Finite Group Databasebeta

← #29689913

Subgroups — #29689913

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840298C211yesmaximal, characteristic
2C22×#14519706C211yesmaximal, characteristic, complemented
3#21448221C211yesmaximal, characteristic, complemented
4#21446775C212yesmaximal, complemented
5#21446748C212yesmaximal, complemented
6#21446247C211yesmaximal, characteristic
7#21446245C211yesmaximal, characteristic
8#21446243C211yesmaximal, characteristic, complemented
9#21445822C212yesmaximal, complemented
10#21445818C212yesmaximal, complemented
11C61×#1892C211yesmaximal, characteristic, complemented
12C23×#92418C2211yescharacteristic
13#16840298—24no—
14#16840298C418yescomplemented
15C2×#14519772—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×#14519761—21no—
19C2×#14519751C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519735—21no—
24C2×#14519733—21no—
25C22×C4×Dic61C2211yescharacteristic
26C2×#14519706—24no—
27C2×#14519706C2212yes—
28C2×#14519706C2212yes—
29C2×#14519701C2212yes—
30C2×#14519699C2212yes—
31C2×#14519697C2212yes—
32#16839631C2212yes—
33#16839626—21no—
34#16839622C2211yescharacteristic
35#16839620—21no—
36C4⋊C4×Dic61C2212yes—
37#16839613—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C23×C8×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×C4×D61—21no—
51C22×#92418—22no—
52C22×#92418—42no—
53C22×#92418C2312yes—
54#14519772—28no—
55#14519768—22no—
56#14519768C2×C414yes—
57#14519766—22no—
58#14519766C2×C414yes—
59#14519761—28no—
60#14519751—22no—
61#14519751C2×C414yes—
62Q8×Dic61—22no—
63Q8×Dic61C2×C414yes—
64#14519739—22no—
65#14519739C2×C414yes—
66#14519737—28no—
67#14519735—28no—
68#14519733—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×C22⋊Dic61—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61C2×C414yes—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D414yes—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519706—24no—
90#14519706—24no—
91#14519701—42no—
92#14519699—42no—
93#14519697—42no—
94#14519684—21no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61—22no—
99C22×C8×C61—42no—
100C22×C8×C61C2312yes—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61C2×C414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108#103×C61—42no—
109#95×C61—21no—
110C24×C61C2×D411yescharacteristic
111C23×D61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61C22×C412yes—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—24no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—24no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61C22⋊C414yes—
134C22×Dic61C22⋊C414yes—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—21no—
142C2×D244C22×C412yes—
143C2×C4×D61—24no—
144C2×C4×D61—44no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—44no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C22×C412yes—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—42no—
156C22⋊Dic61—42no—
157#11758198—24no—
158#11758198—24no—
159#11758198—24no—
160#11758198—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61—24no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C22×C412yes—
167#11758196—24no—
168#11758196—44no—
169#11758196—44no—
170C4×Dic61—24no—
171C4×Dic61—44no—
172C4×Dic61C22⋊C414yes—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22×C412yes—
175C4×Dic61C22×C412yes—
176C4×Dic61C2×D412yes—
177C4×Dic61C2×D412yes—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—44no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61C2×C22⋊C412yes—
185C23×C61AGL(1,ℤ/8)12yes—
186C23×C61#13712yes—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—41no—
192C22×D61—41no—
193Q8×C61—24no—
194Q8×C61—41no—
195Q8×C61—41no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61C2×C22⋊C412yes—
208C2×C4×C61C23×C411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61H(5,𝔽2)12yes—
211C2×C4×C612−1+412yes—
212C22⋊D61—48no—
213C2×Dic61—22no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—28no—
219C2×Dic61—28no—
220C2×Dic61—22no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61C2×C22⋊C412yes—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C2×C22⋊C412yes—
229C2×Dic61C2×C22⋊C412yes—
230C2×Dic61#13514yes—
231D244—24no—
232D244—41no—
233D244—41no—
234C4×D61—48no—
235C8×C61—42no—
236#92418—42no—
237#21887—611nomaximal, Sylow
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—24no—
241C22×C61—24no—
242C22×C61#41714yes—
243C22×C61C22×C22⋊C411yescharacteristic
244C22×C61#53712yes—
245C22×C61C2×#13512yes—
246C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C22×C61C2×#13711yescharacteristic
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—82no—
252C2×D61—82no—
253C4×C61—22no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—82no—
257C4×C61#52012yes—
258C4×C61#53511yescharacteristic
259C4×C61#53611yescharacteristic
260Dic61—22no—
261Dic61—22no—
262Dic61—22no—
263Dic61—42no—
264Dic61—82no—
265Dic61#41614yes—
266C2×#529—611no—
267C22×C4⋊C8—611no—
268#2285—611no—
269#1989—612no—
270#1983—612no—
271#1893—611no—
272#1893—611no—
273#1892—611no—
274#1892D6111yescharacteristic, complemented
275#1803—612no—
276#1802—612no—
