Finite Group Databasebeta

← #29601689

Subgroups — #29601689

Order 15616.

2702 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D61C211yesmaximal, characteristic
2C2×C4×#11758198C212yesmaximal, complemented
3C2×C4×C22⋊Dic61C211yesmaximal, characteristic, complemented
4#21444069C212yesmaximal, complemented
5#21444066C212yesmaximal, complemented
6#21444061C212yesmaximal, complemented
7#21444058C212yesmaximal
8C4×#14519684C212yesmaximal, complemented
9C22×#117×C61C211yesmaximal, characteristic, complemented
10C23×C22⋊D61C2211yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C2×C4×C22⋊D61—24no—
13C2×C4×C22⋊D61—24no—
14C2×C4×C22⋊D61C418yescomplemented
15C22×#11758196C2211yescharacteristic
16C22×C22⋊Dic61C2212yes—
17C22×C22⋊Dic61C2211yescharacteristic
18C22×#11758198C2214yes—
19C22×#11758198C2211yescharacteristic
20C22×C4×Dic61C2212yes—
21C22×C4×Dic61C2211yescharacteristic
22C4×#11758198—28no—
23C4×C22⋊Dic61—24no—
24C2×#14519684C2214yes—
25C2×#14519684C2214yes—
26C2×#14519684C2212yes—
27C2×#14519684C2214yes—
28C24×C4×C61C2211yescharacteristic
29C2×#117×C61—24no—
30C2×#117×C61—28no—
31C22×C4⋊C4×C61C2212yes—
32C22×C22⋊C4×C61C2212yes—
33C22×C42×C61C2212yes—
34C24×D61C2311yescharacteristic
35C22×C22⋊D61—22no—
36C22×C22⋊D61—22no—
37C22×C22⋊D61—24no—
38C22×C22⋊D61—24no—
39C22×C22⋊D61C2×C414yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2314yes—
43C23×Dic61C2311yescharacteristic
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—24no—
47C22×C4×D61—24no—
48C22×C4×D61C2×C414yes—
49C4×C22⋊D61—232no—
50C2×#11758196—22no—
51C2×#11758196—22no—
52C2×#11758196C2×C414yes—
53C2×C22⋊Dic61—22no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—24no—
56C2×C22⋊Dic61—28no—
57C2×C22⋊Dic61C2×C414yes—
58C2×#11758198—28no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×#11758198—28no—
62C2×#11758198—28no—
63C2×#11758198C2×C414yes—
64C2×C4×Dic61—22no—
65C2×C4×Dic61—22no—
66C2×C4×Dic61—24no—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—24no—
69C2×C4×Dic61C2×C414yes—
70#14519684—24no—
71#14519684—28no—
72#14519684—28no—
73#14519684—28no—
74C25×C61C2311yescharacteristic
75C23×C4×C61—24no—
76C23×C4×C61—24no—
77C23×C4×C61C2×C414yes—
78C23×C4×C61D414yes—
79C23×C4×C61D414yes—
80C23×C4×C61C2314yes—
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83#117×C61—232no—
84C2×C4⋊C4×C61—28no—
85C2×C4⋊C4×C61—24no—
86C2×C22⋊C4×C61—24no—
87C2×C22⋊C4×C61—28no—
88C2×C42×C61—24no—
89C2×C42×C61—24no—
90C2×C42×C61D418yescomplemented
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61—24no—
96C24×C61C22×C412yes—
97C24×C61C2×D412yes—
98C24×C61C2×D412yes—
99C24×C61C2411yescharacteristic
100C23×D61—21no—
101C23×D61—21no—
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—24no—
106C23×D61C22×C412yes—
107C23×D61#4114yes—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—216no—
111C22×C4×C61—28no—
112C22×C4×C61—216no—
113C22×C4×C61—28no—
114C22×C4×C61—24no—
115C22×C4×C61—24no—
116C22×C4×C61—24no—
117C22×C4×C61—24no—
118C22×C4×C61—24no—
119C22×C4×C61—24no—
120C22×C4×C61C22⋊C4116yescomplemented
