Finite Group Databasebeta

← #29681655

Subgroups — #29681655

Order 15616.

1742 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840336C211yesmaximal, characteristic
2#21459020C211yesmaximal, characteristic
3C2×#16839735C211yesmaximal, characteristic
4#21448405C211yesmaximal, characteristic, complemented
5#21446229C218yesmaximal, complemented
6#21445552C211yesmaximal, characteristic, complemented
7#21445551C211yesmaximal, characteristic, complemented
8C61×#2310C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10#16840336—24no—
11#16840336C418yescomplemented
12C2×#14519775—21no—
13C2×#14519766—21no—
14C2×#14519760C2214yes—
15C2×#14519753—22no—
16#16840243—22no—
17#16840242—22no—
18#16840239—21no—
19#16840236—21no—
20#16840235C2214yes—
21C4×#11758222—22no—
22#16840230—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25C2×#14519736C2211yescharacteristic
26#117×D61—22no—
27#16839735—24no—
28C2×#14519691C2214yes—
29#16839669—22no—
30#16839669—22no—
31#16839625C2214yes—
32#16839529C2214yes—
33#16839528C2214yes—
34C23×Q8×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#391—22no—
38C61×#391—22no—
39C61×#381C2214yes—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222C2×C414yes—
47C2×#11758222D414yes—
48C22×Dic122D411yescharacteristic
49C22×D244D411yescharacteristic
50C22×C22⋊D61C2312yes—
51C22×C4×D61D412yes—
52#14519775—28no—
53#14519766—28no—
54#14519760—28no—
55#14519760C2×C4116yescomplemented
56#14519753—216no—
57C4×Dic122—24no—
58#14519742—24no—
59C4×D244—24no—
60#14519739—24no—
61C4×C22⋊D61—28no—
62#14519737—28no—
63#14519736—21no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736—22no—
68#14519736—22no—
69#14519736—22no—
70#14519736C2×C414yescomplemented
71#14519735—24no—
72#14519734—24no—
73#14519733—24no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D412yes—
77C2×#11758196—24no—
78C4⋊C4×D61—24no—
79C2×C22⋊Dic61C2312yes—
80C2×C22⋊Dic61C2312yes—
81C2×C22⋊Dic61C2312yes—
82C2×#11758198—24no—
83C22⋊C4×D61—24no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61D412yes—
87C42×D61—24no—
88#14519691—28no—
89#14519684—28no—
90C22×Q8×C61—21no—
91C22×Q8×C61—21no—
92C22×Q8×C61—21no—
93C22×Q8×C61—21no—
94C22×Q8×C61—22no—
95C22×Q8×C61—21no—
96C22×Q8×C61—21no—
97C22×Q8×C61—22no—
98C22×Q8×C61—22no—
99C22×Q8×C61C2×C414yescomplemented
100C23×C4×C61C2312yes—
101C23×C4×C61C2311yescharacteristic
102#117×C61—21no—
103#117×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61—22no—
107#117×C61—22no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61D412yes—
110C2×C22⋊C4×C61C2312yes—
111C2×C22⋊C4×C61C2312yes—
112C2×C42×C61—21no—
113C2×C42×C61D412yes—
114#95×C61—28no—
115C24×C61C2411yescharacteristic
116C23×D61C2×D412yes—
117C2×Q8×C61—24no—
118C2×Q8×C61—24no—
119C2×Q8×C61—24no—
120C2×Q8×C61—24no—
121C2×Q8×C61—24no—
122C2×Q8×C61—24no—
123C2×Q8×C61—41no—
124C2×Q8×C61—41no—
125C2×Q8×C61—41no—
126C2×Q8×C61—41no—
127C2×Q8×C61—42no—
128C2×Q8×C61—42no—
129C2×Q8×C61—42no—
130C2×Q8×C61—42no—
131C2×Q8×C61—42no—
132C2×Q8×C61—42no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61C22×C414yes—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C2×D414yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61#4112yes—
151C22×C4×C61#4112yes—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61C22×C414yes—
157C2×C22⋊D61C2×D414yes—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D414yes—
162#11758222—24no—
163#11758222—24no—
164#11758222—24no—
165#11758222—24no—
166#11758222—24no—
167#11758222—24no—
168#11758222—24no—
169#11758222—24no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—22no—
173C2×D244C22⋊C414yes—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—24no—
177C2×C4×D61—28no—
178C2×C4×D61C22⋊C418yes—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—22no—
182C2×Dic122C22⋊C414yes—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—42no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61—42no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61—22no—
198C22⋊Dic61C22×C414yes—
199#11758198—28no—
200#11758198—216no—
