Finite Group Databasebeta

← #29690594

Subgroups — #29690594

Order 15616.

786 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839817C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic
3#21448212C211yesmaximal, characteristic
4#21446590C212yesmaximal, complemented
5#21446586C212yesmaximal, complemented
6#21446021C212yesmaximal
7#21446019C212yesmaximal
8#21445987C211yesmaximal, characteristic, complemented
9#21445985C211yesmaximal, characteristic, complemented
10#21445952C211yesmaximal, characteristic, complemented
11C61×#1836C211yesmaximal, characteristic, complemented
12C2×#14519735C2212yes—
13C2×C42×D61C2211yescharacteristic
14#16839817C418yescomplemented
15C2×#14519711C2212yes—
16C2×#14519710C2212yes—
17C2×#14519709C2212yes—
18C2×#14519708C2212yes—
19C2×#14519706C2212yes—
20C2×#14519689C2212yes—
21C2×#14519688C2211yescharacteristic
22#16839632C2211yescharacteristic
23#16839629—21no—
24#16839628C2211yescharacteristic
25#16839627C2212yes—
26#16839624—21no—
27#16839619—21no—
28#16839619—21no—
29#16839619C2211yescharacteristic
30#16839619C2211yescharacteristic
31C4⋊C4×Dic61—21no—
32C4⋊C4×Dic61—21no—
33C4⋊C4×Dic61C2211yescharacteristic
34C4⋊C4×Dic61C2211yescharacteristic
35#16839609—21no—
36#16839607—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—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61C2312yes—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×#11758198—22no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—21no—
60C2×C4×Dic61—21no—
61C2×C4×Dic61D411yescharacteristic
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61C2311yescharacteristic
64C42×D61C2×C414yes—
65C42×D61Q814yes—
66#14519711—42no—
67#14519710—42no—
68#14519709—42no—
69#14519708—42no—
70#14519706C2×C418yes—
71#14519689C2×C418yes—
72#14519688C2×C414yes—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79#14519684—21no—
80#14519684—21no—
81C22×C8×C61C2312yes—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61C2312yes—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C42×C61C2311yescharacteristic
88#97×C61C2×C414yes—
89#95×C61—21no—
90#95×C61—21no—
91C23×D61C2×D411yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—22no—
101C22×Dic61C2×D411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61C4⋊C414yes—
111C2×C4×D61C4⋊C414yes—
112C2×C4×D61C22×C412yes—
113C2×C4×D61C2×Q812yes—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C2×C8×C61C22×C414yes—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
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—
134C4×Dic61—42no—
135C4×Dic61—42no—
136C4×Dic61—44no—
137C4×Dic61C4⋊C414yes—
138C4×Dic61C4⋊C414yes—
139C4×Dic61C22×C412yes—
140C4×Dic61C2×Q812yes—
141C2×#92418—22no—
142C2×#92418—22no—
143C2×#92418C22×C414yes—
144C23×C61C22×D411yescharacteristic
145C22×D61—21no—
146C22×D61—21no—
147C22×D61—21no—
148C22×D61—21no—
149C22×D61—21no—
150C22×D61—21no—
151C22×D61C2×C4⋊C412yes—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61C2×C4⋊C412yes—
166C2×C4×C61C2×C4⋊C412yes—
167C2×C4×C61C23×C411yescharacteristic
168C2×C4×C61C22×Q811yescharacteristic
169C2×C4×C61H(5,𝔽2)12yes—
170C2×C4×C612−1+412yes—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61C2×C4⋊C412yes—
187C2×Dic61C2×C4⋊C412yes—
188C2×Dic61C2×C4⋊C412yes—
189C4×D61—24no—
190C4×D61—24no—
191C4×D61—44no—
192C4×D61—44no—
193C8×C61—24no—
194#92418—24no—
195#21989—611nomaximal, Sylow
196C22×C61C22×C4⋊C411yescharacteristic
197C22×C61#53511yescharacteristic
198C22×C61#53611yescharacteristic
199C22×C61#57511yescharacteristic
200C22×C61#57511yescharacteristic
201C22×C61#57511yescharacteristic
202C22×C61#57511yescharacteristic
203C2×D61—22no—
204C2×D61—22no—
205C2×D61—22no—
206C2×D61—41no—
207C2×D61—41no—
208C2×D61—41no—
209C2×D61—41no—
210C4×C61—22no—
211C4×C61—22no—
212C4×C61—82no—
213C4×C61—81no—
214C4×C61—81no—
215C4×C61#52012yes—
216C4×C61#54412yes—
217Dic61—22no—
218Dic61—22no—
219Dic61—22no—
220Dic61—81no—
221Dic61—82no—
222Dic61—81no—
223C2×#432—611no—
224#2285—611no—
225#2283—611no—
226#1949—612no—
227#1949—612no—
228#1854—612no—
229#1854—612no—
230#1847—611no—
231#1847—611no—
232#1836—611no—
233#1836D6111yescharacteristic, complemented
234C2×C61#231711yescharacteristic
235C2×C61#291111yescharacteristic
236C2×C61#291111yescharacteristic
237C2×C61#321911yescharacteristic
238C2×C61#321911yescharacteristic
239C2×C61#324011yescharacteristic
240C2×C61#324011yescharacteristic
241D61—42no—