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C2×D418yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D414yes—
129C22×C4×C61#4114yes—
130C2×C22⋊D61—216no—
131C2×C22⋊D61—216no—
132C2×C22⋊D61—28no—
133C2×C22⋊D61—28no—
134C2×C22⋊D61—28no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—24no—
142C22×Dic61—24no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—24no—
147C22×Dic61—24no—
148C22×Dic61—24no—
149C22×Dic61—24no—
150C22×Dic61—24no—
151C22×Dic61C22×C412yes—
152C22×Dic61C22×C412yes—
153C22×Dic61C22×C412yes—
154C22×Dic61#4118yes—
155C22×Dic61#4114yes—
156C2×C4×D61—28no—
157C2×C4×D61—216no—
158C2×C4×D61—44no—
159C2×C4×D61—48no—
160C2×C4×D61—44no—
161C4⋊C4×C61—216no—
162C22⋊C4×C61—216no—
163C42×C61—216no—
164C22⋊Dic61—216no—
165C22⋊Dic61—28no—
166#11758198—232no—
167#11758198—28no—
168#11758196—28no—
169C4×Dic61—28no—
170C4×Dic61—48no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—24no—
174C23×C61—24no—
175C23×C61—22no—
176C23×C61—24no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—44no—
183C23×C61—44no—
184C23×C61—44no—
185C23×C61—44no—
186C23×C61C2×C22⋊C414yes—
187C23×C61C23×C411yescharacteristic
188C23×C61C22×D411yescharacteristic
189C23×C61C22×D411yescharacteristic
190C23×C61C22×D412yes—
191C23×C61C2×#4112yes—
192C23×C61C2×#4112yes—
193C23×C61C2×#4112yes—
194C23×C61C2×#4114yes—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—24no—
198C22×D61—24no—
199C22×D61—28no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—28no—
203C22×D61—41no—
204C22×D61—42no—
205C22×D61—41no—
206C22×D61—42no—
207C22×D61—44no—
208C22×D61—44no—
209C22×D61—42no—
210C22×D61#11714yescomplemented
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—232no—
216C2×C4×C61—28no—
217C2×C4×C61—28no—
218C2×C4×C61—28no—
219C2×C4×C61—44no—
220C2×C4×C61—44no—
221C2×C4×C61—44no—
222C2×C4×C61—44no—
223C2×C4×C61C2×C22⋊C414yes—
224C2×C4×C61C2×C22⋊C414yes—
225C2×C4×C61C4×D418yes—
226C2×C4×C61#12014yes—
227C2×C4×C61#12118yes—
228C2×C4×C61#12314yes—
229C22⋊D61—232no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—28no—
239C2×Dic61—28no—
240C2×Dic61—28no—
241C2×Dic61—41no—
242C2×Dic61—44no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—44no—
246C2×Dic61—42no—
247C2×Dic61—44no—
248C2×Dic61—44no—
249C2×Dic61—42no—
250C2×Dic61#11714yescomplemented
251C4×D61—416no—
252#13839—611nomaximal, Sylow
253C22×C61—24no—
254C22×C61—24no—
255C22×C61—24no—
256C22×C61—44no—
257C22×C61—48no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61C2×#11711yescharacteristic
263C22×C61C2×#11711yescharacteristic
264C22×C61C2×C4×D412yes—
265C22×C61C4×#4112yes—
266C22×C61C2×#12011yescharacteristic
267C22×C61C2×#12112yes—
268C22×C61C2×#12311yescharacteristic
269C22×C61#52514yes—
270C22×C61#52512yes—
271C22×C61#52512yes—
272C22×C61#52512yes—
273C22×C61#52914yes—
274C22×C61#52914yes—
275C22×C61#53214yes—
276C22×C61#53212yes—
277C2×D61—24no—
278C2×D61—28no—
279C2×D61—44no—
280C2×D61—48no—
281C2×D61—44no—
282C2×D61—42no—
283C2×D61—44no—
284C2×D61—42no—
285C4×C61—24no—
286C4×C61—48no—
287C4×C61#38618yes—
288Dic61—24no—
289Dic61—48no—
290Dic61—42no—
291Dic61—42no—
