Finite Group Databasebeta

← #29729545

Subgroups — #29729545

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839963C211yesmaximal, characteristic
2C2×#16839886C211yesmaximal, characteristic, complemented
3#21451192C211yesmaximal, characteristic, complemented
4#21450821C211yesmaximal, characteristic
5#21447188C212yesmaximal, complemented
6#21447138C212yesmaximal, complemented
7#21446671C212yesmaximal, complemented
8#21446624C211yesmaximal, characteristic
9#21446503C212yesmaximal, complemented
10C4×#14519700C211yesmaximal, characteristic, complemented
11C61×#1948C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519765—21no—
15C2×C4×Dic122C2211yescharacteristic
16C2×C4×Dic122C2211yescharacteristic
17#16839963—24no—
18#16839886—24no—
19C2×#14519706C2212yes—
20C2×#14519700C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519699C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×C4×#92418C2211yescharacteristic
26C2×C8×Dic61C2212yes—
27#16839692—22no—
28#16839680—22no—
29#16839667—21no—
30#16839667C2212yes—
31#16839664—21no—
32#16839663C2212yes—
33#16839663C2212yes—
34#16839661—21no—
35#16839661C2212yes—
36#16839524C2211yescharacteristic
37#16839524C2211yescharacteristic
38C2×#106×C61C2212yes—
39C2×C4⋊C8×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C22×Dic122D411yescharacteristic
44C22×Dic122C2311yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48C2×#11758190—24no—
49C2×#11758190—24no—
50C2×#11758190—42no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519774—22no—
54#14519774—22no—
55#14519765—44no—
56C4×Dic122—22no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×#11758196—21no—
64C2×#11758196D412yes—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2312yes—
71#14519706—24no—
72#14519700—22no—
73#14519700—22no—
74#14519699—24no—
75#14519697—24no—
76#14519697—42no—
77C4×#92418—22no—
78C8×Dic61—24no—
79#14519684—22no—
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×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90#106×C61—42no—
91C4⋊C8×C61D414yes—
92#95×C61—21no—
93C23×D61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61C2×D412yes—
103C22×Dic61C2×D412yes—
104C2×D244—22no—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—42no—
111C2×D244—42no—
112C2×D244—42no—
113C2×D244—42no—
114C2×D244—81no—
115C2×D244—81no—
116C2×Dic122—22no—
117C2×Dic122—22no—
118C2×Dic122—21no—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122SD1614yescomplemented
124C2×Dic122#4112yes—
125C2×C8×C61—22no—
126C2×C8×C61—22no—
127C2×C8×C61C2×D414yes—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—42no—
135C4⋊C4×C61—81no—
136C42×C61—42no—
137C42×C61C2×D412yes—
138C42×C61#4112yes—
139#11758198—44no—
140#11758198—44no—
141#11758198—82no—
142#11758198—82no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—82no—
147#11758196—24no—
148#11758196—42no—
149C4×Dic61—22no—
150C4×Dic61—22no—
151C4×Dic61—42no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418#4112yes—
158#11758190—48no—
159C23×C61C22×D411yescharacteristic
160C22×D61—42no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—42no—
166C22×D61—42no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61#12112yes—
179C2×C4×C61#12112yes—
180C2×C4×C61C2×SD1612yes—
181C2×C4×C61#13512yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61H(5,𝔽2)11yescharacteristic
187C2×C4×C612−1+411yescharacteristic
188C2×Dic61—22no—
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61C2×SD1614yes—
200D244—42no—
201D244—42no—
202D244—42no—
203D244—42no—
204D244—84no—
205D244—84no—
206Dic122—28no—
207Dic122—22no—
208Dic122—22no—
209Dic122—41no—
210Dic122—41no—
211C8×C61—24no—
212#92418—22no—
213#92418—41no—
214#25295—611nomaximal, Sylow
215C22×C61C2×#12111yescharacteristic
216C22×C61#54811yescharacteristic
217C22×C61#55011yescharacteristic
218C22×C61C22×SD1611yescharacteristic
219C22×C61C2×#13511yescharacteristic
220C22×C61#57611yescharacteristic
221C22×C61#57711yescharacteristic
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—82no—
226C2×D61—82no—
227C2×D61—82no—
228C2×D61—82no—
229C4×C61—22no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61—82no—
233C4×C61#46514yes—
234C4×C61#52911yescharacteristic
235C4×C61#54211yescharacteristic
236C4×C61#54611yescharacteristic
237C4×C61#54811yescharacteristic
238Dic61—24no—
239Dic61—42no—
240Dic61—42no—
241Dic61—81no—
242C2×#460—611no—
243C2×C8×Q8—611no—
244#2582—611no—
245#2548—611no—
246#2065—612no—
247#2055—612no—
248#1968—612no—
249#1957—611no—
250#1948D6111yescharacteristic, complemented
251#1934—612no—
252C4×#103—611no—
253C2×C61#242611yescharacteristic
254C2×C61C2×#46511yescharacteristic
255C2×C61#306211yescharacteristic
256C2×C61#336211yescharacteristic
257C2×C61#336511yescharacteristic
258C2×C61#339211yescharacteristic
259C2×C61#339611yescharacteristic
260D61—162no—
261C22×SD16—612no—
262C2×#127—611no—
263C2×#124—1221no—
264C2×C4×Q8—611no—
265C2×C4×Q8—611no—
266#460—1224no—
267C8×Q8—1224no—
