Finite Group Databasebeta

← #29647847

Subgroups — #29647847

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459686C211yesmaximal, characteristic
2#21454902C211yesmaximal, characteristic, complemented
3#21448739C211yesmaximal, characteristic, complemented
4#21444383C211yesmaximal, characteristic, complemented
5#21444380C211yesmaximal, characteristic
6#21444356C214yesmaximal, complemented
7#21444343C214yesmaximal, complemented
8#21444313C211yesmaximal, characteristic
9C61×#1503C211yesmaximal, characteristic, complemented
10#16840497C412yescomplemented
11#16840491C412yescomplemented
12#16840486C412yescomplemented
13#16840485—22no—
14#16840473C412yescomplemented
15#16840450—22no—
16#16840361—22no—
17#16840338—21no—
18#16840312C2211yescharacteristic
19#16840298C2214yes—
20#16840242—21no—
21C2×C4×Dic122—21no—
22C2×C4×Dic122C2211yescharacteristic
23#16839892—22no—
24#16839889—22no—
25#16839816C2214yes—
26#16839815C2211yescharacteristic
27#16839660—21no—
28#16839657—22no—
29#16839628C2211yescharacteristic
30#16839626C2212yes—
31#16839613C2212yes—
32#16839499C2214yes—
33#16839497C2214yes—
34#16839495C2212yes—
35#16839495C2212yes—
36C61×#533C2211yescharacteristic
37C61×#431C2211yescharacteristic
38C4×C4⋊C4×C61C2211yescharacteristic
39C61×#328C2212yes—
40C61×#328C2212yes—
41C2×#11758222—21no—
42C22×Dic122D411yescharacteristic
43#14519777—22no—
44#14519777C2×C411yescharacteristic
45#14519774C2×C411yescharacteristic
46#14519772—24no—
47#14519772—24no—
48#14519768—24no—
49#14519768C2×C414yes—
50#14519766C2×C414yes—
51#14519765—22no—
52#14519765—21no—
53#14519762—24no—
54#14519761—24no—
55#14519757—22no—
56#14519757—22no—
57#14519753—24no—
58#14519751—21no—
59#14519751C2312yes—
60Q8×Dic61C2×C414yes—
61C4×Dic122—22no—
62C4×Dic122—44no—
63C4×Dic122C2×C412yes—
64C4×Dic122C2×C412yes—
65C4×Dic122D412yes—
66C4×Dic122D412yes—
67C4×D244—21no—
68C4×D244—22no—
69C4×D244C2×C412yes—
70#14519739C2×C414yes—
71#14519737—24no—
72#14519735—22no—
73#14519735—24no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×#11758196D412yes—
79C4⋊C4×D61—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61C2312yes—
82C4⋊#92418—21no—
83#14519706—24no—
84C22⋊#92418C2312yes—
85#14519687—24no—
86C4×#92418—21no—
87#14519684—21no—
88#14519684—22no—
89#14519684C2312yes—
90C4⋊Q8×C61C2×C411yescharacteristic
91C4⋊Q8×C61C2×C411yescharacteristic
92#125×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61C2312yes—
95C4×Q8×C61—21no—
96C4×Q8×C61C2×C412yes—
97#117×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—21no—
102C22⋊C8×C61C2312yes—
103C4×C8×C61—21no—
104#95×C61C2311yescharacteristic
105#95×C61C2311yescharacteristic
106C2×Q8×C61—21no—
107C2×Q8×C61C22×C412yes—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C2×C22⋊D61—22no—
114C22×Dic61—22no—
115C22×Dic61C2×D412yes—
116C22×Dic61C2×D412yes—
117#11758222—42no—
118#11758222—42no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244C22×C412yes—
122C2×C4×D61—22no—
123C2×C4×D61—24no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122—21no—
127C2×Dic122C22⋊C412yes—
128C2×Dic122C22⋊C412yes—
129C2×C8×C61—22no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—42no—
136C4⋊C4×C61C22×C412yes—
137C22⋊C4×C61—22no—
138C42×C61—22no—
139C42×C61—42no—
140C42×C61C22×C411yescharacteristic
141C42×C61C22×C411yescharacteristic
