Finite Group Databasebeta

← #29729526

Subgroups — #29729526

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839964C211yesmaximal, characteristic
2C2×C8×Dic122C211yesmaximal, characteristic, complemented
3#21451035C211yesmaximal, characteristic, complemented
4#21450742C211yesmaximal, characteristic
5#21447196C212yesmaximal, complemented
6#21447133C212yesmaximal
7#21446670C212yesmaximal
8#21446580C211yesmaximal, characteristic, complemented
9#21446497C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic
11C61×#1956C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519757—21no—
14C2×C4×Dic122C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16#16839964—24no—
17C8×Dic122—24no—
18C2×#14519708C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519706C2212yes—
21C2×#14519701C2212yes—
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×C8×Dic61C2212yes—
25#16839700—22no—
26#16839684—22no—
27#16839669C2212yes—
28#16839669C2212yes—
29#16839667—21no—
30#16839663C2212yes—
31#16839661C2212yes—
32#16839627—21no—
33#16839619—21no—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839524C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#103×C61C2212yes—
40C2×#103×C61C2212yes—
41C2×C4×C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C22×Dic122D411yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C2×#11758201—24no—
47C2×#11758201—24no—
48C2×#11758201—42no—
49C22×#92418C2312yes—
50#14519757—44no—
51C4×Dic122—22no—
52C4×Dic122—22no—
53C4×D244—22no—
54C4×D244—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61D411yescharacteristic
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196D412yes—
61C2×#11758198—21no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2312yes—
66#14519708—42no—
67C4⋊#92418D414yes—
68#14519706—24no—
69#14519701—24no—
70#14519699—22no—
71#14519699—22no—
72C8×Dic61—24no—
73#14519684—21no—
74#14519684—22no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79C22×Q8×C61C2312yes—
80C22×C8×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2312yes—
89C2×C42×C61C2311yescharacteristic
90#103×C61—24no—
91#103×C61—42no—
92C4×C8×C61—22no—
93C23×D61—21no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2411yescharacteristic
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61C2×D412yes—
109C22×Dic61C2×D412yes—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244#4112yes—
116C2×C4×D61—42no—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×Dic122—22no—
120C2×Dic122—22no—
121C2×Dic122SD1614yescomplemented
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61—21no—
127C2×C8×C61#4112yes—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—42no—
131C4⋊C4×C61—42no—
132C4⋊C4×C61—42no—
133C4⋊C4×C61—81no—
134C42×C61C2×D412yes—
135C42×C61#4112yes—
136#11758201—48no—
137#11758198—42no—
138#11758198—44no—
139#11758198—82no—
140C4⋊Dic61—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—44no—
148#11758196—24no—
149C4×Dic61—22no—
150C4×Dic61—22no—
151C4×Dic61—42no—
152C2×#92418—22no—
153C2×#92418—22no—
154C2×#92418C2×D414yes—
155C23×C61C22×D411yescharacteristic
156C22×D61—22no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163Q8×C61—42no—
164Q8×C61—42no—
165Q8×C61—42no—
166Q8×C61—42no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61#12112yes—
176C2×C4×C61#12112yes—
177C2×C4×C61C2×SD1612yes—
178C2×C4×C61#13512yes—
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C2×C4×C61H(5,𝔽2)11yescharacteristic
184C2×C4×C612−1+411yescharacteristic
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—22no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61C2×SD1614yes—
200D244—22no—
201D244—22no—
202D244—41no—
203D244—41no—
204C4×D61—84no—
205Dic122—28no—
206C8×C61—22no—
207C8×C61—41no—
208#92418—24no—
209#25293—611nomaximal, Sylow
210C22×C61C2×#12111yescharacteristic
211C22×C61#54811yescharacteristic
212C22×C61#55011yescharacteristic
213C22×C61C22×SD1611yescharacteristic
214C22×C61C2×#13511yescharacteristic
215C22×C61#57711yescharacteristic
216C22×C61#57811yescharacteristic
217C2×D61—42no—
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224C4×C61—22no—
225C4×C61—82no—
226C4×C61—82no—
227C4×C61#46514yes—
228C4×C61#52911yescharacteristic
229C4×C61#54211yescharacteristic
230C4×C61#54611yescharacteristic
231C4×C61#54811yescharacteristic
232Dic61—24no—
233Dic61—42no—
234Dic61—81no—
235Dic61—81no—
236Dic61—81no—
237C2×#461—611no—
238C2×C8×Q8—611no—
239#2568—611no—
240#2538—611no—
241#2066—612no—
242#2055—612no—
243#1968—612no—
244#1956D6111yescharacteristic, complemented
245#1948—611no—
246#1933—612no—
247C4×#102—611no—
248C2×C61#242611yescharacteristic
249C2×C61C2×#46511yescharacteristic
250C2×C61#306311yescharacteristic
251C2×C61#335911yescharacteristic
252C2×C61#336011yescharacteristic
