Finite Group Databasebeta

← #29680069

Subgroups — #29680069

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic
3#21448223C211yesmaximal, characteristic
4#21445904C212yesmaximal
5#21445823C212yesmaximal, complemented
6#21445822C212yesmaximal, complemented
7#21445752C212yesmaximal
8#21445652C211yesmaximal, characteristic, complemented
9#21445650C211yesmaximal, characteristic, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C61×#1763C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C2×Q8×Dic61C2211yescharacteristic
14C2×C4×Dic122—21no—
15C2×C4×Dic122C2211yescharacteristic
16C2×#14519742—21no—
17C2×#14519742C2211yescharacteristic
18C4×Q8×D61—24no—
19C4×Q8×D61C418yescomplemented
20C2×#14519734—21no—
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C42×D61—21no—
24C2×C42×D61C2211yescharacteristic
25#16839817—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519689C2212yes—
31C2×#14519688C2211yescharacteristic
32#16839631C2212yes—
33#16839626—21no—
34#16839621—21no—
35#16839619C2212yes—
36C4×#11758196—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#97×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C2×Q8×D61—24no—
49C2×Q8×D61—22no—
50C2×Q8×D61—24no—
51C2×Q8×D61C2×C414yes—
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58Q8×Dic61—22no—
59Q8×Dic61C2×C414yes—
60C4×Dic122—22no—
61C4×Dic122—28no—
62C4×Dic122C2×C414yes—
63#14519742—28no—
64#14519742—22no—
65#14519742C2×C414yes—
66#14519734—44no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C4⋊C4×D61—28no—
76C4⋊C4×D61—22no—
77C4⋊C4×D61C2×C414yes—
78C2×#11758198—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61—28no—
87C42×D61C2×C414yes—
88C42×D61D414yes—
89#14519706—24no—
90#14519702—24no—
91#14519701—24no—
92#14519700—24no—
93#14519689—24no—
94#14519688—22no—
95#14519684—21no—
96#14519684—22no—
97C22×Q8×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C4×Q8×C61—22no—
100C4×Q8×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#103×C61—24no—
108#97×C61—22no—
109#95×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61C22×C412yes—
114C2×Q8×C61#4114yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127Q8×D61—216no—
128Q8×D61—28no—
129Q8×D61—28no—
130C2×C4×D61—28no—
131C2×C4×D61—24no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—24no—
137C2×C4×D61—24no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61C22⋊C414yes—
141C2×C4×D61C22⋊C414yes—
142C2×C4×D61C22×C412yes—
143C2×C4×D61C2×D412yes—
144C2×Dic122—28no—
145C2×Dic122—21no—
146C2×Dic122—22no—
147C2×Dic122—24no—
148C2×Dic122C22×C412yes—
149C2×Dic122#4114yes—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C22×C412yes—
158C42×C61—24no—
159C42×C61—42no—
160C42×C61C22×C412yes—
161C42×C61C2×D412yes—
162#11758198—44no—
163C4⋊Dic61—21no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61C22×C412yes—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—44no—
172#11758196—44no—
173C4×Dic61—24no—
174C4×Dic61—24no—
175C4×Dic61—24no—
176C4×Dic61—42no—
177C4×Dic61C22⋊C414yes—
178C4×Dic61C22⋊C414yes—
179C4×Dic61C22×C412yes—
180C4×Dic61C2×D412yes—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C23×C61C22×D411yescharacteristic
185C22×D61—22no—
186C22×D61—22no—
187C22×D61C2×C22⋊C412yes—
188C22×D61AGL(1,ℤ/8)12yes—
189C22×D61#13712yes—
190Q8×C61—22no—
191Q8×C61—24no—
192Q8×C61#11714yescomplemented
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C2×C22⋊C412yes—
206C2×C4×C61C23×C411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4112yes—
209C2×C4×C61C2×#4112yes—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—24no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61C2×C22⋊C412yes—
224C2×Dic61#13714yes—
225C4×D61—216no—
226C4×D61—28no—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—24no—
230C4×D61—28no—
231C4×D61—48no—
232Dic122—216no—
233Dic122—22no—
234Dic122—24no—
235Dic122#11714yescomplemented
236C8×C61—42no—
237#92418—42no—
238#20740—611nomaximal, Sylow
239C22×C61C22×C22⋊C411yescharacteristic
240C22×C61#52512yes—
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C61C2×#13712yes—
243C22×C61C2×#13711yescharacteristic
244C2×D61—24no—
245C2×D61—24no—
246C2×D61—22no—
247C2×D61—22no—
248C2×D61#41714yes—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—24no—
252C4×C61—42no—
253C4×C61—82no—
254C4×C61C2×#11711yescharacteristic
255C4×C61C2×#11711yescharacteristic
256C4×C61#52512yes—
257Dic61—22no—
258Dic61—22no—
259Dic61—24no—
260Dic61—42no—
261Dic61—82no—
262Dic61#41914yes—
263C22×C4×Q8—611no—
264C2×#432—611no—
265#2285—611no—
266#1823—612no—
267#1803—612no—
268#1803—612no—
269#1787—612no—
270#1763—611no—
271#1763—611no—
272#1763—611no—
273#1763D6111yescharacteristic, complemented
274C2×C61#228011yescharacteristic
275C2×C61C2×#41911yescharacteristic
276C2×C61C2×#41711yescharacteristic
277C2×C61#311212yes—
278C2×C61#311312yes—
279D61—24no—
280C23×Q8—611no—
281C2×C4×Q8—611no—
