Finite Group Databasebeta

← #29606974

Subgroups — #29606974

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839697C211yesmaximal, characteristic
2C2×C4×#11758198C211yesmaximal, characteristic
3#21444087C211yesmaximal, characteristic, complemented
4#21444078C211yesmaximal, characteristic, complemented
5#21444069C212yesmaximal, complemented
6#21444061C212yesmaximal
7#21444061C211yesmaximal, characteristic
8#21444050C212yesmaximal, complemented
9#21444046C211yesmaximal, characteristic, complemented
10#21444040C212yesmaximal
11C2×C61×#389C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758196C2212yes—
14C22×#11758198C2211yescharacteristic
15C22×#11758198C2211yescharacteristic
16C22×#11758198C2212yes—
17C22×#11758198C2212yes—
18C22×C4×Dic61C2211yescharacteristic
19#16839697C418yescomplemented
20C4×#11758198—24no—
21C2×#14519684C2212yes—
22C2×#14519684C2211yescharacteristic
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2211yescharacteristic
32C22×C4⋊C4×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2212yes—
34C22×C4⋊C4×C61C2211yescharacteristic
35C22×C42×C61C2211yescharacteristic
36C61×#389—24no—
37C2×#95×C61C2212yes—
38C24×D61C2311yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—42no—
47C22×C4×D61—42no—
48C22×C4×D61Q814yes—
49C2×#11758196—42no—
50C2×#11758196—42no—
51C2×#11758196C2×C418yescomplemented
52C2×#11758198—22no—
53C2×#11758198—22no—
54C2×#11758198—22no—
55C2×#11758198—22no—
56C2×#11758198—22no—
57C2×#11758198—24no—
58C2×#11758198—42no—
59C2×#11758198—42no—
60C2×#11758198—42no—
61C2×#11758198—42no—
62C2×#11758198C2×C414yes—
63C2×#11758198C2×C418yes—
64C2×C4×Dic61—22no—
65C2×C4×Dic61—42no—
66#14519684—22no—
67#14519684—22no—
68#14519684—22no—
69#14519684—24no—
70#14519684—24no—
71#14519684—42no—
72#14519684—42no—
73#14519684—42no—
74#14519684—42no—
75#14519684—42no—
76#14519684C2×C414yescomplemented
77C23×C4×C61C2312yes—
78C23×C4×C61C2312yes—
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—24no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—42no—
87C2×C4⋊C4×C61—42no—
88C2×C4⋊C4×C61—42no—
89C2×C4⋊C4×C61C2×C414yescomplemented
90C2×C42×C61—42no—
91C2×C42×C61D414yes—
92#95×C61—24no—
93C24×C61C2411yescharacteristic
94C23×D61—21no—
95C23×D61—21no—
96C23×D61—21no—
97C23×D61—21no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61C22×C412yes—
103C23×D61C2×Q812yes—
104C23×D61#4112yes—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61C22×C414yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×Q812yes—
132C22×C4×C61#4112yes—
133C22×C4×C61#4112yes—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—22no—
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×Dic61C22×C414yes—
160C22×Dic61C22×C412yes—
161C22×Dic61C2×Q812yes—
162C22×Dic61#4112yes—
163C2×C4×D61—28no—
164C2×C4×D61—48no—
165C2×C4×D61—44no—
166C2×C4×D61—44no—
167C2×C4×D61—48no—
168C4⋊C4×C61—28no—
169C4⋊C4×C61—44no—
170C4⋊C4×C61—48no—
171C42×C61—44no—
172#11758198—28no—
173#11758198—28no—
174#11758198—48no—
175#11758198—48no—
176#11758196—48no—
177C4×Dic61—44no—
178C23×C61C23×C411yescharacteristic
179C23×C61C22×D411yescharacteristic
180C23×C61C22×Q811yescharacteristic
181C23×C61C2×#4111yescharacteristic
182C23×C61C2×#4111yescharacteristic
183C23×C61C2×#4111yescharacteristic
184C23×C61C2×#4111yescharacteristic
185C23×C61H(5,𝔽2)12yes—
186C23×C61H(5,𝔽2)12yes—
187C23×C61H(5,𝔽2)12yes—
188C23×C612−1+412yes—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—24no—
198C22×D61—24no—
199C22×D61—24no—
