Finite Group Databasebeta

← #29680603

Subgroups — #29680603

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#97×D61C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic
3#21448212C211yesmaximal, characteristic
4#21445988C211yesmaximal, characteristic, complemented
5#21445985C211yesmaximal, characteristic, complemented
6#21445953C211yesmaximal, characteristic, complemented
7#21445847C212yesmaximal
8#21445843C212yesmaximal
9#21445715C212yesmaximal, complemented
10#21445715C212yesmaximal, complemented
11C61×#1836C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519735—21no—
14C2×C42×D61C2211yescharacteristic
15C2×#14519710C2212yes—
16C2×#14519710C2212yes—
17C2×#14519708C2212yes—
18C2×#14519708C2212yes—
19C2×#14519688C2211yescharacteristic
20C2×#14519687C2212yes—
21#97×D61C418yescomplemented
22#16839632C2211yescharacteristic
23#16839629—21no—
24#16839628C2211yescharacteristic
25#16839627—21no—
26#16839624—21no—
27#16839619—21no—
28#16839619C2211yescharacteristic
29#16839619C2211yescharacteristic
30#16839619C2212yes—
31C4⋊C4×Dic61—21no—
32C4⋊C4×Dic61C2211yescharacteristic
33C4⋊C4×Dic61C2211yescharacteristic
34C4⋊C4×Dic61C2212yes—
35#16839609—21no—
36#16839607—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39C2×#97×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#382C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C22×#92418C2312yes—
48C2×C8×D61—24no—
49C2×C8×D61—24no—
50C2×C8×D61C2×C418yes—
51#14519735—44no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758198—22no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61D411yescharacteristic
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61C2311yescharacteristic
66C42×D61C2×C414yes—
67C42×D61Q814yes—
68#14519710—24no—
69#14519710—24no—
70#14519708—24no—
71#14519708—24no—
72#14519688C2×C414yes—
73#14519687C2×C418yes—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—22no—
79#14519684—22no—
80#14519684—21no—
81#14519684—21no—
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—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—42no—
123C4⋊C4×C61—42no—
124C42×C61C22×C412yes—
125C42×C61C2×Q812yes—
126#11758198—42no—
127#11758198—44no—
128#11758198—42no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—42no—
134C4⋊Dic61—42no—
135C4×Dic61—42no—
136C4×Dic61—42no—
137C4×Dic61—44no—
138C4×Dic61C4⋊C414yes—
139C4×Dic61C4⋊C414yes—
140C4×Dic61C22×C412yes—
141C4×Dic61C2×Q812yes—
142C2×#92418C22×C414yes—
143C2×#92418#4114yes—
144C2×#92418#4114yes—
145C8×D61—216no—
146C23×C61C22×D411yescharacteristic
147C22×D61—21no—
148C22×D61—21no—
149C22×D61—21no—
150C22×D61—21no—
151C22×D61C2×C4⋊C412yes—
152C22×D61AGL(1,ℤ/8)12yes—
153C22×D61#13712yes—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61C2×C4⋊C412yes—
168C2×C4×C61C2×C4⋊C412yes—
169C2×C4×C61C23×C411yescharacteristic
170C2×C4×C61C22×Q811yescharacteristic
171C2×C4×C61C2×#4112yes—
172C2×C4×C61C2×#4112yes—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—42no—
180C2×Dic61—41no—
181C2×Dic61—41no—
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—
191C4×D61—28no—
192C4×D61—24no—
193C4×D61—24no—
194C4×D61—44no—
195C8×C61—24no—
196#92418#11718yescomplemented
197#20823—611nomaximal, Sylow
198C22×C61C22×C4⋊C411yescharacteristic
199C22×C61#53211yescharacteristic
200C22×C61#53211yescharacteristic
201C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
202C22×C61C2×#13711yescharacteristic
203C22×C61#57511yescharacteristic
204C22×C61#57511yescharacteristic
205C2×D61—22no—
206C2×D61—22no—
207C2×D61—22no—
208C2×D61—22no—
209C2×D61—22no—
210C2×D61—22no—
211C2×D61#42814yes—
212C4×C61—22no—
213C4×C61—22no—
214C4×C61—82no—
215C4×C61—81no—
216C4×C61—81no—
217C4×C61C2×#11712yes—
218C4×C61#53312yes—
219Dic61—22no—
220Dic61—22no—
221Dic61—22no—
222Dic61—81no—
223Dic61—82no—
224Dic61—81no—
225C22×#97—611no—
226#2285—611no—
227#2283—611no—
228#1847—611no—
229#1847—611no—
230#1836—611no—
231#1836D6111yescharacteristic, complemented
232#1808—612no—
233#1807—612no—
234#1776—612no—
235#1776—612no—
236C2×C61#228511yescharacteristic
237C2×C61C2×#42811yescharacteristic
238C2×C61#291111yescharacteristic
239C2×C61#318011yescharacteristic
240C2×C61#318011yescharacteristic
241C2×C61#318111yescharacteristic
242C2×C61#318111yescharacteristic
243D61—24no—
244C23×C8—612no—
245C2×#117—1221no—
246C22×C42—611no—
247C2×#107—612no—
248C2×#107—612no—
249C2×#106—612no—
250C2×#106—612no—
251C2×#97—612no—
252C2×#97—611no—
253C2×#97—618no—