242C2×#117—612no—
243C22×C42—611no—
244#432—618no—
245C2×#107—612no—
246C2×#107—612no—
247C2×#106—612no—
248C2×#106—612no—
249C2×C4⋊C8—612no—
250C2×C22⋊C8—612no—
251C2×#97—611no—
252C2×#97C2×D6111yescharacteristic
253#384—611no—
254#384C2×D6111yescharacteristic
255#383—1221no—
256#382—612no—
257#382—611no—
258#382C2×D6111yescharacteristic
259#381—1221no—
260#380—611no—
261#380—611no—
262#380—1221no—
263#380—1221no—
264C4×C4⋊C4—611no—
265C4×C4⋊C4—611no—
266C4×C4⋊C4—1221no—
267C4×C4⋊C4—1221no—
268C4×C22⋊C4—1221no—
269#376—1221no—
270#340—612no—
271#340—612no—
272#340C2×D6112yes—
273#340C2×D6112yes—
274C61#2198911yescharacteristic, complemented, Sylow
275C23×C4—611no—
276C23×C4—611no—
277C23×C4—611no—
278C22×C8—612no—
279C22×C8C22×D6112yes—
280#117—2442no—
281#117—2442no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—612no—
285C2×C4⋊C4—611no—
286C2×C4⋊C4—611no—
287C2×C4⋊C4—1221no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4C22×D6112yes—
290C2×C4⋊C4C22×D6111yescharacteristic
291C2×C4⋊C4C22×D6111yescharacteristic
292C2×C22⋊C4—1222no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295C2×C42—611no—
296C2×C42—611no—
297C2×C42—611no—
298C2×C42—614no—
299C2×C42—614no—
300C2×C42—1222no—
301C2×C42—1221no—
302C2×C42—1221no—
303C2×C42C22×D6111yescharacteristic
304#107—2442no—
305#107—2442no—
306#106—2442no—
307#106—2442no—
308C4⋊C8—618no—
309C22⋊C8—618no—
310#97—614no—
311#97C4×D6114yes—
312#95—21no—
313#95—21no—
314#95—1221no—
315#95—1221no—
316#95—1221no—
317#95—1221no—
318#95—1222no—
319#95—1222no—
320#95—1221no—
321#95—1221no—
322C24—611no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—612no—
330C22×C4—612no—
331C22×C4—614no—
332C22×C4—614no—
333C22×C4—1222no—
334C22×C4—1222no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—1222no—
338C22×C4—2441no—
339C22×C4—2441no—
340C22×C4—2441no—
341C22×C4—2441no—
342C22×C4D4×D6111yescharacteristic
343C22×C4D4×D6111yescharacteristic
344C22×C4C23×D6111yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—614no—
348C2×C8—1222no—
349C2×C8—1222no—
350C2×C8C2×C4×D6114yes—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—1222no—
358C4⋊C4—1222no—
359C4⋊C4—1222no—
360C4⋊C4—1222no—
361C4⋊C4—2442no—
362C4⋊C4—2442no—
363C22⋊C4—2444no—
364C22⋊C4—2442no—
365C22⋊C4—2442no—
366C42—612no—
367C42—612no—
368C42—614no—
369C42—614no—
370C42—2442no—
371C42—2442no—
372C42—2444no—
373C42C2×C4×D6112yes—
374C42Q8×D6112yes—
375C23—612no—
376C23—1221no—
377C23—1221no—
378C23—1221no—
379C23—1221no—
380C23—1221no—
381C23—1221no—
382C23C2×D4×D6111yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—612no—
397C2×C4—612no—
398C2×C4—612no—
399C2×C4—1221no—
400C2×C4—1221no—
401C2×C4—1221no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—1224no—
406C2×C4—1224no—
407C2×C4—2444no—
408C2×C4—2444no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4C4⋊C4×D6112yes—
419C2×C4C4⋊C4×D6112yes—
420C2×C4C22×C4×D6111yescharacteristic
421C2×C4C2×Q8×D6111yescharacteristic
422C2×C4#1451982312yes—
423C2×C4#1451982412yes—
424C8—24no—
425C8—1224no—
426C22—1222no—
427C22—1222no—
428C22—1222no—
429C22—2441no—
430C22—2441no—
431C22—2441no—
432C22—2441no—
433C22C2×C4⋊C4×D6111yescharacteristic
434C22#1684033011yescharacteristic
435C22#1684033211yescharacteristic
436C22#1684055811yescharacteristic
437C22#1684055811yescharacteristic
438C22#1684056111yescharacteristic
439C22#1684056111yescharacteristic
440C4—22no—
441C4—22no—
442C4—81no—
443C4—81no—
444C4—82no—
445C4—1222no—
446C4—1222no—
447C4—1222no—
448C4—4881no—
449C4—4881no—
450C4—4882no—
451C4#1684024212yes—
452C4#1684039212yes—
453C2—2442no—
454C2#2144844411yescharacteristic
455C2#2145440611yescharacteristic
456C2#2145440611yescharacteristic
457C2#2145685711yescharacteristic
458C2#2145685911yescharacteristic
459C2#2145705411yescharacteristic
460C2#2145705611yescharacteristic
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#21989
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. #21446590C2semidirect
5. #21446586C2semidirect
8. #21445987C2semidirect
9. #21445985C2semidirect
10. #21445952C2semidirect
11. C61×#1836C2semidirect
14. #16839817C4semidirect
233. #1836D61semidirect
274. C61#21989semidirect

Maximal subgroups

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

subgroupindex
C2×#168398172
#214482242
#214482122
#214465902
#214465862
#214460212
#214460192
#214459872
#214459852
#214459522
C61×#18362
#2198961