Finite Group Databasebeta

← #29689977

Subgroups — #29689977

Order 15616.

1118 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×#16840297C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic, complemented
3#21448215C211yesmaximal, characteristic, complemented
4#21446777C211yesmaximal, characteristic, complemented
5#21446751C211yesmaximal, characteristic, complemented
6#21446749C211yesmaximal, characteristic, complemented
7#21446256C212yesmaximal
8#21446251C212yesmaximal, complemented
9#21445833C212yesmaximal, complemented
10#21445828C212yesmaximal, complemented
11C61×#1989C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
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×C42×D61C2211yescharacteristic
25#16839784—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519706C2212yes—
28C2×#14519702C2212yes—
29C2×#14519700C2212yes—
30C2×#14519699C2212yes—
31C2×#14519697C2212yes—
32#16839629—21no—
33#16839627C2212yes—
34#16839624—21no—
35#16839619C2212yes—
36C4×C4⋊Dic61C2211yescharacteristic
37C4×#11758198—21no—
38#16839608—21no—
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
53C2×#11758189—24no—
54C2×#11758189—42no—
55C2×#11758189—42no—
56C22×#92418C2312yes—
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—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87C4⋊#92418—22no—
88#14519706—24no—
89#14519702—42no—
90#14519700—42no—
91#14519699—42no—
92#14519697—42no—
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97C22×C8×C61C2312yes—
98#125×C61—22no—
99#125×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—22no—
108#95×C61—21no—
109C23×D61—21no—
110C23×D61C2×D411yescharacteristic
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—24no—
125C2×D244—21no—
126C2×D244—42no—
127C2×D244—42no—
128C2×D244C22×C412yes—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—24no—
132C2×C4×D61—42no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61—44no—
136C2×C4×D61C22⋊C414yes—
137C2×C4×D61C22⋊C414yes—
138C2×C4×D61C22×C412yes—
139C2×C4×D61C2×D412yes—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122C22×C412yes—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C22×C412yes—
154C4⋊C4×C61C22×C412yes—
155C42×C61—42no—
156C42×C61C22×C412yes—
157C42×C61C2×D412yes—
158#11758198—24no—
159#11758198—24no—
160#11758198—24no—
161#11758198—24no—
162#11758198—24no—
163#11758198—44no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167#11758196—24no—
168C4×Dic61—24no—
169C4×Dic61—42no—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22⋊C414yes—
172C4×Dic61C22×C412yes—
173C4×Dic61C2×D412yes—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177#11758189—48no—
178C23×C61C22×D411yescharacteristic
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—22no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61C2×C22⋊C412yes—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
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)12yes—
205C2×C4×C612−1+412yes—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—22no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61C2×C22⋊C412yes—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218D244—24no—
219D244—48no—
220D244—41no—
221D244—41no—
222C4×D61—24no—
223C4×D61—22no—
224C4×D61—22no—
225C4×D61—44no—
226C4×D61—44no—
227C4×D61—48no—
228Dic122—24no—
229Dic122—41no—
230Dic122—41no—
231C8×C61—42no—
232#92418—42no—
233#21895—611nomaximal, Sylow
234C22×C61C22×C22⋊C411yescharacteristic
235C22×C61#53712yes—
236C22×C61#57511yescharacteristic
237C22×C61#57512yes—
238C22×C61#57511yescharacteristic
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—82no—
248C2×D61—82no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—82no—
254C4×C61#52012yes—
255C4×C61#53511yescharacteristic
256C4×C61#53611yescharacteristic
257Dic61—22no—
258Dic61—22no—
259Dic61—22no—
260Dic61—42no—
261Dic61—82no—
262C2×#529—611no—
263C2×#423—611no—
264#2283—611no—
265#1989—611no—
266#1989D6111yescharacteristic, complemented
267#1983—611no—
268#1983—611no—
269#1895—612no—
270#1894—612no—
271#1805—612no—
272#1804—612no—
273C2×C61#231111yescharacteristic
274C2×C61#289611yescharacteristic
275C2×C61#289611yescharacteristic
276C2×C61#325612yes—
277C2×C61#325612yes—
278D61—42no—
279D61—82no—
280C22×M16—612no—
281#529—618no—
282#529—1224no—
283C2×#126—1221no—
284C2×#125C2×D6111yescharacteristic
285C2×#124—611no—
286C2×#123—1221no—
287C2×C22⋊Q8—611no—
288C2×#121—611no—
289C2×C4×Q8—611no—
