Finite Group Databasebeta

← #29645821

Subgroups — #29645821

Order 15616.

1314 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758224C211yesmaximal, characteristic
2C2×#16839812C212yesmaximal, complemented
3#21444141C212yesmaximal, complemented
4#21444138C212yesmaximal, complemented
5#21444137C211yesmaximal, characteristic, complemented
6#21444135C212yesmaximal, complemented
7#21444132C212yesmaximal, complemented
8#21444130C212yesmaximal, complemented
9C61×#1459C211yesmaximal, characteristic, complemented
10C22×#11758224C411yescharacteristic
11C4×#11758224—22no—
12C4×#11758224—22no—
13C4×#11758224—22no—
14C4×#11758224—22no—
15C2×C4×Dic122C411yescharacteristic
16C2×#14519742C2211yescharacteristic
17C2×D4×Dic61C411yescharacteristic
18C2×C4×C22⋊D61C411yescharacteristic
19C2×C4×C22⋊D61C2211yescharacteristic
20C2×#14519737C411yescharacteristic
21C2×#14519737C2211yescharacteristic
22C2×#14519735C2211yescharacteristic
23C2×#14519733C411yescharacteristic
24C2×#14519733C2211yescharacteristic
25C22×C4×Dic61C411yescharacteristic
26C22×C4×Dic61C2211yescharacteristic
27C2×C42×D61C411yescharacteristic
28#16839812—22no—
29#16839812—22no—
30#16839812C2214yes—
31#16839812C2214yes—
32C2×#14519706—22no—
33C2×C22⋊#92418C2212yes—
34C2×C8×Dic61—22no—
35#16839488C2212yes—
36#16839487C2214yes—
37#16839486C2212yes—
38#16839485C2214yes—
39C2×C4×D4×C61C2211yescharacteristic
40C2×C22⋊C8×C61C2212yes—
41C61×#324C2212yes—
42C61×#323C2212yes—
43C2×#11758224—22no—
44C2×#11758224—22no—
45C2×#11758224—24no—
46C2×#11758224C818yescomplemented
47C22×Dic122C2×C411yescharacteristic
48C22×C22⋊D61C2×C411yescharacteristic
49C22×C22⋊D61C2×C411yescharacteristic
50C23×Dic61C2×C411yescharacteristic
51C23×Dic61C2×C411yescharacteristic
52C23×Dic61C2×C411yescharacteristic
53C23×Dic61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2×C411yescharacteristic
57C22×#92418—22no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60#14519742—22no—
61#14519742—22no—
62D4×Dic61—22no—
63D4×Dic61—22no—
64C4×C22⋊D61—22no—
65C4×C22⋊D61—22no—
66C4×C22⋊D61—22no—
67C4×C22⋊D61—22no—
68#14519737—22no—
69#14519737—22no—
70#14519737—22no—
71#14519737—22no—
72#14519735—22no—
73#14519735—22no—
74#14519733—22no—
75#14519733—22no—
76#14519733—22no—
77#14519733—22no—
78C2×C4⋊Dic61C2×C411yescharacteristic
79C2×#11758196C2×C411yescharacteristic
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C411yescharacteristic
82C2×#11758196C2311yescharacteristic
83C2×C22⋊Dic61C2×C411yescharacteristic
84C2×C22⋊Dic61C2×C411yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×#11758198C2311yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61C2×C411yescharacteristic
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61C2×C411yescharacteristic
96C2×C4×Dic61C2×C411yescharacteristic
97C2×C4×Dic61C2×C411yescharacteristic
98C2×C4×Dic61D412yes—
99C2×C4×Dic61D412yes—
100C2×C4×Dic61D412yes—
101C2×C4×Dic61D412yes—
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C42×D61—21no—
105C42×D61—21no—
106C42×D61—21no—
107C42×D61—21no—
108#14519706—24no—
109#14519706—24no—
110C22⋊#92418—22no—
111C22⋊#92418C2314yes—
112C8×Dic61—24no—
113C8×Dic61—24no—
114C22×D4×C61C2×C411yescharacteristic
115C23×C4×C61C2×C411yescharacteristic
116C23×C4×C61C2311yescharacteristic
117C22×C8×C61—22no—
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61C2×C411yescharacteristic
122C2×C22⋊C4×C61C2311yescharacteristic
123C2×C42×C61C2×C411yescharacteristic
124C22⋊C8×C61—22no—
125C22⋊C8×C61C2314yes—
126C24×C61C22×C411yescharacteristic
127C24×C61C22×C411yescharacteristic
128C23×D61C22×C411yescharacteristic
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61C2×C814yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—22no—
144C2×C22⋊D61—22no—
145C2×C22⋊D61—22no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61C2×C814yes—
148C2×C22⋊D61C2×C814yescomplemented
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61C22⋊C412yes—
157C22×Dic61C22⋊C412yes—
158C22×Dic61C22⋊C412yes—
159C22×Dic61C22⋊C412yes—
160C22×Dic61C2×C814yes—
161C22×Dic61C2×C814yescomplemented
162C22×Dic61M1614yes—
163C22×Dic61C22×C411yescharacteristic
164C22×Dic61C22×C411yescharacteristic
