Finite Group Databasebeta

← #29686241

Subgroups — #29686241

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×#16839984C211yesmaximal, characteristic
2C2×#16839854C211yesmaximal, characteristic, complemented
3C2×#16839847C211yesmaximal, characteristic
4#21448358C211yesmaximal, characteristic
5C4×Q8×Dic61C211yesmaximal, characteristic, complemented
6#21446749C211yesmaximal, characteristic
7#21446585C211yesmaximal, characteristic, complemented
8#21446451C211yesmaximal, characteristic, complemented
9#21446437C211yesmaximal, characteristic
10#21446172C211yesmaximal, characteristic
11#21446158C211yesmaximal, characteristic, complemented
12C4⋊C8×Dic61C211yesmaximal, characteristic, complemented
13#21445638C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic
15C61×#1805C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16839984C418yescomplemented
24#16839854—24no—
25#16839847—24no—
26C2×#14519710C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839663C2211yescharacteristic
39#16839651—21no—
40#16839651—21no—
41C4⋊C4×Dic61—21no—
42C4⋊C4×Dic61—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×C4⋊Dic61—21no—
46C4×C4⋊Dic61C2211yescharacteristic
47#16839609—21no—
48#16839609—21no—
49C42×Dic61—21no—
50C42×Dic61C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202C2×C414yescomplemented
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201C2×C414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519765—22no—
74#14519765C2×C414yes—
75Q8×Dic61—22no—
76Q8×Dic61—22no—
77Q8×Dic61—22no—
78Q8×Dic61D414yes—
79#14519742—22no—
80#14519742—22no—
81#14519739—22no—
82#14519739—22no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61D411yescharacteristic
100C2×C4×Dic61D411yescharacteristic
101C2×C4×Dic61D411yescharacteristic
102C2×C4×Dic61D411yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105#14519710—22no—
106#14519708—22no—
107C4⋊#92418C2×C414yescomplemented
108#14519702—22no—
109#14519702—22no—
110#14519700C2×C414yescomplemented
111#14519699—22no—
112#14519699—22no—
113#14519697C2×C414yes—
114C8×Dic61—22no—
115C8×Dic61—22no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121C22×Q8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4×Q8×C61—22no—
125C4×Q8×C61C2×C414yescomplemented
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61—21no—
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—22no—
133#103×C61—22no—
134#103×C61—22no—
135#95×C61—21no—
136#95×C61—21no—
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C412yes—
142C2×Q8×C61C2×D412yes—
143C2×Q8×C61#4112yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
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—24no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C22×C412yes—
186C42×C61—24no—
187C42×C61C22×C412yes—
188C42×C61#4112yes—
189#11758202—28no—
190#11758201—28no—
191#11758198—24no—
192#11758198—24no—
193#11758198—42no—
194#11758198—42no—
195#11758198—42no—
196C4⋊Dic61—24no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—24no—
200C4⋊Dic61C2×D412yes—
201C4⋊Dic61#4112yes—
202#11758196—42no—
203C4×Dic61—24no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—24no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—41no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C4×Dic61C2×D412yes—
216C4×Dic61#4112yes—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C22×C412yes—
222C2×#92418C22×C412yes—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—22no—
234Q8×C61C4×D414yes—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
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—41no—
250C2×C4×C61—41no—
251C2×C4×C61C4×D412yes—
252C2×C4×C61C4×D412yes—
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61#12112yes—
276C2×Dic61#12112yes—
277C2×Dic61#13512yes—
278C2×Dic61#13512yes—
279C2×Dic61#13712yes—
280D244—22no—
281D244—22no—
282D244—41no—
283D244—41no—
284C4×D61—84no—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289C8×C61—41no—
290C8×C61—41no—
291#92418—22no—
292#92418—22no—
293C4×#464—611nomaximal, Sylow
294C22×C61C2×C4×D411yescharacteristic
295C22×C61C2×#12111yescharacteristic
296C22×C61#52511yescharacteristic
297C22×C61C2×#13511yescharacteristic
298C22×C61C2×#13511yescharacteristic
299C22×C61C2×#13711yescharacteristic
300C22×C61#57511yescharacteristic
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—24no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61C2×C4×D411yescharacteristic
315C4×C61C4×#4111yescharacteristic
316C4×C61#52911yescharacteristic
317C4×C61#53211yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—41no—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
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#1925—611no—
334#1880—611no—
335#1878—611no—
336#1805D6111yescharacteristic, complemented
337C4×C4⋊C8—611no—
338C4×#103—611no—
339C4×#102—611no—
340C2×C61C4×#12111yescharacteristic
341C2×C61C4×#13511yescharacteristic
342C2×C61C4×#13711yescharacteristic
343C2×C61C2×#46411yescharacteristic
344C2×C61#303511yescharacteristic
345C2×C61#310211yescharacteristic
