Finite Group Databasebeta

← #29689917

Subgroups — #29689917

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840297C211yesmaximal, characteristic
2C2×C4⋊C8×D61C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21446777C211yesmaximal, characteristic, complemented
5#21446751C211yesmaximal, characteristic, complemented
6#21446749C211yesmaximal, characteristic, complemented
7#21446246C212yesmaximal
8#21446241C212yesmaximal, complemented
9#21445823C212yesmaximal, complemented
10#21445818C212yesmaximal, complemented
11C61×#1989C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840297—24no—
14#16840297C418yescomplemented
15C2×#14519772—21no—
16C2×#14519765C2211yescharacteristic
17C2×#14519762—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519752C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519739—21no—
22C2×#14519739C2211yescharacteristic
23C2×#14519735—21no—
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519706C2212yes—
28C4⋊C8×D61—24no—
29C2×#14519702C2212yes—
30C2×#14519700C2212yes—
31C2×#14519699C2212yes—
32C2×#14519697C2212yes—
33#16839631C2212yes—
34#16839626—21no—
35#16839621—21no—
36#16839619C2212yes—
37C4×#11758196—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×D244—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2312yes—
54C2×C8×D61—24no—
55C2×C8×D61—42no—
56C2×C8×D61—42no—
57#14519772—28no—
58#14519765—22no—
59#14519765C2×C414yes—
60#14519762—28no—
61#14519757—22no—
62#14519757C2×C414yes—
63#14519752—22no—
64#14519752C2×C414yes—
65#14519742—22no—
66#14519742C2×C414yes—
67#14519739—28no—
68#14519739—22no—
69#14519739C2×C414yes—
70#14519735—28no—
71#14519734—44no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90C4⋊#92418—22no—
91#14519706—24no—
92#14519702—42no—
93#14519700—42no—
94#14519699—42no—
95#14519697—42no—
96#14519684—21no—
97#14519684—22no—
98C22×C8×C61C2312yes—
99#125×C61—22no—
100#125×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—22no—
109#95×C61—21no—
110C23×D61—21no—
111C23×D61C2×D411yescharacteristic
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—21no—
119C22×Dic61—22no—
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—24no—
126C2×D244—21no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244C22×C412yes—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—24no—
135C2×C4×D61—24no—
136C2×C4×D61—44no—
137C2×C4×D61C22⋊C414yes—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22×C412yes—
140C2×C4×D61C2×D412yes—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—21no—
144C2×Dic122C22×C412yes—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C22×C412yes—
155C4⋊C4×C61C22×C412yes—
156C42×C61—42no—
157C42×C61C22×C412yes—
158C42×C61C2×D412yes—
159#11758198—24no—
160#11758198—24no—
161#11758198—24no—
162#11758198—24no—
163#11758198—24no—
164#11758198—44no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168#11758196—24no—
169#11758196—44no—
170#11758196—44no—
171C4×Dic61—24no—
172C4×Dic61—42no—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22⋊C414yes—
175C4×Dic61C22×C412yes—
176C4×Dic61C2×D412yes—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C8×D61—48no—
181C23×C61C22×D411yescharacteristic
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61C2×C22⋊C412yes—
190C22×D61AGL(1,ℤ/8)12yes—
191C22×D61#13712yes—
192C2×C4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61H(5,𝔽2)12yes—
208C2×C4×C612−1+412yes—
209C2×Dic61—22no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—22no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C2×C22⋊C412yes—
219C2×Dic61C2×C22⋊C412yes—
220C2×Dic61#13514yes—
221D244—24no—
222D244—48no—
223D244—41no—
224D244—41no—
225C4×D61—28no—
226C4×D61—24no—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—28no—
230C4×D61—48no—
231Dic122—24no—
232Dic122—41no—
233Dic122—41no—
234C8×C61—42no—
235#92418—42no—
236#21887—611nomaximal, Sylow
237C22×C61C22×C22⋊C411yescharacteristic
238C22×C61#53712yes—
239C22×C61C2×#13512yes—
240C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C22×C61C2×#13711yescharacteristic
242C2×D61—22no—
243C2×D61—22no—
244C2×D61—24no—
245C2×D61—24no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—82no—
250C2×D61—82no—
251C2×D61#41714yes—
252C4×C61—22no—
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—42no—
263Dic61—82no—
264Dic61#41614yes—
265C2×#529—611no—
266C22×C4⋊C8—611no—
267#2285—611no—
268#1989—611no—
269#1989D6111yescharacteristic, complemented
270#1983—611no—
271#1983—611no—
272#1893—612no—
273#1892—612no—
274#1803—612no—
275#1802—612no—
276C2×C61#231111yescharacteristic
277C2×C61C2×#41611yescharacteristic
278C2×C61C2×#41711yescharacteristic
279C2×C61#324712yes—
280C2×C61#324812yes—
281D61—24no—
282D61—82no—
