Finite Group Databasebeta

← #29689073

Subgroups — #29689073

Order 15616.

1486 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
1C2×#16840272C211yesmaximal, characteristic
2C2×#16839725C211yesmaximal, characteristic, complemented
3#21448201C211yesmaximal, characteristic, complemented
4#21446258C211yesmaximal, characteristic, complemented
5#21446252C211yesmaximal, characteristic, complemented
6#21446250C211yesmaximal, characteristic, complemented
7#21446138C212yesmaximal, complemented
8#21446130C212yesmaximal, complemented
9#21445913C212yesmaximal, complemented
10#21445756C212yesmaximal, complemented
11C61×#1895C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15#16840272—24no—
16#16840272C418yescomplemented
17C2×#14519759—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×#14519745—21no—
22C2×D4×Dic61—21no—
23C2×#14519737—21no—
24C2×#14519735C2211yescharacteristic
25C2×#14519735C2212yes—
26C2×#14519702C2212yescomplemented
27C2×#14519700C2212yescomplemented
28C2×#14519698C2212yescomplemented
29#16839725—24no—
30C2×#14519689C2212yes—
31C2×#14519689C2212yes—
32#16839624—21no—
33#16839622C2211yescharacteristic
34#16839619C2212yes—
35#16839617—21no—
36C22⋊C4×Dic61—21no—
37#16839609—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×M16×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×#102×C61C2212yescomplemented
44C61×#380C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C2×#11758224—22no—
48C2×#11758224—42no—
49C2×#11758224—42no—
50C2×#11758224C2×C414yes—
51C22×Dic122C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61D411yescharacteristic
56C22×C4×D61—42no—
57C22×C4×D61—42no—
58C22×C4×D61C2×C414yes—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D414yes—
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C2×#11758194—22no—
65C2×#11758194—42no—
66C22×#92418C2312yes—
67#14519759—28no—
68#14519757—22no—
69#14519757C2×C414yes—
70#14519756—22no—
71#14519756C2×C414yes—
72#14519748—22no—
73#14519748C2×C414yes—
74#14519745—28no—
75D4×Dic61—28no—
76#14519737—28no—
77#14519735—22no—
78#14519735—42no—
79#14519735—42no—
80#14519735C2×C414yes—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—22no—
91C2×#11758198—22no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—22no—
94#14519702—42no—
95#14519700—42no—
96#14519698—42no—
97#14519689—24no—
98#14519689—24no—
99#14519684—21no—
100#14519684—22no—
101C22×D4×C61—21no—
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—22no—
105C2×M16×C61—42no—
106C22×C8×C61C2312yes—
107#121×C61—22no—
108#121×C61C2×C414yes—
109C2×C4⋊C4×C61C2312yes—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113#102×C61—42no—
114#95×C61—21no—
115C24×C61—21no—
116C24×C61C2×D411yescharacteristic
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—41no—
120C23×D61—41no—
121C23×D61C22⋊C414yes—
122C23×D61C2×D411yescharacteristic
123C23×D61C2×D411yescharacteristic
124C2×D4×C61—24no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—44no—
129C2×D4×C61C22×C412yes—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—24no—
139C2×C22⋊D61—24no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C2×C22⋊D61—44no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—24no—
146C22×Dic61—22no—
147C22×Dic61—42no—
148C22×Dic61—44no—
149C22×Dic61C22⋊C414yes—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152#11758224—48no—
153#11758224—44no—
154#11758224—44no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—48no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
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×Dic122C22×C412yes—
174M16×C61—44no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C22×C412yes—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—42no—
184C22⋊Dic61—24no—
185C22⋊Dic61—24no—
186C22⋊Dic61—42no—
187#11758198—24no—
188#11758198—24no—
189#11758198—44no—
190#11758198—44no—
