Finite Group Databasebeta

← #29686517

Subgroups — #29686517

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840001C211yesmaximal, characteristic
2C2×#16839854C211yesmaximal, characteristic, complemented
3C22×C8×Dic61C211yesmaximal, characteristic
4C22⋊Q8×Dic61C211yesmaximal, characteristic
5#21448347C211yesmaximal, characteristic
6#21446677C211yesmaximal, characteristic, complemented
7#21446551C211yesmaximal, characteristic, complemented
8#21446496C211yesmaximal, characteristic, complemented
9#21446492C211yesmaximal, characteristic, complemented
10#21446242C211yesmaximal, characteristic
11#21445846C211yesmaximal, characteristic
12#21445711C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445638C211yesmaximal, characteristic, complemented
15C61×#1892C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22#16840001C418yescomplemented
23#16839854—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×C8×Dic61—24no—
32C2×C8×Dic61C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839669C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839648—21no—
41#16839644—21no—
42#16839637—21no—
43#16839634—21no—
44#16839622C2211yescharacteristic
45C4⋊C4×Dic61—21no—
46C4⋊C4×Dic61C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48#16839613—21no—
49C22⋊C4×Dic61—21no—
50C22⋊C4×Dic61—21no—
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C23×C8×C61C2211yescharacteristic
56C2×C22⋊Q8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61D411yescharacteristic
64C23×Dic61D411yescharacteristic
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202C2×C414yescomplemented
68C22×#92418—22no—
69C22×#92418—22no—
70C22×#92418C2×C414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519756—22no—
74#14519756C2×C414yes—
75Q8×Dic61—22no—
76Q8×Dic61—22no—
77#14519742—22no—
78#14519742—22no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C4×Dic61—22no—
90C2×C4×Dic61—22no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61D411yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99#14519710—22no—
100#14519710—22no—
101#14519710C2×C414yescomplemented
102#14519702—22no—
103#14519702—22no—
104#14519701C2×C414yescomplemented
105#14519700C2×C414yescomplemented
106C8×Dic61—22no—
107C8×Dic61—22no—
108C8×Dic61—22no—
109C8×Dic61D414yes—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×Q8×C61C2311yescharacteristic
120C23×C4×C61C2311yescharacteristic
121C22×C8×C61—22no—
122C22×C8×C61—22no—
123C22×C8×C61—22no—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22⋊Q8×C61—22no—
127C22⋊Q8×C61C2×C414yes—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61—21no—
134#103×C61—22no—
135#103×C61—22no—
136#95×C61—21no—
137C24×C61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61#4112yes—
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61Q1614yescomplemented
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C2×C8×C61—28no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D412yes—
182C2×C8×C61#4112yes—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—42no—
192C22⋊Dic61—24no—
193C22⋊Dic61—42no—
194#11758202—28no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—42no—
201C4⋊Dic61—42no—
202C4⋊Dic61C22×C412yes—
203#11758196—24no—
204#11758196—42no—
205#11758196—42no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C4×Dic61C2×D412yes—
215C4×Dic61#4112yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—28no—
219C2×#92418C22×C412yes—
220C2×#92418C22×C412yes—
221C2×#92418C2×D412yes—
222C2×#92418#4112yes—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61C4×D412yes—
228C23×C61C2×Q1612yes—
229C23×C61#13512yes—
230C23×C61C22×D411yescharacteristic
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—41no—
234Q8×C61—41no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
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×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61C4×D412yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×Dic61—28no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—28no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—41no—
271C2×Dic61—41no—
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#12112yes—
281C2×Dic61#12112yes—
282C2×Dic61C2×Q1612yes—
283C2×Dic61C2×Q1612yes—
284C2×Dic61#13512yes—
285C2×Dic61#13512yes—
286Dic122—22no—
287Dic122—22no—
288Dic122—41no—
289Dic122—41no—
290C8×C61—22no—
291C8×C61—22no—
292#92418—22no—
293#92418C4×D414yes—
294C4×#467—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61C4×Q1614yescomplemented
302C22×C61C2×C4×D411yescharacteristic
303C22×C61C2×#12111yescharacteristic
304C22×C61#53211yescharacteristic
305C22×C61C22×Q1611yescharacteristic
306C22×C61C2×#13511yescharacteristic
307C22×C61C2×#13511yescharacteristic
308C22×C61C2×#13511yescharacteristic
309C4×C61—22no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61C2×C4×D411yescharacteristic
315C4×C61C4×#4111yescharacteristic
316C4×C61#52511yescharacteristic
317C4×C61#52911yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—22no—
321Dic61—42no—
322Dic61—42no—
323Dic61—81no—
324Dic61—81no—
325Dic61#46714yescomplemented
326C2×#467—611no—
327C2×C4×Q16—611no—
328C22×C4×C8—611no—
329C4×C22⋊Q8—611no—
330C4×C22⋊Q8—611no—
331#1969—611no—
332#1942—611no—
333#1933—611no—
334#1933—611no—
335#1892—611no—
336#1892D6111yescharacteristic
337#1808—611no—
338C4×#107—611no—
339C4×#103—611no—
340C4×#103—611no—
341C2×C61—24no—
342C2×C61C4×#12111yescharacteristic
343C2×C61C2×C4×Q1611yescharacteristic
