Finite Group Databasebeta

← #29699165

Subgroups — #29699165

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D488C211yesmaximal, characteristic
2#21448770C211yesmaximal, characteristic, complemented
3#21448719C211yesmaximal, characteristic, complemented
4#21446514C212yesmaximal, complemented
5#21446410C212yesmaximal, complemented
6#21446168C212yesmaximal
7#21445816C211yesmaximal, characteristic, complemented
8#21445816C211yesmaximal, characteristic, complemented
9#21445686C211yesmaximal, characteristic, complemented
10#21445643C212yesmaximal, complemented
11C61×#1834C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15C2×#14519739C2212yes—
16C2×#14519735—21no—
17C2×C4⋊C4×D61—21no—
18C4×D488C418yescomplemented
19C2×#14519711C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×#14519687C2212yes—
26#16839663C2212yes—
27#16839658C2212yes—
28#16839652—22no—
29#16839633—22no—
30#16839619—21no—
31#16839619—21no—
32C4⋊C4×Dic61C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839516C2212yes—
37C2×C4×C8×C61C2211yescharacteristic
38C61×#384C2211yescharacteristic
39C61×#384C2211yescharacteristic
40C61×#340C2212yes—
41C61×#340C2212yes—
42C22×D244C2311yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×D488—22no—
48C2×D488—22no—
49C2×D488C2×C414yes—
50C22×#92418C2312yes—
51C4×D244—22no—
52C4×D244—22no—
53C4×D244C2×C414yes—
54C4×D244C2×C414yescomplemented
55#14519739—24no—
56#14519739—42no—
57#14519735—44no—
58C2×C4⋊Dic61D411yescharacteristic
59C2×C4⋊Dic61D411yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758196—22no—
63C4⋊C4×D61—44no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61C2312yes—
71#14519711C2×C414yescomplemented
72C4⋊#92418—22no—
73#14519699C2×C414yes—
74#14519699C2×C414yescomplemented
75#14519697—24no—
76#14519697—42no—
77#14519687—42no—
78#14519684—22no—
79#14519684—22no—
80#14519684D412yes—
81C22×C8×C61C2311yescharacteristic
82C22×C8×C61C2311yescharacteristic
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61C2312yes—
88C2×C4⋊C4×C61C2312yes—
89C2×C42×C61C2311yescharacteristic
90C4×C8×C61C2×C414yescomplemented
91C23×D61—21no—
92C23×D61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61C2×D412yes—
101C22×Dic61C2×D412yes—
102C2×D244—21no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244C22×C412yes—
111C2×D244C22×C412yes—
112C2×D244#4112yes—
113C2×D244#4112yes—
114C2×C4×D61—24no—
115C2×C4×D61—24no—
116C2×C4×D61—42no—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61—44no—
121C2×C4×D61—44no—
122C2×C4×D61—82no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61—21no—
127C2×C8×C61C22×C412yes—
128C2×C8×C61C22×C412yes—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—42no—
134C4⋊C4×C61—42no—
135C4⋊C4×C61—81no—
136C4⋊C4×C61—81no—
137C42×C61C22×C412yes—
138C42×C61#4112yes—
139#11758198—24no—
140#11758198—24no—
141#11758198—44no—
142#11758198—44no—
143#11758198—82no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61C22×C412yes—
149C4⋊Dic61C22×C412yes—
150#11758196—44no—
151C4×Dic61—22no—
152C4×Dic61—22no—
153C4×Dic61—44no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157D488—28no—
158C23×C61C22×D411yescharacteristic
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—22no—
162C22×D61—22no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61C4×D412yes—
182C2×C4×C61C4×D412yes—
183C2×C4×C61AGL(1,ℤ/8)12yes—
184C2×C4×C61AGL(1,ℤ/8)12yes—