292C22×C4×D4—611no—
293C22×#117D6111yescharacteristic, complemented
294C2×C4×C22⋊C4—612no—
295C2×C4×C22⋊C4—611no—
296#1441—612no—
297#1440—612no—
298#1439—612no—
299#1438—612no—
300C4×#95—612no—
301C2×C61—44no—
302C2×C61#228011yescharacteristic
303C2×C61C2×#38611yescharacteristic
304C2×C61#229012yes—
305C2×C61#229312yes—
306C2×C61#229711yescharacteristic
307C2×C61#238812yes—
308C2×C61#239214yes—
309C2×C61#239512yes—
310D61—44no—
311C23×D4—611no—
312C24×C4—611no—
313C24×C4C2×D6111yescharacteristic
314C2×C4×D4—618no—
315C2×C4×D4—1224no—
316C2×C4×D4—1224no—
317C2×#117—24no—
318C2×#117—28no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4C2×D6112yes—
321C22×C22⋊C4—611no—
322C22×C22⋊C4—611no—
323C22×C22⋊C4—612no—
324C22×C22⋊C4—614no—
325C22×C22⋊C4C2×D6112yes—
326C22×C42—611no—
327C22×C42—612no—
328C22×C42C2×D6112yes—
329C4×C22⋊C4—1224no—
330C4×C22⋊C4—1228no—
331C2×#95—614no—
332C2×#95—614no—
333C2×#95—612no—
334C2×#95—614no—
335C61#1383911yescharacteristic, complemented, Sylow
336C25—611no—
337C25C22×D6111yescharacteristic
338C22×D4—614no—
339C22×D4—1224no—
340C22×D4—1224no—
341C22×D4—1222no—
342C22×D4—1222no—
343C23×C4—24no—
344C23×C4—24no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—614no—
348C23×C4—611no—
349C23×C4—614no—
350C23×C4—1222no—
351C23×C4—1224no—
352C23×C4—1222no—
353C23×C4—1224no—
354C23×C4C4×D6114yes—
355C23×C4D24414yes—
356C23×C4C22⋊D6114yes—
357C23×C4C22×D6114yes—
358C23×C4C22×D6111yescharacteristic
359C23×C4C22×D6111yescharacteristic
360C4×D4—12232no—
361#117—232no—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—614no—
370C2×C22⋊C4—614no—
371C2×C22⋊C4—1228no—
372C2×C22⋊C4—1228no—
373C2×C22⋊C4—1228no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1228no—
380C2×C42—24no—
381C2×C42—24no—
382C2×C42—614no—
383C2×C42—1222no—
384C2×C42—1222no—
385C2×C42—1224no—
386C2×C42—1224no—
387C2×C42—1224no—
388C2×C42C22⋊D6118yescomplemented
389#95—1228no—
390#95—1228no—
391#95—1224no—
392#95—1228no—
393C24—22no—
394C24—24no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—612no—
399C24—614no—
400C24—1222no—
401C24—1221no—
402C24—1221no—
403C24—1224no—
404C24—1222no—
405C24—1222no—
406C24C2×C4×D6112yes—
407C24C2×D24412yes—
408C24C2×C22⋊D6112yes—
409C24C23×D6111yescharacteristic
410C2×D4—1228no—
411C2×D4—1228no—
412C2×D4—12216no—
413C2×D4—12216no—
414C2×D4—1228no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—216no—
418C22×C4—28no—
419C22×C4—216no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—612no—
428C22×C4—614no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—618no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1224no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1224no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1222no—
448C22×C4—1228no—
449C22×C4—12216no—
450C22×C4—2444no—
451C22×C4—2448no—
452C22×C4—2444no—
453C22×C4#11758198116yescomplemented
454C22×C4C2×C4×D6112yes—
455C22×C4C2×C4×D6112yes—
456C22×C4C2×D24412yes—
457C22×C4C2×D24412yes—
458C22×C4#1175822214yes—
459C22×C4C2×C22⋊D6118yes—
460C22×C4C2×C22⋊D6112yes—
461C22×C4C2×C22⋊D6112yes—