201C4⋊Dic61—24no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61—42no—
205#11758196—28no—
206#11758196—216no—
207C4×Dic61—28no—
208C4×Dic61—24no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—42no—
212C23×C61—42no—
213C23×C61C23×C411yescharacteristic
214C23×C61C22×D411yescharacteristic
215C23×C61C22×D411yescharacteristic
216C23×C61C22×D411yescharacteristic
217C23×C61C22×D411yescharacteristic
218C23×C61C2×#4111yescharacteristic
219C23×C61C2×#4111yescharacteristic
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—44no—
223C22×D61C2×C22⋊C414yes—
224Q8×C61—44no—
225Q8×C61—44no—
226Q8×C61—44no—
227Q8×C61—44no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—44no—
237C2×C4×C61—44no—
238C2×C4×C61—42no—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61C4×D412yes—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12112yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61#12312yes—
254C2×C4×C61#12312yes—
255C22⋊D61—24no—
256C22⋊D61—24no—
257C22⋊D61—24no—
258C22⋊D61—24no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—24no—
262C2×Dic61—24no—
263C2×Dic61—44no—
264C2×Dic61—44no—
265C2×Dic61—44no—
266C2×Dic61—42no—
267C2×Dic61C2×C22⋊C414yes—
268D244—24no—
269D244—24no—
270C4×D61—28no—
271C4×D61—28no—
272C4×D61—88no—
273Dic122—24no—
274Dic122—24no—
275#20962—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61C2×C4×D411yescharacteristic
285C22×C61C2×C4×D411yescharacteristic
286C22×C61C2×#12011yescharacteristic
287C22×C61C2×#12111yescharacteristic
288C22×C61C2×#12111yescharacteristic
289C22×C61C2×#12311yescharacteristic
290C2×D61—24no—
291C2×D61—44no—
292C2×D61—44no—
293C2×D61—82no—
294C4×C61—44no—
295C4×C61—44no—
296C4×C61—82no—
297C4×C61#38612yes—
298C4×C61#38612yes—
299C4×C61#38612yes—
300C4×C61#38612yes—
301Dic61—24no—
302Dic61—44no—
303Dic61—44no—
304Dic61—82no—
305C2×#536—611no—
306#3428—611no—
307C2×#410—611no—
308#2310—611no—
309#2310D6111yescharacteristic, complemented
310#1890—618no—
311#1740—611no—
312#1740—611no—
313C2×C61—21no—
314C2×C61—21no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—21no—
319C2×C61C2×#38611yescharacteristic
320C2×C61#302812yes—
321D61—44no—
322C22×#41—611no—
323C23×Q8C2×D6111yescharacteristic
324#536—618no—
325#536—1224no—
326C2×C4⋊Q8—1221no—
327C2×#124—1221no—
328C2×#123—614no—
329C2×C22⋊Q8—1222no—
330#520—1222no—
331#520—1222no—
332#519—1221no—
333#518—614no—
334#518—1221no—
335C4×#41—1222no—
336C4×#41—1222no—
337C2×#117—611no—
338C2×#117—611no—
339C2×#117—611no—
340C2×#117—1222no—
341C2×#117C2×D6111yescharacteristic
342C2×#117C2×D6111yescharacteristic
343#410—1224no—
344C2×#99—614no—
345#391—22no—
346#391—22no—
347#391—1222no—
348#391—1222no—
349#381—614no—
350#381C2×D6114yes—
351#342—614no—
352#342—614no—
353C61#2096211yescharacteristic, complemented, Sylow
354C2×#41—614no—
355C2×#41—614no—
356C2×#41—1222no—
357C2×#41—1221no—
358C2×#41—1221no—
359C2×#41—1222no—
360C2×#41—1221no—
361C2×#41—1221no—
362C22×Q8—22no—
363C22×Q8—21no—
364C22×Q8—21no—
365C22×Q8—22no—
366C22×Q8—21no—
367C22×Q8—21no—
368C22×Q8—22no—
369C22×Q8—21no—
370C22×Q8—21no—
371C22×Q8—611no—
372C22×Q8C4×D6114yescomplemented
373C22×D4—612no—
374C22×D4—611no—
375C23×C4—612no—
376C23×C4C22×D6112yes—
377C23×C4C22×D6111yescharacteristic
378C4⋊Q8—1228no—
379#124—1228no—
380#123—6116no—
381#123—1228no—
382C22⋊Q8—12216no—
383C4×Q8—1224no—
384C4×Q8—1224no—
385C4×D4—1224no—
386C4×D4—1228no—
387C4×D4—1224no—
388C4×D4—1228no—
389#117—21no—
390#117—21no—
391#117—21no—
392#117—21no—
393#117—22no—
394#117—22no—
395#117—614no—
396#117—1224no—
397#117—1224no—
398#117—1224no—
399#117—1221no—
400#117—1221no—
401#117—1222no—
402#117—1221no—
403#117—1221no—
404#117—1222no—
405#117—1222no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—1224no—
409C2×C4⋊C4—1224no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4C22⋊D6112yes—
413C2×C22⋊C4—612no—
414C2×C22⋊C4—612no—
415C2×C22⋊C4—612no—
416C2×C22⋊C4—1224no—
417C2×C22⋊C4—1224no—
418C2×C22⋊C4C22×D6112yes—
419C2×C22⋊C4C22×D6112yes—
420C2×C42—21no—
421C2×C42—612no—
422C2×C42—1224no—
