Finite Group Databasebeta

← #29677808

Subgroups — #29677808

Order 15616.

1706 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758224C211yesmaximal, characteristic
2C2×#16839760C212yesmaximal
3C2×#16839714C211yesmaximal, characteristic, complemented
4C4×C4⋊C4×D61C211yesmaximal, characteristic, complemented
5#21445652C211yesmaximal, characteristic, complemented
6#21445650C211yesmaximal, characteristic, complemented
7#21445644C211yesmaximal, characteristic, complemented
8#21445638C212yesmaximal, complemented
9#102×Dic61C212yesmaximal, complemented
10#21445537C212yesmaximal, complemented
11C61×#1762C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×#11758189C2212yes—
14C4×#11758224—24no—
15C4×#11758224C418yescomplemented
16C2×C4×Dic122C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×C4×C22⋊D61—21no—
20C2×#14519737—21no—
21C2×#14519735C2211yescharacteristic
22C2×#14519733—21no—
23C2×C4⋊C4×D61C2212yes—
24C22×C4×Dic61—21no—
25C2×C42×D61—21no—
26C2×C42×D61C2211yescharacteristic
27#16839760C4116yescomplemented
28C2×#14519702C2212yes—
29C2×#14519700C2212yes—
30C2×#14519698C2212yes—
31#16839714—24no—
32C2×#14519688C2211yescharacteristic
33C2×C8×Dic61C2212yes—
34C4⋊C4×Dic61C2212yes—
35C4×#11758196—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839613—21no—
38#95×D61—21no—
39#16839525C2212yes—
40#16839524C2212yes—
41#16839524C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C4×C4⋊C4×C61C2211yescharacteristic
46C61×#340C2212yes—
47C2×#11758224—22no—
48C2×#11758224—24no—
49C2×#11758224C2×C414yes—
50C2×#11758224C2×C418yes—
51C22×Dic122C2311yescharacteristic
52C22×C22⋊D61—21no—
53C23×Dic61—21no—
54C23×Dic61—21no—
55C22×C4×D61—22no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C2×#11758189—24no—
60C2×#11758189—24no—
61C2×#11758189C2×C418yescomplemented
62C22×#92418C2312yes—
63C4×Dic122—22no—
64C4×Dic122C2×C414yescomplemented
65#14519742—22no—
66#14519742C2×C414yescomplemented
67D4×Dic61—22no—
68D4×Dic61C2×C414yescomplemented
69C4×C22⋊D61—28no—
70#14519737—28no—
71#14519735—22no—
72#14519735C2×C414yes—
73#14519733—28no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—22no—
78C2×#11758196—21no—
79C4⋊C4×D61—24no—
80C4⋊C4×D61C2×C418yescomplemented
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—28no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—22no—
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C42×D61—44no—
93C42×D61C2×C414yes—
94C42×D61D414yes—
95#14519702C2×C418yescomplemented
96#14519700C2×C418yescomplemented
97#14519698C2×C418yescomplemented
98#14519688—22no—
99C8×Dic61—24no—
100#14519684—21no—
101#14519684—22no—
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C22×C8×C61C2312yes—
105C4×D4×C61—22no—
106C4×D4×C61C2×C414yescomplemented
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—21no—
110C2×C42×C61—21no—
111C2×C42×C61C2311yescharacteristic
112#102×C61C2×C418yescomplemented
113#97×C61—22no—
114#95×C61—21no—
115C24×C61—21no—
116C23×D61C2×D411yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61C22×C414yes—
120C2×D4×C61C22×C412yes—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—24no—
124C22×C4×C61—44no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—24no—