268C2×#106C2×D6112yes—
269C2×C4⋊C8—612no—
270C2×C4⋊C8C2×D6111yescharacteristic
271C2×#103—611no—
272C2×#103—611no—
273C2×#102—612no—
274C2×#102—612no—
275C2×#102—612no—
276C2×C4×C8—612no—
277C2×C4×C8—611no—
278#398—1222no—
279#394—1222no—
280#391—612no—
281#391—1221no—
282#390—612no—
283#390—612no—
284#390—1221no—
285#389—612no—
286#389—1221no—
287#384C2×D6111yescharacteristic
288C4×C4⋊C4C2×D6111yescharacteristic
289#340—611no—
290#340—611no—
291#340C2×D6112yes—
292C61#2529511yescharacteristic, complemented, Sylow
293C22×Q8—611no—
294C22×Q8—611no—
295C22×D4—612no—
296C22×D4—1221no—
297C22×D4—1221no—
298C2×SD16—1224no—
299C2×SD16—1224no—
300C2×SD16—2442no—
301C22×C8—611no—
302C22×C8—611no—
303C22×C8C22×D6112yes—
304#127—1222no—
305#127—1222no—
306#124—2444no—
307C4×Q8—1222no—
308C4×Q8—1222no—
309C4×Q8—1222no—
310C4×Q8—1222no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4—612no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4C22×D6111yescharacteristic
319C2×C4⋊C4C22×D6111yescharacteristic
320C2×C4⋊C4C22×D6112yes—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C42—21no—
326C2×C42—612no—
327C2×C42—1221no—
328C2×C42C22×D6111yescharacteristic
329#106—42no—
330C4⋊C8—1224no—
331C4⋊C8D24414yes—
332#103—1222no—
333#103—1222no—
334#102—1224no—
335#102—1224no—
336#102—2442no—
337C4×C8—1224no—
338C4×C8—1222no—
339#95—21no—
340#95—1222no—
341#95—1222no—
342#95—1222no—
343C24—1222no—
344C2×Q8—614no—
345C2×Q8—612no—
346C2×Q8—1221no—
347C2×Q8—1222no—
348C2×Q8—1222no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8—1221no—
352C2×Q8—1221no—
353C2×D4—1222no—
354C2×D4—1222no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1222no—
358C2×D4—1222no—
359C2×D4—4881no—
360C2×D4—4881no—
361C2×D4—2442no—
362C2×D4—2442no—
363C2×D4—2442no—
364C2×D4—2442no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—612no—
369C22×C4—612no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4D4×D6111yescharacteristic
373C22×C4D4×D6111yescharacteristic
374C22×C4C23×D6111yescharacteristic
375SD16—2448no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—612no—
379C2×C8—1221no—
380C2×C8—1221no—
381C2×C8—1221no—
382C2×C8—1221no—
383C2×C8—1221no—
384C2×C8C2×D24414yes—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—42no—
392C4⋊C4—81no—
393C4⋊C4—1224no—
394C4⋊C4—1222no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—4882no—
398C4⋊C4—2442no—
399C22⋊C4—4882no—
400C22⋊C4—4882no—
401C22⋊C4—2444no—
402C22⋊C4—2444no—
403C42—42no—
404C42—1222no—
405C42—1222no—
406C42—2442no—
407C42C2×D24412yes—
408C42#1175822612yes—
409C23—2442no—
410C23—2442no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414C23—2442no—
415C23—2442no—
416C23C2×D4×D6111yescharacteristic
417Q8—1228no—
418Q8—1222no—
419Q8—1222no—
420Q8—2441no—
421Q8—2441no—
422D4—4884no—
423D4—4884no—
424D4—2442no—
425D4—2442no—
426D4—2442no—
427D4—2442no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—614no—
440C2×C4—1222no—
441C2×C4—1222no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4#1451975212yes—
452C2×C4#1451975212yes—
453C2×C4SD16×D6112yes—
454C2×C4#1451979812yes—
455C2×C4C22×D24411yescharacteristic
456C2×C4#41×D6111yescharacteristic
457C2×C4#41×D6111yescharacteristic
458C2×C4C2×#1175822611yescharacteristic
459C2×C4#1451982211yescharacteristic
460C2×C4#1451982511yescharacteristic
461C8—24no—
462C8—1222no—
463C8—2441no—
464C22—4882no—
465C22—4882no—
466C22—4882no—
467C22—4882no—
468C22—4882no—
469C22—4882no—
470C22—4882no—
471C22C2×#1451975211yescharacteristic
472C22#1684042211yescharacteristic
473C22#1684043011yescharacteristic
474C22C2×SD16×D6111yescharacteristic
475C22C2×#1451979811yescharacteristic
476C22#1684056611yescharacteristic
477C22#1684057111yescharacteristic
478C4—22no—
479C4—81no—
480C4—81no—
481C4—82no—
482C4—1224no—
483C4—4881no—
484C4—2442no—
485C4—2442no—
486C4#1683999214yes—
487C4#1684029711yescharacteristic
488C4#1684038611yescharacteristic
489C4#1684040711yescharacteristic
490C4#1684042411yescharacteristic
491C2—9762no—
492C2#2144938311yescharacteristic
493C2C2×#1683999211yescharacteristic
494C2#2145550011yescharacteristic
495C2#2145848911yescharacteristic
496C2#2145851411yescharacteristic
497C2#2145879511yescharacteristic
498C2#2145883011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1948C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#1948D61
hypocenterC61#25295
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. C2×#16839886C2semidirect
3. #21451192C2semidirect
5. #21447188C2semidirect
6. #21447138C2semidirect
7. #21446671C2semidirect
9. #21446503C2semidirect
10. C4×#14519700C2semidirect
11. C61×#1948C2semidirect
123. C2×Dic122SD16semidirect
250. #1948D61semidirect
292. C61#25295semidirect

Maximal subgroups

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

subgroupindex
C2×#168399632
C2×#168398862
#214511922
#214508212
#214471882
#214471382
#214466712
#214466242
#214465032
C4×#145197002
C61×#19482
#2529561