Finite Group Databasebeta

← #29690014

Subgroups — #29690014

Order 15616.

1242 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459115C211yesmaximal, characteristic
2C2×C4×C4⋊Dic61C211yesmaximal, characteristic, complemented
3#21446763C212yesmaximal, complemented
4#21446738C212yesmaximal, complemented
5#21446260C214yesmaximal, complemented
6#21445836C214yesmaximal, complemented
7C61×#2920C211yesmaximal, characteristic, complemented
8#16840319C411yescharacteristic, complemented
9#16840319C411yescharacteristic, complemented
10#16840319C2211yescharacteristic
11#16840312C411yescharacteristic, complemented
12#16840312C411yescharacteristic, complemented
13#16840312C2211yescharacteristic
14#16840296—21no—
15#16840296—21no—
16#16840296—21no—
17#16840296—21no—
18#16840296C412yescomplemented
19#16840296C412yescomplemented
20#16840273—21no—
21#16840273—21no—
22#16840273C2212yes—
23C4×#11758222—21no—
24C4×#11758222—21no—
25C4×#11758222C2212yes—
26C22×C4⋊Dic61—21no—
27C22×C4×Dic61C2212yes—
28C2×#14519703C2218yes—
29C4×C4⋊Dic61—21no—
30C4×C4⋊Dic61—21no—
31C4×C4⋊Dic61C2212yes—
32C4×C4⋊Dic61C2212yes—
33C2×#14519684—21no—
34#16839523C2218yes—
35C22×C42×C61C2211yescharacteristic
36C61×#432—21no—
37C61×#432—21no—
38C61×#431—21no—
39C61×#431—21no—
40C61×#423C2212yes—
41C61×#407C2212yes—
42C4×M16×C61C2212yes—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222C2312yes—
46C23×Dic61—22no—
47#14519777C2×C411yescharacteristic, complemented
48#14519777C2×C411yescharacteristic, complemented
49#14519777C2×C411yescharacteristic, complemented
50#14519774C2×C411yescharacteristic, complemented
51#14519771—22no—
52#14519771—22no—
53#14519769—21no—
54#14519769—21no—
55#14519769C2×C412yescomplemented
56#14519765—21no—
57#14519765—21no—
58#14519765C2×C412yescomplemented
59#14519760—22no—
60#14519760—22no—
61#14519760—22no—
62#14519756—21no—
63#14519756—21no—
64#14519756—21no—
65#14519756—21no—
66#14519756C2×C412yes—
67#14519756C2×C412yes—
68#14519751—21no—
69#14519751—21no—
70#14519751—21no—
71#14519751—21no—
72#14519751C2×C412yes—
73#14519751C2×C412yes—
74#14519746—22no—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×Dic122—21no—
79C4×Dic122C2×C412yescomplemented
80C4×Dic122C2×C412yescomplemented
81C4×D244—21no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244—21no—
85C4×D244C2×C412yescomplemented
86C4×D244C2×C412yescomplemented
87C4×C22⋊D61—22no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—22no—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92C4×C22⋊D61—22no—
93#14519736—21no—
94#14519736—21no—
95#14519736C2312yes—
96#14519734—22no—
97#14519734—22no—
98#14519734—22no—
99C2×C4⋊Dic61—28no—
100C2×C4⋊Dic61—22no—
101C2×C4⋊Dic61—22no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61—44no—
104C2×C4×Dic61C2314yes—
105C42×D61—22no—
106#14519703—48no—
107#14519684—22no—
108#14519684—22no—
109C23×C4×C61D411yescharacteristic
110C23×C4×C61D411yescharacteristic
111C23×C4×C61C2311yescharacteristic
112C2×M16×C61C2314yes—
113C2×C42×C61C2×C411yescharacteristic
114C2×C42×C61C2×C411yescharacteristic
115C2×C42×C61C2×C411yescharacteristic
116C2×C42×C61C2×C411yescharacteristic
117C2×C42×C61D411yescharacteristic
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61D411yescharacteristic
122C2×C42×C61D411yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C4⋊C8×C61—22no—
126C4⋊C8×C61—22no—
127C22⋊C8×C61—22no—
128C22⋊C8×C61—22no—
129#97×C61—22no—
130C4×C8×C61—22no—
131C24×C61C2×D411yescharacteristic
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—42no—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C22×Dic61—28no—
158C22×Dic61—24no—
159C22×Dic61—44no—
160C22×Dic61—44no—
161#11758222—24no—
162#11758222—24no—
163#11758222—44no—
164#11758222—44no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244C22×C412yes—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174M16×C61—24no—
175M16×C61—24no—
176C2×C8×C61—24no—
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22⋊C411yescharacteristic
184C42×C61C22⋊C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C22×C411yescharacteristic
187C42×C61C22×C411yescharacteristic
188C42×C61C22×C411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202C4⋊Dic61—28no—