129C2×C22⋊D61—28no—
130C22×Dic61—21no—
131C22×Dic61—24no—
132C22×Dic61—28no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—24no—
136C22×Dic61—42no—
137C22×Dic61—42no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141#11758224—28no—
142#11758224—216no—
143#11758224C42116yescomplemented
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—28no—
147C2×C4×D61—42no—
148C2×C4×D61—44no—
149C2×C4×D61—44no—
150C2×C4×D61C22⋊C414yes—
151C2×C4×D61C22⋊C414yes—
152C2×C4×D61C22×C412yes—
153C2×C4×D61C22×C414yes—
154C2×C4×D61C2×D412yes—
155C2×C4×D61#4114yes—
156C2×Dic122—21no—
157C2×Dic122—22no—
158C2×Dic122C22×C414yes—
159C2×Dic122C22×C412yes—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61C22×C414yes—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61C22×C414yes—
166C4⋊C4×C61C22×C412yes—
167C22⋊C4×C61—24no—
168C42×C61—42no—
169C42×C61C22×C412yes—
170C42×C61C2×D412yes—
171C22⋊Dic61—24no—
172#11758198—24no—
173C4⋊Dic61—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61C22×C414yes—
176C4⋊Dic61C22×C412yes—
177#11758196—24no—
178#11758196—24no—
179#11758196—28no—
180C4×Dic61—24no—
181C4×Dic61—24no—
182C4×Dic61—24no—
183C4×Dic61—42no—
184C4×Dic61—44no—
185C4×Dic61C22⋊C414yes—
186C4×Dic61C22⋊C414yes—
187C4×Dic61C22×C412yes—
188C4×Dic61C2×D412yes—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418C22×C414yes—
192#11758189—216no—
193C23×C61—22no—
194C23×C61—24no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61C22×D411yescharacteristic
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—22no—
202C22×D61C2×C22⋊C412yes—
203C22×D61C4×D414yes—
204D4×C61—22no—
205D4×C61—24no—
206D4×C61C2×C4214yescomplemented
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—48no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C23×C412yes—
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C22⋊D61—216no—
223C2×Dic61—22no—
224C2×Dic61—24no—
225C2×Dic61—22no—
226C2×Dic61—216no—
227C2×Dic61—22no—
228C2×Dic61—24no—
229C2×Dic61—42no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—48no—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C4×D414yes—
238C2×Dic61AGL(1,ℤ/8)12yes—
239C2×Dic61#13712yes—
240C4×D61—28no—
241C4×D61—44no—
242C4×D61—416no—
243C4×D61C2×C4214yes—
244C4×D61#11714yes—
245C4×D61C4×D418yes—
246Dic122—22no—
247Dic122—24no—
248Dic122C2×C4214yescomplemented
249C8×C61—24no—
250#92418—24no—
251C4×#417—611nomaximal, Sylow
252C22×C61—24no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61C22×C22⋊C411yescharacteristic
258C22×C61C2×C4×D412yes—
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C22×C61#57512yes—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—24no—
265C2×D61—24no—
266C2×D61C4×C22⋊C414yes—
267C4×C61—22no—
268C4×C61—22no—
269C4×C61—42no—
270C4×C61—44no—
271C4×C61C22×C4211yescharacteristic
272C4×C61C2×#11711yescharacteristic
273C4×C61C2×C4×D412yes—
274Dic61—22no—
275Dic61—24no—
276Dic61—42no—
277Dic61—44no—
278Dic61C4×C22⋊C414yes—
279Dic61#41714yescomplemented
280C2×C4×#41—611no—
281C2×#417—612no—
282C2×C4×M16—611no—
283C2×C4×C4⋊C4—611no—
284#1763—611no—
285#1763—611no—
286#1762—611no—
287#1762D6111yescharacteristic, complemented
