Finite Group Databasebeta

← #29699252

Subgroups — #29699252

Order 15616.

990 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×#11758199C211yesmaximal, characteristic
2#21448766C211yesmaximal, characteristic, complemented
3#21446501C212yesmaximal, complemented
4#21446387C212yesmaximal, complemented
5#21446167C212yesmaximal
6#21445946C211yesmaximal, characteristic, complemented
7#21445819C211yesmaximal, characteristic, complemented
8#21445816C211yesmaximal, characteristic, complemented
9C4⋊C4×#92418C211yesmaximal, characteristic, complemented
10C4×#14519698C212yesmaximal, complemented
11D4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14C2×#14519735—21no—
15C4×#11758199C418yescomplemented
16C2×#14519710C2211yescharacteristic
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519698C2212yes—
19C2×#14519698C2211yescharacteristic
20C2×#14519697C2212yes—
21C2×#14519697C2211yescharacteristic
22C2×C4×#92418C2212yes—
23C2×C4×#92418C2211yescharacteristic
24#16839664C2212yes—
25#16839654—22no—
26#16839632C2211yescharacteristic
27#16839619—21no—
28C4×C4⋊Dic61C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C2×C4×D4×C61C2212yes—
35C2×C4×D4×C61C2211yescharacteristic
36C22×C4⋊C4×C61—21no—
37C61×#389C2212yes—
38C61×#385—22no—
39C61×#384C2211yescharacteristic
40C61×#380—21no—
41C4×C4⋊C4×C61C2211yescharacteristic
42C22×D244C2311yescharacteristic
43C22×C4×D61—21no—
44C2×#11758199—22no—
45C2×#11758199—22no—
46C2×#11758199C2×C414yes—
47C22×#92418C2312yes—
48C22×#92418C2311yescharacteristic
49C22×#92418C2311yescharacteristic
50C4×D244—22no—
51C4×D244C2×C414yescomplemented
52#14519735—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758198—21no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×C4×Dic61—22no—
61#14519710C2×C414yescomplemented
62C4⋊#92418—22no—
63#14519698—24no—
64#14519698C2×C414yes—
65#14519697—42no—
66#14519697C2×C414yescomplemented
67C4×#92418—42no—
68C4×#92418C2×C414yescomplemented
69#14519684—22no—
70C22×D4×C61C2311yescharacteristic
71C23×C4×C61—21no—
72C23×C4×C61—22no—
73C4×D4×C61—24no—
74C4×D4×C61—22no—
75C4×D4×C61—42no—
76C4×D4×C61C2×C414yes—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—44no—
81C2×C4⋊C4×C61D411yescharacteristic
82C2×C4⋊C4×C61D411yescharacteristic
83C2×C4⋊C4×C61C2312yes—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C22⋊C4×C61—21no—
86C2×C22⋊C4×C61—22no—
87C2×C42×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89#95×C61—22no—
90#95×C61D412yes—
91C24×C61—21no—
92C23×D61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61C22×C412yes—
97C2×D4×C61#4112yes—
98C2×D4×C61#4112yes—
99C22×C4×C61—21no—
100C22×C4×C61—24no—
101C22×C4×C61—22no—
102C22×C4×C61—44no—
103C22×C4×C61—42no—
104C22×C4×C61—42no—
105C22×C4×C61—44no—
106C22×C4×C61—82no—
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—22no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244C22×C412yes—
120C2×C4×D61—24no—
121C2×C4×D61—42no—
122C2×C4×D61—42no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—44no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—81no—
130C4⋊C4×C61D814yescomplemented
131C4⋊C4×C61C22×C412yes—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—44no—
134C42×C61—22no—
135C42×C61—22no—
136C42×C61C22×C412yes—
137C42×C61#4112yes—
138#11758199—28no—
139#11758198—24no—
140#11758198—44no—
141#11758198—82no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—22no—
145C4⋊Dic61—42no—
146C4⋊Dic61—81no—
147C4⋊Dic61C22×C412yes—
148C4×Dic61—44no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—21no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418C22×C412yes—
