Finite Group Databasebeta

← #29689929

Subgroups — #29689929

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×C4⋊C8×D61C211yesmaximal, characteristic
3#21448212C211yesmaximal, characteristic
4#21446769C212yesmaximal, complemented
5#21446740C212yesmaximal, complemented
6#21446241C212yesmaximal
7#21446239C212yesmaximal
8#21445835C211yesmaximal, characteristic, complemented
9#21445834C211yesmaximal, characteristic, complemented
10#21445831C211yesmaximal, characteristic, complemented
11C61×#1805C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840316—24no—
14#16840316C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519775C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519755—21no—
20C2×#14519753C2211yescharacteristic
21C2×C4×Dic122—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519734—21no—
24C2×C42×D61C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519706C2212yes—
27C4⋊C8×D61—24no—
28C2×#14519702C2212yes—
29C2×#14519701C2212yes—
30C2×#14519700C2212yes—
31#16839629—21no—
32#16839628C2211yescharacteristic
33#16839624—21no—
34#16839619C2212yes—
35C4⋊C4×Dic61C2212yes—
36#16839609—21no—
37#16839607—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52C2×C8×D61—24no—
53C2×C8×D61—42no—
54C2×C8×D61—42no—
55#14519776—22no—
56#14519776C2×C414yes—
57#14519775—22no—
58#14519775C2×C414yes—
59#14519767—28no—
60#14519757—22no—
61#14519757C2×C414yes—
62#14519755—28no—
63#14519753—22no—
64#14519753C2×C414yes—
65C4×Dic122—22no—
66C4×Dic122—28no—
67C4×Dic122C2×C414yes—
68#14519734—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87C4⋊#92418—22no—
88#14519706—24no—
89#14519702—42no—
90#14519701—42no—
91#14519700—42no—
92#14519684—21no—
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C4×Q8×C61—22no—
99C4×Q8×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#103×C61—42no—
107#95×C61—21no—
108#95×C61—21no—
109C23×D61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61C22×C412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C2×C4×D61—24no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—24no—
134C2×C4×D61C22⋊C414yes—
135C2×C4×D61C22⋊C414yes—
136C2×C4×D61C22×C412yes—
137C2×C4×D61C2×D412yes—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—24no—
144C2×Dic122—44no—
145C2×Dic122C22×C412yes—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C22×C412yes—
154C42×C61—24no—
155C42×C61C22×C412yes—
156C42×C61C2×D412yes—
157#11758198—24no—
158#11758198—24no—
159#11758198—44no—
160C4⋊Dic61—21no—
161C4⋊Dic61—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61C22×C412yes—
165#11758196—24no—
166#11758196—24no—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170C4×Dic61—24no—
171C4×Dic61—44no—
172C4×Dic61C22⋊C414yes—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22×C412yes—
175C4×Dic61C2×D412yes—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C8×D61—48no—
180C23×C61C22×D411yescharacteristic
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—22no—
184C22×D61C2×C22⋊C412yes—
185C22×D61#13714yes—
186Q8×C61—22no—
187Q8×C61—22no—
188Q8×C61—41no—
189Q8×C61—41no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C23×C411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61H(5,𝔽2)11yescharacteristic
205C2×C4×C61H(5,𝔽2)11yescharacteristic
206C2×C4×C612−1+411yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61C2×C22⋊C412yes—
220C2×Dic61C2×C22⋊C412yes—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61#13514yes—
223C4×D61—28no—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—28no—
229Dic122—22no—
230Dic122—22no—
231Dic122—41no—
232Dic122—41no—
233Dic122—48no—
234C8×C61—42no—
235#92418—42no—
236#21889—611nomaximal, Sylow
237C22×C61C22×C22⋊C411yescharacteristic
238C22×C61#53711yescharacteristic
239C22×C61#53711yescharacteristic
240C22×C61C2×#13512yes—
241C22×C61C2×#13712yes—
242C2×D61—22no—
243C2×D61—22no—
244C2×D61—24no—
245C2×D61—24no—
246C2×D61#41914yes—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61—42no—
250C4×C61—42no—
251C4×C61—82no—
252C4×C61#52011yescharacteristic
253C4×C61#52011yescharacteristic
254C4×C61#53511yescharacteristic
255C4×C61#53611yescharacteristic
256Dic61—22no—
257Dic61—22no—
258Dic61—42no—
259Dic61—42no—
260Dic61—82no—
261Dic61#41614yes—
262C2×#533—611no—
263C22×C4⋊C8—611no—
264#2283—611no—
265#1988—612no—
266#1981—612no—
267#1892—612no—
268#1892—612no—
269#1805—611no—
270#1805—611no—
271#1805—611no—
272#1805D6111yescharacteristic, complemented
273C2×C61#231111yescharacteristic
274C2×C61C2×#41611yescharacteristic
275C2×C61C2×#41911yescharacteristic
276C2×C61#324812yes—
277C2×C61#324812yes—
278D61—24no—
279C23×C8—612no—