254C2×#97C2×D6111yescharacteristic
255#384—611no—
256#384C2×D6111yescharacteristic
257#383—1221no—
258#382—611no—
259#382—1221no—
260#382C2×D6111yescharacteristic
261#381—1221no—
262#380—611no—
263#380—612no—
264#380—611no—
265#380—1221no—
266C4×C4⋊C4—611no—
267C4×C4⋊C4—611no—
268C4×C4⋊C4—612no—
269C4×C4⋊C4—1221no—
270C4×C22⋊C4—1221no—
271#376—1221no—
272#340—612no—
273#340—612no—
274#340C2×D6112yes—
275#340C2×D6112yes—
276C61#2082311yescharacteristic, complemented, Sylow
277C23×C4—611no—
278C23×C4—611no—
279C23×C4—611no—
280C22×C8—612no—
281C22×C8—618no—
282C22×C8—1224no—
283C22×C8—1224no—
284C22×C8C22×D6112yes—
285#117—2444no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—611no—
289C2×C4⋊C4—611no—
290C2×C4⋊C4—612no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4—1221no—
293C2×C4⋊C4C22×D6111yescharacteristic
294C2×C4⋊C4C22×D6111yescharacteristic
295C2×C4⋊C4C22×D6112yes—
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1221no—
298C2×C22⋊C4—1221no—
299C2×C42—611no—
300C2×C42—611no—
301C2×C42—611no—
302C2×C42—614no—
303C2×C42—614no—
304C2×C42—1222no—
305C2×C42—1221no—
306C2×C42—1221no—
307C2×C42C22×D6111yescharacteristic
308#107—1224no—
309#107—1224no—
310#106—1224no—
311#106—1224no—
312#97—618no—
313#97—614no—
314#97C4×D6114yes—
315#95—21no—
316#95—21no—
317#95—1221no—
318#95—1221no—
319#95—1222no—
320#95—1221no—
321#95—1221no—
322#95—1222no—
323#95—1221no—
324#95—1221no—
325C24—611no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—611no—
330C22×C4—611no—
331C22×C4—611no—
332C22×C4—612no—
333C22×C4—612no—
334C22×C4—614no—
335C22×C4—614no—
336C22×C4—1222no—
337C22×C4—1222no—
338C22×C4—1222no—
339C22×C4—1222no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—1222no—
343C22×C4—2441no—
344C22×C4—2441no—
345C22×C4D4×D6111yescharacteristic
346C22×C4D4×D6111yescharacteristic
347C22×C4C23×D6111yescharacteristic
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—614no—
351C2×C8—614no—
352C2×C8—614no—
353C2×C8—12216no—
354C2×C8C2×C4×D6114yes—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—1222no—
362C4⋊C4—1222no—
363C4⋊C4—1222no—
364C4⋊C4—1222no—
365C4⋊C4—2442no—
366C4⋊C4—2442no—
367C22⋊C4—2444no—
368C22⋊C4—2442no—
369C22⋊C4—2442no—
370C42—612no—
371C42—612no—
372C42—614no—
373C42—614no—
374C42—2442no—
375C42—2442no—
376C42—2444no—
377C42C2×C4×D6112yes—
378C42Q8×D6112yes—
379C23—612no—
380C23—612no—
381C23—612no—
382C23—1221no—
383C23—1221no—
384C23—1221no—
385C23—1221no—
386C23C2×D4×D6111yescharacteristic
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—612no—
401C2×C4—612no—
402C2×C4—612no—
403C2×C4—1228no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—1221no—
407C2×C4—1221no—
408C2×C4—1221no—
409C2×C4—1221no—
410C2×C4—1224no—
411C2×C4—1224no—
412C2×C4—2444no—
413C2×C4—2442no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4—2442no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2441no—
422C2×C4C4⋊C4×D6112yes—
423C2×C4C4⋊C4×D6112yes—
424C2×C4C22×C4×D6111yescharacteristic
425C2×C4C2×Q8×D6111yescharacteristic
426C2×C4#41×D6112yes—
427C2×C4#41×D6112yes—
428C8—24no—
429C8—618no—
430C22—614no—
431C22—1222no—
432C22—1222no—
433C22—1222no—
434C22—1222no—
435C22—1222no—
436C22—1222no—
437C22C2×C4⋊C4×D6111yescharacteristic
438C22#1684031111yescharacteristic
439C22#1684031111yescharacteristic
440C22AGL(1,ℤ/8)×D6111yescharacteristic
441C22#137×D6111yescharacteristic
442C22#1684055811yescharacteristic
443C22#1684056111yescharacteristic
444C4—22no—
445C4—22no—
446C4—82no—
447C4—81no—
448C4—81no—
449C4—1222no—
450C4—1222no—
451C4—1222no—
452C4—4881no—
453C4—4882no—
454C4—4881no—
455C4#117×D6112yes—
456C4#1684031812yes—
457C2—1224no—
458C2#2144822511yescharacteristic
459C2#428×D6111yescharacteristic
460C2#2145440611yescharacteristic
461C2#2145651611yescharacteristic
462C2#2145652011yescharacteristic
463C2#2145652711yescharacteristic
464C2#2145653111yescharacteristic
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#20823
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. #21445988C2semidirect
5. #21445985C2semidirect
6. #21445953C2semidirect
9. #21445715C2semidirect
10. #21445715C2semidirect
11. C61×#1836C2semidirect
21. #97×D61C4semidirect
196. #92418#117semidirect
231. #1836D61semidirect
276. C61#20823semidirect

Maximal subgroups

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

subgroupindex
C2×#97×D612
#214482242
#214482122
#214459882
#214459852
#214459532
#214458472
#214458432
#214457152
C61×#18362
#2082361