Finite Group Databasebeta

← #29689036

Subgroups — #29689036

Order 15616.

1842 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#121×D61C211yesmaximal, characteristic
2C22×#14519689C211yesmaximal, characteristic, complemented
3#21448201C211yesmaximal, characteristic, complemented
4#21446248C211yesmaximal, characteristic, complemented
5#21446247C211yesmaximal, characteristic, complemented
6#21446245C211yesmaximal, characteristic, complemented
7#21446140C212yesmaximal, complemented
8#21446138C212yesmaximal, complemented
9#21445908C212yesmaximal, complemented
10#21445748C212yesmaximal, complemented
11C61×#1893C211yesmaximal, characteristic, complemented
12C22×D4×D61—21no—
13C22×D4×D61C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C23×#92418C2211yescharacteristic, complemented
16C2×#14519752C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×#14519750—21no—
19C2×#14519749—21no—
20C2×#14519748C2211yescharacteristic
21#121×D61—24no—
22#121×D61C418yescomplemented
23C2×#14519735C2212yescomplemented
24C2×C4⋊C4×D61C2211yescharacteristic
25C2×C22⋊C4×D61—21no—
26C2×#14519699C2212yescomplemented
27C2×#14519698C2212yescomplemented
28C2×#14519697C2212yescomplemented
29C2×#14519689—24no—
30C2×#14519689C2212yes—
31C2×#14519689C2212yescomplemented
32#16839624—21no—
33#16839622C2211yescharacteristic, complemented
34#16839619C2212yes—
35#16839617—21no—
36C22⋊C4×Dic61—21no—
37#16839609—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C23×C8×C61C2211yescharacteristic, complemented
42C2×#121×C61C2211yescharacteristic
43C2×#102×C61C2212yescomplemented
44C61×#380C2211yescharacteristic, complemented
45C61×#340C2212yescomplemented
46C24×D61—21no—
47C24×D61D411yescharacteristic
48C22×D244—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C2×D4×D61—28no—
54C2×D4×D61—22no—
55C2×D4×D61—42no—
56C2×D4×D61—42no—
57C2×D4×D61—44no—
58C2×D4×D61—44no—
59C2×D4×D61C2×C414yes—
60C23×Dic61D411yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—42no—
63C22×C4×D61—42no—
64C22×C4×D61C2×C414yes—
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61D414yes—
68C22×C4×D61C2311yescharacteristic
69C22×C4×D61C2311yescharacteristic
70C22×#92418—22no—
71C22×#92418—42no—
72C22×#92418C2312yes—
73#14519752—22no—
74#14519752C2×C414yes—
75#14519751—22no—
76#14519751C2×C414yes—
77#14519750—28no—
78#14519749—28no—
79#14519748—22no—
80#14519748C2×C414yes—
81#14519735—42no—
82#14519735—42no—
83C2×C4⋊Dic61C2312yes—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61C2×C414yes—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—22no—
93C2×#11758198—22no—
94C22⋊C4×D61—28no—
95C2×C4×Dic61—22no—
96#14519699—42no—
97#14519698—42no—
98#14519697—42no—
99#14519689—24no—
100#14519689—24no—
101#14519684—21no—
102#14519684—22no—
103C22×D4×C61—21no—
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C22×C8×C61—22no—
107C22×C8×C61—42no—
108C22×C8×C61C2312yes—
109#121×C61—22no—
110#121×C61C2×C414yes—
111C2×C4⋊C4×C61C2312yes—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115#102×C61—42no—
116#95×C61—21no—
117C24×C61—21no—
118C24×C61C2×D411yescharacteristic
119C23×D61—22no—
120C23×D61—22no—
121C23×D61—24no—
122C23×D61—21no—
123C23×D61—24no—
124C23×D61—42no—
125C23×D61—44no—
126C23×D61—42no—
127C23×D61—42no—
128C23×D61—44no—
129C23×D61C22⋊C414yes—
130C23×D61C2×D411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C2×D4×C61—24no—
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61C22×C412yes—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—24no—
148C2×C22⋊D61—24no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C22×Dic61—21no—
