Finite Group Databasebeta

← #29729870

Subgroups — #29729870

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839964C211yesmaximal, characteristic
2#21450993C211yesmaximal, characteristic
3#21448879C211yesmaximal, characteristic, complemented
4#21447240C212yesmaximal, complemented
5#21447133C212yesmaximal
6#21446768C211yesmaximal, characteristic, complemented
7#21446537C211yesmaximal, characteristic, complemented
8#21446172C212yesmaximal
9#21445832C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic
11C2×C61×#487C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×Q8×Dic61C2212yes—
14C2×C4×D244C2211yescharacteristic
15#16839964—24no—
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519706C2212yes—
18C2×#14519701C2212yes—
19C2×#14519699C2211yescharacteristic
20C2×#14519699C2211yescharacteristic
21#16839673—22no—
22#16839669C2212yes—
23#16839663C2212yes—
24#16839661—21no—
25#16839648—22no—
26#16839627C2212yes—
27#16839619—21no—
28#16839619—21no—
29C4⋊C4×Dic61—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31C4×C4⋊Dic61C2211yescharacteristic
32#16839525C2211yescharacteristic
33#16839525C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839516C2212yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×C4⋊Q8×C61C2211yescharacteristic
39C61×#487—24no—
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C4×D61—21no—
45C2×#11758201—24no—
46C2×#11758201—24no—
47C2×#11758201—42no—
48C22×#92418C2312yes—
49Q8×Dic61—24no—
50Q8×Dic61—42no—
51C4×D244—22no—
52C4×D244—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61D411yescharacteristic
55C2×C4⋊Dic61D411yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×#11758198—21no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—22no—
64C2×C4×Dic61C2312yes—
65C4⋊#92418—22no—
66#14519706—42no—
67#14519701—24no—
68#14519699—22no—
69#14519699—22no—
70#14519684—21no—
71#14519684—22no—
72#14519684—22no—
73#14519684—22no—
74#14519684—22no—
75#14519684D412yes—
76C22×Q8×C61—21no—
77C22×Q8×C61C2312yes—
78C22×C8×C61C2311yescharacteristic
79C22×C8×C61C2311yescharacteristic
80C4⋊Q8×C61—22no—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61D414yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89#103×C61—24no—
90#103×C61—24no—
91C4×C8×C61—22no—
92C23×D61—21no—
93C2×Q8×C61—22no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—42no—
99C2×Q8×C61—81no—
100C2×Q8×C61—81no—
101C2×Q8×C61C2×D414yes—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61C2×D412yes—
111C22×Dic61C2×D412yes—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244#4112yes—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61—21no—
127C4⋊C4×C61—24no—
128C4⋊C4×C61—24no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—42no—
132C42×C61C2×D412yes—
133C42×C61#4112yes—
134#11758201—48no—
135#11758198—42no—
136#11758198—82no—
137#11758198—82no—
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—44no—
146C4⋊Dic61#4112yes—
147C4×Dic61—22no—
148C4×Dic61—22no—
149C4×Dic61—44no—
150C4×Dic61—42no—
151C2×#92418—22no—
152C2×#92418—22no—
153C2×#92418—22no—
154C23×C61C22×D411yescharacteristic
155C22×D61—22no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162Q8×C61—24no—
163Q8×C61—24no—
164Q8×C61—42no—
165Q8×C61—42no—
166Q8×C61—84no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—24no—
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×C61#13512yes—
178C2×C4×C61#13712yes—
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—22no—
191C2×Dic61—22no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200D244—22no—
201D244—22no—
202D244—41no—
203D244—41no—
204C4×D61—84no—
205C8×C61—41no—
206C8×C61—41no—
207#92418—42no—
208#25319—611nomaximal, Sylow
209C22×C61C2×#12111yescharacteristic
210C22×C61#54611yescharacteristic
211C22×C61#54811yescharacteristic
212C22×C61C2×#13511yescharacteristic
213C22×C61C2×#13711yescharacteristic
214C22×C61#57711yescharacteristic
215C22×C61#57711yescharacteristic
216C2×D61—42no—
217C2×D61—42no—
218C2×D61—42no—
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—22no—
224C4×C61—44no—
225C4×C61—82no—
226C4×C61#46414yes—
227C4×C61#52911yescharacteristic
228C4×C61#54211yescharacteristic
229C4×C61#54611yescharacteristic
230C4×C61#54711yescharacteristic
231Dic61—42no—
232Dic61—42no—
233Dic61—42no—
234Dic61—81no—
235Dic61—81no—
236C2×#487D6111yescharacteristic, complemented
237C2×#461—611no—
238#2563—611no—
239#2365—611no—
240#2074—612no—
241#2055—612no—
242#1987—611no—
243#1939—611no—
244#1880—612no—
245#1805—612no—
246C4×#102—611no—
247C2×C61#242711yescharacteristic
248C2×C61C2×#46411yescharacteristic
249C2×C61#304311yescharacteristic
250C2×C61#330311yescharacteristic
251C2×C61#332911yescharacteristic