185C2×C4×C61C23×C411yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61H(5,𝔽2)11yescharacteristic
190C2×C4×C612−1+411yescharacteristic
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203D244—22no—
204D244—22no—
205D244—22no—
206D244—22no—
207D244—22no—
208D244—41no—
209D244—41no—
210D244#11714yescomplemented
211C4×D61—48no—
212C4×D61—48no—
213C4×D61—88no—
214C8×C61—22no—
215C8×C61—22no—
216#92418—42no—
217#22758—611nomaximal, Sylow
218C22×C61C2×C4×D411yescharacteristic
219C22×C61#54611yescharacteristic
220C22×C61#54811yescharacteristic
221C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
222C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C61#57611yescharacteristic
224C22×C61#57811yescharacteristic
225C2×D61—24no—
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—42no—
231C2×D61—42no—
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—22no—
240C4×C61—82no—
241C4×C61—82no—
242C4×C61#44214yes—
243C4×C61C2×#11711yescharacteristic
244C4×C61#52011yescharacteristic
245C4×C61#55211yescharacteristic
246C4×C61#55311yescharacteristic
247Dic61—42no—
248Dic61—42no—
249Dic61—42no—
250Dic61—42no—
251C2×C4×D8—611no—
252#2354—611no—
253D4×C4⋊C4—611no—
254#1936—612no—
255#1921—612no—
256#1879—612no—
257#1834D6111yescharacteristic, complemented
258#1802—611no—
259#1802—611no—
260#1770—611no—
261#1762—612no—
262C2×C61#235211yescharacteristic
263C2×C61C2×#44211yescharacteristic
264C2×C61#295111yescharacteristic
265C2×C61#332511yescharacteristic
266C2×C61#333011yescharacteristic
267C2×C61#336611yescharacteristic
268C2×C61#336711yescharacteristic
269D61—44no—
270D61—82no—
271C22×D8—611no—
272C2×C4×D4—612no—
273C2×C4×D4—611no—
274C2×C4×D4—611no—
275C2×#117—1221no—
276C22×C4⋊C4—1221no—
277C4×D8—618no—
278C2×#107—611no—
279C2×C4⋊C8—611no—
280C2×#102—611no—
281C2×#102—611no—
282C2×#102—612no—
283C2×#102—612no—
284C2×#97—612no—
285C2×C4×C8C2×D6111yescharacteristic
286#390—612no—
287#389—612no—
288#388—1222no—
289#385—1222no—
290#384C2×D6111yescharacteristic
291#384C2×D6111yescharacteristic
292#380—1221no—
293#380—1221no—
294C4×C4⋊C4—611no—
295C4×C4⋊C4—612no—
296#340—612no—
297#340—612no—
298#340C2×D6112yes—
299#340C2×D6112yes—
300#336—612no—
301C61#2275811yescharacteristic, complemented, Sylow
302C22×D4—611no—
303C22×D4—611no—
304C23×C4—1222no—
305C23×C4—1221no—
306C23×C4—1221no—
307C2×D8—614no—
308C2×D8—1222no—
309C2×D8—1222no—
310C22×C8—612no—
311C22×C8C22×D6111yescharacteristic
312C22×C8C22×D6111yescharacteristic
313C4×D4—614no—
314C4×D4—614no—
315C4×D4—1224no—
316C4×D4—1222no—
317C4×D4—1222no—
318C4×D4—2442no—
319#117—2444no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—2444no—
330C2×C4⋊C4C22×D6112yes—
331C2×C4⋊C4C22×D6112yes—
332C2×C22⋊C4—1221no—
333C2×C22⋊C4—1221no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1222no—
337C2×C42—612no—
338C2×C42—1222no—
339C2×C42C22×D6111yescharacteristic
340#107—614no—
341C4⋊C8—1222no—
342#102—614no—
343#102—614no—
344#102—1224no—
345#102—2442no—
346#97—2442no—
347C4×C8C4×D6114yescomplemented
348#95—612no—
349#95—1222no—
350#95—1222no—
351C24—1221no—
352C24—1221no—
353C2×D4—612no—
354C2×D4—612no—
355C2×D4—612no—
356C2×D4—612no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—1221no—
364C2×D4—1221no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—612no—
368C22×C4—612no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1224no—
372C22×C4—1224no—
373C22×C4—4882no—
