
415C22×C8—1222no—
416C22×C8—2442no—
417C22×C8C22×D6112yes—
418C4⋊Q8—1222no—
419C4⋊Q8—1222no—
420#123—1228no—
421#123—1228no—
422C22⋊Q8—1228no—
423#121—22no—
424#121—1222no—
425#121—1222no—
426#121—1224no—
427#121—2442no—
428#121C22⋊D6114yescomplemented
429#120—2444no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1222no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—611no—
448C2×C22⋊C4—611no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1222no—
452C2×C22⋊C4—1222no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1221no—
455C2×C42—611no—
456#102—24no—
457#102—1224no—
458#102—1224no—
459#102—2442no—
460C22⋊C8—22no—
461C22⋊C8—2442no—
462#95—1221no—
463C24—21no—
464C24—22no—
465C24—44no—
466C24—42no—
467C24—42no—
468C24—42no—
469C24—82no—
470C24—611no—
471C24D4×D6111yescharacteristic
472#41—24432no—
473C2×Q8—1224no—
474C2×Q8—2442no—
475C2×Q8—2442no—
476C2×D4—28no—
477C2×D4—24no—
478C2×D4—22no—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—48no—
487C2×D4—44no—
488C2×D4—44no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—614no—
493C2×D4—1222no—
494C2×D4—1222no—
495C2×D4—1224no—
496C2×D4—4882no—
497C2×D4—2444no—
498C2×D4—2444no—
499C2×D4—2442no—
500C2×D4—2442no—
501C2×D4#1175822414yescomplemented
502C2×D4C2×C22⋊D6112yes—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—1224no—
510C22×C4—1224no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1224no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4D4×D6112yes—
526C22×C4D4×D6111yescharacteristic
527C22×C4C2×C22⋊D6112yes—
528C22×C4C23×D6111yescharacteristic
529D8—48no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—1222no—
534C2×C8—1222no—
535C2×C8—1222no—
536C2×C8—2444no—
537C4⋊C4—21no—
538C4⋊C4—612no—
539C4⋊C4—612no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1224no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4C2×C22⋊D6112yes—
549C22⋊C4—24no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—4882no—
555C22⋊C4—2444no—
556C22⋊C4—2444no—
557C22⋊C4—2442no—
558C42—1221no—
559C42—1221no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—48no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—88no—
576C23—88no—
577C23—84no—
578C23—84no—
579C23—612no—
580C23—1221no—
581C23—1221no—
582C23—1222no—
583C23—1222no—
584C23#1451974512yes—
585C23D8×D6112yes—
586C23#1451980112yes—
587C23C2×D4×D6111yescharacteristic
588Q8—2448no—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—48no—
593D4—48no—
594D4—41no—
595D4—41no—
596D4—42no—
597D4—42no—
598D4—1228no—
599D4—4888no—
600D4—2448no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—612no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—2448no—
618C2×C4—2448no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2448no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2442no—
630C2×C4#1451974512yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C22×C22⋊D6111yescharacteristic
633C2×C4C2×#1175822412yes—
634C2×C4#1451982212yes—
635C8—42no—
636C8—2442no—
637C22—22no—
638C22—24no—
639C22—22no—
640C22—24no—
641C22—42no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—42no—
646C22—44no—
647C22—816no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—1222no—
655C22—1224no—
656C22—2441no—
657C22—2442no—
658C22—2441no—
659C22#1683989414yes—
660C22C2×#1451974511yescharacteristic
661C22#1684039912yes—
662C22C2×D8×D6111yescharacteristic
663C22C2×#1451980111yescharacteristic
664C22#1684057212yes—
665C4—22no—
666C4—42no—
667C4—1222no—
668C4—4882no—
669C4—2441no—
670C4—2442no—
671C4—2442no—
672C4—2441no—
673C4#1684026811yescharacteristic
674C4#1684032711yescharacteristic
675C4#1684040612yes—
676C2—24no—
677C2—84no—
678C2—82no—
679C2—2442no—
680C2#2144913511yescharacteristic
681C2C2×#1683989411yescharacteristic
682C2#2145510411yescharacteristic
683C2#2145762512yes—
684C2#2145778412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#447C2
socleC23×C61C22×D4
hypercenterC2×#447D61
hypocenterC61#23960
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C2×#16840412C2semidirect
2. C22×#14519698C2semidirect
3. #21449169C2semidirect
4. #21446953C2semidirect
5. #21446814C2semidirect
6. #21446248C2semidirect
7. #21446169C2semidirect
8. #21446133C2semidirect
9. #21446109C2semidirect
10. #21445906C2semidirect
11. C2×C61×#447C2semidirect
16. C2×#14519776C22semidirect
21. C2×#14519748C22semidirect
23. C2×D4×Dic61C22semidirect
26. C2×C4⋊C4×D61C22semidirect
30. C2×#14519698C22semidirect
31. C2×#14519689C22semidirect
37. #16839525C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839516C22semidirect
77. C2×C4⋊Dic61D4semidirect
115. #121×C61D4semidirect
146. C2×D4×C61#41semidirect
162. C2×C22⋊D61D8semidirect
335. C2×#447D61semidirect
389. C61#23960semidirect
428. #121C22⋊D61semidirect
501. C2×D4#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168404122
C22×#145196982
#214491692
#214469532
#214468142
#214462482
#214461692
#214461332
#214461092
#214459062
C2×C61×#4472
#2396061