Finite Group Databasebeta

← #29677796

Subgroups — #29677796

Order 15616.

1586 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic
2C2×#16839769C212yesmaximal
3C2×#16839714C211yesmaximal, characteristic
4C4×#14519735C211yesmaximal, characteristic
5#21445652C211yesmaximal, characteristic, complemented
6#21445650C211yesmaximal, characteristic, complemented
7#21445649C211yesmaximal, characteristic, complemented
8#103×Dic61C212yesmaximal, complemented
9#21445638C212yesmaximal, complemented
10#21445537C212yesmaximal
11C61×#1763C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×#11758189C2212yes—
14C2×Q8×Dic61C2211yescharacteristic
15C2×C4×Dic122—21no—
16C2×C4×Dic122C2211yescharacteristic
17C2×#14519742—21no—
18C2×#14519742C2211yescharacteristic
19C4×Q8×D61—24no—
20C4×Q8×D61C418yescomplemented
21C2×#14519735C2212yes—
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C42×D61—21no—
25C2×C42×D61C2211yescharacteristic
26#16839769C4116yescomplemented
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30#16839714—24no—
31C2×#14519688C2211yescharacteristic
32C2×C8×Dic61C2212yes—
33C4⋊C4×Dic61C2212yes—
34C4×C4⋊Dic61C2211yescharacteristic
35C4×#11758198—21no—
36#16839609—21no—
37#16839607—21no—
38C42×Dic61—21no—
39#16839525C2212yes—
40#16839524C2212yes—
41#16839524C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C4×C4⋊C4×C61C2211yescharacteristic
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—24no—
51C2×Q8×D61C2×C414yes—
52C2×Q8×D61C2×C418yes—
53C22×C4×D61—21no—
54C22×C4×D61—21no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C2×#11758189—24no—
59C2×#11758189—24no—
60C2×#11758189C2×C418yes—
61C22×#92418C2312yes—
62Q8×Dic61—22no—
63Q8×Dic61C2×C414yes—
64C4×Dic122—22no—
65C4×Dic122—28no—
66C4×Dic122C2×C414yes—
67#14519742—28no—
68#14519742—22no—
69#14519742C2×C414yes—
70#14519735—24no—
71#14519735C2×C418yes—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C4⋊C4×D61—28no—
79C4⋊C4×D61—22no—
80C4⋊C4×D61C2×C414yes—
81C2×#11758198—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—22no—
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91C42×D61—28no—
92C42×D61C2×C414yes—
93C42×D61D414yes—
94#14519702C2×C418yescomplemented
95#14519701C2×C418yescomplemented
96#14519700C2×C418yescomplemented
97#14519688—22no—
98C8×Dic61—24no—
99#14519684—21no—
100#14519684—22no—
101C22×Q8×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103C4×Q8×C61—22no—
104C4×Q8×C61C2×C414yes—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C42×C61—21no—
109C2×C42×C61—21no—
110C2×C42×C61C2311yescharacteristic
111#103×C61C2×C418yescomplemented
112#97×C61—22no—
113#95×C61—21no—
114C23×D61C2×D411yescharacteristic
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61C22×C414yes—
118C2×Q8×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131Q8×D61—28no—
132Q8×D61—216no—
133Q8×D61C42116yescomplemented
134C2×C4×D61—24no—
135C2×C4×D61—28no—
136C2×C4×D61—24no—
137C2×C4×D61—24no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×C4×D61C22⋊C414yes—
143C2×C4×D61C22⋊C414yes—
144C2×C4×D61C22×C412yes—
145C2×C4×D61C22×C414yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61#4114yes—
148C2×Dic122—21no—
149C2×Dic122—22no—
150C2×Dic122—24no—
151C2×Dic122—28no—
152C2×Dic122C22×C414yes—
153C2×Dic122C22×C412yes—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61C22×C414yes—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C22×C414yes—
161C4⋊C4×C61C22×C412yes—
162C42×C61—24no—
163C42×C61—42no—
164C42×C61C22×C412yes—
165C42×C61C2×D412yes—
166#11758198—28no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4⋊Dic61—24no—
170C4⋊Dic61C22×C414yes—
171C4⋊Dic61C22×C412yes—
172#11758196—24no—
173#11758196—24no—
174#11758196—24no—
175C4×Dic61—24no—
176C4×Dic61—24no—
177C4×Dic61—24no—
178C4×Dic61—28no—
179C4×Dic61—42no—