346C2×C61#310811yescharacteristic
347D61—82no—
348C22×Q16—611no—
349C22×SD16—611no—
350C2×#124—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8—611no—
353C2×C4×Q8—611no—
354C2×C4×Q8C2×D6111yescharacteristic
355C2×C4×D4—611no—
356#464—618no—
357C4×Q16—1224no—
358C4×SD16—1224no—
359C2×#107—611no—
360C2×#106—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8C2×D6111yescharacteristic
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103C2×D6111yescharacteristic
367C2×#103C2×D6111yescharacteristic
368C2×#102—611no—
369C2×#102—611no—
370C2×#102—611no—
371C2×C4×C8—611no—
372C2×C4×C8—611no—
373#391—611no—
374#390—611no—
375#388—1221no—
376#388—1221no—
377#382C2×D6111yescharacteristic
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—1221no—
382C4×C4⋊C4—1221no—
383C4×C4⋊C4—1221no—
384C4×C22⋊C4—1221no—
385C4×C22⋊C4—1221no—
386C43—611no—
387C43—1221no—
388#340—611no—
389#340—611no—
390#340C2×D6111yescharacteristic
391#340C2×D6111yescharacteristic
392#336—611no—
393#336—611no—
394C61C4×#46411yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×Q8C22×D6111yescharacteristic
397C22×D4—611no—
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—22no—
412C4×Q8—614no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×Q8—1222no—
418C4×Q8C4×D6114yescomplemented
419C4×D4—1222no—
420C4×D4—1222no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
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—1221no—
431C2×C22⋊C4—1221no—
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—1222no—
450#106—1222no—
451C4⋊C8—22no—
452C4⋊C8—614no—
453#103—22no—
454#103—22no—
455#103—614no—
456#103—1222no—
457#103—1222no—
458#102—614no—
459#102—1222no—
460#102—1222no—
461C4×C8—1222no—
462C4×C8—1222no—
463#95—21no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470C24—1221no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8C2×C4×D6112yes—
481C2×Q8D4×D6112yes—
482C2×Q8#1175822612yes—
483C2×D4—612no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
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—21no—
514C2×C8—612no—
515C2×C8—612no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C4⋊C4—21no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—612no—
527C4⋊C4—612no—
528C4⋊C4—1224no—
529C4⋊C4—1224no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2442no—
533C4⋊C4C2×C4×D6112yes—
534C22⋊C4—1224no—
535C22⋊C4—1224no—
536C22⋊C4—2442no—
537C22⋊C4—2442no—
538C22⋊C4—2442no—
539C42—24no—
540C42—612no—
541C42—612no—
542C42—1222no—
543C42—1222no—
544C42—1222no—
545C42—1224no—
546C42—1224no—
547C42—1221no—
548C42—1221no—
549C42—2441no—
550C42—2442no—
551C42—2442no—
552C42—2442no—
553C42—2442no—
554C42C2×C4×D6112yes—
555C42#1175822412yes—
556C23—1222no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23C2×D4×D6111yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—1222no—
568Q8—1222no—
569Q8—2441no—
570Q8—2441no—
571Q8#1451973914yes—
572D4—1222no—
573D4—1222no—
574D4—2441no—
575D4—2441no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—4884no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4#1451973712yes—
615C2×C4#1451973712yes—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4C2×#1175822411yescharacteristic
621C2×C4C2×#1175822611yescharacteristic
622C8—41no—
623C8—41no—
624C8—1222no—
625C8—1222no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22—4881no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22C2×#1451973711yescharacteristic
634C22#121×D6111yescharacteristic
635C22#1684027511yescharacteristic
636C22#135×D6111yescharacteristic
637C22#135×D6111yescharacteristic
638C22#137×D6111yescharacteristic
639C22#1684056111yescharacteristic
640C4—22no—
641C4—24no—
642C4—22no—
643C4—42no—
644C4—81no—
645C4—81no—
646C4—614no—
647C4—1222no—
648C4—1222no—
649C4—2442no—
650C4—2442no—
651C4—2441no—
652C4—2442no—
653C4C2×#1451973911yescharacteristic
654C4#1684023011yescharacteristic
655C4#1684029811yescharacteristic
656C4#1684030711yescharacteristic
657C2—4882no—
658C2#2144832511yescharacteristic
659C2#2145458711yescharacteristic
660C2#2145463411yescharacteristic
661C2#464×D6111yescharacteristic
662C2#2145527311yescharacteristic
663C2#2145584411yescharacteristic
664C2#2145590411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1805D61
hypocenterC61C4×#464
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#16839854C2semidirect
5. C4×Q8×Dic61C2semidirect
7. #21446585C2semidirect
8. #21446451C2semidirect
11. #21446158C2semidirect
12. C4⋊C8×Dic61C2semidirect
13. #21445638C2semidirect
15. C61×#1805C2semidirect
23. #16839984C4semidirect
67. C2×#11758202C2×C4semidirect
107. C4⋊#92418C2×C4semidirect
110. #14519700C2×C4semidirect
125. C4×Q8×C61C2×C4semidirect
324. Dic61#464semidirect
336. #1805D61semidirect
394. C61C4×#464semidirect
418. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399842
C2×#168398542
C2×#168398472
#214483582
C4×Q8×Dic612
#214467492
#214465852
#214464512
#214464372
#214461722
#214461582
C4⋊C8×Dic612
#214456382
#214456332
C61×#18052
C4×#46461