191C4⋊Dic61—21no—
192C4⋊Dic61—24no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22×C412yes—
196#11758196—24no—
197C4×Dic61—24no—
198C4×Dic61—44no—
199#11758194—44no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—42no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—42no—
212C23×C61C2×C22⋊C412yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—22no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—44no—
223C22×D61—44no—
224C22×D61—44no—
225C22×D61—48no—
226C22×D61—48no—
227C22×D61C2×C22⋊C412yes—
228C22×D61C2×C22⋊C412yes—
229D4×C61—24no—
230D4×C61—48no—
231D4×C61—41no—
232D4×C61—41no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61H(5,𝔽2)12yes—
246C2×C4×C61H(5,𝔽2)12yes—
247C22⋊D61—48no—
248C22⋊D61—48no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—44no—
254C2×Dic61—48no—
255C2×Dic61—42no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C4×D61—24no—
264C4×D61—24no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—44no—
270C4×D61—44no—
271Dic122—24no—
272Dic122—41no—
273Dic122—41no—
274C8×C61—42no—
275#92418—42no—
276#21766—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—82no—
286C22×C61—82no—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61#53412yes—
289C22×C61#57511yescharacteristic
290C22×C61#57512yes—
291C22×C61#57511yescharacteristic
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—416no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C4×C61—22no—
304C4×C61—42no—
305C4×C61—82no—
306C4×C61#51812yes—
307C4×C61#53411yescharacteristic
308C4×C61#53411yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—42no—
312Dic61—42no—
313Dic61—82no—
314C2×#525—611no—
315C2×#407—611no—
316#2280—611no—
317#1895—611no—
318#1895D6111yescharacteristic, complemented
319#1894—611no—
320#1894—611no—
321#1875—612no—
322#1873—612no—
323#1825—612no—
324#1788—612no—
325C2×C61—42no—
326C2×C61—82no—
327C2×C61#230811yescharacteristic
328C2×C61#289611yescharacteristic
329C2×C61#289611yescharacteristic
330C2×C61#319712yes—
331C2×C61#319712yes—
332D61—42no—
333D61—42no—
334C22×#41—611no—
335C24×C4—611no—
336C22×M16—611no—
337C22×M16C2×D6111yescharacteristic
338#525—618no—
339#525—1224no—
340C2×#123—1221no—
341C2×C22⋊Q8—611no—
342C2×C22⋊Q8—611no—
343C2×#121—611no—
344C2×#121C2×D6111yescharacteristic
345C2×#120—1221no—
346C2×C4×D4—1221no—
347C2×C4×D4—1221no—
348C2×#117—611no—
349C2×#117—612no—
350C2×#103—612no—
351C2×#103—612no—
352C2×#102—612no—
353C2×#102C2×D6112yescomplemented
354#407—1224no—
355C2×C22⋊C8—612no—
356C2×C22⋊C8—612no—
357#381—1221no—
358#380—611no—
359#380—612no—
360#380C2×D6111yescharacteristic
361#379—1221no—
362C4×C22⋊C4—1221no—
363C4×C22⋊C4—1221no—
364#340—612no—
365#340—612no—
366#340—612no—
367#340C2×D6112yescomplemented
368C61#2176611yescharacteristic, complemented, Sylow
369C25—611no—
370C2×#41—614no—
371C2×#41—1222no—
372C2×#41—2442no—
373C2×#41—2442no—
374C22×Q8—611no—
375C22×D4—21no—
376C22×D4—1221no—
377C22×D4—1221no—
378C22×D4C22×D6111yescharacteristic
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—614no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—614no—
386C23×C4—1221no—
387C23×C4—2442no—
388C23×C4—2442no—
389C23×C4C22×D6111yescharacteristic
390C2×M16—22no—
391C2×M16—42no—
392C2×M16—1222no—
393C2×M16—2442no—
394C22×C8—612no—
395C22×C8C22×D6112yes—
396#123—1228no—
397C22⋊Q8—614no—
398C22⋊Q8—614no—
399C22⋊Q8—1222no—
400C22⋊Q8—1222no—
401#121—22no—
402#121—614no—
403#121—1222no—
404#121C4×D6114yes—
405#120—1228no—
406C4×D4—1228no—
407C4×D4—1228no—
408#117—614no—
409#117—1222no—
410#117—2442no—
411#117—2442no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6112yes—
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1222no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1221no—
427C2×C42—1221no—
428C2×C42—1222no—
429#103—2442no—
430#103—2442no—
431#102—42no—
432#102—2442no—