157C2×#92418C22×C412yes—
158C23×C61—22no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61C22×D411yescharacteristic
166C22×D61—22no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173D4×C61—22no—
174D4×C61—22no—
175D4×C61—22no—
176D4×C61#11714yescomplemented
177C2×C4×C61—22no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—48no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—88no—
192C2×C4×C61C4×D412yes—
193C2×C4×C61C4×D412yes—
194C2×C4×C61C2×D814yes—
195C2×C4×C61C2×D812yes—
196C2×C4×C61#13512yes—
197C2×C4×C61C23×C411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)11yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C2×Dic61—22no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209D244—22no—
210D244—22no—
211D244—41no—
212D244—41no—
213C4×D61—48no—
214#92418—22no—
215#92418—22no—
216#92418—42no—
217#22766—611nomaximal, Sylow
218C22×C61—24no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61C2×C4×D411yescharacteristic
226C22×C61#54611yescharacteristic
227C22×C61#54811yescharacteristic
228C22×C61C22×D811yescharacteristic
229C22×C61C2×#13511yescharacteristic
230C22×C61#57711yescharacteristic
231C22×C61#57711yescharacteristic
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—24no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—82no—
244C4×C61C4×D814yes—
245C4×C61C2×#11711yescharacteristic
246C4×C61#52011yescharacteristic
247C4×C61#55211yescharacteristic
248C4×C61#55311yescharacteristic
249Dic61—42no—
250Dic61—82no—
251C2×C4×D8—611no—
252#2354—611no—
253D4×C4⋊C4D6111yescharacteristic, complemented
254#1934—612no—
255#1918—612no—
256#1879—612no—
257#1834—611no—
258#1802—611no—
259#1802—611no—
260C8×C4⋊C4—611no—
261C4×#102—612no—
262C2×C61—44no—
263C2×C61#235211yescharacteristic
264C2×C61C2×C4×D811yescharacteristic
265C2×C61#295211yescharacteristic
266C2×C61#329411yescharacteristic
267C2×C61#330311yescharacteristic
268C2×C61#335811yescharacteristic
269C2×C61#336011yescharacteristic
270D61—82no—
271C22×D8—611no—
272C2×C4×D4—611no—
273C2×C4×D4C2×D6112yes—
274C2×C4×D4C2×D6111yescharacteristic
275C2×#117—1221no—
276C22×C4⋊C4—21no—
277C4×D8—618no—
278C2×#107—611no—
279C2×C4⋊C8—611no—
280C2×#102—611no—
281C2×#102—611no—
282C2×#102—612no—
283C2×#102—612no—
284C2×C4×C8—611no—
285C2×C4×C8—612no—
286#390—612no—
287#389C2×D6112yes—
288#388—1222no—
289#385—22no—
290#384—611no—
291#384C2×D6111yescharacteristic
292#380—21no—
293#380—1221no—
294C4×C4⋊C4—612no—
295C4×C4⋊C4C2×D6111yescharacteristic
296#340—612no—
297#340—612no—
298#340—612no—
299#340—612no—
300#336—612no—
301C61#2276611yescharacteristic, complemented, Sylow
302C22×D4—611no—
303C22×D4C22×D6111yescharacteristic
304C23×C4—21no—
305C23×C4—22no—
306C23×C4—1221no—
307C2×D8—614no—
308C2×D8—1222no—
309C2×D8—1222no—
310C22×C8—611no—
311C22×C8—611no—
312C22×C8—612no—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4—42no—
316C4×D4—614no—
317C4×D4—1222no—
318C4×D4C4×D6114yes—
319#117—2444no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—612no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4C22⋊D6111yescharacteristic
329C2×C4⋊C4C22⋊D6111yescharacteristic
330C2×C4⋊C4C22×D6112yes—
331C2×C4⋊C4C22×D6111yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1222no—
337C2×C42—1222no—
338C2×C42C22×D6112yes—
339C2×C42C22×D6111yescharacteristic
340#107—614no—
341C4⋊C8—1222no—
342#102—614no—
343#102—614no—
344#102—1224no—
345#102—2442no—
346C4×C8—614no—
347C4×C8—2442no—
348#95—22no—
349#95—1222no—
350#95C22⋊D6112yes—
351C24—21no—
352C24—1221no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—612no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4C2×C4×D6112yes—
