Finite Group Databasebeta

← #29681657

Subgroups — #29681657

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×#16840338C211yesmaximal, characteristic
2#21459020C211yesmaximal, characteristic
3C2×#16839735C211yesmaximal, characteristic
4#21448403C211yesmaximal, characteristic, complemented
5#21448403C211yesmaximal, characteristic, complemented
6#21446231C218yesmaximal, complemented
7#21445551C211yesmaximal, characteristic, complemented
8C61×#1740C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×Dic122C2211yescharacteristic
11#16840338—24no—
12#16840338C418yescomplemented
13C2×#14519777—21no—
14C2×#14519765—21no—
15C2×#14519761C2214yes—
16C2×#14519757—22no—
17#16840243—22no—
18#16840242—22no—
19#16840239—21no—
20#16840236—21no—
21#16840235C2214yes—
22C4×#11758222—22no—
23#16840230—22no—
24C2×#14519736C2211yescharacteristic
25C2×#14519736C2211yescharacteristic
26#117×D61—22no—
27#16839735—24no—
28C2×#14519691C2214yes—
29#16839668—22no—
30#16839668—22no—
31#16839667—22no—
32#16839667—22no—
33#16839623C2214yes—
34#16839623C2214yes—
35#16839528C2214yes—
36C2×#117×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#342C2214yes—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222C2×C414yes—
47C2×#11758222D414yes—
48C22×Dic122—21no—
49C22×Dic122—21no—
50C22×Dic122—22no—
51C22×Dic122—21no—
52C22×Dic122—21no—
53C22×Dic122—21no—
54C22×Dic122—21no—
55C22×Dic122—22no—
56C22×Dic122—22no—
57C22×Dic122C2×C414yescomplemented
58C22×Dic122D411yescharacteristic
59C22×D244D411yescharacteristic
60C22×C22⋊D61C2312yes—
61C23×Dic61C2312yes—
62C22×C4×D61D412yes—
63#14519777—28no—
64#14519765—28no—
65#14519761—28no—
66#14519761C2×C4116yescomplemented
67#14519757—216no—
68C4×Dic122—24no—
69#14519742—24no—
70C4×D244—24no—
71#14519739—24no—
72C4×C22⋊D61—28no—
73#14519737—28no—
74#14519736—21no—
75#14519736—21no—
76#14519736—21no—
77#14519736—21no—
78#14519736—22no—
79#14519736—22no—
80#14519735—24no—
81#14519734—24no—
82#14519733—24no—
83C2×C4⋊Dic61—24no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61D412yes—
86C4⋊C4×D61—24no—
87C2×C22⋊Dic61C2312yes—
88C2×C22⋊Dic61C2312yes—
89C2×#11758198—24no—
90C22⋊C4×D61—24no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61D412yes—
93C42×D61—24no—
94#14519691—28no—
95#14519684—28no—
96#14519684—28no—
97C23×C4×C61C2311yescharacteristic
98#117×C61—21no—
99#117×C61—21no—
100#117×C61—21no—
101#117×C61—21no—
102#117×C61—22no—
103#117×C61—22no—
104#117×C61—22no—
105#117×C61C2×C414yescomplemented
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61D412yes—
109C2×C22⋊C4×C61C2312yes—
110C2×C22⋊C4×C61C2312yes—
111C2×C22⋊C4×C61C2312yes—
112C2×C42×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61D412yes—
115C24×C61C2411yescharacteristic
116C23×D61C2×D412yes—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D414yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61C22×C414yes—
135C2×C22⋊D61C2×D414yes—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61C22×C414yes—
144C22×Dic61C2×D412yes—
145C22×Dic61C2×D414yes—
146#11758222—24no—
147#11758222—24no—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—24no—
152#11758222—24no—
153#11758222—24no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—22no—
157C2×D244C22⋊C414yes—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—24no—
161C2×C4×D61—28no—
162C2×C4×D61C22⋊C418yes—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—22no—
166C2×Dic122—24no—
167C2×Dic122—24no—
168C2×Dic122—24no—
169C2×Dic122—24no—
170C2×Dic122—24no—
171C2×Dic122—24no—
172C2×Dic122—41no—
173C2×Dic122—41no—
174C2×Dic122—41no—
175C2×Dic122—41no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×Dic122C22⋊C414yes—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—42no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61C22×C414yes—
193C42×C61—24no—
194C42×C61—22no—
195C42×C61—22no—
196C42×C61—42no—
197C22⋊Dic61—22no—
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201#11758198—28no—
202#11758198—216no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61—216no—
206C4⋊Dic61—42no—
207#11758196—28no—
208C4×Dic61—28no—
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—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—44no—
227C2×C4×C61—44no—
228C2×C4×C61—44no—
229C2×C4×C61—42no—
230C2×C4×C61C2×C22⋊C412yes—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61C4×D412yes—
234C2×C4×C61C4×D412yes—
235C2×C4×C61C4×D412yes—
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12112yes—
241C2×C4×C61#12112yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12312yes—
245C2×C4×C61#12312yes—
246C22⋊D61—24no—
247C22⋊D61—24no—
248C22⋊D61—24no—
249C22⋊D61—24no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—44no—
261C2×Dic61—44no—
262C2×Dic61—42no—
263C2×Dic61C2×C22⋊C414yes—
264D244—24no—
265D244—24no—
266C4×D61—28no—
267C4×D61—28no—
268C4×D61—88no—
269Dic122—24no—
270Dic122—24no—
271Dic122—44no—
272Dic122—44no—
273Dic122—44no—
274Dic122—44no—
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#2310—611no—
310#1890—618no—
311#1740—611no—
312#1740D6111yescharacteristic, complemented