433C22⋊C8—1224no—
434C22⋊C8—1224no—
435#95—21no—
436#95—1222no—
437#95—1221no—
438C24—21no—
439C24—611no—
440C24—611no—
441C24—614no—
442C24—2442no—
443C24—2441no—
444C24—2442no—
445C24—2441no—
446C24D4×D6111yescharacteristic
447#41—2448no—
448#41—2444no—
449#41—2444no—
450C2×Q8—612no—
451C2×Q8—1221no—
452C2×Q8—1222no—
453C2×Q8—1222no—
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4—1224no—
460C2×D4—1224no—
461C2×D4—2444no—
462C2×D4—2442no—
463C2×D4—2442no—
464C2×D4C2×C4×D6112yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—614no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—614no—
477C22×C4—614no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—2448no—
487C22×C4—2444no—
488C22×C4—2444no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2444no—
493C22×C4C2×C4×D6112yes—
494C22×C4D4×D6112yes—
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497M16—44no—
498M16—2444no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—612no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1224no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4C2×C4×D6112yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—1224no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—2444no—
522C22⋊C4—2444no—
523C22⋊C4—2442no—
524C42—1224no—
525C42—2444no—
526C23—21no—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—612no—
536C23—612no—
537C23—1222no—
538C23—1221no—
539C23—1222no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1222no—
544C23—1222no—
545C23—2444no—
546C23—2444no—
547C23—2444no—
548C23—2448no—
549C23—2448no—
550C23C22⋊C4×D6112yes—
551C23C2×D4×D6111yescharacteristic
552Q8—1224no—
553Q8—2441no—
554Q8—2441no—
555D4—24no—
556D4—48no—
557D4—41no—
558D4—41no—
559D4—2448no—
560D4—2448no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4—2448no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2442no—
592C2×C4C22⋊C4×D6112yes—
593C2×C4C22×C4×D6111yescharacteristic
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4#1451982212yes—
596C2×C4#1451982312yes—
597C8—42no—
598C8—2442no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—82no—
608C22—82no—
609C22—1222no—
610C22—1222no—
611C22—1222no—
612C22—1222no—
613C22—1222no—
614C22—1222no—
615C22—24416no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22C2×C22⋊C4×D6111yescharacteristic
621C22#1684032012yes—
622C22#1684055811yescharacteristic
623C22#1684055812yes—
624C22#1684056111yescharacteristic
625C4—22no—
626C4—42no—
627C4—82no—
628C4—1222no—
629C4—1222no—
630C4—4882no—
631C4—2442no—
632C4—2442no—
633C4#1684023412yes—
634C4#1684032211yescharacteristic
635C4#1684032311yescharacteristic
636C2—42no—
637C2—82no—
638C2—2442no—
639C2—2442no—
640C2#2144839311yescharacteristic
641C2#2145433311yescharacteristic
642C2#2145433511yescharacteristic
643C2#2145665512yes—
644C2#2145665712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1895C2
socleC23×C61C22×D4
prime radicalC2×#16840272C2
cosocleC22×C4×C61C24
hypercenter#1895D61
hypocenterC61#21766
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
2. C2×#16839725C2semidirect
3. #21448201C2semidirect
4. #21446258C2semidirect
5. #21446252C2semidirect
6. #21446250C2semidirect
7. #21446138C2semidirect
8. #21446130C2semidirect
9. #21445913C2semidirect
10. #21445756C2semidirect
11. C61×#1895C2semidirect
16. #16840272C4semidirect
26. C2×#14519702C22semidirect
27. C2×#14519700C22semidirect
28. C2×#14519698C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#102×C61C22semidirect
45. C61×#340C22semidirect
318. #1895D61semidirect
353. C2×#102C2×D61semidirect
367. #340C2×D61semidirect
368. C61#21766semidirect

Maximal subgroups

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

subgroupindex
C2×#168402722
C2×#168397252
#214482012
#214462582
#214462522
#214462502
#214461382
#214461302
#214459132
#214457562
C61×#18952
#2176661