Finite Group Databasebeta

← #29601595

Subgroups — #29601595

Order 15616.

1906 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×#16839655C212yesmaximal
2C2×C4×C4⋊Dic61C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Dic61C211yesmaximal, characteristic
4#21444056C212yesmaximal
5#21444053C212yesmaximal, complemented
6#21444046C211yesmaximal, characteristic, complemented
7#21444046C211yesmaximal, characteristic, complemented
8#21444042C211yesmaximal, characteristic
9C4×#14519684C211yesmaximal, characteristic, complemented
10#95×Dic61C212yesmaximal, complemented
11C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
12C22×C4⋊Dic61C2212yes—
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2212yes—
15C22×C4×Dic61C2212yes—
16C22×C4×Dic61C2212yes—
17C22×C4×Dic61C2212yes—
18#16839655C4116yescomplemented
19C4×C4⋊Dic61—24no—
20C4×C22⋊Dic61—24no—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2211yescharacteristic
31C24×C4×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2211yescharacteristic
34C22×C42×C61C2211yescharacteristic
35C22×C42×C61C2211yescharacteristic
36C4×C22⋊C4×C61C418yescomplemented
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C2×C4⋊Dic61—24no—
44C2×C4⋊Dic61—24no—
45C2×C4⋊Dic61C2×C418yescomplemented
46C2×C22⋊Dic61—24no—
47C2×C22⋊Dic61—24no—
48C2×C22⋊Dic61—42no—
49C2×C22⋊Dic61—42no—
50C2×C22⋊Dic61C2×C418yes—
51C2×C22⋊Dic61C2×C418yes—
52C2×C4×Dic61—24no—
53C2×C4×Dic61—24no—
54C2×C4×Dic61—24no—
55C2×C4×Dic61—24no—
56C2×C4×Dic61—42no—
57C2×C4×Dic61—42no—
58C2×C4×Dic61—42no—
59C2×C4×Dic61—42no—
60C2×C4×Dic61C2×C418yescomplemented
61#14519684—22no—
62#14519684—22no—
63#14519684—24no—
64#14519684—24no—
65#14519684—22no—
66#14519684—24no—
67#14519684—42no—
68#14519684C2×C418yescomplemented
69#14519684C2×C418yescomplemented
70#14519684D414yes—
71C25×C61C2311yescharacteristic
72C23×C4×C61—22no—
73C23×C4×C61—24no—
74C23×C4×C61—22no—
75C23×C4×C61—42no—
76C23×C4×C61C2×C414yes—
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C2×C22⋊C4×C61—22no—
84C2×C22⋊C4×C61—22no—
85C2×C22⋊C4×C61C2×C418yes—
86C2×C22⋊C4×C61C2×C414yes—
87C2×C22⋊C4×C61C2×C414yes—
88C2×C42×C61—22no—
89C2×C42×C61—22no—
90C2×C42×C61—42no—
91C2×C42×C61C2×C414yes—
92C2×C42×C61C2×C414yes—
93#95×C61C2×C414yes—
94#95×C61C2×C414yes—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—21no—
98C24×C61—22no—
99C24×C61—21no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—21no—
103C24×C61—21no—
104C24×C61C22×C414yes—
105C24×C61C22×C412yes—
106C24×C61C2411yescharacteristic
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—28no—
127C22×C4×C61—28no—
128C22×C4×C61—44no—
129C22×C4×C61—48no—
130C22×C4×C61—48no—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C414yes—
133C22×C4×C61C22×C414yes—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C22×C412yes—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61#4112yes—
141C22×C4×C61#4112yes—
142C22×C4×C61#4114yes—
143C22×C4×C61#4112yes—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61C22×C414yes—
167C22×Dic61C22×C414yes—
168C22×Dic61C22×C414yes—
169C22×Dic61C22×C414yes—
170C22×Dic61C2×D414yes—
171C22×Dic61#4114yes—
172C22⋊C4×C61—28no—
173C22⋊C4×C61C42116yescomplemented
174C42×C61—28no—
175C42×C61—44no—
176C22⋊Dic61—216no—
177C22⋊Dic61—48no—
178C4⋊Dic61—216no—
179C4×Dic61—48no—
180C4×Dic61—48no—
181C4×Dic61—48no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—24no—
185C23×C61—22no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—22no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—24no—
194C23×C61—42no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61—41no—
199C23×C61—42no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61C2×C4214yes—
205C23×C61C23×C411yescharacteristic
206C23×C61C23×C412yes—
207C23×C61C22×D411yescharacteristic
208C23×C61C2×#4112yes—
