Finite Group Databasebeta

← #29689066

Subgroups — #29689066

Order 15616.

1538 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D61C211yesmaximal, characteristic
2C2×#16839725C211yesmaximal, characteristic
3#380×D61C211yesmaximal, characteristic
4#21446253C211yesmaximal, characteristic, complemented
5#21446252C211yesmaximal, characteristic, complemented
6#21446250C211yesmaximal, characteristic, complemented
7#21446125C212yesmaximal, complemented
8#21446123C212yesmaximal, complemented
9#21445904C212yesmaximal, complemented
10#21445752C212yesmaximal, complemented
11C61×#1894C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15C2×#14519757C2211yescharacteristic
16C2×#14519756C2211yescharacteristic
17C2×#14519755—21no—
18C2×#14519754—21no—
19C2×#14519753C2211yescharacteristic
20C22⋊Q8×D61—24no—
21C22⋊Q8×D61C418yescomplemented
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C4⋊C4×D61C2212yes—
25C2×C22⋊C4×D61—21no—
26C2×C22⋊C4×D61—21no—
27C2×#14519702C2212yescomplemented
28C2×#14519701C2212yescomplemented
29C2×#14519700C2212yescomplemented
30#16839725—24no—
31C2×#14519689C2212yes—
32C2×#14519689C2212yes—
33#16839622C2211yescharacteristic
34#16839621—21no—
35#16839620—21no—
36#16839619C2212yes—
37#95×D61—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×M16×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yescomplemented
44C61×#380C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—42no—
51C2×Q8×D61—42no—
52C2×Q8×D61C2×C414yescomplemented
53C23×Dic61D411yescharacteristic
54C22×C4×D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—22no—
57C22×C4×D61—42no—
58C22×C4×D61—42no—
59C22×C4×D61C2×C414yes—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D414yes—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758194—22no—
66C2×#11758194—42no—
67C22×#92418C2312yes—
68#14519757—22no—
69#14519757C2×C414yes—
70#14519756—22no—
71#14519756C2×C414yescomplemented
72#14519755—28no—
73#14519754—28no—
74#14519753—22no—
75#14519753C2×C414yes—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—22no—
83C4⋊C4×D61—28no—
84C4⋊C4×D61—22no—
85C4⋊C4×D61—42no—
86C4⋊C4×D61—42no—
87C4⋊C4×D61C2×C414yescomplemented
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—22no—
93C22⋊C4×D61—28no—
94C22⋊C4×D61—44no—
95#14519702—42no—
96#14519701—42no—
97#14519700—42no—
98#14519689—24no—
99#14519689—24no—
100#14519684—21no—
101#14519684—22no—
102C22×Q8×C61C2311yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—22no—
105C2×M16×C61—42no—
106C22×C8×C61C2312yes—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61C2×C414yescomplemented
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2312yes—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114#103×C61—42no—
115#95×C61—21no—
116C24×C61C2×D411yescharacteristic
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—24no—
120C23×D61—42no—
121C23×D61C22⋊C414yes—
122C23×D61C2×D411yescharacteristic
123C23×D61C2×D411yescharacteristic
124C2×Q8×C61—21no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61C22×C412yes—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61—42no—
141C22×Dic61C22⋊C414yes—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144Q8×D61—48no—
145Q8×D61—44no—
146Q8×D61—44no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—48no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—44no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C22×C412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D412yes—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—21no—
173C2×Dic122—24no—
174C2×Dic122—44no—
175C2×Dic122C22×C412yes—
176M16×C61—44no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—42no—
187C22⋊Dic61—24no—
188C22⋊Dic61—42no—
189#11758198—24no—
190#11758198—24no—
191#11758198—44no—
192C4⋊Dic61—21no—
193C4⋊Dic61—24no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22×C412yes—
197#11758196—24no—
198#11758196—24no—
199#11758196—24no—
200#11758196—44no—