200C22×D61—24no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61C4×Q814yes—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61C4×D414yes—
240C2×C4×C61C22⋊Q814yes—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C4×D61—416no—
271#14317—611nomaximal, Sylow
272C22×C61C2×C4×D411yescharacteristic
273C22×C61C2×C4×Q811yescharacteristic
274C22×C61C4×#4111yescharacteristic
275C22×C61#51812yes—
276C22×C61#52012yes—
277C22×C61C2×C22⋊Q811yescharacteristic
278C22×C61#52511yescharacteristic
279C22×C61#52911yescharacteristic
280C22×C61#53311yescharacteristic
281C22×C61#53411yescharacteristic
282C22×C61#53511yescharacteristic
283C22×C61#53511yescharacteristic
284C22×C61#53611yescharacteristic
285C22×C61#53812yes—
286C22×C61#53812yes—
287C22×C61#53912yes—
288C22×C61#53912yes—
289C22×C61#53912yes—
290C22×C61#54012yes—
291C22×C61#54012yes—
292C22×C61#54212yes—
293C22×C61#54312yes—
294C22×C61#54412yes—
295C2×D61—28no—
296C2×D61—24no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—44no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—44no—
305C2×D61—44no—
306C4×C61—24no—
307C4×C61—24no—
308C4×C61—44no—
309C4×C61—42no—
310C4×C61—82no—
311Dic61—42no—
312Dic61—44no—
313Dic61—42no—
314Dic61—42no—
315Dic61—82no—
316C2×#399—611no—
317C2×#389D6111yescharacteristic, complemented
318C2×C4×C22⋊C4—611no—
319#1445—611no—
320#1442—611no—
321#1441—612no—
322#1439—611no—
323#1439—612no—
324#1436—612no—
325#1434—611no—
326C22⋊#95—612no—
327C2×C61C4×C22⋊Q811yescharacteristic
328C2×C61#232211yescharacteristic
329C2×C61#232811yescharacteristic
330C2×C61#232912yes—
331C2×C61#233512yes—
332C2×C61#262611yescharacteristic
333C2×C61#262811yescharacteristic
334C2×C61#263311yescharacteristic
335C2×C61#263511yescharacteristic
336C2×C61#264912yes—
337C2×C61#266712yes—
338D61—44no—
339C24×C4—611no—
340C22×C4⋊C4—612no—
341C22×C4⋊C4C2×D6111yescharacteristic
342C22×C4⋊C4C2×D6112yes—
343C22×C4⋊C4C2×D6111yescharacteristic
344C22×C22⋊C4—611no—
345C22×C22⋊C4—612no—
346C22×C22⋊C4—611no—
347C22×C22⋊C4—612no—
348C22×C42—611no—
349C22×C42C2×D6111yescharacteristic
350#399—618no—
351#389—24no—
352C4×C22⋊C4—1224no—
353C2×#95—611no—
354C2×#95—612no—
355C2×#95—611no—
356C2×#95—612no—
357C2×#95—611no—
358C2×#95—611no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95—612no—
363C2×#95—612no—
364C2×#95C2×D6112yes—
365C61#1431711yescharacteristic, complemented, Sylow
366C25—611no—
367C23×C4—611no—
368C23×C4—612no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—614no—
372C23×C4—612no—
373C23×C4—1222no—
374C23×C4—1222no—
375C23×C4—2442no—
376C23×C4—2442no—
377C23×C4C22×D6112yes—
378C23×C4C22×D6112yes—
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6111yescharacteristic
381C23×C4C22×D6111yescharacteristic
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—42no—
387C2×C4⋊C4—42no—
388C2×C4⋊C4—42no—
389C2×C4⋊C4—618no—
390C2×C4⋊C4—2442no—
391C2×C4⋊C4—2442no—
392C2×C4⋊C4C4×D6114yescomplemented
393C2×C22⋊C4—614no—
394C2×C22⋊C4—618no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1224no—
401C2×C22⋊C4—2442no—
402C2×C22⋊C4—2442no—
403C2×C22⋊C4—2442no—
404C2×C22⋊C4—2442no—
405C2×C42—42no—
406C2×C42—1222no—
407C2×C42—2442no—
408C2×C42C22⋊D6114yes—
409#95—24no—
410#95—614no—
411#95—1224no—
412#95—1222no—
413#95—1224no—
414#95—1222no—
415#95—1222no—
416#95—2442no—
417#95—2442no—
418#95—2442no—
419#95—2442no—
420#95—2442no—
421C24—612no—
422C24—612no—
423C24—612no—
424C24—1221no—
425C24—1221no—
426C24—1221no—
427C24—1221no—
428C24—1222no—
429C24—1222no—
430C24—1222no—
431C24—1222no—