203C4⋊Dic61—28no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C22×C412yes—
209C4⋊Dic61C22×C412yes—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—28no—
220C4×Dic61—44no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61C2×C22⋊C411yescharacteristic
226C23×C61C2×C22⋊C411yescharacteristic
227C23×C61C22×D411yescharacteristic
228C22×D61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61#10414yescomplemented
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61H(5,𝔽2)12yes—
260C2×C4×C612−1+412yes—
261C22⋊D61—42no—
262C22⋊D61—42no—
263C22⋊D61—44no—
264C22⋊D61—44no—
265C2×Dic61—28no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—416no—
270C2×Dic61—44no—
271C2×Dic61—44no—
272D244—24no—
273D244—42no—
274D244—42no—
275C4×D61—44no—
276C4×D61—44no—
277C4×D61—44no—
278Dic122—24no—
279Dic122—42no—
280Dic122—42no—
281C8×C61—44no—
282#21899—611nomaximal, Sylow
283C22×C61—24no—
284C22×C61—24no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61C2×#10412yes—
292C22×C61#42112yes—
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61#53511yescharacteristic
295C22×C61#53611yescharacteristic
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61C2×#10412yes—
308C4×C61C2×#10412yes—
309C4×C61#52012yes—
310C4×C61#53712yes—
311Dic61—48no—
312Dic61—42no—
313Dic61—42no—
314Dic61—42no—
315#3444—611no—
316#2920D6111yescharacteristic, complemented
317C2×C4×C4⋊C4—611no—
318#1986—612no—
319#1980—612no—
320#1896—614no—
321#1806—614no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—22no—
325C2×C61—22no—
326C2×C61#231111yescharacteristic
327C2×C61C22×#10411yescharacteristic
328C2×C61C2×#42111yescharacteristic
329D61—82no—
330#533—611no—
331#533—611no—
332#533—611no—
333#532—611no—
334#532—611no—
335#532—611no—
336#529—612no—
337#529—612no—
338#529—1221no—
339#529—1221no—
340#529—1221no—
341#529—1221no—
342#525—612no—
343#525—1221no—
344#525—1221no—
345C4×#41—612no—
346C4×#41—1221no—
347C4×#41—1221no—
348C22×C4⋊C4—1221no—
349C22×C42—612no—
350C22×C42C2×D6111yescharacteristic
351#432—21no—
352#432—21no—
353#431—21no—
354#431—21no—
355#423C2×D6112yes—
356C2×#104—618no—
357#407C2×D6112yes—
358C4×M16C2×D6112yes—
359C4×C4⋊C4—612no—
360C4×C4⋊C4—612no—
361C4×C4⋊C4—1221no—
362C4×C4⋊C4—1221no—
363C2×#95—1221no—
364#339—618no—
365C61#2189911yescharacteristic, complemented, Sylow
366C2×#41—612no—
367C2×#41—1221no—
368C2×#41—1221no—
369C23×C4—1222no—
370C23×C4D24411yescharacteristic
371C23×C4D24411yescharacteristic
372C23×C4C22×D6111yescharacteristic
373C2×M16C22×D6114yes—
374C4⋊Q8—611no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377#127—611no—
378#126—1222no—
379#126—1222no—
380#125—612no—
381#125—1221no—
382#125—1221no—
383#124—612no—
384#124—1221no—
385#124—1221no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389C22⋊Q8—612no—
390C22⋊Q8—612no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395#121—612no—
396#121—612no—
397#121—1221no—
398#121—1221no—
399#121—1221no—
400#121—1221no—
401#120—1222no—
402C4×Q8—612no—
403C4×Q8—612no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×Q8—1221no—
408C4×D4—612no—
409C4×D4—612no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1221no—
420#117—612no—
421#117—1221no—
422#117—1221no—
423#117—1222no—
424#117—1222no—
425#117—1222no—
426C2×C4⋊C4—1228no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1222no—
429C2×C42—614no—
430C2×C42—1222no—
431C2×C42—1222no—
432C2×C42—2444no—
433C2×C42C4×D6111yescharacteristic
434C2×C42C4×D6111yescharacteristic
435C2×C42C4×D6111yescharacteristic
436C2×C42C4×D6111yescharacteristic
437C2×C42D24411yescharacteristic
438C2×C42D24411yescharacteristic
439C2×C42C22⋊D6111yescharacteristic
440C2×C42C22⋊D6111yescharacteristic
441C2×C42C22⋊D6111yescharacteristic
442C2×C42C22⋊D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445C4⋊C8—22no—
446C4⋊C8—22no—
447#104—2448no—
448C22⋊C8—22no—
449C22⋊C8—22no—
450#97—22no—
451C4×C8—22no—
452#95—1222no—
453#95—1222no—
454C24C2×D24411yescharacteristic
455#41—1224no—
456#41—1224no—
457#41—2444no—
458#41—2444no—
459C2×Q8—612no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×D4—612no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1221no—
467C2×D4—1221no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—42no—
472C22×C4—1228no—
