Finite Group Databasebeta

← #29686255

Subgroups — #29686255

Order 15616.

1016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839983C211yesmaximal, characteristic
2C2×#16839856C211yesmaximal, characteristic, complemented
3C2×#16839853C211yesmaximal, characteristic
4#21448366C211yesmaximal, characteristic
5C4×#14519742C211yesmaximal, characteristic, complemented
6#21446750C211yesmaximal, characteristic
7#21446584C211yesmaximal, characteristic, complemented
8#21446453C211yesmaximal, characteristic, complemented
9#21446177C211yesmaximal, characteristic
10#21446159C211yesmaximal, characteristic, complemented
11#21445833C211yesmaximal, characteristic, complemented
12#21445667C211yesmaximal, characteristic, complemented
13#21445638C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#1925C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24#16839983C418yescomplemented
25#16839856—24no—
26#16839853—24no—
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839650—21no—
40#16839650—21no—
41#16839627C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196—21no—
45#16839613—21no—
46#16839613—21no—
47C22⋊C4×Dic61—21no—
48C22⋊C4×Dic61—21no—
49C42×Dic61—21no—
50C42×Dic61C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839517C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C23×Dic61—21no—
65C2×Dic244—22no—
66C2×Dic244—22no—
67C2×Dic244C2×C414yescomplemented
68C2×#11758200—22no—
69C2×#11758200—22no—
70C2×#11758200C2×C414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519764—22no—
74#14519764C2×C414yes—
75C4×Dic122—22no—
76C4×Dic122—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—22no—
80#14519742—22no—
81#14519742C2×C414yescomplemented
82#14519742D414yes—
83D4×Dic61—22no—
84D4×Dic61—22no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
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×Dic61D411yescharacteristic
101C2×C4×Dic61D411yescharacteristic
102C2×C4×Dic61D411yescharacteristic
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106#14519708—22no—
107#14519706—22no—
108#14519706C2×C414yescomplemented
109#14519702—22no—
110#14519702—22no—
111#14519700—22no—
112#14519700—22no—
113#14519700C2×C414yescomplemented
114#14519698—22no—
115C8×Dic61—22no—
116C8×Dic61—22no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×D4×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61—21no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#107×C61—22no—
134#102×C61—22no—
135#102×C61C2×C414yes—
136#95×C61—21no—
137C24×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
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×C61C22×C412yes—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C412yes—
184C4⋊C4×C61C2×D412yes—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—42no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—42no—
190C22⋊Dic61—24no—
191C22⋊Dic61—24no—
192C22⋊Dic61—42no—
193#11758200—28no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61#4112yes—
198#11758196—24no—
199#11758196—24no—
200#11758196—24no—
201#11758196—42no—
202C4×Dic61—24no—
203C4×Dic61—22no—
204C4×Dic61—24no—
205C4×Dic61—24no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—41no—
209C4×Dic61—42no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61C22×C412yes—
213C4×Dic61C2×D412yes—
214C4×Dic61#4112yes—
215C4×Dic61#4112yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C22×C412yes—
222Dic244—28no—
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C22×D411yescharacteristic
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—41no—
234D4×C61—41no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61#13512yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—84no—
275C2×Dic61C4×D412yes—
276C2×Dic61C4×D412yes—
277C2×Dic61#12112yes—
278C2×Dic61#12112yes—
279C2×Dic61#13512yes—
280C2×Dic61#13712yes—
281Dic122—22no—
282Dic122—22no—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122C4×D414yes—
289C8×C61—22no—
290C8×C61—41no—
291#92418—22no—
292#92418—41no—
293C4×#464—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61C2×C4×D411yescharacteristic
302C22×C61C2×#12111yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61C2×#13511yescharacteristic
305C22×C61C2×#13511yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57511yescharacteristic
308C4×C61—22no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61C2×C4×D411yescharacteristic
314C4×C61C4×#4111yescharacteristic
315C4×C61#52911yescharacteristic
316C4×C61#53211yescharacteristic
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—24no—
321Dic61—42no—
322Dic61—42no—
323Dic61—81no—
324Dic61#46414yescomplemented
325C2×#464—611no—
326C2×C4×Q16—611no—
327C2×C4×SD16—611no—
328C4×#124—611no—
329C42×Q8—611no—
330#1983—611no—
331#1949—611no—
332#1927—611no—
333#1925D6111yescharacteristic
334#1880—611no—
335#1878—611no—
336#1805—611no—
337C4×C4⋊C8—611no—
338C4×#103—611no—
339C4×#102—611no—
340C2×C61—82no—
341C2×C61C4×#12111yescharacteristic
342C2×C61C4×#13511yescharacteristic