374C22×C4—2444no—
375C22×C4—2444no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4—2442no—
379C22×C4—2442no—
380C22×C4D4×D6111yescharacteristic
381C22×C4D4×D6111yescharacteristic
382C22×C4C23×D6111yescharacteristic
383D8—1228no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—1222no—
389C2×C8—1222no—
390C2×C8—1222no—
391C2×C8C2×C4×D6112yes—
392C2×C8C2×C4×D6112yes—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—81no—
400C4⋊C4—81no—
401C4⋊C4—612no—
402C4⋊C4—612no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—2444no—
408C22⋊C4—1224no—
409C22⋊C4—1224no—
410C22⋊C4—4882no—
411C22⋊C4—2444no—
412C22⋊C4—2444no—
413C42—1222no—
414C42—1222no—
415C42—2444no—
416C42C2×C4×D6112yes—
417C42#1175822612yes—
418C23—1222no—
419C23—1222no—
420C23—1222no—
421C23—1222no—
422C23—2441no—
423C23—2441no—
424C23—2441no—
425C23—2441no—
426C23—2441no—
427C23—2441no—
428C23—2441no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23C2×D4×D6111yescharacteristic
433D4—614no—
434D4—1222no—
435D4—1222no—
436D4—1222no—
437D4—1222no—
438D4—1222no—
439D4—2441no—
440D4—2441no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—4888no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2448no—
463C2×C4—2448no—
464C2×C4#1451973912yes—
465C2×C4#1451973912yes—
466C2×C4#1451980412yes—
467C2×C4#1451980412yes—
468C2×C4C22×C4×D6111yescharacteristic
469C2×C4#41×D6111yescharacteristic
470C2×C4#41×D6111yescharacteristic
471C2×C4C2×#1175822611yescharacteristic
472C2×C4#1451982311yescharacteristic
473C2×C4#1451982411yescharacteristic
474C8—22no—
475C8—22no—
476C8—2442no—
477C22—1224no—
478C22—4881no—
479C22—4881no—
480C22—4881no—
481C22—4881no—
482C22—2442no—
483C22—2442no—
484C22—2442no—
485C22—2442no—
486C22—2442no—
487C22—2442no—
488C22—2442no—
489C22—2442no—
490C22—2442no—
491C22C2×#1451973911yescharacteristic
492C22#1684040511yescharacteristic
493C22#1684042011yescharacteristic
494C22C2×#1451980411yescharacteristic
495C22C2×#1451980411yescharacteristic
496C22#1684056311yescharacteristic
497C22#1684057511yescharacteristic
498C4—22no—
499C4—82no—
500C4—82no—
501C4—2442no—
502C4—2442no—
503C4—2442no—
504C4—2442no—
505C4#1683987314yes—
506C4#117×D6111yescharacteristic
507C4#1684024211yescharacteristic
508C4#1684045011yescharacteristic
509C4#1684046511yescharacteristic
510C2—4882no—
511C2—2444no—
512C2#2144874511yescharacteristic
513C2C2×#1683987311yescharacteristic
514C2#2145469311yescharacteristic
515C2#2145806011yescharacteristic
516C2#2145812311yescharacteristic
517C2#2145853111yescharacteristic
518C2#2145854211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1834C2
socleC23×C61C22×D4
prime radicalC22×D488C22
cosocleC22×C4×C61C24
hypercenter#1834D61
hypocenterC61#22758
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
2. #21448770C2semidirect
3. #21448719C2semidirect
4. #21446514C2semidirect
5. #21446410C2semidirect
7. #21445816C2semidirect
8. #21445816C2semidirect
9. #21445686C2semidirect
10. #21445643C2semidirect
11. C61×#1834C2semidirect
18. C4×D488C4semidirect
54. C4×D244C2×C4semidirect
71. #14519711C2×C4semidirect
74. #14519699C2×C4semidirect
90. C4×C8×C61C2×C4semidirect
210. D244#117semidirect
257. #1834D61semidirect
301. C61#22758semidirect
347. C4×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4882
#214487702
#214487192
#214465142
#214464102
#214461682
#214458162
#214456862
#214456432
C61×#18342
#2275861