Finite Group Databasebeta

← #29690587

Subgroups — #29690587

Order 15616.

830 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839817C211yesmaximal, characteristic
2#21448225C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic
4#21446581C212yesmaximal, complemented
5#21446576C212yesmaximal, complemented
6#21446021C212yesmaximal
7#21446019C212yesmaximal
8#21445988C211yesmaximal, characteristic, complemented
9#21445987C211yesmaximal, characteristic, complemented
10#21445954C211yesmaximal, characteristic, complemented
11C61×#1836C211yesmaximal, characteristic, complemented
12C2×#14519735—21no—
13C2×#14519735C2211yescharacteristic
14C2×#14519735C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16839817C418yescomplemented
17C2×#14519711C2212yes—
18C2×#14519710C2212yes—
19C2×#14519709C2212yes—
20C2×#14519708C2212yes—
21C2×#14519706C2212yes—
22C2×#14519689C2212yes—
23C2×#14519688C2211yescharacteristic
24#16839632C2211yescharacteristic
25#16839631C2211yescharacteristic
26#16839631C2211yescharacteristic
27#16839628C2211yescharacteristic
28#16839627—21no—
29#16839621—21no—
30#16839621—21no—
31#16839619—21no—
32#16839619C2212yes—
33C4⋊C4×Dic61—21no—
34C4⋊C4×Dic61C2212yes—
35#16839613—21no—
36#16839613—21no—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×#97×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#382C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C22×#92418C2312yes—
48#14519735—42no—
49#14519735—42no—
50#14519735—44no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2312yes—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758198—22no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×C4×Dic61—22no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61D411yescharacteristic
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C42×D61C2×C414yes—
70C42×D61Q814yes—
71#14519711—42no—
72#14519710—42no—
73#14519709—42no—
74#14519708—42no—
75#14519706C2×C418yes—
76#14519689C2×C418yes—
77#14519688C2×C414yes—
78#14519684—21no—
79#14519684—21no—
80#14519684—22no—
81#14519684—22no—
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89#97×C61C2×C414yes—
90#95×C61—21no—
91#95×C61—21no—
92C23×D61C2×D411yescharacteristic
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61C4⋊C414yes—
112C2×C4×D61C4⋊C414yes—
113C2×C4×D61C22×C412yes—
114C2×C4×D61C2×Q812yes—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C2×C8×C61C22×C414yes—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—24no—
121C4⋊C4×C61—42no—
122C4⋊C4×C61—42no—
123C42×C61C22×C412yes—
124C42×C61C2×Q812yes—
125#11758198—42no—
126#11758198—42no—
127#11758198—44no—
128C4⋊Dic61—22no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—42no—
133C4⋊Dic61—42no—
134#11758196—42no—
135#11758196—42no—
136#11758196—42no—
137#11758196—42no—
138C4×Dic61—42no—
139C4×Dic61—42no—
140C4×Dic61—44no—
141C4×Dic61C4⋊C414yes—
142C4×Dic61C4⋊C414yes—
143C4×Dic61C22×C412yes—
144C4×Dic61C2×Q812yes—
145C2×#92418—22no—
146C2×#92418—22no—
147C2×#92418C22×C414yes—
148C23×C61C22×D411yescharacteristic
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61C2×C4⋊C412yes—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61C2×C4⋊C412yes—
170C2×C4×C61C2×C4⋊C412yes—
171C2×C4×C61C23×C411yescharacteristic
172C2×C4×C61C22×Q811yescharacteristic
173C2×C4×C61H(5,𝔽2)12yes—
174C2×C4×C612−1+412yes—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61C2×C4⋊C412yes—
189C2×Dic61C2×C4⋊C412yes—
190C2×Dic61C2×C4⋊C412yes—
191C2×Dic61AGL(1,ℤ/8)12yes—
192C2×Dic61#13712yes—
193C4×D61—24no—
194C4×D61—24no—
195C4×D61—28no—
196C4×D61—44no—
197C8×C61—24no—
198#92418—24no—
199#21988—611nomaximal, Sylow
200C22×C61C22×C4⋊C411yescharacteristic
201C22×C61#53511yescharacteristic
202C22×C61#53611yescharacteristic
203C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
204C22×C61C2×#13711yescharacteristic
205C22×C61#57511yescharacteristic
206C22×C61#57511yescharacteristic
207C2×D61—22no—
208C2×D61—22no—
209C2×D61—22no—
210C2×D61—41no—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—41no—
214C4×C61—22no—
215C4×C61—22no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61—82no—
219C4×C61#52012yes—
220C4×C61#54412yes—
221Dic61—22no—
222Dic61—22no—
223Dic61—81no—
224Dic61—82no—
225Dic61—81no—
226Dic61#42814yes—
227C2×#432—611no—
228#2285—611no—
229#2285—611no—
230#1948—612no—
231#1947—612no—
232#1854—612no—
233#1854—612no—
234#1847—611no—
235#1847—611no—
236#1836—611no—
237#1836D6111yescharacteristic, complemented
238C2×C61#231711yescharacteristic
239C2×C61C2×#42811yescharacteristic
240C2×C61#291111yescharacteristic
241C2×C61#321411yescharacteristic
242C2×C61#321611yescharacteristic
243C2×C61#323511yescharacteristic