288C4×#103—612no—
289C4×#102—612no—
290#1736—612no—
291C2×C61—44no—
292C2×C61C2×C4×C22⋊C411yescharacteristic
293C2×C61C2×#41711yescharacteristic
294C2×C61#289611yescharacteristic
295C2×C61C4×AGL(1,ℤ/8)12yes—
296C2×C61C4×#13712yes—
297D61—24no—
298C22×#41—611no—
299C22×M16—612no—
300C4×#41—618no—
301C4×#41—1224no—
302C2×C4×Q8—611no—
303C2×C4×Q8—611no—
304C2×C4×D4—611no—
305C2×C4×D4—1221no—
306C2×C4×D4—1221no—
307C2×C4×D4C2×D6111yescharacteristic
308C2×#117—611no—
309C2×#117—1221no—
310C22×C4⋊C4—612no—
311C22×C42—611no—
312C22×C42—1221no—
313C22×C42—1221no—
314#417—6116no—
315C2×#103—612no—
316C2×#103—612no—
317C2×#102—612no—
318C2×#102C2×D6112yes—
319C4×M16—1224no—
320C2×#97—611no—
321C2×#97C2×D6111yescharacteristic
322C2×C4×C8—612no—
323C4×C4⋊C4—612no—
324C4×C4⋊C4—611no—
325C4×C4⋊C4—1221no—
326C4×C4⋊C4—1221no—
327C4×C4⋊C4C2×D6111yescharacteristic
328C2×#95—1221no—
329#340—612no—
330#340—612no—
331#340—612no—
332#340C2×D6112yes—
333C61C4×#41711yescharacteristic, complemented, Sylow
334C2×#41—614no—
335C2×#41—618no—
336C2×#41—1222no—
337C2×#41—1224no—
338C22×Q8—611no—
339C22×D4—1221no—
340C22×D4C22×D6111yescharacteristic
341C23×C4—21no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—1222no—
346C23×C4—1221no—
347C23×C4—1221no—
348C2×M16—618no—
349C2×M16—1224no—
350C2×M16—1224no—
351C22×C8—612no—
352C22×C8C22×D6112yes—
353C4×Q8—614no—
354C4×Q8—614no—
355C4×Q8—1222no—
356C4×Q8—1222no—
357C4×D4—22no—
358C4×D4—614no—
359C4×D4—1222no—
360C4×D4—1228no—
361C4×D4—1228no—
362C4×D4C4×D6114yescomplemented
363#117—614no—
364#117—1228no—
365#117—1222no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—618no—
369C2×C4⋊C4—1224no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C22×D6112yes—
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C42—21no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—614no—
383C2×C42—614no—
384C2×C42—1228no—
385C2×C42—1221no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1222no—
389C2×C42—1221no—
390C2×C42—2444no—
391C2×C42C22×D6111yescharacteristic
392#103—618no—
393#103—618no—
394#102—618no—
395#102C4×D6118yescomplemented
396#97—22no—
397#97—1222no—
398C4×C8—1224no—
399#95—21no—
400#95—1222no—
401#95—1221no—
402C24—21no—
403C24—611no—
404#41—6116no—
405#41—1228no—
406#41—12216no—
407C2×Q8—614no—
408C2×Q8—612no—
409C2×Q8—1221no—
410C2×Q8—1222no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—1224no—
414C2×D4—1228no—
415C2×D4C2×C4×D6114yes—
416C2×D4C2×C4×D6112yes—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—614no—
425C22×C4—612no—
426C22×C4—614no—
427C22×C4—611no—
428C22×C4—614no—
429C22×C4—614no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1228no—
433C22×C4—1224no—
434C22×C4—1228no—
435C22×C4—1224no—
436C22×C4—1221no—
437C22×C4—1222no—
438C22×C4—1221no—
439C22×C4—2442no—
440C22×C4—2444no—
441C22×C4—2444no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4D4×D6111yescharacteristic
445C22×C4D4×D6111yescharacteristic
446C22×C4C23×D6111yescharacteristic
447M16—12216no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—614no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8C2×C4×D6114yes—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—614no—