277C2×C61—24no—
278C2×C61#231111yescharacteristic
279C2×C61C2×#41611yescharacteristic
280C2×C61C2×#41711yescharacteristic
281C2×C61#324712yes—
282C2×C61#324812yes—
283D61—82no—
284C23×C8—611no—
285C23×C8C2×D6111yescharacteristic
286#529—618no—
287#529—1224no—
288C2×#126—1221no—
289C2×#125—611no—
290C2×#124—611no—
291C2×#123—1221no—
292C2×C22⋊Q8C2×D6111yescharacteristic
293C2×#121—611no—
294C2×C4×Q8—611no—
295C2×C4×D4—611no—
296C2×C4×D4—1221no—
297C2×#117—1221no—
298C2×#117—1221no—
299C22×C42—611no—
300C2×C4⋊C8—612no—
301C2×C4⋊C8—612no—
302C2×C4⋊C8—1224no—
303C2×#103—612no—
304C2×#103C2×D6112yes—
305C2×#102—612no—
306C2×#102—612no—
307#384—612no—
308#382—1221no—
309#380—611no—
310#380—1221no—
311#380C2×D6111yescharacteristic
312C4×C4⋊C4—612no—
313C4×C4⋊C4—1221no—
314#340—612no—
315#340—612no—
316#340—612no—
317#340C2×D6112yes—
318C61#2188711yescharacteristic, complemented, Sylow
319C22×Q8C22×D6111yescharacteristic
320C22×D4—611no—
321C22×D4—1221no—
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—1221no—
325C23×C4C22×D6111yescharacteristic
326C22×C8—22no—
327C22×C8—42no—
328C22×C8—612no—
329C22×C8—1222no—
330C22×C8—2442no—
331C22×C8C22×D6112yes—
332#126—1228no—
333#125—614no—
334#125—1222no—
335#124—614no—
336#124—1222no—
337#123—1228no—
338C22⋊Q8—22no—
339C22⋊Q8C4×D6114yes—
340#121—614no—
341#121—1222no—
342C4×Q8—614no—
343C4×Q8—1222no—
344C4×D4—614no—
345C4×D4—1222no—
346C4×D4—1228no—
347#117—1228no—
348#117—2444no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4C22×D6112yes—
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1221no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C42—614no—
367C2×C42—611no—
368C2×C42—611no—
369C2×C42—614no—
370C2×C42—611no—
371C2×C42—611no—
372C2×C42—1221no—
373C2×C42—1222no—
374C4⋊C8—1224no—
375C4⋊C8—1224no—
376#103—42no—
377#103—2442no—
378#102—2442no—
379#102—2442no—
380#95—21no—
381#95—1222no—
382#95—1221no—
383C24—1221no—
384C24D4×D6111yescharacteristic
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8C2×C4×D6112yes—
389C2×D4—612no—
390C2×D4—1224no—
391C2×D4—1221no—
392C2×D4—1222no—
393C2×D4—1222no—
394C2×D4—2442no—
395C2×D4—2442no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—614no—
406C22×C4—614no—
407C22×C4—1224no—
408C22×C4—1224no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—2444no—
414C22×C4C2×C4×D6112yes—
415C22×C4D4×D6112yes—
416C22×C4D4×D6111yescharacteristic
417C22×C4C23×D6111yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—44no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C2×C8—2444no—
426C4⋊C4—21no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—612no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1221no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—2444no—
438C4⋊C4—2444no—
439C4⋊C4C2×C4×D6112yes—
440C22⋊C4—24no—
441C22⋊C4—42no—
442C22⋊C4—1224no—
443C22⋊C4—1224no—
444C22⋊C4—1224no—
445C22⋊C4—1224no—
446C22⋊C4—2442no—
447C42—612no—
448C42—612no—
449C42—612no—
450C42—612no—
451C42—614no—
452C42—614no—
453C42—1224no—
454C42—2444no—
455C23—22no—
456C23—22no—
457C23—1222no—
458C23—2442no—
459C23—2441no—
460C23—2441no—
461C23—2442no—
462C23C22⋊C4×D6112yes—
463C23#1451980412yes—
464C23#1451980912yes—
465C23C2×D4×D6111yescharacteristic
466Q8—24no—
467Q8—41no—
468Q8—41no—
469D4—1224no—
470D4—2448no—
471D4—2441no—
472D4—2441no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—612no—
485C2×C4—614no—
486C2×C4—612no—
487C2×C4—612no—
488C2×C4—612no—
489C2×C4—1228no—
490C2×C4—1228no—
491C2×C4—1222no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1222no—
497C2×C4—2448no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4C22⋊C4×D6112yes—
504C2×C4C22×C4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4#1451982212yes—
507C2×C4#1451982412yes—
508C8—42no—
509C8—2442no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—24no—
514C22—4882no—
515C22—4882no—
516C22—2442no—
517C22—2442no—
518C22—2442no—
519C22#1683976214yes—
520C22C2×C22⋊C4×D6111yescharacteristic
521C22#1684033912yes—
522C22#135×D6112yes—
523C22C2×#1451980411yescharacteristic
524C22C2×#1451980911yescharacteristic
525C4—22no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4—614no—
530C4—1222no—
531C4—1222no—
532C4—1222no—
533C4—4882no—
534C4—2442no—
535C4#1684024112yes—
536C4#1684032711yescharacteristic
537C4#1684033411yescharacteristic
538C2—24no—
539C2—4882no—
540C2#2144840611yescharacteristic
541C2#416×D6111yescharacteristic
542C2C2×#1683976211yescharacteristic
543C2#2145714012yes—
544C2#2145715112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1892D61
hypocenterC61#21887
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C22×#14519706C2semidirect
3. #21448221C2semidirect
4. #21446775C2semidirect
5. #21446748C2semidirect
8. #21446243C2semidirect
9. #21445822C2semidirect
10. #21445818C2semidirect
11. C61×#1892C2semidirect
14. #16840298C4semidirect
274. #1892D61semidirect
318. C61#21887semidirect

Maximal subgroups

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

subgroupindex
C2×#168402982
C22×#145197062
#214482212
#214467752
#214467482
#214462472
#214462452
#214462432
#214458222
#214458182
C61×#18922
#2188761