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144#11758198—22no—
145#11758198—22no—
146#11758198—24no—
147#11758198—24no—
148#11758198—24no—
149C4⋊Dic61—21no—
150C4⋊Dic61—24no—
151C4⋊Dic61—24no—
152C4⋊Dic61—42no—
153C4⋊Dic61C22⋊C412yes—
154C4⋊Dic61C22⋊C412yes—
155C4⋊Dic61C22×C412yes—
156#11758196—24no—
157#11758196—44no—
158#11758196—44no—
159#11758196C22⋊C414yes—
160#11758196C22⋊C414yes—
161C4×Dic61—24no—
162C4×Dic61—44no—
163C4×Dic61C22×C414yes—
164C4×Dic61C2×D414yes—
165C2×#92418—22no—
166C23×C61C22×D411yescharacteristic
167C22×D61—22no—
168Q8×C61—42no—
169Q8×C61—81no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61C2×C22⋊C411yescharacteristic
181C2×C4×C61C2×C22⋊C411yescharacteristic
182C2×C4×C61C2×C22⋊C411yescharacteristic
183C2×C4×C61C2×C22⋊C411yescharacteristic
184C2×C4×C61AGL(1,ℤ/8)12yes—
185C2×C4×C61#13712yes—
186C2×C4×C61C23×C411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C22⋊D61—42no—
189C22⋊D61—42no—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61C2×C22⋊C414yes—
198C2×Dic61C2×C22⋊C414yes—
199D244—42no—
200D244—42no—
201D244—81no—
202C4×D61—44no—
203C4×D61—84no—
204Dic122—24no—
205Dic122—24no—
206Dic122—42no—
207C8×C61—42no—
208#92418—42no—
209#17628—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61C22×C22⋊C411yescharacteristic
213C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
214C22×C61C2×#13711yescharacteristic
215C2×D61—42no—
216C2×D61—42no—
217C2×D61—42no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—42no—
223C4×C61—42no—
224C4×C61—42no—
225C4×C61—81no—
226C4×C61#41712yes—
227C4×C61#41712yes—
228Dic61—24no—
229Dic61—24no—
230Dic61—42no—
231Dic61—82no—
232#3507—611no—
233#2983—611no—
234#2351—611no—
235#1518—611no—
236#1517—611no—
237#1512—614no—
238#1509—614no—
239#1503D6111yescharacteristic, complemented
240#1502—611no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61#288411yescharacteristic
244C2×C61C2×#41711yescharacteristic
245C2×C61#290111yescharacteristic
246D61—82no—
247#559—612no—
248#557—612no—
249#556—612no—
250#556—1222no—
251#554—612no—
252#552—1222no—
253#539—1222no—
254#536—1221no—
255#533C2×D6111yescharacteristic
256#532—611no—
257#529—614no—
258#520—1221no—
259C2×C4×Q8—611no—
260C2×C4×Q8—1221no—
261#446—1222no—
262#446—1222no—
263#432—614no—
264#431—611no—
265#431C2×D6111yescharacteristic
266#389—1221no—
267#389—1222no—
268#382—611no—
269#382—612no—
270C4×C4⋊C4—612no—
271C4×C4⋊C4C2×D6111yescharacteristic
272#328—614no—
273#328C2×D6112yes—
274#328C2×D6112yes—
275#327—614no—
276#327—612no—
277#327—612no—
278C61#1762811yescharacteristic, complemented, Sylow
279C2×#41—1221no—
280C22×Q8—611no—
281C4⋊Q8—611no—
282C4⋊Q8—1222no—
283C4⋊Q8C4×D6111yescharacteristic
284C4⋊Q8C4×D6111yescharacteristic
285#127—611no—
286#126—1224no—
287#126—1224no—
288#125—21no—
289#125—614no—
290#125—1224no—
291#124—614no—
292#124—1222no—
293#124—1221no—
294#123—1224no—
295#123—1224no—
296C22⋊Q8—21no—
297C22⋊Q8—1222no—
298C22⋊Q8—1222no—
299C22⋊Q8—1224no—
300C22⋊Q8C22×D6112yes—
301#121—612no—
302#121—1221no—
303C4×Q8—21no—
304C4×Q8—612no—
305C4×Q8—612no—
306C4×Q8—612no—
307C4×Q8—612no—
308C4×Q8—614no—
309C4×Q8—1222no—
310C4×Q8—2444no—
311C4×Q8C4×D6112yes—
312C4×D4—614no—
313C4×D4—612no—
314C4×D4—1222no—
315C4×D4—1224no—
316C4×D4—1221no—
317#117—21no—
318#117—1224no—
319#117—1222no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—611no—