473C22×C4—1224no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—2444no—
478C22×C4—2444no—
479C22×C4#1175819811yescharacteristic
480C22×C4#1175819811yescharacteristic
481C22×C4#1175819811yescharacteristic
482C22×C4#1175819811yescharacteristic
483C22×C4#1175819811yescharacteristic
484C22×C4#1175819811yescharacteristic
485C22×C4#1175819811yescharacteristic
486C22×C4#1175819811yescharacteristic
487C22×C4C2×C4×D6111yescharacteristic
488C22×C4C2×C4×D6111yescharacteristic
489C22×C4C2×D24411yescharacteristic
490C22×C4C2×D24411yescharacteristic
491C22×C4C2×D24411yescharacteristic
492C22×C4C2×D24411yescharacteristic
493C22×C4C2×D24411yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497M16—24no—
498M16—24no—
499C2×C8—24no—
500C4⋊C4—612no—
501C4⋊C4—612no—
502C4⋊C4—1228no—
503C4⋊C4—1228no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C42—1228no—
527C42—2444no—
528C42#1175819811yescharacteristic
529C42#1175819811yescharacteristic
530C42#1175819811yescharacteristic
531C42#1175819811yescharacteristic
532C42#1175819811yescharacteristic
533C42#1175819811yescharacteristic
534C42#1175819811yescharacteristic
535C42#1175819811yescharacteristic
536C42C2×C4×D6111yescharacteristic
537C42C2×C4×D6111yescharacteristic
538C42C2×C4×D6111yescharacteristic
539C42C2×C4×D6111yescharacteristic
540C42C2×C22⋊D6111yescharacteristic
541C42C2×C22⋊D6111yescharacteristic
542C42C2×C22⋊D6111yescharacteristic
543C42C2×C22⋊D6111yescharacteristic
544C23—22no—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—1222no—
549C23C2×#1175819811yescharacteristic
550C23C2×#1175819811yescharacteristic
551C23C22×D24411yescharacteristic
552Q8—1224no—
553Q8—2442no—
554Q8—2442no—
555D4—1224no—
556D4—2442no—
557D4—2444no—
558D4—2444no—
559D4—2442no—
560D4—2442no—
561D4—2442no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—1228no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—2444no—
584C2×C4—24416no—
585C2×C4—2444no—
586C2×C4—2444no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4#1451970314yescomplemented
590C2×C4C2×#1175819811yescharacteristic
591C2×C4C2×#1175819811yescharacteristic
592C2×C4C2×#1175819811yescharacteristic
593C2×C4C2×#1175819811yescharacteristic
594C2×C4C2×#1175819811yescharacteristic
595C2×C4C2×#1175819811yescharacteristic
596C2×C4C2×#1175819811yescharacteristic
597C2×C4C2×#1175819811yescharacteristic
598C2×C4C2×#1175819811yescharacteristic
599C2×C4C2×#1175819811yescharacteristic
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C2×C4#1451982312yes—
603C2×C4#1451982412yes—
604C8—44no—
605C22—24no—
606C22—24no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22C2×#1451970312yes—
617C22#1683977812yes—
618C22C22×#1175819811yescharacteristic
619C22#1684033111yescharacteristic
620C22#1684033811yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—2448no—
630C4—2442no—
631C4—2442no—
632C4—2442no—
633C4C2×#1451970312yes—
634C4C2×#1451970312yes—
635C4#1684024212yes—
636C4#1684034012yes—
637C2—21no—
638C2—22no—
639C2—22no—
640C2—21no—
641C2—4882no—
642C2#2144840711yescharacteristic
643C2C22×#1451970311yescharacteristic
644C2C2×#1683977811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2920C2
socleC22×C61C22×C22⋊C4
prime radical#21459115C2
cosocleC22×C4×C61C24
hypercenter#2920D61
hypocenterC61#21899
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ #2920

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Dic61C2semidirect
3. #21446763C2semidirect
4. #21446738C2semidirect
5. #21446260C2semidirect
6. #21445836C2semidirect
7. C61×#2920C2semidirect
8. #16840319C4semidirect
9. #16840319C4semidirect
11. #16840312C4semidirect
12. #16840312C4semidirect
18. #16840296C4semidirect
19. #16840296C4semidirect
47. #14519777C2×C4semidirect
48. #14519777C2×C4semidirect
49. #14519777C2×C4semidirect
50. #14519774C2×C4semidirect
55. #14519769C2×C4semidirect
58. #14519765C2×C4semidirect
79. C4×Dic122C2×C4semidirect
80. C4×Dic122C2×C4semidirect
85. C4×D244C2×C4semidirect
86. C4×D244C2×C4semidirect
246. C2×C4×C61#104semidirect
316. #2920D61semidirect
365. C61#21899semidirect
589. C2×C4#14519703semidirect

Maximal subgroups

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

subgroupindex
#214591152
C2×C4×C4⋊Dic612
#214467632
#214467382
#214462602
#214458362
C61×#29202
#2189961