280#533—618no—
281#533—1224no—
282C2×C4⋊Q8—611no—
283C2×C4⋊Q8—611no—
284C2×#125—1221no—
285C2×C22⋊Q8—611no—
286C2×C22⋊Q8—611no—
287C2×C22⋊Q8—1221no—
288C2×C4×Q8—611no—
289C2×C4×Q8—1221no—
290C2×C4×Q8C2×D6111yescharacteristic
291C2×#117—1221no—
292C22×C42—611no—
293C2×C4⋊C8—612no—
294C2×C4⋊C8—611no—
295C2×C4⋊C8—1224no—
296C2×C4⋊C8C2×D6111yescharacteristic
297C2×#103—612no—
298C2×#103—612no—
299C2×#103—612no—
300C2×#103C2×D6112yes—
301#383—1221no—
302#382—611no—
303#382C2×D6111yescharacteristic
304#381—1221no—
305#380—612no—
306C4×C4⋊C4—612no—
307C4×C22⋊C4—1221no—
308#376—1221no—
309#340—612no—
310#340—612no—
311#340—612no—
312#340C2×D6112yes—
313C61#2188911yescharacteristic, complemented, Sylow
314C22×Q8—611no—
315C22×Q8—1221no—
316C22×Q8C22×D6111yescharacteristic
317C23×C4—611no—
318C23×C4—611no—
319C23×C4—611no—
320C22×C8—612no—
321C22×C8—1224no—
322C22×C8—2442no—
323C22×C8—2442no—
324C22×C8C22×D6112yes—
325C4⋊Q8—614no—
326C4⋊Q8—614no—
327C4⋊Q8—1222no—
328C4⋊Q8—1222no—
329#125—1228no—
330C22⋊Q8—614no—
331C22⋊Q8—614no—
332C22⋊Q8—1222no—
333C22⋊Q8—1228no—
334C22⋊Q8—1222no—
335C4×Q8—22no—
336C4×Q8—614no—
337C4×Q8—1228no—
338C4×Q8—1222no—
339C4×Q8C4×D6114yes—
340#117—1228no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4C22×D6112yes—
351C2×C4⋊C4C22×D6111yescharacteristic
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1222no—
355C2×C42—21no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—614no—
360C2×C42—614no—
361C2×C42—1221no—
362C2×C42—1222no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—22no—
365C4⋊C8—1224no—
366C4⋊C8—1222no—
367#103—42no—
368#103—2442no—
369#103—2442no—
370#103—2442no—
371#95—21no—
372#95—21no—
373#95—1222no—
374#95—1222no—
375#95—1221no—
376#95—1221no—
377C24—611no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—612no—
384C2×Q8—1224no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—2444no—
391C2×Q8C2×C4×D6112yes—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—614no—
401C22×C4—614no—
402C22×C4—1224no—
403C22×C4—1224no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4D4×D6111yescharacteristic
412C22×C4D4×D6111yescharacteristic
413C22×C4C23×D6111yescharacteristic
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—2448no—
421C4⋊C4—21no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—612no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1224no—
431C4⋊C4—1221no—
432C4⋊C4—1224no—
433C4⋊C4—1222no—
434C4⋊C4—1222no—
435C4⋊C4C2×C4×D6112yes—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C22⋊C4—2444no—
439C42—24no—
440C42—612no—
441C42—612no—
442C42—614no—
443C42—614no—
444C42—1224no—
445C42—2444no—
446C42C2×C4×D6112yes—
447C42D4×D6112yes—
448C23—612no—
449C23—614no—
450C23—1222no—
451C23—1221no—
452C23—1221no—
453C23C2×D4×D6111yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—41no—
457Q8—41no—
458Q8—1222no—
459Q8—1222no—
460Q8—2448no—
461Q8—2441no—
462Q8—2441no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—612no—
474C2×C4—614no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—1228no—
478C2×C4—1228no—
479C2×C4—1222no—
480C2×C4—1221no—
481C2×C4—1221no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4C22⋊C4×D6112yes—
495C2×C4C22⋊C4×D6112yes—
496C2×C4C22×C4×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4#1451982211yescharacteristic
499C2×C4#1451982311yescharacteristic
500C2×C4#1451982411yescharacteristic
501C2×C4#1451982511yescharacteristic
502C8—42no—
503C8—2442no—
504C22—614no—
505C22—1224no—
506C22—1224no—
507C22—1222no—
508C22—1222no—
509C22C2×C22⋊C4×D6111yescharacteristic
510C22#1684034111yescharacteristic
511C22#1684034211yescharacteristic
512C22#135×D6112yes—
513C22#137×D6112yes—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—614no—
520C4—1222no—
521C4—1222no—
522C4—4882no—
523C4—2442no—
524C4—2442no—
525C4#1684024211yescharacteristic
526C4#1684024311yescharacteristic
527C4#1684032811yescharacteristic
528C4#1684033511yescharacteristic
529C2—1224no—
530C2#2144840811yescharacteristic
531C2#416×D6111yescharacteristic
532C2#419×D6111yescharacteristic
533C2#2145715312yes—
534C2#2145715512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1805D61
hypocenterC61#21889
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #21446769C2semidirect
5. #21446740C2semidirect
8. #21445835C2semidirect
9. #21445834C2semidirect
10. #21445831C2semidirect
11. C61×#1805C2semidirect
14. #16840316C4semidirect
272. #1805D61semidirect
313. C61#21889semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×C4⋊C8×D612
#214482122
#214467692
#214467402
#214462412
#214462392
#214458352
#214458342
#214458312
C61×#18052
#2188961