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C411yescharacteristic
167C22×Dic61C22×C411yescharacteristic
168C22×Dic61C22×C411yescharacteristic
169C22×Dic61C22×C411yescharacteristic
170#11758224—216no—
171#11758224—216no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—22no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—42no—
180C2×C4×D61—42no—
181C2×C4×D61C2×C814yescomplemented
182C2×Dic122—22no—
183C2×Dic122—22no—
184C2×Dic122C2×C814yescomplemented
185C2×C8×C61—24no—
186C2×C8×C61—44no—
187C4⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C42×C61—21no—
191C42×C61—21no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194#11758198—22no—
195#11758198—22no—
196C4⋊Dic61—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C412yes—
216C4×Dic61C22⋊C412yes—
217C4×Dic61C2×D412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D412yes—
221C2×#92418—24no—
222C2×#92418—44no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61C2×C22⋊C412yes—
230C23×C61C22×C812yes—
231C23×C61C22×C812yes—
232C23×C61C2×M1612yes—
233C23×C61C23×C411yescharacteristic
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61C22×C812yes—
239D4×C61—28no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—24no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—42no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C22×C812yes—
255C2×C4×C61C22×D412yes—
256C22⋊D61—28no—
257C22⋊D61—28no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—24no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61C22⋊C818yes—
273C2×Dic61C22⋊C818yes—
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C22×C812yes—
281C2×Dic61C22×C812yes—
282C2×Dic61C22×C812yes—
283C2×Dic61C2×M1612yes—
284C2×Dic61C2×M1612yes—
285C4×D61—24no—
286C4×D61—24no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61—42no—
290C4×D61—42no—
291Dic122—28no—
292C8×C61—44no—
293#92418—44no—
294#17349—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—22no—
297C22×C61—21no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61C2×C22⋊C814yes—
307C22×C61C22×C22⋊C411yescharacteristic
308C22×C61C23×C811yescharacteristic
309C22×C61C22×M1611yescharacteristic
310C2×D61—22no—
311C2×D61—42no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—42no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—21no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61#41012yes—
321Dic61—22no—
322Dic61—21no—
323Dic61—22no—
324Dic61—41no—
325Dic61—41no—
326Dic61C2×C22⋊C814yes—
327Dic61C2×C22⋊C812yes—
328Dic61C2×C22⋊C812yes—
329Dic61#41312yes—
330C2×C4×#41—611no—
331C2×#431—612no—
332#1461—612no—
333#1460—612no—
334#1459—612no—
335#1459—611no—
336#1459D6111yescharacteristic, complemented
337#1458—612no—
338#1458—612no—
339C2×C61—21no—
340C2×C61—21no—
341C2×C61—24no—
342C2×C61—42no—
343C2×C61C22×C22⋊C811yescharacteristic
344C2×C61C2×#41011yescharacteristic
345C2×C61C2×#41311yescharacteristic
346D61—42no—
347C22×#41—611no—
348C4×#41—1222no—
349C4×#41—1222no—
350C4×#41—1222no—
351C4×#41—1222no—
352C2×C4×Q8—611no—
353C2×C4×Q8—611no—
354C2×C4×D4—611no—
355C2×C4×D4—611no—
356C2×C4×D4—611no—
357C2×C4×D4—611no—
358C2×C4×D4—611no—
359C2×C4×D4C2×D6111yescharacteristic
360C2×#117—611no—
361C2×#117—611no—
362C2×#117—611no—
363C22×C42—611no—
364C22×C42—611no—
365C22×C42—611no—
366#431—614no—
367#431—614no—
368#431—1222no—
369#431—1222no—
370C2×C4⋊C8—1222no—
371C2×C22⋊C8—612no—
372C2×C22⋊C8C2×D6112yes—
373C2×C4×C8—1222no—
374#324—612no—
375#324—614no—
376#324C2×D6112yes—
377#323—612no—
378#323—614no—
379#323C2×D6112yes—
380C61#1734911yescharacteristic, complemented, Sylow
381C2×#41—618no—
382C2×#41—1222no—
383C2×#41—1222no—
384C2×#41—1224no—
385C22×Q8—611no—
386C22×D4—611no—
387C22×D4—611no—
388C22×D4C4×D6111yescharacteristic
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4C4×D6111yescharacteristic
397C23×C4C22×D6111yescharacteristic
398C22×C8—22no—
399C22×C8—1222no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416#117—1222no—
417#117—1222no—
418#117—1222no—
419#117—1222no—
420#117—1222no—
421#117—1222no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—611no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—611no—