363C2×D4#1175822412yes—
364C2×D4#1175822612yes—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—82no—
373C22×C4—1221no—
374C22×C4—1224no—
375C22×C4—1222no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4C2×C22⋊D6112yes—
379C22×C4C2×C22⋊D6111yescharacteristic
380C22×C4C2×C22⋊D6112yes—
381C22×C4C2×C22⋊D6111yescharacteristic
382C22×C4C23×D6111yescharacteristic
383D8—1228no—
384C2×C8—612no—
385C2×C8—612no—
386C2×C8—1221no—
387C2×C8—1221no—
388C2×C8—1221no—
389C2×C8—1221no—
390C2×C8—1222no—
391C2×C8—1222no—
392C2×C8—1222no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—81no—
400C4⋊C4—612no—
401C4⋊C4—1221no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4—4881no—
405C4⋊C4—2442no—
406C4⋊C4#1175819914yescomplemented
407C4⋊C4C2×C4×D6112yes—
408C22⋊C4—24no—
409C22⋊C4—44no—
410C22⋊C4—1224no—
411C22⋊C4—4882no—
412C22⋊C4—2444no—
413C42—22no—
414C42—22no—
415C42—2444no—
416C42C2×C4×D6112yes—
417C42#1175822212yes—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—1222no—
426C23—2441no—
427C23—2441no—
428C23—2441no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23C22×C22⋊D6111yescharacteristic
433D4—22no—
434D4—22no—
435D4—22no—
436D4—1222no—
437D4—1222no—
438D4—2441no—
439D4—2441no—
440D4#1451973514yescomplemented
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—88no—
456C2×C4—1222no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2448no—
463C2×C4C4×C22⋊D6112yes—
464C2×C4C4×C22⋊D6112yes—
465C2×C4C2×#1175819914yes—
466C2×C4C2×#1175819912yes—
467C2×C4#1451979712yes—
468C2×C4C22×C4×D6111yescharacteristic
469C2×C4C2×#1175822411yescharacteristic
470C2×C4C2×#1175822211yescharacteristic
471C2×C4C2×#1175822611yescharacteristic
472C2×C4#1451982211yescharacteristic
473C2×C4#1451982511yescharacteristic
474C8—1222no—
475C8—1222no—
476C8—2442no—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22—4881no—
488C22—2442no—
489C22—2442no—
490C22—2442no—
491C22C2×C4×C22⋊D6111yescharacteristic
492C22#1684040611yescharacteristic
493C22#1684042311yescharacteristic
494C22C22×#1175819911yescharacteristic
495C22C2×#1451979711yescharacteristic
496C22#1684057011yescharacteristic
497C22#1684057311yescharacteristic
498C4—22no—
499C4—24no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—4882no—
504C4—2442no—
505C4C4×#1175819914yes—
506C4C2×#1451973511yescharacteristic
507C4#1684024311yescharacteristic
508C4#1684044711yescharacteristic
509C4#1684046011yescharacteristic
510C2—44no—
511C2—4882no—
512C2#2144874411yescharacteristic
513C2C2×C4×#1175819911yescharacteristic
514C2#2145470011yescharacteristic
515C2#2145763211yescharacteristic
516C2#2145774711yescharacteristic
517C2#2145845411yescharacteristic
518C2#2145846811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingD4×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenterD4×C4⋊C4D61
hypocenterC61#22766
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21448766C2semidirect
3. #21446501C2semidirect
4. #21446387C2semidirect
6. #21445946C2semidirect
7. #21445819C2semidirect
8. #21445816C2semidirect
9. C4⋊C4×#92418C2semidirect
10. C4×#14519698C2semidirect
11. D4×C4⋊C4×C61C2semidirect
15. C4×#11758199C4semidirect
51. C4×D244C2×C4semidirect
61. #14519710C2×C4semidirect
66. #14519697C2×C4semidirect
68. C4×#92418C2×C4semidirect
130. C4⋊C4×C61D8semidirect
176. D4×C61#117semidirect
253. D4×C4⋊C4D61semidirect
301. C61#22766semidirect
406. C4⋊C4#11758199semidirect
440. D4#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117581992
#214487662
#214465012
#214463872
#214461672
#214459462
#214458192
#214458162
C4⋊C4×#924182
C4×#145196982
D4×C4⋊C4×C612
#2276661