283C23×C8—612no—
284#529—618no—
285#529—1224no—
286C2×#126—1221no—
287C2×#125C2×D6111yescharacteristic
288C2×#124—611no—
289C2×#123—1221no—
290C2×C22⋊Q8—611no—
291C2×#121—611no—
292C2×C4×Q8—611no—
293C2×C4×D4—611no—
294C2×C4×D4—1221no—
295C2×#117—1221no—
296C2×#117—1221no—
297C22×C42—611no—
298C2×C4⋊C8—612no—
299C2×C4⋊C8—611no—
300C2×C4⋊C8—1224no—
301C2×C4⋊C8C2×D6111yescharacteristic
302C2×#103—612no—
303C2×#103—612no—
304C2×#102—612no—
305C2×#102—612no—
306#384—612no—
307#382—1221no—
308#380—612no—
309#380—1221no—
310C4×C4⋊C4—611no—
311C4×C4⋊C4—1221no—
312C4×C4⋊C4C2×D6111yescharacteristic
313#340—612no—
314#340—612no—
315#340C2×D6112yes—
316#340C2×D6112yes—
317C61#2188711yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×D4—611no—
320C22×D4—1221no—
321C23×C4—611no—
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—1221no—
325C22×C8—612no—
326C22×C8—1224no—
327C22×C8—2442no—
328C22×C8—2442no—
329C22×C8C22×D6112yes—
330#126—1228no—
331#125—22no—
332#125C4×D6114yes—
333#124—614no—
334#124—1222no—
335#123—1228no—
336C22⋊Q8—614no—
337C22⋊Q8—1222no—
338#121—614no—
339#121—1222no—
340C4×Q8—614no—
341C4×Q8—1222no—
342C4×D4—614no—
343C4×D4—1222no—
344C4×D4—1228no—
345#117—1228no—
346#117—2444no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4C22×D6112yes—
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1222no—
363C2×C42—21no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—614no—
368C2×C42—614no—
369C2×C42—1221no—
370C2×C42—1221no—
371C2×C42C22×D6111yescharacteristic
372C4⋊C8—22no—
373C4⋊C8—1224no—
374C4⋊C8—1222no—
375#103—2442no—
376#103—2442no—
377#102—2442no—
378#102—2442no—
379#95—21no—
380#95—1222no—
381#95—1221no—
382C24—611no—
383C24—1221no—
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×D4—612no—
389C2×D4—1224no—
390C2×D4—1221no—
391C2×D4—1222no—
392C2×D4—1222no—
393C2×D4—2442no—
394C2×D4—2442no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—612no—
402C22×C4—612no—
403C22×C4—614no—
404C22×C4—614no—
405C22×C4—1224no—
406C22×C4—1224no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—2444no—
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—2448no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—2444no—
435C4⋊C4—2444no—
436C4⋊C4C2×C4×D6112yes—
437C4⋊C4C2×C4×D6112yes—
438C22⋊C4—1224no—
439C22⋊C4—1224no—
440C22⋊C4—1224no—
441C22⋊C4—1224no—
442C22⋊C4—1224no—
443C22⋊C4—2444no—
444C42—42no—
445C42—612no—
446C42—612no—
447C42—614no—
448C42—614no—
449C42—1224no—
450C42—2442no—
451C42C2×C4×D6112yes—
452C42D4×D6112yes—
453C23—612no—
454C23—612no—
455C23—612no—
456C23—1222no—
457C23—1222no—
458C23—1222no—
459C23—2442no—
460C23—2441no—
461C23—2441no—
462C23—2442no—
463C23C2×D4×D6111yescharacteristic
464Q8—1224no—
465Q8—2441no—
466Q8—2441no—
467D4—1224no—
468D4—2448no—
469D4—2441no—
470D4—2441no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—612no—
483C2×C4—614no—
484C2×C4—612no—
485C2×C4—612no—
486C2×C4—1228no—
487C2×C4—1228no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1222no—
491C2×C4—1224no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—2448no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4C22⋊C4×D6112yes—
501C2×C4C22⋊C4×D6112yes—
502C2×C4C22×C4×D6111yescharacteristic
503C2×C4C2×D4×D6111yescharacteristic
504C2×C4#1451982312yes—
505C2×C4#1451982512yes—
506C8—42no—
507C8—2442no—
508C22—614no—
509C22—1224no—
510C22—1222no—
511C22—1222no—
512C22—1224no—
513C22—4882no—
514C22—4882no—
515C22—2442no—
516C22—2442no—
517C22—2442no—
518C22C2×C22⋊C4×D6111yescharacteristic
519C22#1684034212yes—
520C22#135×D6112yes—
521C22AGL(1,ℤ/8)×D6111yescharacteristic
522C22#137×D6111yescharacteristic
523C4—22no—
524C4—22no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—614no—
529C4—1222no—
530C4—1222no—
531C4—4882no—
532C4—2442no—
533C4#1684024412yes—
534C4#1684033011yescharacteristic
535C4#1684033711yescharacteristic
536C2—1224no—
537C2—4882no—
538C2#2144841011yescharacteristic
539C2#416×D6111yescharacteristic
540C2#417×D6111yescharacteristic
541C2#2145714412yes—
542C2#2145715512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1989D61
hypocenterC61#21887
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C8×D61C2semidirect
3. #21448223C2semidirect
4. #21446777C2semidirect
5. #21446751C2semidirect
6. #21446749C2semidirect
8. #21446241C2semidirect
9. #21445823C2semidirect
10. #21445818C2semidirect
11. C61×#1989C2semidirect
14. #16840297C4semidirect
269. #1989D61semidirect
317. C61#21887semidirect

Maximal subgroups

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

subgroupindex
C2×#168402972
C2×C4⋊C8×D612
#214482232
#214467772
#214467512
#214467492
#214462462
#214462412
#214458232
#214458182
C61×#19892
#2188761