322C2×C4⋊C4—612no—
323C2×C4⋊C4—1222no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4—1222no—
326C2×C4⋊C4—1221no—
327C2×C42—21no—
328C2×C42—612no—
329C2×C42—1222no—
330C2×C42C22×D6111yescharacteristic
331C4⋊C8—21no—
332C4⋊C8—1224no—
333C4⋊C8—1221no—
334C22⋊C8—612no—
335C22⋊C8C22×D6112yes—
336#97—1224no—
337C4×C8—21no—
338C4×C8—1221no—
339#95—612no—
340#95—1222no—
341#95—1221no—
342#95C22×D6111yescharacteristic
343#95C22×D6111yescharacteristic
344#41—2442no—
345#41—2442no—
346C2×Q8—21no—
347C2×Q8—612no—
348C2×Q8—612no—
349C2×Q8—1222no—
350C2×Q8—1221no—
351C2×Q8—1222no—
352C2×Q8C2×C4×D6112yes—
353C2×D4—612no—
354C2×D4—1222no—
355C2×D4—1221no—
356C2×D4—1221no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—612no—
360C22×C4—612no—
361C22×C4—1224no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C23×D6111yescharacteristic
367C2×C8—22no—
368C2×C8—1222no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—612no—
376C4⋊C4—614no—
377C4⋊C4—614no—
378C4⋊C4—612no—
379C4⋊C4—612no—
380C4⋊C4—1224no—
381C4⋊C4—1224no—
382C4⋊C4—1224no—
383C4⋊C4—1221no—
384C4⋊C4—2444no—
385C4⋊C4—2444no—
386C4⋊C4—2442no—
387C4⋊C4C2×C4×D6112yes—
388C22⋊C4—22no—
389C22⋊C4—1224no—
390C22⋊C4—1224no—
391C22⋊C4—1224no—
392C22⋊C4—1222no—
393C22⋊C4—1222no—
394C42—22no—
395C42—42no—
396C42—614no—
397C42—614no—
398C42—1224no—
399C42—2444no—
400C42C2×C4×D6111yescharacteristic
401C42C2×C4×D6111yescharacteristic
402C42D4×D6111yescharacteristic
403C42D4×D6111yescharacteristic
404C23—1222no—
405C23C2×D4×D6111yescharacteristic
406Q8—42no—
407Q8—81no—
408Q8—1224no—
409Q8—1224no—
410Q8—2442no—
411D4—4881no—
412D4—2442no—
413D4—2442no—
414D4—2442no—
415D4—2442no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—614no—
427C2×C4—614no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—4884no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2444no—
437C2×C4C22⋊C4×D6111yescharacteristic
438C2×C4C22⋊C4×D6111yescharacteristic
439C2×C4C22⋊C4×D6111yescharacteristic
440C2×C4C22⋊C4×D6111yescharacteristic
441C2×C4#1451980412yes—
442C2×C4#1451980912yes—
443C2×C4C22×C4×D6111yescharacteristic
444C2×C4C2×D4×D6111yescharacteristic
445C8—42no—
446C8—2442no—
447C22—21no—
448C22—21no—
449C22—2442no—
450C22—2442no—
451C22—2442no—
452C22C2×C22⋊C4×D6111yescharacteristic
453C22C2×#1451980411yescharacteristic
454C22C2×#1451980911yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—81no—
463C4—1224no—
464C4—1224no—
465C4—4882no—
466C4—2442no—
467C4#1683976212yes—
468C4#1683976212yes—
469C2—21no—
470C2—21no—
471C2—4882no—
472C2#2145428111yescharacteristic
473C2C2×#1683976211yescharacteristic
474C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1503C2
socleC22×C61C22×C22⋊C4
prime radical#16840312C22
cosocleC22×C4×C61C24
hypercenter#1503D61
hypocenterC61#17628
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #21454902C2semidirect
3. #21448739C2semidirect
4. #21444383C2semidirect
6. #21444356C2semidirect
7. #21444343C2semidirect
9. C61×#1503C2semidirect
10. #16840497C4semidirect
11. #16840491C4semidirect
12. #16840486C4semidirect
14. #16840473C4semidirect
239. #1503D61semidirect
278. C61#17628semidirect

Maximal subgroups

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

subgroupindex
#214596862
#214549022
#214487392
#214443832
#214443802
#214443562
#214443432
#214443132
C61×#15032
#1762861