290C2×C4×D4—611no—
291C2×C4×D4—1221no—
292C2×#117—1221no—
293C22×C42—611no—
294#423—1224no—
295C2×C4⋊C8—612no—
296C2×C4⋊C8—611no—
297C2×C4⋊C8C2×D6111yescharacteristic
298C2×#103—612no—
299C2×#103—612no—
300C2×#102—612no—
301C2×#102—612no—
302#383—1221no—
303#382—612no—
304#381—1221no—
305#380—612no—
306C4×C4⋊C4—611no—
307C4×C4⋊C4C2×D6111yescharacteristic
308C4×C22⋊C4—1221no—
309#376—1221no—
310#340—612no—
311#340—612no—
312#340C2×D6112yes—
313#340C2×D6112yes—
314C61#2189511yescharacteristic, complemented, Sylow
315C22×Q8—611no—
316C22×D4—611no—
317C22×D4—1221no—
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—1221no—
322C2×M16—1224no—
323C2×M16—2442no—
324C2×M16—2442no—
325C22×C8—612no—
326C22×C8C22×D6112yes—
327#126—1228no—
328#125—22no—
329#125C4×D6114yes—
330#124—614no—
331#124—1222no—
332#123—1228no—
333C22⋊Q8—614no—
334C22⋊Q8—1222no—
335#121—614no—
336#121—1222no—
337C4×Q8—614no—
338C4×Q8—1222no—
339C4×D4—614no—
340C4×D4—1222no—
341C4×D4—1228no—
342#117—1228no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4C22×D6112yes—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1222no—
357C2×C42—21no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—614no—
362C2×C42—614no—
363C2×C42—1221no—
364C2×C42—1221no—
365C2×C42C22×D6111yescharacteristic
366C4⋊C8—22no—
367C4⋊C8—1224no—
368C4⋊C8—1222no—
369#103—2442no—
370#103—2442no—
371#102—2442no—
372#102—2442no—
373#95—21no—
374#95—1222no—
375#95—1222no—
376#95—1222no—
377#95—1221no—
378C24—611no—
379C24—1221no—
380C2×Q8—612no—
381C2×Q8—1221no—
382C2×Q8—1222no—
383C2×Q8—1222no—
384C2×D4—612no—
385C2×D4—1224no—
386C2×D4—1221no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—2442no—
390C2×D4—2442no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—614no—
400C22×C4—614no—
401C22×C4—1224no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1221no—
405C22×C4—1222no—
406C22×C4—2441no—
407C22×C4—2442no—
408C22×C4—2441no—
409C22×C4—2444no—
410C22×C4D4×D6111yescharacteristic
411C22×C4D4×D6111yescharacteristic
412C22×C4C23×D6111yescharacteristic
413M16—2448no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C4⋊C4C2×C4×D6112yes—
431C4⋊C4C2×C4×D6112yes—
432C22⋊C4—1224no—
433C22⋊C4—1224no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—2444no—
438C42—42no—
439C42—612no—
440C42—612no—
441C42—614no—
442C42—614no—
443C42—1224no—
444C42—2442no—
445C42C2×C4×D6112yes—
446C42D4×D6112yes—
447C23—612no—
448C23—1221no—
449C23—1222no—
450C23—1221no—
451C23—1222no—
452C23—1222no—
453C23—2441no—
454C23—2442no—
455C23—2441no—
456C23—2442no—
457C23C2×D4×D6111yescharacteristic
458Q8—1224no—
459Q8—2441no—
460Q8—2441no—
461D4—1224no—
462D4—2448no—
463D4—2441no—
464D4—2441no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—1221no—
480C2×C4—1222no—
481C2×C4—1221no—
482C2×C4—1222no—
483C2×C4—1224no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—2444no—
488C2×C4—2444no—
489C2×C4—2448no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4C22⋊C4×D6112yes—
495C2×C4C22⋊C4×D6112yes—
496C2×C4C22×C4×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4#1451982312yes—
499C2×C4#1451982512yes—
500C8—42no—
501C8—2442no—
502C22—1222no—
503C22—1222no—
504C22—1222no—
505C22—4882no—
506C22—4882no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22—2442no—
512C22C2×C22⋊C4×D6111yescharacteristic
513C22#1684034212yes—
514C22#1684055811yescharacteristic
515C22#1684055912yes—
516C22#1684056111yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—1222no—
523C4—1222no—
524C4—1222no—
525C4—4882no—
526C4—2442no—
527C4#1684024412yes—
528C4#1684033011yescharacteristic
529C4#1684033711yescharacteristic
530C2—4882no—
531C2—2442no—
532C2#2144841011yescharacteristic
533C2#2145433411yescharacteristic
534C2#2145433511yescharacteristic
535C2#2145721912yes—
536C2#2145722212yes—
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#21895
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×#16839784C2semidirect
3. #21448215C2semidirect
4. #21446777C2semidirect
5. #21446751C2semidirect
6. #21446749C2semidirect
8. #21446251C2semidirect
9. #21445833C2semidirect
10. #21445828C2semidirect
11. C61×#1989C2semidirect
14. #16840297C4semidirect
266. #1989D61semidirect
314. C61#21895semidirect

Maximal subgroups

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

subgroupindex
C2×#168402972
C2×#168397842
#214482152
#214467772
#214467512
#214467492
#214462562
#214462512
#214458332
#214458282
C61×#19892
#2189561