Finite Group Databasebeta

← #29686385

Subgroups — #29686385

Order 15616.

980 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×#16839981C211yesmaximal, characteristic
2C2×#16839859C211yesmaximal, characteristic
3C2×#16839854C211yesmaximal, characteristic, complemented
4#21448532C211yesmaximal, characteristic
5#21448183C211yesmaximal, characteristic, complemented
6#21446577C211yesmaximal, characteristic, complemented
7#21446463C211yesmaximal, characteristic, complemented
8#21446442C211yesmaximal, characteristic
9#21446171C211yesmaximal, characteristic
10#21446157C211yesmaximal, characteristic, complemented
11#21445825C211yesmaximal, characteristic, complemented
12#21445678C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#1983C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×D4×Dic61C2211yescharacteristic
23#16839981C418yescomplemented
24#16839859—24no—
25#16839854—24no—
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839669C2211yescharacteristic
36#16839665C2211yescharacteristic
37#16839655—21no—
38#16839655—21no—
39#16839637—21no—
40#16839637—21no—
41#16839632C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196—21no—
45C4×#11758196—21no—
46#16839613—21no—
47#16839608C2211yescharacteristic
48#16839607—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×#124×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C4×C4⋊C4×C61C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202C2×C414yescomplemented
67C2×#11758200—22no—
68C2×#11758200—22no—
69C2×#11758200C2×C414yes—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74C4×Dic122—22no—
75C4×Dic122C2×C414yescomplemented
76#14519742—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742D414yes—
80D4×Dic61—22no—
81D4×Dic61—22no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102#14519710—22no—
103#14519710—22no—
104C4⋊#92418C2×C414yescomplemented
105#14519702—22no—
106#14519702—22no—
107#14519701C2×C414yescomplemented
108#14519698C2×C414yes—
109#14519687—22no—
110C8×Dic61—22no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×Q8×C61C2311yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123#124×C61—22no—
124#124×C61C2×C414yes—
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—22no—
132#103×C61—22no—
133#103×C61—22no—
134#102×C61—22no—
135#102×C61—22no—
136#95×C61—21no—
137C24×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—81no—
162C22×Dic61—81no—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C22×C412yes—
171C2×Dic122C2×D412yes—
172C2×Dic122#4112yes—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D412yes—
182C4⋊C4×C61#4112yes—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C42×C61—81no—
186C42×C61C22×C412yes—
187C42×C61#4112yes—
188C22⋊Dic61—42no—
189C22⋊Dic61—81no—
190C22⋊Dic61—81no—
191#11758202—28no—
192#11758200—28no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4⋊Dic61—81no—
200C4⋊Dic61—81no—
201C4⋊Dic61C22×C412yes—
202#11758196—24no—
203#11758196—24no—
204#11758196—24no—
205C4×Dic61—24no—
206C4×Dic61—24no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—24no—
210C4×Dic61—42no—
211C4×Dic61—81no—
212C4×Dic61—81no—
213C4×Dic61C2×D412yes—
214C4×Dic61#4112yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418C22×C412yes—
220C2×#92418C22×C412yes—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C22×D411yescharacteristic
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—41no—
232Q8×C61—41no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—41no—
236D4×C61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61C4×D412yes—
251C2×C4×C61C4×D412yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—84no—
281C2×Dic61#12112yes—
282C2×Dic61#12112yes—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122C4×D414yes—
287C8×C61—41no—
288C8×C61—41no—
289#92418—22no—
290#92418—22no—
291#21512—611nomaximal, Sylow
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61C2×C4×D411yescharacteristic
300C22×C61C2×#12111yescharacteristic
301C22×C61#52511yescharacteristic
302C22×C61#57611yescharacteristic
303C22×C61#57711yescharacteristic
304C22×C61#57811yescharacteristic
305C22×C61#57811yescharacteristic
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61C2×C4×D411yescharacteristic
312C4×C61C4×#4111yescharacteristic
313C4×C61#52911yescharacteristic
314C4×C61#53211yescharacteristic
315Dic61—22no—
316Dic61—22no—
317Dic61—24no—
318Dic61—41no—
319Dic61—41no—
320Dic61—42no—
321Dic61—81no—
322Dic61—81no—
323C2×#464—611no—
324C2×#440—611no—
325C2×C4×Q16—611no—
326#2326—611no—
327#2276—611no—
328#1983D6111yescharacteristic
329#1947—611no—
330#1928—611no—
331#1925—611no—
332#1880—611no—
333#1878—611no—
334#1803—611no—
335#1768—611no—
336#1763—611no—
337C4×#102—611no—
338C2×C61—82no—
339C2×C61C4×#12111yescharacteristic
340C2×C61#294711yescharacteristic
341C2×C61#295111yescharacteristic
342C2×C61#304411yescharacteristic
343C2×C61#304511yescharacteristic