156C22×Dic61—22no—
157C22×Dic61—42no—
158C22×Dic61C22⋊C414yes—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161D4×D61—48no—
162D4×D61—44no—
163D4×D61—44no—
164D4×D61—432no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—24no—
168C2×D244—21no—
169C2×D244—42no—
170C2×D244—42no—
171C2×D244C22×C412yes—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—24no—
177C2×C4×D61—48no—
178C2×C4×D61—44no—
179C2×C4×D61—44no—
180C2×C4×D61—42no—
181C2×C4×D61—42no—
182C2×C4×D61—42no—
183C2×C4×D61—42no—
184C2×C4×D61C22⋊C414yes—
185C2×C4×D61C22⋊C414yes—
186C2×C4×D61C22×C412yes—
187C2×C4×D61C22×C412yes—
188C2×C4×D61C2×D412yes—
189C2×C4×D61C2×D412yes—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C2×C8×C61—22no—
193C2×C8×C61—44no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61C22×C412yes—
198C22⋊C4×C61—24no—
199C22⋊C4×C61—42no—
200C22⋊Dic61—24no—
201C22⋊Dic61—42no—
202#11758198—24no—
203#11758198—24no—
204#11758198—44no—
205#11758198—44no—
206C4⋊Dic61—21no—
207C4⋊Dic61—22no—
208C4⋊Dic61—22no—
209C4⋊Dic61C22×C412yes—
210#11758196—24no—
211C4×Dic61—44no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—22no—
215C2×#92418—44no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C2×C22⋊C412yes—
224C23×C61AGL(1,ℤ/8)12yes—
225C23×C61AGL(1,ℤ/8)12yes—
226C23×C61C22×D411yescharacteristic
227C22×D61—22no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—28no—
235C22×D61—28no—
236C22×D61—22no—
237C22×D61—44no—
238C22×D61—48no—
239C22×D61—48no—
240C22×D61—48no—
241C22×D61—48no—
242C22×D61—42no—
243C22×D61—42no—
244C22×D61—41no—
245C22×D61—41no—
246C22×D61—48no—
247C22×D61—88no—
248C22×D61—88no—
249C22×D61C2×C22⋊C412yes—
250C22×D61C2×C22⋊C412yes—
251C22×D61#13514yes—
252D4×C61—24no—
253D4×C61—48no—
254D4×C61—41no—
255D4×C61—41no—
256C2×C4×C61—22no—
257C2×C4×C61—24no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61C2×C22⋊C412yes—
267C2×C4×C61C23×C411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61H(5,𝔽2)12yes—
270C2×C4×C61H(5,𝔽2)12yes—
271C22⋊D61—48no—
272C22⋊D61—48no—
273C22⋊D61—48no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—44no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283C2×Dic61C2×C22⋊C412yes—
284C2×Dic61C2×C22⋊C412yes—
285D244—24no—
286D244—48no—
287D244—41no—
288D244—41no—
289C4×D61—24no—
290C4×D61—24no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—44no—
296C4×D61—44no—
297C4×D61—48no—
298C8×C61—42no—
299#92418—42no—
300#21761—611nomaximal, Sylow
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—24no—
304C22×C61—24no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—82no—
309C22×C61—82no—
310C22×C61#41814yes—
311C22×C61C22×C22⋊C411yescharacteristic
312C22×C61#53412yes—
313C22×C61C2×#13512yes—
314C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—24no—
319C2×D61—24no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—22no—
323C2×D61—416no—
324C2×D61—42no—
325C2×D61—42no—
326C2×D61—42no—
327C2×D61—42no—
328C2×D61—42no—
329C2×D61—816no—
330C2×D61—82no—
331C2×D61—82no—
332C2×D61#41614yes—
333C4×C61—22no—
334C4×C61—42no—
335C4×C61—82no—
336C4×C61#51812yes—
337C4×C61#53411yescharacteristic
338C4×C61#53411yescharacteristic
339Dic61—22no—
340Dic61—22no—
341Dic61—42no—
342Dic61—82no—
343C22×#121—611no—
344C22×C22⋊C8—611no—
345#2280—611no—
346#1893—611no—
347#1893—611no—
348#1893—611no—
349#1893D6111yescharacteristic, complemented
350#1875—612no—
351#1875—612no—
352#1824—612no—
353#1786—612no—
354C2×C61—24no—
355C2×C61—82no—
356C2×C61#230811yescharacteristic
357C2×C61C2×#41611yescharacteristic