432C2×C22⋊C4C4×D6111yescharacteristic
433C2×C22⋊C4C22×D6111yescharacteristic
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—612no—
437C2×C42—612no—
438C2×C42—612no—
439C2×C42—612no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—1221no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42C4×D6111yescharacteristic
456C4⋊C8—1224no—
457C4⋊C8—1224no—
458C22⋊C8—22no—
459C22⋊C8—614no—
460C22⋊C8—1222no—
461C22⋊C8C22×D6114yes—
462C4×C8—1224no—
463C4×C8—1224no—
464C24—611no—
465C24C2×C4×D6111yescharacteristic
466C24C2×C4×D6111yescharacteristic
467#41—12216no—
468#41—12216no—
469C2×Q8—614no—
470C2×Q8—1222no—
471C2×Q8—1222no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—614no—
475C2×D4—614no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4C8×D6114yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—614no—
490C22×C4—614no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—612no—
494C22×C4—614no—
495C22×C4—611no—
496C22×C4—612no—
497C22×C4—614no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1222no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1222no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4C2×C4×D6111yescharacteristic
517C22×C4C2×C4×D6111yescharacteristic
518C22×C4C2×C4×D6111yescharacteristic
519C22×C4C2×C4×D6111yescharacteristic
520C22×C4D4×D6112yes—
521C22×C4D4×D6112yes—
522C22×C4C23×D6111yescharacteristic
523C2×C8—24no—
524C2×C8—44no—
525C2×C8—1224no—
526C2×C8—2444no—
527C4⋊C4—22no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C42—21no—
540C42—21no—
541C42—612no—
542C42—612no—
543C42—612no—
544C42—612no—
545C42—611no—
546C42—611no—
547C42—612no—
548C42—611no—
549C42—611no—
550C42—612no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C42—1221no—
557C42—1221no—
558C42—1221no—
559C42—1221no—
560C42—1221no—
561C23—21no—
562C23—21no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—612no—
568C23—1221no—
569C23—1221no—
570C23—2441no—
571C23—2441no—
572C23C22⋊C4×D6112yes—
573C23C2×C8×D6112yes—
574C23C2×C8×D6112yes—
575C23M16×D6112yes—
576C23C22×C4×D6111yescharacteristic
577Q8—1228no—
578D4—28no—
579D4—1228no—
580D4—1228no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—611no—
596C2×C4—612no—
597C2×C4—618no—
598C2×C4—618no—
599C2×C4—612no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—611no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—1224no—
607C2×C4—1224no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2441no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4C22⋊C4×D6111yescharacteristic
627C2×C4C22⋊C4×D6111yescharacteristic
628C2×C4C2×C8×D6112yes—
629C2×C4C2×D4×D6112yes—
630C8—44no—
631C8—2444no—
632C22—21no—
633C22—22no—
634C22—21no—
635C22—22no—
636C22—22no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—1222no—
644C22—2442no—
645C22—2442no—
646C22—2441no—
647C22—2441no—
648C22C22⋊C8×D6114yes—
649C22C2×C22⋊C4×D6111yescharacteristic
650C22C22×C8×D6111yescharacteristic
651C22C2×M16×D6111yescharacteristic
652C4—22no—
653C4—22no—
654C4—21no—
655C4—41no—
656C4—41no—
657C4—612no—
658C4—612no—
659C4—614no—
660C4—612no—
661C4—1222no—
662C4—1222no—
663C4—1221no—
664C4—2441no—
665C4—2441no—
666C4#1683973412yes—
667C2—21no—
668C2—21no—
669C2—24no—
670C2—42no—
671C2—2442no—
672C2C2×C22⋊C8×D6111yescharacteristic
673C2C2×#1683973411yescharacteristic
674C2#413×D6111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC22×C61C23×C8
FrattiniC22×C4C23×D61
FittingC61×#1459C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758224C4
cosocleC22×C4×C61C24
hypercenter#1459D61
hypocenterC61#17349
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. C2×#16839812C2semidirect
3. #21444141C2semidirect
4. #21444138C2semidirect
5. #21444137C2semidirect
6. #21444135C2semidirect
7. #21444132C2semidirect
8. #21444130C2semidirect
9. C61×#1459C2semidirect
46. C2×#11758224C8semidirect
148. C2×C22⋊D61C2×C8semidirect
161. C22×Dic61C2×C8semidirect
181. C2×C4×D61C2×C8semidirect
184. C2×Dic122C2×C8semidirect
336. #1459D61semidirect
380. C61#17349semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582242
C2×#168398122
#214441412
#214441382
#214441372
#214441352
#214441322
#214441302
C61×#14592
#1734961