209C23×C61C2×#4111yescharacteristic
210C23×C61C2×#4111yescharacteristic
211C23×C61C2×#4111yescharacteristic
212C23×C61C2×#4112yes—
213C23×C61H(5,𝔽2)12yes—
214C23×C61H(5,𝔽2)11yescharacteristic
215C23×C612−1+411yescharacteristic
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—24no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—416no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61C2×C4214yes—
245C2×C4×C61C2×C4214yes—
246C2×C4×C61#11714yes—
247C2×C4×C61#11714yes—
248C2×C4×C61C4×D414yes—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—24no—
263C2×Dic61—24no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61C4×D418yes—
277#13822—611nomaximal, Sylow
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—24no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—44no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61C22×C4211yescharacteristic
290C22×C61C2×#11711yescharacteristic
291C22×C61C2×#11711yescharacteristic
292C22×C61C2×C4×D412yes—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61C4×#4112yes—
295C22×C61C4×#4112yes—
296C22×C61C4×#4112yes—
297C22×C61C4×#4111yescharacteristic
298C22×C61#51812yes—
299C22×C61#51811yescharacteristic
300C22×C61#52012yes—
301C22×C61#52011yescharacteristic
302C22×C61#54612yes—
303C22×C61#54611yescharacteristic
304C22×C61#54811yescharacteristic
305C22×C61#55112yes—
306C22×C61#55211yescharacteristic
307C22×C61#55312yes—
308C22×C61#55311yescharacteristic
309C22×C61#55312yes—
310C22×C61#55612yes—
311C22×C61#55912yes—
312C4×C61—24no—
313C4×C61—24no—
314C4×C61—24no—
315C4×C61—24no—
316C4×C61—24no—
317C4×C61—42no—
318Dic61—28no—
319Dic61—44no—
320Dic61—44no—
321Dic61—44no—
322C2×#388—612no—
323C2×C4×C4⋊C4—611no—
324C2×C4×C22⋊C4—611no—
325C2×C4×C22⋊C4D6111yescharacteristic
326#1437—612no—
327#1436—612no—
328#1434—611no—
329#1434—611no—
330C22⋊#95—611no—
331C4×#95—611no—
332C4×#95—612no—
333C2×C61—44no—
334C2×C61C4×#11711yescharacteristic
335C2×C61C42×D411yescharacteristic
336C2×C61#227511yescharacteristic
337C2×C61#234211yescharacteristic
338C2×C61#234711yescharacteristic
339C2×C61#235211yescharacteristic
340C2×C61#235711yescharacteristic
341C2×C61#236412yes—
342C2×C61#236712yes—
343C2×C61#237312yes—
344C2×C61#237412yes—
345C24×C4C2×D6111yescharacteristic
346C22×C4⋊C4—612no—
347C22×C22⋊C4—612no—
348C22×C22⋊C4—612no—
349C22×C22⋊C4C2×D6111yescharacteristic
350C22×C22⋊C4C2×D6111yescharacteristic
351C22×C42—612no—
352C22×C42—612no—
353C22×C42—612no—
354C22×C42C2×D6111yescharacteristic
355C22×C42C2×D6111yescharacteristic
356#388—6116no—
357C4×C4⋊C4—1224no—
358C4×C22⋊C4—1224no—
359C4×C22⋊C4Dic6118yescomplemented
360C2×#95—612no—
361C2×#95—611no—
362C2×#95—612no—
363C2×#95—611no—
364C2×#95—612no—
365C2×#95—612no—
366C2×#95—612no—
367C2×#95—611no—
368C2×#95—611no—
369C2×#95—612no—
370C2×#95C2×D6111yescharacteristic
371C2×#95C2×D6111yescharacteristic
372C61#1382211yescharacteristic, complemented, Sylow
373C25C22×D6111yescharacteristic
374C23×C4—24no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—42no—
378C23×C4—612no—
379C23×C4—612no—
380C23×C4—612no—
381C23×C4—612no—
382C23×C4C2×Dic6114yes—
383C23×C4C22×D6111yescharacteristic
384C23×C4C22×D6111yescharacteristic
385C23×C4C22×D6111yescharacteristic
386C23×C4C22×D6111yescharacteristic
387C23×C4C22×D6111yescharacteristic
388C23×C4C22×D6111yescharacteristic
389C2×C4⋊C4—618no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1224no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—618no—
395C2×C22⋊C4—618no—
396C2×C22⋊C4—1224no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—2442no—
399C2×C22⋊C4—2442no—
400C2×C22⋊C4C4×D6118yes—
401C2×C22⋊C4C2×Dic6114yes—
402C2×C22⋊C4C2×Dic6114yes—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—42no—
406C2×C42—618no—
407C2×C42—1224no—
408C2×C42—1224no—