344C2×C61C4×#13511yescharacteristic
345C2×C61C2×#46711yescharacteristic
346C2×C61#305011yescharacteristic
347C2×C61#316811yescharacteristic
348C2×C61#317011yescharacteristic
349C22×Q16—611no—
350C23×C8—611no—
351C23×C8C2×D6111yescharacteristic
352C2×C22⋊Q8—611no—
353C2×C22⋊Q8C2×D6111yescharacteristic
354C2×C4×Q8—611no—
355C2×C4×Q8—611no—
356C22×C42—611no—
357#467—618no—
358C4×Q16—1224no—
359C2×#107—611no—
360C2×#107—611no—
361C2×#107—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103C2×D6111yescharacteristic
367C2×#103C2×D6111yescharacteristic
368C2×C4×C8—611no—
369C2×C4×C8—611no—
370C2×C4×C8—611no—
371C2×C4×C8—611no—
372C2×C4×C8—1224no—
373#391—611no—
374#391—611no—
375#389—611no—
376#389—611no—
377#387—1221no—
378#387—1221no—
379#385—1221no—
380#385—1221no—
381#380—611no—
382#380C2×D6111yescharacteristic
383C4×C4⋊C4—611no—
384C4×C4⋊C4—611no—
385C4×C4⋊C4—1221no—
386C4×C4⋊C4—1221no—
387C4×C22⋊C4—1221no—
388C4×C22⋊C4—1221no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340C2×D6111yescharacteristic
394#340C2×D6111yescharacteristic
395C61C4×#46711yescharacteristic, complemented, Sylow
396C22×Q8—611no—
397C22×Q8C22×D6111yescharacteristic
398C23×C4—611no—
399C23×C4—611no—
400C23×C4C22×D6111yescharacteristic
401C2×Q16—614no—
402C2×Q16—1222no—
403C2×Q16—1222no—
404C22×C8—22no—
405C22×C8—22no—
406C22×C8—22no—
407C22×C8—611no—
408C22×C8—614no—
409C22×C8—611no—
410C22×C8—1222no—
411C22×C8—1222no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C22⋊Q8—22no—
415C22⋊Q8—614no—
416C22⋊Q8—1222no—
417C22⋊Q8C4×D6114yes—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—1222no—
443C2×C42—1222no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448#107—614no—
449#107—1222no—
450#107—1222no—
451#103—22no—
452#103—22no—
453#103—614no—
454#103—614no—
455#103—1222no—
456#103—1222no—
457C4×C8—614no—
458C4×C8—1222no—
459C4×C8—1222no—
460C4×C8—1222no—
461#95—21no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24D4×D6111yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—612no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8C2×C4×D6112yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—614no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4C2×C4×D6112yes—
507C22×C4D4×D6111yescharacteristic
508C22×C4#1175822412yes—
509C22×C4C23×D6111yescharacteristic
510Q16—1228no—
511C2×C8—28no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—612no—
517C2×C8—612no—
518C2×C8—612no—
519C2×C8—612no—
520C2×C8—1228no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8D4×D6112yes—
524C2×C8#1175822412yes—
525C4⋊C4—21no—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—612no—
532C4⋊C4—1221no—
533C4⋊C4—1224no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—2442no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C4⋊C4C2×C4×D6112yes—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—1224no—
546C22⋊C4—2442no—
547C42—612no—
548C42—612no—
549C42—1222no—
550C42—1222no—
551C42—1221no—
552C42—1221no—
553C42—2442no—
554C42—2442no—
555C42—2442no—
556C42—2442no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23#1451973712yes—
562C23Q16×D6112yes—
563C23#1451979812yes—
564C23C2×D4×D6111yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—1222no—
570Q8—1222no—
571Q8—2441no—
572Q8—2441no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—1228no—
598C2×C4—1228no—
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—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4#1451973712yes—
618C2×C4C22×C4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4C2×#1175822411yescharacteristic
623C2×C4C2×#1175822411yescharacteristic
624C8—22no—
625C8—22no—
626C8—614no—
627C8—1222no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22#1683985414yescomplemented
635C22C2×#1451973711yescharacteristic
636C22#121×D6111yescharacteristic
637C22#1684030711yescharacteristic
638C22C2×Q16×D6111yescharacteristic
639C22#135×D6111yescharacteristic
640C22#135×D6111yescharacteristic
641C22C2×#1451979811yescharacteristic
642C4—22no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—614no—
648C4—1222no—
649C4—1222no—
650C4—1222no—
651C4—4881no—
652C4—4881no—
653C4—2442no—
654C4—2442no—
655C4C4×D4×D6111yescharacteristic
656C4C4×#1175822411yescharacteristic
657C4#1684026811yescharacteristic
658C4#1684028911yescharacteristic
659C2—24no—
660C2#2144832911yescharacteristic
661C2C2×#1683985411yescharacteristic
662C2#2145458711yescharacteristic
663C2#467×D6111yescharacteristic
664C2#2145539511yescharacteristic
665C2#2145638211yescharacteristic
666C2#2145640011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1892D61
hypocenterC61C4×#467
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C2×#16839854C2semidirect
6. #21446677C2semidirect
7. #21446551C2semidirect
8. #21446496C2semidirect
9. #21446492C2semidirect
12. #21445711C2semidirect
13. #103×Dic61C2semidirect
14. #21445638C2semidirect
22. #16840001C4semidirect
67. C2×#11758202C2×C4semidirect
101. #14519710C2×C4semidirect
104. #14519701C2×C4semidirect
105. #14519700C2×C4semidirect
165. C22×Dic61Q16semidirect
301. C22×C61C4×Q16semidirect
325. Dic61#467semidirect
395. C61C4×#467semidirect
634. C22#16839854semidirect

Maximal subgroups

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

subgroupindex
C2×#168400012
C2×#168398542
C22×C8×Dic612
C22⋊Q8×Dic612
#214483472
#214466772
#214465512
#214464962
#214464922
#214462422
#214458462
#214457112
#103×Dic612
#214456382
C61×#18922
C4×#46761