201#11758194—44no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61C2×C22⋊C412yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—21no—
214C22×D61—22no—
215C22×D61—21no—
216C22×D61—28no—
217C22×D61—28no—
218C22×D61—44no—
219C22×D61—48no—
220C22×D61—48no—
221C22×D61C2×C22⋊C412yes—
222C22×D61C2×C22⋊C412yes—
223C22×D61AGL(1,ℤ/8)12yes—
224C22×D61#13712yes—
225C22×D61#13714yes—
226Q8×C61—24no—
227Q8×C61—41no—
228Q8×C61—41no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61H(5,𝔽2)12yes—
245C2×C4×C61H(5,𝔽2)12yes—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—44no—
251C2×Dic61—42no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259C4×D61—24no—
260C4×D61—24no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—44no—
266C4×D61—44no—
267C4×D61—48no—
268C4×D61—48no—
269C4×D61—88no—
270Dic122—24no—
271Dic122—48no—
272Dic122—41no—
273Dic122—41no—
274C8×C61—42no—
275#92418—42no—
276#21765—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61C22×C22⋊C411yescharacteristic
283C22×C61#53412yes—
284C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C61C2×#13712yes—
286C22×C61C2×#13711yescharacteristic
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—24no—
294C2×D61—24no—
295C2×D61—416no—
296C2×D61#41714yes—
297C2×D61#41914yes—
298C4×C61—22no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—82no—
302C4×C61#51812yes—
303C4×C61#53411yescharacteristic
304C4×C61#53411yescharacteristic
305Dic61—22no—
306Dic61—22no—
307Dic61—42no—
308Dic61—42no—
309Dic61—82no—
310C22×C22⋊Q8—611no—
311C2×#407—611no—
312C2×#380—611no—
313#1894—611no—
314#1894—611no—
315#1894—611no—
316#1894D6111yescharacteristic, complemented
317#1872—612no—
318#1872—612no—
319#1823—612no—
320#1787—612no—
321C2×C61—42no—
322C2×C61#230811yescharacteristic
323C2×C61C2×#41911yescharacteristic
324C2×C61C2×#41711yescharacteristic
325C2×C61#319312yes—
326C2×C61#319512yes—
327D61—24no—
328D61—24no—
329C23×Q8—611no—
330C24×C4—611no—
331C22×M16—611no—
332C22×M16C2×D6111yescharacteristic
333C2×C22⋊Q8—611no—
334C2×C22⋊Q8—611no—
335C2×C22⋊Q8—618no—
336C2×C22⋊Q8—611no—
337C2×C22⋊Q8—1221no—
338C2×C22⋊Q8—1221no—
339C2×C22⋊Q8—1224no—
340C2×C22⋊Q8C2×D6111yescharacteristic
341C22×C4⋊C4—611no—
342C22×C4⋊C4—612no—
343C22×C4⋊C4—1221no—
344C22×C22⋊C4—1221no—
345C22×C22⋊C4—1221no—
346C2×#103—612no—
347C2×#103—612no—
348C2×#103—612no—
349C2×#103C2×D6112yescomplemented
350#407—1224no—
351C2×C22⋊C8—612no—
352C2×C22⋊C8—612no—
353#380—611no—
354#380—612no—
355#380—1221no—
356#380—1221no—
357#380C2×D6111yescharacteristic
358C2×#95—1221no—
359#340—612no—
360#340—612no—
361#340—612no—
362#340C2×D6112yescomplemented
363C61#2176511yescharacteristic, complemented, Sylow
364C25—611no—
365C22×Q8—614no—
366C22×Q8—611no—
367C22×Q8—1221no—
368C22×Q8—1222no—
369C22×Q8—2442no—
370C22×Q8—2442no—
371C22×Q8C22×D6111yescharacteristic
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—614no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—614no—
379C23×C4—1221no—
380C23×C4—1221no—
381C23×C4—1222no—
382C23×C4—2442no—
383C23×C4—2442no—
384C23×C4C22×D6111yescharacteristic
385C2×M16—22no—
386C2×M16—42no—
387C2×M16—1222no—
388C2×M16—2442no—
389C22×C8—612no—
390C22×C8C22×D6112yes—
391C22⋊Q8—22no—
392C22⋊Q8—614no—
393C22⋊Q8—614no—
394C22⋊Q8—614no—
395C22⋊Q8—1228no—
396C22⋊Q8—1228no—
397C22⋊Q8—1222no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400C22⋊Q8C4×D6114yescomplemented
401C2×C4⋊C4—21no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—614no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1228no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—2442no—
413C2×C4⋊C4—2442no—
414C2×C4⋊C4C22×D6112yes—
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1228no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—2444no—
425#103—42no—
426#103—2442no—
427#103—2442no—
428#103—2442no—
429C22⋊C8—1224no—
430C22⋊C8—1224no—
431#95—21no—
432#95—1222no—
433#95—1221no—
434C24—611no—