252C2×C61#336011yescharacteristic
253C2×C61#337211yescharacteristic
254D61—82no—
255C22×SD16—612no—
256C2×C4⋊Q8C2×D6111yescharacteristic
257C2×C4⋊Q8C2×D6111yescharacteristic
258C2×C4×Q8—612no—
259C2×C4×D4—611no—
260#487—24no—
261#461—1224no—
262C2×C4⋊C8—611no—
263C2×C4⋊C8—612no—
264C2×#103—612no—
265C2×#103C2×D6112yes—
266C2×#103C2×D6112yes—
267C2×#102—611no—
268C2×#102—611no—
269C2×C4×C8C2×D6111yescharacteristic
270#393—1222no—
271#391—612no—
272#390—612no—
273#389—1221no—
274#387—1222no—
275#382—612no—
276#380—1221no—
277#380—1221no—
278C4×C4⋊C4—611no—
279C4×C4⋊C4—611no—
280C4×C4⋊C4—1221no—
281#340—611no—
282#340—611no—
283#340—612no—
284#340—612no—
285#336—612no—
286C61#2531911yescharacteristic, complemented, Sylow
287C22×Q8—21no—
288C22×Q8C22×D6112yes—
289C22×D4—611no—
290C23×C4—1221no—
291C2×SD16—1224no—
292C2×SD16—1224no—
293C2×SD16—2442no—
294C22×C8—612no—
295C22×C8C22×D6111yescharacteristic
296C22×C8C22×D6111yescharacteristic
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8C22⋊D6114yes—
301C4×Q8—1224no—
302C4×Q8—2442no—
303C4×D4—1222no—
304C4×D4—1222no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—611no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4C22×D6112yes—
315C2×C22⋊C4—1222no—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1221no—
318C2×C42—612no—
319C2×C42—1222no—
320C2×C42—1221no—
321C2×C42C22×D6111yescharacteristic
322C4⋊C8—1222no—
323C4⋊C8—2442no—
324#103—24no—
325#103—24no—
326#103—1224no—
327#102—1222no—
328#102—1222no—
329C4×C8—22no—
330#95—612no—
331#95—1222no—
332#95—1222no—
333#95—1222no—
334#95—1222no—
335#95—1221no—
336C24—1221no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8C2×C22⋊D6114yes—
346C2×D4—612no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—1221no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—612no—
355C22×C4—612no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—2442no—
360C22×C4—2442no—
361C22×C4—2442no—
362C22×C4D4×D6111yescharacteristic
363C22×C4D4×D6111yescharacteristic
364C22×C4C23×D6111yescharacteristic
365SD16—2448no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—1222no—
373C2×C8—1222no—
374C2×C8—1222no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—42no—
380C4⋊C4—612no—
381C4⋊C4—1222no—
382C4⋊C4—1221no—
383C4⋊C4—1222no—
384C4⋊C4—1221no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—2444no—
389C22⋊C4—4882no—
390C22⋊C4—4882no—
391C22⋊C4—2442no—
392C42—1222no—
393C42—1222no—
394C42—2444no—
395C42—2442no—
396C42#1175822412yes—
397C42C2×C22⋊D6112yes—
398C23—1222no—
399C23—2441no—
400C23—2441no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23C2×D4×D6111yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—84no—
411D4—1222no—
412D4—1222no—
413D4—2441no—
414D4—2441no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—4884no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4#1451974912yes—
440C2×C4#1451974912yes—
441C2×C4#1451980012yes—
442C2×C4#1451980912yes—
443C2×C4C22×C22⋊D6111yescharacteristic
444C2×C4#41×D6111yescharacteristic
445C2×C4#41×D6111yescharacteristic
446C2×C4C2×#1175822411yescharacteristic
447C2×C4#1451982311yescharacteristic
448C2×C4#1451982411yescharacteristic
449C8—41no—
450C8—41no—
451C8—2442no—
452C22—4881no—
453C22—4881no—
454C22—4881no—
455C22—4881no—
456C22—2442no—
457C22—2442no—
458C22—2442no—
459C22C2×#1451974911yescharacteristic
460C22#1684040511yescharacteristic
461C22#1684042011yescharacteristic
462C22C2×#1451980011yescharacteristic
463C22C2×#1451980911yescharacteristic
464C22#1684056811yescharacteristic
465C22#1684056811yescharacteristic
466C4—22no—
467C4—44no—
468C4—82no—
469C4—4881no—
470C4—4881no—
471C4—2442no—
472C4—2442no—
473C4—2442no—
474C4#1683998614yes—
475C4#1684029111yescharacteristic
476C4#1684038111yescharacteristic
477C4#1684040011yescharacteristic
478C4#1684041111yescharacteristic
479C2—4882no—
480C2#2144939911yescharacteristic
481C2C2×#1683998611yescharacteristic
482C2#2145533711yescharacteristic
483C2#2145774611yescharacteristic
484C2#2145811211yescharacteristic
485C2#2145847311yescharacteristic
486C2#2145860511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#487C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#487D61
hypocenterC61#25319
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
3. #21448879C2semidirect
4. #21447240C2semidirect
6. #21446768C2semidirect
7. #21446537C2semidirect
9. #21445832C2semidirect
11. C2×C61×#487C2semidirect
236. C2×#487D61semidirect
286. C61#25319semidirect

Maximal subgroups

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

subgroupindex
C2×#168399642
#214509932
#214488792
#214472402
#214471332
#214467682
#214465372
#214461722
#214458322
C4×#145196992
C2×C61×#4872
#2531961