180C4×Dic61—44no—
181C4×Dic61C22⋊C414yes—
182C4×Dic61C22⋊C414yes—
183C4×Dic61C22×C412yes—
184C4×Dic61C2×D412yes—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418C22×C414yes—
188#11758189—216no—
189C23×C61C22×D411yescharacteristic
190C22×D61—22no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61C2×C22⋊C412yes—
194C22×D61C4×D414yes—
195Q8×C61—22no—
196Q8×C61—24no—
197Q8×C61C2×C4214yes—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—24no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61C2×C22⋊C412yes—
210C2×C4×C61C2×C22⋊C412yes—
211C2×C4×C61C23×C411yescharacteristic
212C2×C4×C61C23×C412yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4112yes—
215C2×Dic61—22no—
216C2×Dic61—24no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—22no—
220C2×Dic61—24no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61C2×C22⋊C412yes—
226C2×Dic61C2×C22⋊C412yes—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C4×D414yes—
229C2×Dic61#13714yes—
230C4×D61—216no—
231C4×D61—28no—
232C4×D61—44no—
233C4×D61—48no—
234C4×D61C2×C4214yes—
235C4×D61#11714yes—
236C4×D61C4×D418yes—
237Dic122—22no—
238Dic122—24no—
239Dic122—216no—
240Dic122C2×C4214yes—
241C8×C61—24no—
242#92418—24no—
243C4×#419—611nomaximal, Sylow
244C22×C61C22×C22⋊C411yescharacteristic
245C22×C61C2×C4×D412yes—
246C22×C61C2×#13712yes—
247C22×C61#57511yescharacteristic
248C22×C61#57511yescharacteristic
249C2×D61—22no—
250C2×D61—22no—
251C2×D61—24no—
252C2×D61—24no—
253C2×D61C4×C22⋊C414yes—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—24no—
257C4×C61—42no—
258C4×C61—44no—
259C4×C61C22×C4211yescharacteristic
260C4×C61C2×#11711yescharacteristic
261C4×C61C2×C4×D412yes—
262Dic61—22no—
263Dic61—24no—
264Dic61—42no—
265Dic61—44no—
266Dic61C4×C22⋊C414yes—
267Dic61#41914yescomplemented
268C22×C4×Q8—611no—
269C2×#419—612no—
270C2×C4×M16—611no—
271C4×#117—611no—
272#1763—611no—
273#1763—611no—
274#1763—611no—
275#1763D6111yescharacteristic, complemented
276C4×#103—612no—
277C4×#103—612no—
278#1736—612no—
279C2×C61C2×C4×C22⋊C411yescharacteristic
280C2×C61C2×#41911yescharacteristic
281C2×C61#289611yescharacteristic
282C2×C61C4×#13712yes—
283C2×C61C4×#13712yes—
284D61—24no—
285C23×Q8—611no—
286C22×M16—612no—
287C2×C4×Q8—611no—
288C2×C4×Q8—618no—
289C2×C4×Q8—611no—
290C2×C4×Q8—611no—
291C2×C4×Q8—1221no—
292C2×C4×Q8—1221no—
293C2×C4×Q8—1224no—
294C2×C4×Q8C2×D6111yescharacteristic
295C2×#117—612no—
296C22×C4⋊C4—611no—
297C22×C4⋊C4—1221no—
298C22×C42—611no—
299C22×C42—1221no—
300#419—6116no—
301C2×#103—612no—
302C2×#103—612no—
303C2×#103—612no—
304C2×#103C2×D6112yes—
305C4×M16—1224no—
306C2×#97—611no—
307C2×#97C2×D6111yescharacteristic
308C2×C4×C8—612no—
309C4×C4⋊C4—612no—
310C4×C4⋊C4—611no—
311C4×C4⋊C4C2×D6111yescharacteristic
312C4×C22⋊C4—1221no—
313C4×C22⋊C4—1221no—
314#376—1221no—
315C43—1221no—
316#340—612no—
317#340—612no—
318#340—612no—
319#340C2×D6112yes—
320C61C4×#41911yescharacteristic, complemented, Sylow
321C22×Q8—614no—
322C22×Q8—618no—
323C22×Q8—611no—
324C22×Q8—1221no—
325C22×Q8—1222no—
326C22×Q8—1224no—
327C22×Q8C22×D6111yescharacteristic
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—1221no—
332C23×C4—1221no—
333C2×M16—618no—
334C2×M16—1224no—
335C2×M16—1224no—
336C22×C8—612no—
337C22×C8C22×D6112yes—
338C4×Q8—22no—
339C4×Q8—614no—
340C4×Q8—614no—
341C4×Q8—614no—
342C4×Q8—1228no—
343C4×Q8—1222no—
344C4×Q8—1228no—
345C4×Q8—1222no—
346C4×Q8—1222no—
347C4×Q8C4×D6114yes—
348#117—618no—
349#117—1224no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—614no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1228no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C4⋊C4C22×D6112yes—
362C2×C22⋊C4—1222no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—611no—
368C2×C42—614no—
369C2×C42—614no—
370C2×C42—1228no—
371C2×C42—1222no—