244C2×C61#323711yescharacteristic
245D61—42no—
246C2×#117—611no—
247C2×#117—611no—
248C2×#117—1221no—
249C22×C42—611no—
250#432—618no—
251C2×#107—612no—
252C2×#107—612no—
253C2×#106—612no—
254C2×#106—612no—
255C2×C4⋊C8—612no—
256C2×C22⋊C8—612no—
257C2×#97—611no—
258C2×#97C2×D6111yescharacteristic
259#384—611no—
260#384—611no—
261#384—611no—
262#384C2×D6111yescharacteristic
263#382—611no—
264#382—1221no—
265#382C2×D6111yescharacteristic
266#380—612no—
267#380—1221no—
268#380—1221no—
269#380—1221no—
270C4×C4⋊C4—612no—
271C4×C4⋊C4—1221no—
272C4×C4⋊C4—1221no—
273C4×C4⋊C4—1221no—
274#340—612no—
275#340—612no—
276#340C2×D6112yes—
277#340C2×D6112yes—
278C61#2198811yescharacteristic, complemented, Sylow
279C23×C4—611no—
280C23×C4—611no—
281C23×C4—611no—
282C22×C8—612no—
283C22×C8C22×D6112yes—
284#117—2442no—
285#117—2442no—
286#117—2444no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—611no—
290C2×C4⋊C4—611no—
291C2×C4⋊C4—612no—
292C2×C4⋊C4—1221no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4C22×D6111yescharacteristic
299C2×C4⋊C4C22×D6111yescharacteristic
300C2×C4⋊C4C22×D6112yes—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1221no—
303C2×C22⋊C4—1222no—
304C2×C42—611no—
305C2×C42—611no—
306C2×C42—611no—
307C2×C42—614no—
308C2×C42—614no—
309C2×C42—1221no—
310C2×C42—1221no—
311C2×C42—1222no—
312C2×C42C22×D6111yescharacteristic
313#107—2442no—
314#107—2442no—
315#106—2442no—
316#106—2442no—
317C4⋊C8—618no—
318C22⋊C8—618no—
319#97—614no—
320#97C4×D6114yes—
321#95—21no—
322#95—21no—
323#95—1222no—
324#95—1222no—
325#95—1221no—
326#95—1221no—
327C24—611no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—611no—
332C22×C4—611no—
333C22×C4—611no—
334C22×C4—612no—
335C22×C4—612no—
336C22×C4—614no—
337C22×C4—614no—
338C22×C4—1222no—
339C22×C4—1222no—
340C22×C4—1222no—
341C22×C4—1222no—
342C22×C4—1222no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—2441no—
346C22×C4—2441no—
347C22×C4D4×D6111yescharacteristic
348C22×C4D4×D6111yescharacteristic
349C22×C4C23×D6111yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—614no—
353C2×C8—1222no—
354C2×C8—1222no—
355C2×C8C2×C4×D6114yes—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—1222no—
362C4⋊C4—1222no—
363C4⋊C4—1222no—
364C4⋊C4—1222no—
365C4⋊C4—2442no—
366C4⋊C4—2442no—
367C4⋊C4—2442no—
368C4⋊C4—2442no—
369C4⋊C4—2442no—
370C4⋊C4—2442no—
371C22⋊C4—2444no—
372C22⋊C4—2442no—
373C22⋊C4—2442no—
374C42—612no—
375C42—612no—
376C42—614no—
377C42—614no—
378C42—2444no—
379C42—2442no—
380C42—2442no—
381C42C2×C4×D6112yes—
382C42Q8×D6112yes—
383C23—612no—
384C23—1221no—
385C23—1221no—
386C23—1221no—
387C23—1221no—
388C23—1221no—
389C23—1221no—
390C23C2×D4×D6111yescharacteristic
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—612no—
405C2×C4—612no—
406C2×C4—612no—
407C2×C4—612no—
408C2×C4—612no—
409C2×C4—1228no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1224no—
415C2×C4—1224no—
416C2×C4—2444no—
417C2×C4—2442no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4—2442no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2442no—
426C2×C4C4⋊C4×D6112yes—
427C2×C4C4⋊C4×D6112yes—
428C2×C4C22×C4×D6111yescharacteristic
429C2×C4C2×Q8×D6111yescharacteristic
430C2×C4#1451982312yes—
431C2×C4#1451982412yes—
432C8—24no—
433C8—1224no—
434C22—1222no—
435C22—1222no—
436C22—1222no—
437C22—2441no—
438C22—2441no—
439C22—2441no—
440C22—2441no—
441C22C2×C4⋊C4×D6111yescharacteristic
442C22#1684033011yescharacteristic
443C22#1684033211yescharacteristic
444C22AGL(1,ℤ/8)×D6111yescharacteristic
445C22#137×D6111yescharacteristic
446C22#1684055811yescharacteristic
447C22#1684056111yescharacteristic
448C4—22no—
449C4—22no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—614no—
454C4—1222no—
455C4—1222no—
456C4—4882no—
457C4—4881no—
458C4—4881no—
459C4#1684024212yes—
460C4#1684039212yes—
461C2—2442no—
462C2#2144844411yescharacteristic
463C2#428×D6111yescharacteristic
464C2#2145440611yescharacteristic
465C2#2145681511yescharacteristic
466C2#2145683111yescharacteristic
467C2#2145701211yescharacteristic
468C2#2145702811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1836C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1836D61
hypocenterC61#21988
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
4. #21446581C2semidirect
5. #21446576C2semidirect
8. #21445988C2semidirect
9. #21445987C2semidirect
10. #21445954C2semidirect
11. C61×#1836C2semidirect
16. #16839817C4semidirect
237. #1836D61semidirect
278. C61#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#168398172
#214482252
#214482242
#214465812
#214465762
#214460212
#214460192
#214459882
#214459872
#214459542
C61×#18362
#2198861