358C2×C61C2×#41811yescharacteristic
359C2×C61#318812yes—
360C2×C61#318812yes—
361D61—24no—
362D61—42no—
363D61—82no—
364C23×D4—611no—
365C23×D4—1221no—
366C24×C4—611no—
367C23×C8—611no—
368C23×C8C2×D6111yescharacteristic, complemented
369C2×#121—611no—
370C2×#121—611no—
371C2×#121—618no—
372C2×#121—611no—
373C2×#121—1221no—
374C2×#121—1224no—
375C2×#121—1221no—
376C2×#121C2×D6111yescharacteristic
377C2×#117—612no—
378C22×C4⋊C4—611no—
379C22×C22⋊C4—1221no—
380C2×#102—612no—
381C2×#102—612no—
382C2×#102—612no—
383C2×#102C2×D6112yescomplemented
384C2×C22⋊C8—612no—
385C2×C22⋊C8—612no—
386C2×C22⋊C8—1224no—
387#381—1221no—
388#380—611no—
389#380—612no—
390#380C2×D6111yescharacteristic, complemented
391#379—1221no—
392C4×C22⋊C4—1221no—
393C4×C22⋊C4—1221no—
394#340—612no—
395#340—612no—
396#340—612no—
397#340C2×D6112yescomplemented
398C61#2176111yescharacteristic, complemented, Sylow
399C25—611no—
400C25—1221no—
401C22×D4—21no—
402C22×D4—614no—
403C22×D4—611no—
404C22×D4—1221no—
405C22×D4—1222no—
406C22×D4—1221no—
407C22×D4—1228no—
408C22×D4—1221no—
409C22×D4—1221no—
410C22×D4—2442no—
411C22×D4—2442no—
412C22×D4—2444no—
413C22×D4—2444no—
414C22×D4C22×D6111yescharacteristic
415C23×C4—611no—
416C23×C4—611no—
417C23×C4—611no—
418C23×C4—614no—
419C23×C4—611no—
420C23×C4—611no—
421C23×C4—614no—
422C23×C4—1221no—
423C23×C4—2442no—
424C23×C4—2442no—
425C23×C4C22×D6111yescharacteristic
426C22×C8—22no—
427C22×C8—42no—
428C22×C8—612no—
429C22×C8—1222no—
430C22×C8—2442no—
431C22×C8C22×D6112yes—
432#121—22no—
433#121—614no—
434#121—614no—
435#121—614no—
436#121—1228no—
437#121—1222no—
438#121—1222no—
439#121—1228no—
440#121—1222no—
441#121C4×D6114yes—
442#117—2442no—
443#117—2442no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—614no—
446C2×C4⋊C4—612no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4C22×D6112yes—
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1228no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1222no—
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1221no—
460C2×C42—1222no—
461#102—42no—
462#102—2442no—
463#102—2442no—
464#102—2442no—
465C22⋊C8—1224no—
466C22⋊C8—1224no—
467#95—21no—
468#95—1222no—
469#95—1221no—
470C24—21no—
471C24—611no—
472C24—611no—
473C24—614no—
474C24—1224no—
475C24—1224no—
476C24—1222no—
477C24—1222no—
478C24—1221no—
479C24—2442no—
480C24—2444no—
481C24—2442no—
482C24—2442no—
483C24—2444no—
484C24D4×D6111yescharacteristic
485C2×D4—24no—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4—612no—
492C2×D4—1224no—
493C2×D4—1224no—
494C2×D4—1224no—
495C2×D4—1224no—
496C2×D4—1221no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—2442no—
500C2×D4—2442no—
501C2×D4—2448no—
502C2×D4—2444no—
503C2×D4—2444no—
504C2×D4—2444no—
505C2×D4—24432no—
506C2×D4—2442no—
507C2×D4—2442no—
508C2×D4—2442no—
509C2×D4—2442no—
510C2×D4C2×C4×D6112yes—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—614no—
518C22×C4—612no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—612no—
522C22×C4—614no—
523C22×C4—614no—
524C22×C4—1224no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—1221no—
530C22×C4—1222no—
531C22×C4—2448no—
532C22×C4—2444no—
533C22×C4—2444no—
534C22×C4—2442no—
535C22×C4—2442no—
536C22×C4—2442no—
537C22×C4—2442no—
538C22×C4—2442no—
539C22×C4C2×C4×D6112yes—
540C22×C4D4×D6112yes—
541C22×C4D4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—44no—
547C2×C8—1222no—
548C2×C8—1222no—
549C2×C8—1222no—
550C2×C8—2444no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—612no—
555C4⋊C4—1224no—
556C4⋊C4—1221no—
557C4⋊C4—1222no—