409C2×C42—1224no—
410C2×C42—1224no—
411C2×C42—2442no—
412C2×C42—2442no—
413C2×C42—2442no—
414C2×C42—2442no—
415C2×C42C2×Dic6114yes—
416C2×C42C2×Dic6114yes—
417#95—618no—
418#95—618no—
419#95—614no—
420#95—1224no—
421#95—1224no—
422#95—1224no—
423#95—1222no—
424#95—1222no—
425#95—1222no—
426#95—2442no—
427#95C2×Dic6114yes—
428#95C2×Dic6114yes—
429C24—21no—
430C24—22no—
431C24—21no—
432C24—21no—
433C24—21no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24C2×C4×D6114yes—
439C24C22×Dic6112yes—
440C24C23×D6111yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—614no—
466C22×C4—614no—
467C22×C4—614no—
468C22×C4—614no—
469C22×C4—614no—
470C22×C4—614no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4C2×C4×D6114yes—
494C22×C4C2×C4×D6114yes—
495C22×C4#1175822212yes—
496C22×C4#1175822214yes—
497C22×C4#1175822212yes—
498C22×C4D4×D6112yes—
499C22×C4#1175822412yes—
500C22×C4C22×Dic6112yes—
501C22×C4C22×Dic6112yes—
502C22×C4C22×Dic6112yes—
503C22×C4C22×Dic6112yes—
504C22×C4C22×Dic6112yes—
505C22×C4C22×Dic6112yes—
506C4⋊C4—12216no—
507C22⋊C4—28no—
508C22⋊C4—12216no—
509C22⋊C4—2448no—
510C22⋊C4C4×Dic61116yescomplemented
511C42—28no—
512C42—44no—
513C42—2448no—
514C42—2448no—
515C42—2448no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23C2×C4×Dic6114yes—
539C23C22×C4×D6112yes—
540C23C23×Dic6111yescharacteristic
541C23C2×D4×D6111yescharacteristic
542C23#41×D6112yes—
543C23C2×#1175822411yescharacteristic
544C23C2×#1175822211yescharacteristic
545C23C2×#1175822212yes—
546C23C2×#1175822211yescharacteristic
547C23#1451982212yes—
548C23#1451982311yescharacteristic
549C23#1451982411yescharacteristic
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—618no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4C2×C4×Dic6114yes—
607C2×C4C2×C4×Dic6114yes—
608C2×C4#1451973614yes—
609C2×C4#1451973614yes—
610C2×C4D4×Dic6114yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22C22×C4×Dic6111yescharacteristic
623C22C2×#1451973611yescharacteristic
624C22C2×#1451973611yescharacteristic
625C22C4×D4×D6112yes—
626C22C2×D4×Dic6111yescharacteristic
627C22C4×#1175822412yes—
628C22C4×#1175822212yes—
629C22C4×#1175822212yes—
630C22#41×Dic6111yescharacteristic
631C22#1684023412yes—
632C22#1684023711yescharacteristic
633C22#1684024112yes—
634C22#1684024511yescharacteristic
635C22#1684040212yes—
636C22#1684040811yescharacteristic
637C22#1684042111yescharacteristic
638C22#1684043812yes—
639C22#1684045211yescharacteristic
640C22#1684045612yes—
641C22#1684045811yescharacteristic
642C22#1684046212yes—
643C22#1684048012yes—
644C22#1684049512yes—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—1228no—
652C4—2444no—
653C4—2444no—
654C4—2444no—
655C2—44no—
656C2C4×#1451973611yescharacteristic
657C2C4×D4×Dic6111yescharacteristic
658C2#2144818011yescharacteristic
659C2#2144866811yescharacteristic
660C2#2144871211yescharacteristic
661C2#2144874811yescharacteristic
662C2#2144879211yescharacteristic
663C2#2144887012yes—
664C2#2144889112yes—
665C2#2144894112yes—
666C2#2144895312yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC2×C4×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#13822
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Dic61C2semidirect
5. #21444053C2semidirect
6. #21444046C2semidirect
7. #21444046C2semidirect
9. C4×#14519684C2semidirect
10. #95×Dic61C2semidirect
18. #16839655C4semidirect
36. C4×C22⋊C4×C61C4semidirect
45. C2×C4⋊Dic61C2×C4semidirect
60. C2×C4×Dic61C2×C4semidirect
68. #14519684C2×C4semidirect
69. #14519684C2×C4semidirect
173. C22⋊C4×C61C42semidirect
359. C4×C22⋊C4Dic61semidirect
372. C61#13822semidirect
510. C22⋊C4C4×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396552
C2×C4×C4⋊Dic612
C2×C4×C22⋊Dic612
#214440562
#214440532
#214440462
#214440422
C4×#145196842
#95×Dic612
C2×C4×C22⋊C4×C612
#1382261