372C2×C42—1221no—
373C2×C42—1221no—
374C2×C42—1221no—
375C2×C42—1222no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378#103—618no—
379#103—618no—
380#103—618no—
381#103C4×D6118yescomplemented
382#97—22no—
383#97—1222no—
384C4×C8—1224no—
385#95—21no—
386#95—1222no—
387#95—1221no—
388C24—611no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—6116no—
392C2×Q8—614no—
393C2×Q8—612no—
394C2×Q8—1224no—
395C2×Q8—1228no—
396C2×Q8—1228no—
397C2×Q8—12216no—
398C2×Q8—1221no—
399C2×Q8—1222no—
400C2×Q8C2×C4×D6114yes—
401C2×Q8C2×C4×D6112yes—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—612no—
408C22×C4—614no—
409C22×C4—612no—
410C22×C4—614no—
411C22×C4—611no—
412C22×C4—614no—
413C22×C4—614no—
414C22×C4—1224no—
415C22×C4—1224no—
416C22×C4—1228no—
417C22×C4—1224no—
418C22×C4—1221no—
419C22×C4—1222no—
420C22×C4—1221no—
421C22×C4—2441no—
422C22×C4—2441no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4D4×D6111yescharacteristic
426C22×C4D4×D6111yescharacteristic
427C22×C4C23×D6111yescharacteristic
428M16—12216no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—614no—
432C2×C8—1222no—
433C2×C8—1222no—
434C2×C8C2×C4×D6114yes—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—614no—
439C4⋊C4—612no—
440C4⋊C4—1224no—
441C4⋊C4—1224no—
442C4⋊C4—1224no—
443C4⋊C4—1221no—
444C4⋊C4—1222no—
445C4⋊C4—1224no—
446C4⋊C4C2×C4×D6114yes—
447C4⋊C4C2×C4×D6112yes—
448C22⋊C4—1228no—
449C42—24no—
450C42—42no—
451C42—612no—
452C42—612no—
453C42—614no—
454C42—614no—
455C42—1224no—
456C42—1224no—
457C42—1228no—
458C42—1224no—
459C42—2442no—
460C42—2444no—
461C42C2×C4×D6112yes—
462C42D4×D6112yes—
463C23—612no—
464C23—614no—
465C23—1222no—
466C23—1221no—
467C23—1221no—
468C23C2×D4×D6111yescharacteristic
469Q8—22no—
470Q8—24no—
471Q8—614no—
472Q8—12216no—
473Q8—1222no—
474Q8—1224no—
475Q8C42×D6114yes—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—612no—
488C2×C4—612no—
489C2×C4—614no—
490C2×C4—614no—
491C2×C4—614no—
492C2×C4—614no—
493C2×C4—618no—
494C2×C4—612no—
495C2×C4—1228no—
496C2×C4—12216no—
497C2×C4—1222no—
498C2×C4—1224no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1222no—
502C2×C4—1224no—
503C2×C4—2444no—
504C2×C4—2448no—
505C2×C4—2442no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4C22⋊C4×D6112yes—
510C2×C4C22⋊C4×D6112yes—
511C2×C4C22×C4×D6111yescharacteristic
512C2×C4C22×C4×D6112yes—
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4#41×D6112yes—
515C8—24no—
516C8—1224no—
517C22—614no—
518C22—1224no—
519C22—1224no—
520C22—1222no—
521C22—1222no—
522C22C2×C22⋊C4×D6111yescharacteristic
523C22C4×D4×D6112yes—
524C22#137×D6112yes—
525C22#1684055811yescharacteristic
526C22#1684056111yescharacteristic
527C4—24no—
528C4—22no—
529C4—22no—
530C4—44no—
531C4—42no—
532C4—614no—
533C4—614no—
534C4—1224no—
535C4—1222no—
536C4—2444no—
537C4—2442no—
538C4C2×C42×D6111yescharacteristic
539C4#117×D6111yescharacteristic
540C4C4×D4×D6112yes—
541C2—1224no—
542C2C4×C22⋊C4×D6111yescharacteristic
543C2#419×D6111yescharacteristic
544C2#2145433511yescharacteristic
545C2#2145463512yes—
546C2#2145464112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61C4×#419
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
5. #21445652C2semidirect
6. #21445650C2semidirect
7. #21445649C2semidirect
8. #103×Dic61C2semidirect
9. #21445638C2semidirect
11. C61×#1763C2semidirect
20. C4×Q8×D61C4semidirect
26. #16839769C4semidirect
94. #14519702C2×C4semidirect
95. #14519701C2×C4semidirect
96. #14519700C2×C4semidirect
111. #103×C61C2×C4semidirect
133. Q8×D61C42semidirect
267. Dic61#419semidirect
275. #1763D61semidirect
320. C61C4×#419semidirect
381. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
C2×#168397692
C2×#168397142
C4×#145197352
#214456522
#214456502
#214456492
#103×Dic612
#214456382
#214455372
C61×#17632
C4×#41961