462C22×C4C2×C22⋊D6114yes—
463C4⋊C4—216no—
464C4⋊C4—1228no—
465C22⋊C4—216no—
466C22⋊C4—12232no—
467C22⋊C4—12216no—
468C22⋊C4—1228no—
469C22⋊C4—1228no—
470C42—216no—
471C42—1228no—
472C42—2448no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—614no—
489C23—1222no—
490C23—1222no—
491C23—1224no—
492C23—1224no—
493C23—1228no—
494C23—1224no—
495C23—1224no—
496C23—1228no—
497C23—2441no—
498C23—2442no—
499C23—2441no—
500C23—2442no—
501C23—2444no—
502C23—2444no—
503C23—2442no—
504C23C2×#1175819814yes—
505C23C22×C4×D6111yescharacteristic
506C23C22×C22⋊D6111yescharacteristic
507C23C22×C22⋊D6112yes—
508C23C22×D24411yescharacteristic
509C23#41×D6112yes—
510C23#41×D6112yes—
511C23#41×D6114yes—
512C23C2×#1175822212yes—
513D4—12232no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—232no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—614no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1228no—
536C2×C4—1228no—
537C2×C4—1228no—
538C2×C4—2441no—
539C2×C4—2444no—
540C2×C4—2442no—
541C2×C4—2441no—
542C2×C4—2444no—
543C2×C4—2442no—
544C2×C4—2444no—
545C2×C4—2444no—
546C2×C4—2442no—
547C2×C4—24416no—
548C2×C4C2×#1175819814yes—
549C2×C4C2×#1175819814yes—
550C2×C4C4×C22⋊D6118yes—
551C2×C4#1451974614yes—
552C2×C4#1451975118yes—
553C2×C4#1451976014yes—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—48no—
561C22—44no—
562C22—1224no—
563C22—1228no—
564C22—2442no—
565C22—2444no—
566C22—2444no—
567C22—2448no—
568C22—2442no—
569C22—2444no—
570C22C22×#1175819811yescharacteristic
571C22#117×D6111yescharacteristic
572C22#117×D6111yescharacteristic
573C22C2×C4×C22⋊D6112yes—
574C22#1684023012yes—
575C22C2×#1451974611yescharacteristic
576C22C2×#1451975112yes—
577C22C2×#1451976011yescharacteristic
578C22#1684026912yes—
579C22#1684026914yes—
580C22#1684027412yes—
581C22#1684027412yes—
582C22#1684029014yes—
583C22#1684029014yes—
584C22#1684030814yes—
585C22#1684030812yes—
586C4—24no—
587C4—48no—
588C4—1224no—
589C4—2442no—
590C4—2448no—
591C4—2442no—
592C4#1683964118yes—
593C2—44no—
594C2—2444no—
595C2#2144819811yescharacteristic
596C2C2×#1683964111yescharacteristic
597C2#2144825612yes—
598C2#2144827812yes—
599C2#2144830711yescharacteristic
600C2#2144903212yes—
601C2#2144905514yes—
602C2#2144907612yes—
subgroupG/H
centerC23×C4C22×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC22×#117×C61C2
socle, cosocleC24×C61C24
prime radicalC23×C22⋊D61C22
hypercenterC22×#117D61
hypocenterC61#13839
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758198C2semidirect
3. C2×C4×C22⋊Dic61C2semidirect
4. #21444069C2semidirect
5. #21444066C2semidirect
6. #21444061C2semidirect
8. C4×#14519684C2semidirect
9. C22×#117×C61C2semidirect
14. C2×C4×C22⋊D61C4semidirect
90. C2×C42×C61D4semidirect
120. C22×C4×C61C22⋊C4semidirect
210. C22×D61#117semidirect
250. C2×Dic61#117semidirect
293. C22×#117D61semidirect
335. C61#13839semidirect
388. C2×C42C22⋊D61semidirect
453. C22×C4#11758198semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D612
C2×C4×#117581982
C2×C4×C22⋊Dic612
#214440692
#214440662
#214440612
#214440582
C4×#145196842
C22×#117×C612
#1383961