457C4⋊C4—612no—
458C4⋊C4—1228no—
459C4⋊C4—1224no—
460C4⋊C4—1221no—
461C4⋊C4—1222no—
462C4⋊C4—1224no—
463C4⋊C4C2×C4×D6114yes—
464C4⋊C4C2×C4×D6112yes—
465C22⋊C4—24no—
466C22⋊C4—1224no—
467C22⋊C4—1224no—
468C42—42no—
469C42—612no—
470C42—612no—
471C42—614no—
472C42—614no—
473C42—1224no—
474C42—1224no—
475C42—1224no—
476C42—2442no—
477C42—2444no—
478C42C2×C4×D6112yes—
479C42D4×D6112yes—
480C23—22no—
481C23—24no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—612no—
486C23—614no—
487C23—1221no—
488C23—1221no—
489C23—1222no—
490C23C2×D4×D6111yescharacteristic
491Q8—614no—
492Q8—1222no—
493Q8—1224no—
494D4—22no—
495D4—24no—
496D4—12216no—
497D4C42×D6114yescomplemented
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—614no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—614no—
513C2×C4—614no—
514C2×C4—618no—
515C2×C4—612no—
516C2×C4—1228no—
517C2×C4—12216no—
518C2×C4—1222no—
519C2×C4—1224no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1224no—
523C2×C4—2444no—
524C2×C4—24416no—
525C2×C4—2448no—
526C2×C4—2442no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2442no—
530C2×C4C22⋊C4×D6112yes—
531C2×C4C22⋊C4×D6112yes—
532C2×C4C22×C4×D6111yescharacteristic
533C2×C4C22×C4×D6112yes—
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4#41×D6112yes—
536C8—24no—
537C8—1224no—
538C22—24no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22—614no—
544C22—1224no—
545C22—1224no—
546C22—1222no—
547C22—1222no—
548C22C2×C22⋊C4×D6111yescharacteristic
549C22C4×D4×D6112yes—
550C22AGL(1,ℤ/8)×D6111yescharacteristic
551C22#137×D6111yescharacteristic
552C22#1684055812yes—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—42no—
557C4—614no—
558C4—614no—
559C4—1224no—
560C4—1222no—
561C4—2444no—
562C4—2442no—
563C4C2×C42×D6111yescharacteristic
564C4#117×D6111yescharacteristic
565C4C4×D4×D6112yes—
566C2—44no—
567C2—1224no—
568C2C4×C22⋊C4×D6111yescharacteristic
569C2#417×D6111yescharacteristic
570C2#2145433311yescharacteristic
571C2#2145463012yes—
572C2#2145464112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61C4×#417
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
3. C2×#16839714C2semidirect
4. C4×C4⋊C4×D61C2semidirect
5. #21445652C2semidirect
6. #21445650C2semidirect
7. #21445644C2semidirect
8. #21445638C2semidirect
9. #102×Dic61C2semidirect
10. #21445537C2semidirect
11. C61×#1762C2semidirect
15. C4×#11758224C4semidirect
27. #16839760C4semidirect
61. C2×#11758189C2×C4semidirect
64. C4×Dic122C2×C4semidirect
66. #14519742C2×C4semidirect
68. D4×Dic61C2×C4semidirect
80. C4⋊C4×D61C2×C4semidirect
95. #14519702C2×C4semidirect
96. #14519700C2×C4semidirect
97. #14519698C2×C4semidirect
106. C4×D4×C61C2×C4semidirect
112. #102×C61C2×C4semidirect
143. #11758224C42semidirect
206. D4×C61C2×C42semidirect
248. Dic122C2×C42semidirect
279. Dic61#417semidirect
287. #1762D61semidirect
333. C61C4×#417semidirect
362. C4×D4C4×D61semidirect
395. #102C4×D61semidirect
497. D4C42×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582242
C2×#168397602
C2×#168397142
C4×C4⋊C4×D612
#214456522
#214456502
#214456442
#214456382
#102×Dic612
#214455372
C61×#17622
C4×#41761