282C2×C4×Q8—618no—
283C2×C4×Q8—611no—
284C2×C4×Q8—611no—
285C2×C4×Q8—1221no—
286C2×C4×Q8—1221no—
287C2×C4×Q8—1224no—
288C2×C4×Q8C2×D6111yescharacteristic
289C2×#117—1221no—
290C22×C4⋊C4—611no—
291C22×C4⋊C4—1221no—
292C22×C42—611no—
293C22×C42—1221no—
294#432—1224no—
295C2×C4⋊C8—612no—
296C2×#103—612no—
297C2×#103—612no—
298C2×#103—612no—
299C2×#103C2×D6112yes—
300C2×C22⋊C8—612no—
301C2×#97—611no—
302C2×#97C2×D6111yescharacteristic
303#384—612no—
304#382—1221no—
305#380—612no—
306#380—1221no—
307C4×C4⋊C4—611no—
308C4×C4⋊C4—1221no—
309C4×C4⋊C4C2×D6111yescharacteristic
310#340—612no—
311#340—612no—
312#340—612no—
313#340C2×D6112yes—
314C61#2074011yescharacteristic, complemented, Sylow
315C22×Q8—614no—
316C22×Q8—611no—
317C22×Q8—1221no—
318C22×Q8—1222no—
319C22×Q8—1224no—
320C22×Q8—1224no—
321C22×Q8C22×D6111yescharacteristic
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—611no—
325C23×C4—1221no—
326C23×C4—1221no—
327C22×C8—612no—
328C22×C8C22×D6112yes—
329C4×Q8—22no—
330C4×Q8—614no—
331C4×Q8—614no—
332C4×Q8—614no—
333C4×Q8—1228no—
334C4×Q8—1222no—
335C4×Q8—1228no—
336C4×Q8—1222no—
337C4×Q8—1222no—
338C4×Q8C4×D6114yes—
339#117—2444no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—614no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4—1228no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4C22×D6112yes—
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C22⋊C4—1222no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—614no—
361C2×C42—614no—
362C2×C42—1228no—
363C2×C42—1221no—
364C2×C42—1221no—
365C2×C42—1221no—
366C2×C42—1221no—
367C2×C42C22×D6111yescharacteristic
368C4⋊C8—1224no—
369#103—24no—
370#103—1224no—
371#103—1224no—
372#103—1224no—
373C22⋊C8—1224no—
374#97—22no—
375#97—1222no—
376#95—21no—
377#95—1222no—
378#95—1221no—
379C24—611no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—612no—
383C2×Q8—614no—
384C2×Q8—1224no—
385C2×Q8—1228no—
386C2×Q8—1228no—
387C2×Q8—12216no—
388C2×Q8—1228no—
389C2×Q8—1221no—
390C2×Q8—1222no—
391C2×Q8C2×C4×D6112yes—
392C2×Q8#1175822214yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—612no—
400C22×C4—612no—
401C22×C4—614no—
402C22×C4—614no—
403C22×C4—1224no—
404C22×C4—1224no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1228no—
408C22×C4—1224no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1221no—
413C22×C4—1222no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C4⋊C4—21no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—612no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1221no—
434C4⋊C4—1224no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—2444no—
438C4⋊C4—2444no—
439C4⋊C4C2×C4×D6112yes—
440C22⋊C4—2444no—
441C42—24no—
442C42—42no—
443C42—612no—
444C42—612no—
445C42—614no—
446C42—614no—
447C42—1224no—
448C42—1224no—
449C42—1224no—
450C42—2442no—
451C42C2×C4×D6112yes—
452C42D4×D6112yes—
453C23—612no—
454C23—612no—
455C23—612no—
456C23—1222no—
457C23—1222no—
458C23C2×D4×D6111yescharacteristic
459Q8—22no—
460Q8—24no—
461Q8—614no—
462Q8—12216no—
463Q8—1222no—
464Q8—1224no—
465Q8#1451973414yescomplemented
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—612no—
478C2×C4—614no—
479C2×C4—612no—
480C2×C4—612no—
481C2×C4—1228no—
482C2×C4—1228no—
483C2×C4—12216no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1224no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1224no—
492C2×C4—1222no—
493C2×C4—2448no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4C22⋊C4×D6112yes—
500C2×C4C22⋊C4×D6112yes—
501C2×C4C22×C4×D6111yescharacteristic
502C2×C4C2×D4×D6111yescharacteristic
503C2×C4#41×D6112yes—
504C2×C4C2×#1175822212yes—
505C8—42no—
506C8—2442no—
507C22—614no—
508C22—1222no—
509C22—1224no—
510C22—1224no—
511C22—1222no—
512C22C2×C22⋊C4×D6111yescharacteristic
513C22#1684027112yes—
514C22AGL(1,ℤ/8)×D6111yescharacteristic
515C22#137×D6112yes—
516C22#137×D6111yescharacteristic
517C4—24no—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—82no—
522C4—614no—
523C4—1222no—
524C4—1224no—
525C4—1222no—
526C4—4882no—
527C4—2442no—
528C4#117×D6111yescharacteristic
529C4C2×#1451973411yescharacteristic
530C4#1684027112yes—
531C2—1224no—
532C2#2144820011yescharacteristic
533C2#419×D6111yescharacteristic
534C2#417×D6111yescharacteristic
535C2#2145593912yes—
536C2#2145594712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61#20740
LCSG ▷ C2×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. #21445823C2semidirect
6. #21445822C2semidirect
8. #21445652C2semidirect
9. #21445650C2semidirect
10. #21445649C2semidirect
11. C61×#1763C2semidirect
19. C4×Q8×D61C4semidirect
192. Q8×C61#117semidirect
235. Dic122#117semidirect
273. #1763D61semidirect
314. C61#20740semidirect
465. Q8#14519734semidirect

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×#168398172
#214482232
#214459042
#214458232
#214458222
#214457522
#214456522
#214456502
#214456492
C61×#17632
#2074061