344C2×C61#312111yescharacteristic
345C2×C61#312411yescharacteristic
346C22×Q16—611no—
347C22×SD16—611no—
348C2×#124C2×D6111yescharacteristic
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8—611no—
353C2×C4×D4—611no—
354#464—618no—
355#440—1224no—
356C4×Q16—1224no—
357C2×#107—611no—
358C2×#107—611no—
359C2×C4⋊C8—611no—
360C2×C4⋊C8C2×D6111yescharacteristic
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103C2×D6111yescharacteristic
365C2×#103C2×D6111yescharacteristic
366C2×#102—611no—
367C2×#102C2×D6111yescharacteristic
368C2×#102C2×D6111yescharacteristic
369C2×#97—611no—
370C2×C4×C8—611no—
371#391—611no—
372#390—611no—
373#388—1221no—
374#388—1221no—
375#385—1221no—
376#385—1221no—
377#384—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—1221no—
381C4×C4⋊C4—1221no—
382C4×C4⋊C4—1221no—
383C4×C4⋊C4C2×D6111yescharacteristic
384#376—611no—
385#376—1221no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#336—611no—
391#336—611no—
392C61#2151211yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×Q8C22×D6111yescharacteristic
395C22×D4C22×D6111yescharacteristic
396C23×C4—1221no—
397C2×Q16—614no—
398C2×Q16—1222no—
399C2×Q16—1222no—
400C2×SD16—614no—
401C2×SD16—1222no—
402C2×SD16—1222no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407#124—22no—
408#124C4×D6114yes—
409C4×Q8—614no—
410C4×Q8—614no—
411C4×Q8—1222no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—21no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#107—1222no—
447C4⋊C8—22no—
448C4⋊C8—614no—
449#103—22no—
450#103—22no—
451#103—614no—
452#103—1222no—
453#103—1222no—
454#102—22no—
455#102—22no—
456#102—614no—
457#97—1222no—
458C4×C8—1222no—
459#95—21no—
460#95—611no—
461#95—611no—
462#95—611no—
463#95—611no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—612no—
475C2×Q8—612no—
476C2×Q8—612no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8C2×C4×D6112yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D6112yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—4881no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504Q16—1228no—
505SD16—1228no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—612no—
513C2×C8—612no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—612no—
521C4⋊C4—1224no—
522C4⋊C4—1224no—
523C4⋊C4—1221no—
524C4⋊C4—1224no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—4881no—
530C4⋊C4—4881no—
531C4⋊C4—2442no—
532C4⋊C4D4×D6112yes—
533C4⋊C4#1175822412yes—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—4881no—
537C22⋊C4—4881no—
538C22⋊C4—2442no—
539C42—81no—
540C42—612no—
541C42—612no—
542C42—1224no—
543C42—1224no—
544C42—1221no—
545C42—1221no—
546C42—1224no—
547C42—4881no—
548C42—4881no—
549C42—2442no—
550C42C2×C4×D6112yes—
551C42#1175822412yes—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—614no—
565Q8—1222no—
566Q8—1222no—
567Q8—1222no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1222no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—4884no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4#1451973712yes—
611C2×C4#1451973712yes—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4C2×#1175822411yescharacteristic
617C2×C4C2×#1175822411yescharacteristic
618C8—41no—
619C8—41no—
620C8—1222no—
621C8—1222no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×#1451973711yescharacteristic
630C22#121×D6111yescharacteristic
631C22#1684027211yescharacteristic
632C22#1684056611yescharacteristic
633C22#1684056911yescharacteristic
634C22#1684057611yescharacteristic
635C22#1684057811yescharacteristic
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—1224no—
642C4—1222no—
643C4—1222no—
644C4—4881no—
645C4—4881no—
646C4—2442no—
647C4—2441no—
648C4—2441no—
649C4C4×D4×D6111yescharacteristic
650C4C4×#1175822411yescharacteristic
651C4#1684029111yescharacteristic
652C4#1684031011yescharacteristic
653C2—82no—
654C2#2144833511yescharacteristic
655C2#2145466511yescharacteristic
656C2#2145469011yescharacteristic
657C2#2145534711yescharacteristic
658C2#2145535911yescharacteristic
659C2#2145604511yescharacteristic
660C2#2145607811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1983D61
hypocenterC61#21512
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
3. C2×#16839854C2semidirect
5. #21448183C2semidirect
6. #21446577C2semidirect
7. #21446463C2semidirect
10. #21446157C2semidirect
11. #21445825C2semidirect
12. #21445678C2semidirect
13. #21445651C2semidirect
23. #16839981C4semidirect
66. C2×#11758202C2×C4semidirect
75. C4×Dic122C2×C4semidirect
104. C4⋊#92418C2×C4semidirect
107. #14519701C2×C4semidirect
392. C61#21512semidirect

Maximal subgroups

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

subgroupindex
C2×#168399812
C2×#168398592
C2×#168398542
#214485322
#214481832
#214465772
#214464632
#214464422
#214461712
#214461572
#214458252
#214456782
#214456512
#102×Dic612
C61×#19832
#2151261