253C2×C61#339211yescharacteristic
254C2×C61#339311yescharacteristic
255D61—82no—
256C22×SD16—612no—
257C2×C4⋊Q8C2×D6111yescharacteristic
258C2×C22⋊Q8—1221no—
259C2×C4×Q8—611no—
260C2×C4×D4—611no—
261#461—1224no—
262C8×Q8—1224no—
263C2×#106—612no—
264C2×C4⋊C8—611no—
265C2×C4⋊C8—612no—
266C2×#103—612no—
267C2×#103C2×D6112yes—
268C2×#103C2×D6112yes—
269C2×#102—611no—
270C2×#102—611no—
271C2×C4×C8—612no—
272C2×C4×C8C2×D6111yescharacteristic
273#400—1222no—
274#396—1222no—
275#391—612no—
276#391—612no—
277#391—1221no—
278#390—612no—
279#389—612no—
280#384C2×D6111yescharacteristic
281#382—1221no—
282#380—1221no—
283C4×C4⋊C4—611no—
284#340—611no—
285#340—611no—
286#340—612no—
287C61#2529311yescharacteristic, complemented, Sylow
288C22×Q8—611no—
289C22×Q8C22×D6112yes—
290C22×D4—611no—
291C23×C4—1221no—
292C2×SD16—1224no—
293C2×SD16—1224no—
294C2×SD16—2442no—
295C22×C8—612no—
296C22×C8C22×D6111yescharacteristic
297C22×C8C22×D6111yescharacteristic
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C22⋊Q8—2444no—
301C4×Q8—1222no—
302C4×Q8—1222no—
303C4×D4—1222no—
304C4×D4—1222no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—612no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4C22×D6112yes—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1222no—
318C2×C22⋊C4—1221no—
319C2×C42—612no—
320C2×C42—1221no—
321C2×C42C22×D6111yescharacteristic
322#106—2442no—
323C4⋊C8—614no—
324C4⋊C8—1224no—
325#103—24no—
326#103—42no—
327#103—1224no—
328#102—1222no—
329#102—1222no—
330C4×C8—22no—
331C4×C8—1224no—
332#95—1222no—
333#95—1222no—
334#95—1222no—
335#95—1222no—
336#95—1222no—
337#95—1221no—
338C24—1221no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—614no—
346C2×Q8—1222no—
347C2×Q8—1222no—
348C2×D4—612no—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—1221no—
352C2×D4—1221no—
353C2×D4—1221no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—612no—
357C22×C4—612no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—2442no—
362C22×C4—2442no—
363C22×C4—2442no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C23×D6111yescharacteristic
367SD16—2448no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—614no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8#1175822412yes—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—81no—
383C4⋊C4—1224no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—2444no—
392C22⋊C4—4882no—
393C22⋊C4—2444no—
394C22⋊C4—2442no—
395C42—1222no—
396C42—1222no—
397C42—2442no—
398C42#1175822212yes—
399C42D4×D6112yes—
400C23—1222no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23—2441no—
406C23—2441no—
407C23C2×D4×D6111yescharacteristic
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—1228no—
413D4—1222no—
414D4—1222no—
415D4—2441no—
416D4—2441no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—614no—
426C2×C4—1222no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—4884no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4#1451975012yes—
442C2×C4#1451975012yes—
443C2×C4SD16×D6112yes—
444C2×C4#1451980012yes—
445C2×C4C2×D4×D6111yescharacteristic
446C2×C4#41×D6111yescharacteristic
447C2×C4C2×#1175822411yescharacteristic
448C2×C4C2×#1175822211yescharacteristic
449C2×C4#1451982211yescharacteristic
450C2×C4#1451982411yescharacteristic
451C8—22no—
452C8—41no—
453C8—1224no—
454C22—4881no—
455C22—4881no—
456C22—4881no—
457C22—4881no—
458C22—2442no—
459C22—2442no—
460C22—2442no—
461C22C2×#1451975011yescharacteristic
462C22#1684042011yescharacteristic
463C22#1684043211yescharacteristic
464C22C2×SD16×D6111yescharacteristic
465C22C2×#1451980011yescharacteristic
466C22#1684056811yescharacteristic
467C22#1684057511yescharacteristic
468C4—22no—
469C4—82no—
470C4—82no—
471C4—1224no—
472C4—4881no—
473C4—4881no—
474C4—4881no—
475C4—2442no—
476C4#1683999414yes—
477C4#1684029411yescharacteristic
478C4#1684037711yescharacteristic
479C4#1684040911yescharacteristic
480C4#1684041811yescharacteristic
481C2—4882no—
482C2#2144938611yescharacteristic
483C2C2×#1683999411yescharacteristic
484C2#2145550811yescharacteristic
485C2#2145845811yescharacteristic
486C2#2145847311yescharacteristic
487C2#2145879711yescharacteristic
488C2#2145880911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1956D61
hypocenterC61#25293
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic122C2semidirect
3. #21451035C2semidirect
5. #21447196C2semidirect
8. #21446580C2semidirect
9. #21446497C2semidirect
11. C61×#1956C2semidirect
121. C2×Dic122SD16semidirect
244. #1956D61semidirect
287. C61#25293semidirect

Maximal subgroups

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

subgroupindex
C2×#168399642
C2×C8×Dic1222
#214510352
#214507422
#214471962
#214471332
#214466702
#214465802
#214464972
C4×#145196992
C61×#19562
#2529361