435C24—611no—
436C24—614no—
437C24—1222no—
438C24—1224no—
439C24—1222no—
440C24—2442no—
441C24D4×D6111yescharacteristic
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—612no—
446C2×Q8—1224no—
447C2×Q8—1221no—
448C2×Q8—1222no—
449C2×Q8—1222no—
450C2×Q8—2444no—
451C2×Q8—2448no—
452C2×Q8—2444no—
453C2×Q8—2444no—
454C2×Q8C2×C4×D6112yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—614no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—614no—
468C22×C4—614no—
469C22×C4—1224no—
470C22×C4—1224no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1222no—
478C22×C4—2448no—
479C22×C4—2444no—
480C22×C4—2444no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2444no—
484C22×C4—2444no—
485C22×C4—2444no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2444no—
489C22×C4—2442no—
490C22×C4C2×C4×D6112yes—
491C22×C4D4×D6112yes—
492C22×C4D4×D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494M16—44no—
495M16—2444no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C4⋊C4—21no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—612no—
507C4⋊C4—1224no—
508C4⋊C4—1224no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1224no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—2444no—
515C4⋊C4C2×C4×D6112yes—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—2444no—
522C22⋊C4—2442no—
523C23—21no—
524C23—22no—
525C23—21no—
526C23—22no—
527C23—612no—
528C23—612no—
529C23—612no—
530C23—614no—
531C23—612no—
532C23—1228no—
533C23—1228no—
534C23—1221no—
535C23—1222no—
536C23—1221no—
537C23—1222no—
538C23—1222no—
539C23—2444no—
540C23—2448no—
541C23—2448no—
542C23C22⋊C4×D6112yes—
543C23C2×D4×D6111yescharacteristic
544Q8—24no—
545Q8—41no—
546Q8—41no—
547Q8—1224no—
548Q8—2448no—
549Q8—2441no—
550Q8—2441no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—4888no—
576C2×C4—2444no—
577C2×C4—2444no—
578C2×C4—2448no—
579C2×C4—2448no—
580C2×C4—2444no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2442no—
587C2×C4C22⋊C4×D6112yes—
588C2×C4C22×C4×D6111yescharacteristic
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4#1451982212yes—
591C2×C4#1451982312yes—
592C8—42no—
593C8—2442no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—42no—
599C22—614no—
600C22—614no—
601C22—1222no—
602C22—1222no—
603C22—1224no—
604C22—1224no—
605C22—1222no—
606C22—1222no—
607C22—1224no—
608C22—1224no—
609C22—24416no—
610C22C2×C22⋊C4×D6111yescharacteristic
611C22#1684032012yes—
612C22AGL(1,ℤ/8)×D6111yescharacteristic
613C22#137×D6112yes—
614C22#137×D6111yescharacteristic
615C4—22no—
616C4—42no—
617C4—42no—
618C4—82no—
619C4—1222no—
620C4—1222no—
621C4—4882no—
622C4—2442no—
623C4—2442no—
624C4#1684023412yes—
625C4#1684032211yescharacteristic
626C4#1684032311yescharacteristic
627C2—42no—
628C2—1224no—
629C2—1224no—
630C2#2144839311yescharacteristic
631C2#419×D6111yescharacteristic
632C2#417×D6111yescharacteristic
633C2#2145662012yes—
634C2#2145663912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1894C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1894D61
hypocenterC61#21765
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
4. #21446253C2semidirect
5. #21446252C2semidirect
6. #21446250C2semidirect
7. #21446125C2semidirect
8. #21446123C2semidirect
9. #21445904C2semidirect
10. #21445752C2semidirect
11. C61×#1894C2semidirect
21. C22⋊Q8×D61C4semidirect
27. C2×#14519702C22semidirect
28. C2×#14519701C22semidirect
29. C2×#14519700C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#103×C61C22semidirect
45. C61×#340C22semidirect
52. C2×Q8×D61C2×C4semidirect
71. #14519756C2×C4semidirect
87. C4⋊C4×D61C2×C4semidirect
108. C22⋊Q8×C61C2×C4semidirect
316. #1894D61semidirect
349. C2×#103C2×D61semidirect
362. #340C2×D61semidirect
363. C61#21765semidirect
400. C22⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D612
C2×#168397252
#380×D612
#214462532
#214462522
#214462502
#214461252
#214461232
#214459042
#214457522
C61×#18942
#2176561