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×Q8—611no—
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—1222no—
340C2×#117C2×D6111yescharacteristic
341C2×#117C2×D6111yescharacteristic
342C2×#117C2×D6111yescharacteristic
343#410—1224no—
344C2×#99—614no—
345#391—1222no—
346#391—1222no—
347#391—1222no—
348#391—1222no—
349#381—614no—
350#381—614no—
351#342—614no—
352#342C2×D6114yes—
353C61#2096211yescharacteristic, complemented, Sylow
354C2×#41—614no—
355C2×#41—614no—
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1222no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1222no—
362C22×Q8—614no—
363C22×Q8—611no—
364C22×Q8—1221no—
365C22×Q8—1221no—
366C22×Q8—1222no—
367C22×Q8—1221no—
368C22×Q8—1221no—
369C22×Q8—1222no—
370C22×Q8—1221no—
371C22×Q8—1221no—
372C22×Q8—1222no—
373C22×D4—612no—
374C22×D4—611no—
375C23×C4—612no—
376C23×C4—612no—
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—22no—
396#117—1224no—
397#117—1224no—
398#117—1224no—
399#117—1221no—
400#117—1221no—
401#117—1221no—
402#117—1221no—
403#117—1222no—
404#117—1222no—
405#117C4×D6114yescomplemented
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—612no—
409C2×C4⋊C4—1224no—
410C2×C4⋊C4—1224no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4D24412yes—
413C2×C22⋊C4—612no—
414C2×C22⋊C4—612no—
415C2×C22⋊C4—1224no—
416C2×C22⋊C4—1224no—
417C2×C22⋊C4C22×D6112yes—
418C2×C22⋊C4C22×D6112yes—
419C2×C22⋊C4C22×D6112yes—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—612no—
423C2×C42—1224no—
424C2×C42—1221no—
425C2×C42D24412yes—
426#99—1228no—
427#95—1228no—
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—614no—
440C2×Q8—1224no—
441C2×Q8—1224no—
442C2×Q8—1224no—
443C2×Q8—1224no—
444C2×Q8—1224no—
445C2×Q8—1224no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1222no—
449C2×Q8—2441no—
450C2×Q8—2441no—
451C2×Q8—2441no—
452C2×Q8—2441no—
453C2×Q8—2442no—
454C2×Q8—2442no—
455C2×Q8—2442no—
456C2×Q8—2442no—
457C2×Q8—2442no—
458C2×Q8—2442no—
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—612no—
472C22×C4—618no—
473C22×C4—614no—
474C22×C4—614no—
475C22×C4—1228no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4C2×C4×D6112yes—
487C22×C4C2×D24414yes—
488C22×C4C2×D24411yescharacteristic
489C22×C4C2×D24411yescharacteristic
490C22×C4#1175822212yes—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4D4×D6112yes—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4#1175822612yes—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—12216no—
502C4⋊C4—1228no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—2442no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—12216no—
512C22⋊C4—1228no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4C2×C4×D6114yes—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—42no—
522C42—1228no—
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×D24411yescharacteristic
536C23C2×#1175822211yescharacteristic
537C23C2×#1175822611yescharacteristic
538Q8—1224no—
539Q8—1224no—
540Q8—2444no—
541Q8—2444no—
542Q8—2444no—
543Q8—2444no—
544D4—1224no—
545D4—1224no—
546D4—1224no—
547D4—1224no—
548D4—1224no—
549D4—1224no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—614no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1228no—
560C2×C4—1228no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—4888no—
566C2×C4—2444no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2444no—
573C2×C4C22⋊C4×D6112yes—
574C2×C4C22⋊C4×D6112yes—
575C2×C4#1451973912yes—
576C2×C4#1451973912yes—
577C2×C4C4×D24412yes—
578C2×C4C4×D24412yes—
579C2×C4#1451974711yescharacteristic
580C2×C4#1451974711yescharacteristic
581C2×C4#1451974711yescharacteristic
582C2×C4#1451974711yescharacteristic
583C2×C4#1451975012yes—
584C2×C4#1451975012yes—
585C2×C4#1451975212yes—
586C2×C4#1451975212yes—
587C2×C4#1451976212yes—
588C2×C4#1451976212yes—
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×#1451973911yescharacteristic
602C22C2×C4×D24411yescharacteristic
603C22C2×#1451974711yescharacteristic
604C22C2×#1451975011yescharacteristic
605C22C2×#1451975211yescharacteristic
606C22C2×#1451976211yescharacteristic
607C4—44no—
608C4—44no—
609C4—82no—
610C4—1224no—
611C4—4882no—
612C4—2444no—
613C4—2444no—
614C4#1683964012yes—
615C4#1683964012yes—
616C4#1683964012yes—
617C4#1683964012yes—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—2444no—
625C2C2×#1683964011yescharacteristic
626C2#2145522612yes—
subgroupG/H
centerC22C2×#14519747
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1740C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC24×C61C24
hypercenter#1740D61
hypocenterC61#20962
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. #21448403C2semidirect
5. #21448403C2semidirect
6. #21446231C2semidirect
7. #21445551C2semidirect
8. C61×#1740C2semidirect
12. #16840338C4semidirect
57. C22×Dic122C2×C4semidirect
66. #14519761C2×C4semidirect
105. #117×C61C2×C4semidirect
312. #1740D61semidirect
353. C61#20962semidirect
405. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403382
#214590202
C2×#168397352
#214484032
#214462312
#214455512
C61×#17402
#2096261