558C4⋊C4—1222no—
559C4⋊C4C2×C4×D6112yes—
560C22⋊C4—24no—
561C22⋊C4—42no—
562C22⋊C4—1224no—
563C22⋊C4—1224no—
564C22⋊C4—1224no—
565C22⋊C4—2444no—
566C22⋊C4—2444no—
567C22⋊C4—2442no—
568C42—2444no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—42no—
576C23—612no—
577C23—614no—
578C23—612no—
579C23—1228no—
580C23—1221no—
581C23—1221no—
582C23—1222no—
583C23—1228no—
584C23—1222no—
585C23—1221no—
586C23—1221no—
587C23—1222no—
588C23—1222no—
589C23—4888no—
590C23—4888no—
591C23—2444no—
592C23—2448no—
593C23—2448no—
594C23—2448no—
595C23—2448no—
596C23—2448no—
597C23—2442no—
598C23—2441no—
599C23—2441no—
600C23—2442no—
601C23C22⋊C4×D6112yes—
602C23#1451980112yes—
603C23#1451980412yes—
604C23C2×D4×D6111yescharacteristic
605D4—24no—
606D4—48no—
607D4—41no—
608D4—41no—
609D4—1224no—
610D4—2448no—
611D4—2448no—
612D4—2448no—
613D4—2448no—
614D4—2441no—
615D4—2441no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—612no—
627C2×C4—612no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1224no—
631C2×C4—1224no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—2444no—
638C2×C4—2444no—
639C2×C4—2448no—
640C2×C4—2444no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4C22⋊C4×D6112yes—
647C2×C4C22×C4×D6111yescharacteristic
648C2×C4C2×D4×D6111yescharacteristic
649C2×C4#1451982212yes—
650C2×C4#1451982312yes—
651C8—42no—
652C8—2442no—
653C22—24no—
654C22—22no—
655C22—24no—
656C22—22no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—82no—
661C22—82no—
662C22—614no—
663C22—1224no—
664C22—1222no—
665C22—1224no—
666C22—1222no—
667C22—1222no—
668C22—1222no—
669C22—1222no—
670C22—48816no—
671C22—4882no—
672C22—4882no—
673C22—24416no—
674C22—2442no—
675C22—2442no—
676C22—2442no—
677C22—2442no—
678C22—2442no—
679C22#1683976514yes—
680C22C2×C22⋊C4×D6111yescharacteristic
681C22#1684032012yes—
682C22#135×D6112yes—
683C22C2×#1451980111yescharacteristic
684C22C2×#1451980411yescharacteristic
685C4—22no—
686C4—42no—
687C4—82no—
688C4—1222no—
689C4—1222no—
690C4—4882no—
691C4—2442no—
692C4#1684023412yes—
693C4#1684032211yescharacteristic
694C4#1684032311yescharacteristic
695C2—24no—
696C2—82no—
697C2—1224no—
698C2—4882no—
699C2—2442no—
700C2#2144839311yescharacteristic
701C2#416×D6111yescharacteristic
702C2C2×#1683976511yescharacteristic
703C2#2145657912yes—
704C2#2145658512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1893C2
socleC23×C61C22×D4
prime radicalC2×#121×D61C2
cosocleC22×C4×C61C24
hypercenter#1893D61
hypocenterC61#21761
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#14519689C2semidirect
3. #21448201C2semidirect
4. #21446248C2semidirect
5. #21446247C2semidirect
6. #21446245C2semidirect
7. #21446140C2semidirect
8. #21446138C2semidirect
9. #21445908C2semidirect
10. #21445748C2semidirect
11. C61×#1893C2semidirect
15. C23×#92418C22semidirect
22. #121×D61C4semidirect
23. C2×#14519735C22semidirect
26. C2×#14519699C22semidirect
27. C2×#14519698C22semidirect
28. C2×#14519697C22semidirect
31. C2×#14519689C22semidirect
33. #16839622C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
41. C23×C8×C61C22semidirect
43. C2×#102×C61C22semidirect
44. C61×#380C22semidirect
45. C61×#340C22semidirect
349. #1893D61semidirect
368. C23×C8C2×D61semidirect
383. C2×#102C2×D61semidirect
390. #380C2×D61semidirect
397. #340C2×D61semidirect
398. C61#21761semidirect

Maximal subgroups

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

subgroupindex
C2×#121×D612
C22×#145196892
#214482012
#214462482
#214462472
#214462452
#214461402
#214461382
#214459082
#214457482
C61×#18932
#2176161