432C24C23×D6111yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—614no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1228no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—2448no—
485C22×C4—2444no—
486C22×C4—2444no—
487C22×C4—2448no—
488C22×C4C2×C4×D6114yes—
489C22×C4C2×C4×D6112yes—
490C22×C4#1175822212yes—
491C22×C4Q8×D6112yes—
492C22×C4#1175822612yes—
493C22×C4C2×C22⋊D6112yes—
494C22×C4C2×C22⋊D6112yes—
495C22×C4C2×C22⋊D6112yes—
496C4⋊C4—28no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—2448no—
500C22⋊C4—1228no—
501C22⋊C4—1228no—
502C22⋊C4—2448no—
503C22⋊C4—2448no—
504C42—44no—
505C42—2444no—
506C23—614no—
507C23—1222no—
508C23—1222no—
509C23—1224no—
510C23—1224no—
511C23—1222no—
512C23—1222no—
513C23—1222no—
514C23—1222no—
515C23—1224no—
516C23—1224no—
517C23—1224no—
518C23—1224no—
519C23—2441no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523C23—2442no—
524C23—2442no—
525C23—2442no—
526C23—2442no—
527C23—2442no—
528C23—2442no—
529C23C22×C4×D6111yescharacteristic
530C23C22×C22⋊D6111yescharacteristic
531C23C2×Q8×D6111yescharacteristic
532C23#41×D6111yescharacteristic
533C23#41×D6111yescharacteristic
534C23C2×#1175822211yescharacteristic
535C23C2×#1175822611yescharacteristic
536C23#1451982212yes—
537C23#1451982212yes—
538C23#1451982312yes—
539C23#1451982512yes—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—1222no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—24416no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4C4×C22⋊D6114yes—
598C2×C4#1451975314yes—
599C22—1228no—
600C22—1224no—
601C22—2442no—
602C22—2444no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22—2444no—
607C22—2442no—
608C22—2442no—
609C22—2444no—
610C22C2×C4×C22⋊D6111yescharacteristic
611C22C4×Q8×D6111yescharacteristic
612C22C4×#1175822611yescharacteristic
613C22#1684023412yes—
614C22#1684024312yes—
615C22C2×#1451975311yescharacteristic
616C22#1684026911yescharacteristic
617C22#1684029911yescharacteristic
618C22#1684031611yescharacteristic
619C22#1684032111yescharacteristic
620C22#1684032611yescharacteristic
621C22#1684032911yescharacteristic
622C22#1684033611yescharacteristic
623C22#1684034412yes—
624C22#1684035212yes—
625C22#1684035912yes—
626C22#1684036012yes—
627C22#1684036012yes—
628C22#1684036212yes—
629C22#1684036312yes—
630C22#1684038612yes—
631C22#1684038712yes—
632C22#1684038912yes—
633C4—24no—
634C4—24no—
635C4—44no—
636C4—42no—
637C4—82no—
638C4—4882no—
639C4—2442no—
640C4—2442no—
641C4—2444no—
642C4—2442no—
643C2—2444no—
644C2C4×#1451975311yescharacteristic
645C2#2144848811yescharacteristic
646C2#2144854111yescharacteristic
647C2#2144855312yes—
648C2#2144861212yes—
649C2#2145162611yescharacteristic
650C2#2145164111yescharacteristic
651C2#2145168811yescharacteristic
652C2#2145170411yescharacteristic
653C2#2145182812yes—
654C2#2145201712yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#389C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenterC2×#389D61
hypocenterC61#14317
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. #21444087C2semidirect
4. #21444078C2semidirect
5. #21444069C2semidirect
8. #21444050C2semidirect
9. #21444046C2semidirect
11. C2×C61×#389C2semidirect
19. #16839697C4semidirect
51. C2×#11758196C2×C4semidirect
76. #14519684C2×C4semidirect
89. C2×C4⋊C4×C61C2×C4semidirect
317. C2×#389D61semidirect
365. C61#14317semidirect
392. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396972
C2×C4×#117581982
#214440872
#214440782
#214440692
#214440612
#214440502
#214440462
#214440402
C2×C61×#3892
#1431761