343C2×C61C4×#13711yescharacteristic
344C2×C61C2×#46411yescharacteristic
345C2×C61#303511yescharacteristic
346C2×C61#310211yescharacteristic
347C2×C61#310811yescharacteristic
348C22×Q16—611no—
349C22×SD16—611no—
350C2×#124—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8—611no—
353C2×C4×Q8—611no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356#464—618no—
357C4×Q16—1224no—
358C4×SD16—1224no—
359C2×#107C2×D6111yescharacteristic
360C2×#106—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#102—611no—
369C2×#102C2×D6111yescharacteristic
370C2×#102C2×D6111yescharacteristic
371C2×C4×C8—611no—
372C2×C4×C8—611no—
373#391—611no—
374#390C2×D6111yescharacteristic
375#388—1221no—
376#388—1221no—
377#382—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
384C4×C22⋊C4—1221no—
385C4×C22⋊C4—1221no—
386C43—611no—
387C43—1221no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340C2×D6111yescharacteristic
392#336—611no—
393#336C2×D6111yescharacteristic
394C61C4×#46411yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×Q8—611no—
397C22×D4C22×D6111yescharacteristic
398C23×C4—1221no—
399C2×Q16—614no—
400C2×Q16—1222no—
401C2×Q16—1222no—
402C2×SD16—614no—
403C2×SD16—1222no—
404C2×SD16—1222no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409#124—614no—
410#124—1222no—
411C4×Q8—614no—
412C4×Q8—614no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×Q8—1222no—
418C4×Q8—1222no—
419C4×D4—1222no—
420C4×D4—1222no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
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—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#107—22no—
450#106—1222no—
451C4⋊C8—614no—
452C4⋊C8—1222no—
453#103—614no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#103—1222no—
458#102—22no—
459#102—1222no—
460#102C4×D6114yes—
461C4×C8—1222no—
462C4×C8—1222no—
463#95—21no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470C24—21no—
471C2×Q8—612no—
472C2×Q8—612no—
473C2×Q8—612no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D6112yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C22×C4C23×D6111yescharacteristic
506Q16—1228no—
507SD16—1228no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—612no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8C2×C4×D6112yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—612no—
525C4⋊C4—1224no—
526C4⋊C4—1224no—
527C4⋊C4—1224no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—2442no—
532C4⋊C4C2×C4×D6112yes—
533C4⋊C4D4×D6112yes—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—1224no—
537C22⋊C4—1224no—
538C22⋊C4—2442no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—612no—
543C42—612no—
544C42—612no—
545C42—612no—
546C42—1222no—
547C42—1222no—
548C42—1224no—
549C42—1224no—
550C42—1224no—
551C42—1222no—
552C42—2441no—
553C42—2442no—
554C42—2442no—
555C42—2442no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D4×D6111yescharacteristic
564Q8—614no—
565Q8—1222no—
566Q8—1222no—
567Q8—1222no—
568Q8—1222no—
569Q8—1222no—
570Q8—2441no—
571Q8—2441no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—4884no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4#1451979612yes—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C8—22no—
623C8—41no—
624C8—1222no—
625C8—2441no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C4×D4×D6111yescharacteristic
634C22#121×D6111yescharacteristic
635C22#1684027011yescharacteristic
636C22#135×D6111yescharacteristic
637C22C2×#1451979611yescharacteristic
638C22#137×D6111yescharacteristic
639C22#1684055811yescharacteristic
640C4—22no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—81no—
645C4—614no—
646C4—1222no—
647C4—1224no—
648C4—1222no—
649C4—1222no—
650C4—4881no—
651C4—2442no—
652C4—2442no—
653C4C4×D4×D6111yescharacteristic
654C4#1684023011yescharacteristic
655C4#1684029511yescharacteristic
656C4#1684031111yescharacteristic
657C2—82no—
658C2#2144833111yescharacteristic
659C2C4×#1451979611yescharacteristic
660C2#2145464111yescharacteristic
661C2#464×D6111yescharacteristic
662C2#2145526711yescharacteristic
663C2#2145583211yescharacteristic
664C2#2145589611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1925C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1925D61
hypocenterC61C4×#464
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
2. C2×#16839856C2semidirect
5. C4×#14519742C2semidirect
7. #21446584C2semidirect
8. #21446453C2semidirect
10. #21446159C2semidirect
11. #21445833C2semidirect
12. #21445667C2semidirect
13. #21445638C2semidirect
24. #16839983C4semidirect
67. C2×Dic244C2×C4semidirect
81. #14519742C2×C4semidirect
108. #14519706C2×C4semidirect
113. #14519700C2×C4semidirect
324. Dic61#464semidirect
394. C61C4×#464semidirect

Maximal subgroups

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

subgroupindex
C2×#168399832
C2×#168398562
C2×#168398532
#214483662
C4×#145197422
#214467502
#214465842
#214464532
#214461772
#214461592
#214458332
#214456672
#214456382
#102×Dic612
C61×#19252
C4×#46461