423C2×C42—1221no—
424C2×C42—1221no—
425C2×C42C22⋊D6112yes—
426#99—1228no—
427#95—28no—
428#95—1228no—
429C24—612no—
430C24C23×D6111yescharacteristic
431#41—1224no—
432#41—1224no—
433#41—1224no—
434#41—1224no—
435#41—1224no—
436#41—1224no—
437#41—1224no—
438#41—1224no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8—41no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—614no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1222no—
459C2×D4—614no—
460C2×D4—614no—
461C2×D4—614no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1222no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—612no—
477C22×C4—618no—
478C22×C4—614no—
479C22×C4—1228no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1224no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4C2×C4×D6114yes—
486C22×C4C2×C4×D6112yes—
487C22×C4#1175822212yes—
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6111yescharacteristic
490C22×C4D4×D6112yes—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4#1175822412yes—
494C22×C4C2×C22⋊D6114yes—
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—12216no—
501C4⋊C4—1228no—
502C4⋊C4—1224no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—2442no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—614no—
511C22⋊C4—12216no—
512C22⋊C4—1228no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C42—22no—
519C42—22no—
520C42—42no—
521C42—1228no—
522C42—1224no—
523C42—1222no—
524C42—1222no—
525C42—2442no—
526C23—42no—
527C23—614no—
528C23—1222no—
529C23—1222no—
530C23—2444no—
531C23C22×C4×D6111yescharacteristic
532C23C2×D4×D6111yescharacteristic
533C23C2×D4×D6111yescharacteristic
534C23C2×D4×D6111yescharacteristic
535C23C22×C22⋊D6111yescharacteristic
536C23C2×#1175822411yescharacteristic
537C23C2×#1175822211yescharacteristic
538Q8—44no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542Q8—1224no—
543Q8—1224no—
544D4—1224no—
545D4—1224no—
546D4—1224no—
547D4—1224no—
548D4—1224no—
549D4—1224no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—614no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1228no—
565C2×C4—1228no—
566C2×C4—1224no—
567C2×C4—1224no—
568C2×C4—4888no—
569C2×C4—2444no—
570C2×C4—2442no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4C22⋊C4×D6112yes—
574C2×C4C22⋊C4×D6112yes—
575C2×C4#1451973712yes—
576C2×C4#1451973712yes—
577C2×C4C4×C22⋊D6112yes—
578C2×C4C4×C22⋊D6112yes—
579C2×C4#1451974511yescharacteristic
580C2×C4#1451974511yescharacteristic
581C2×C4#1451974511yescharacteristic
582C2×C4#1451974511yescharacteristic
583C2×C4#1451974812yes—
584C2×C4#1451974812yes—
585C2×C4#1451975012yes—
586C2×C4#1451975012yes—
587C2×C4#1451975912yes—
588C2×C4#1451975912yes—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—1224no—
597C22—4882no—
598C22—2444no—
599C22—2444no—
600C22C2×C22⋊C4×D6111yescharacteristic
601C22C2×#1451973711yescharacteristic
602C22C2×C4×C22⋊D6111yescharacteristic
603C22C2×#1451974511yescharacteristic
604C22C2×#1451974811yescharacteristic
605C22C2×#1451975011yescharacteristic
606C22C2×#1451975911yescharacteristic
607C4—44no—
608C4—44no—
609C4—82no—
610C4—1224no—
611C4—4882no—
612C4—2444no—
613C4—2444no—
614C4#1683963812yes—
615C4#1683963812yes—
616C4#1683963812yes—
617C4#1683963812yes—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—2444no—
625C2C2×#1683963811yescharacteristic
626C2#2145522412yes—
subgroupG/H
centerC22C2×#14519745
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#2310C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC24×C61C24
hypercenter#2310D61
hypocenterC61#20962
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. #21448405C2semidirect
5. #21446229C2semidirect
6. #21445552C2semidirect
7. #21445551C2semidirect
8. C61×#2310C2semidirect
11. #16840336C4semidirect
55. #14519760C2×C4semidirect
70. #14519736C2×C4semidirect
99. C22×Q8×C61C2×C4semidirect
309. #2310D61semidirect
353. C61#20962semidirect
372. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403362
#214590202
C2×#168397352
#214484052
#214462292
#214455522
#214455512
C61×#23102
#2096261