Finite Group Databasebeta

← #29719015

Subgroups — #29719015

Order 15616.

1064 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519701C211yesmaximal, characteristic, complemented
2#21450291C211yesmaximal, characteristic
3Q8×C22⋊Dic61C211yesmaximal, characteristic, complemented
4#21447230C212yesmaximal, complemented
5#21447132C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic
7#21446176C212yesmaximal, complemented
8#21445996C211yesmaximal, characteristic, complemented
9#21445906C211yesmaximal, characteristic, complemented
10#21445822C212yesmaximal, complemented
11C2×C61×#448C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×#14519758—21no—
14C2×#14519758—21no—
15C2×Q8×Dic61C2212yes—
16C22×C4⋊Dic61C2211yescharacteristic
17C2×#14519706C2212yes—
18C2×#14519701—24no—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519698C2212yes—
22#16839698—22no—
23#16839669—22no—
24#16839669C2212yes—
25#16839665C2212yes—
26#16839631C2212yes—
27#16839622—21no—
28#16839622C2211yescharacteristic
29C4×C22⋊Dic61—21no—
30C4×C22⋊Dic61C2211yescharacteristic
31#16839525C2212yes—
32#16839525C2212yes—
33#16839524C2212yes—
34#16839516C2212yes—
35C23×Q8×C61C2211yescharacteristic
36C22×SD16×C61C2211yescharacteristic
37C22×SD16×C61C2211yescharacteristic
38C2×#121×C61C2211yescharacteristic
39C61×#448—24no—
40C2×#103×C61C2211yescharacteristic
41C2×#103×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C23×Dic61—21no—
44C22×#92418—22no—
45C22×#92418—42no—
46C22×#92418C2312yes—
47#14519758—44no—
48#14519758—44no—
49Q8×Dic61—24no—
50Q8×Dic61—42no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2312yes—
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61D411yescharacteristic
63C2×C22⋊Dic61D411yescharacteristic
64C2×C4×Dic61—22no—
65C2×C4×Dic61C2312yes—
66#14519706—42no—
67#14519701—24no—
68#14519701—24no—
69#14519698—42no—
70#14519684—21no—
71#14519684—22no—
72#14519684—22no—
73#14519684D412yes—
74C22×Q8×C61—21no—
75C22×Q8×C61—22no—
76C22×Q8×C61—24no—
77C22×Q8×C61—42no—
78C22×Q8×C61D414yes—
79C22×Q8×C61C2311yescharacteristic
80C22×Q8×C61C2311yescharacteristic
81C22×D4×C61—21no—
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85C2×SD16×C61—22no—
86C2×SD16×C61—22no—
87C2×SD16×C61—24no—
88C2×SD16×C61—42no—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91#121×C61—22no—
92#121×C61—22no—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C22⋊C4×C61—21no—
95#103×C61—22no—
96#103×C61—22no—
97C22⋊C8×C61—22no—
98C24×C61—21no—
99C24×C61C2×D411yescharacteristic
100C2×Q8×C61—24no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—28no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—44no—
108C2×Q8×C61—44no—
109C2×Q8×C61—48no—
110C2×Q8×C61C2×D412yes—
111C2×Q8×C61C2×D412yes—
112C2×Q8×C61#4114yes—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—24no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—44no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61#4112yes—
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61—42no—
136C22×Dic61C2×D412yes—
137C22×Dic61C2×D412yes—
138SD16×C61—44no—
139SD16×C61—44no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C22⋊C4×C61—42no—
147C22⋊Dic61—22no—
148C22⋊Dic61—44no—
149C22⋊Dic61—42no—
150C22⋊Dic61—44no—
151C22⋊Dic61SD1614yescomplemented
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—44no—
157C4⋊Dic61—42no—
158#11758196—44no—
159#11758196—44no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C4×Dic61—44no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—44no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—41no—
170C23×C61—42no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—42no—
174C23×C61#12112yes—
175C23×C61C2×SD1612yes—
176C23×C61AGL(1,ℤ/8)12yes—
177C23×C61C22×D411yescharacteristic
178Q8×C61—22no—
179Q8×C61—24no—
180Q8×C61—22no—
181Q8×C61—22no—
182Q8×C61—48no—
183Q8×C61—42no—
184Q8×C61#12314yescomplemented
185D4×C61—41no—
186D4×C61—41no—
187D4×C61—42no—
188D4×C61—84no—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—48no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61#12112yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4112yes—
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61H(5,𝔽2)12yes—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—84no—
218C2×Dic61C2×SD1614yes—
219C8×C61—41no—
220C8×C61—41no—
221#92418—42no—
222#24528—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—24no—
225C22×C61—22no—
226C22×C61—24no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—82no—
230C22×C61—81no—
231C22×C61—82no—
232C22×C61#46014yes—
233C22×C61C2×#12111yescharacteristic
234C22×C61#54612yes—
235C22×C61C22×SD1611yescharacteristic
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C61#57812yes—
238C4×C61—22no—
239C4×C61—24no—
240C4×C61—42no—
241C4×C61—81no—
242C4×C61C2×#12311yescharacteristic
243C4×C61#53411yescharacteristic
244C4×C61#55112yes—
245Dic61—24no—
246Dic61—42no—
247Dic61—82no—
248Dic61—82no—
249C2×#448D6111yescharacteristic
250C22×#103—611no—
251#2501—611no—
252Q8×C22⋊C4—611no—
253#2072—612no—
254#2055—612no—
255#1893—611no—
256#1880—612no—
257#1850—611no—
258#1823—611no—
259#1803—612no—
260C2×C61—24no—
261C2×C61—161no—
262C2×C61#241211yescharacteristic
263C2×C61C2×#46011yescharacteristic
264C2×C61#304411yescharacteristic
265C2×C61#329812yes—
266C2×C61#330812yes—
267C23×Q8C2×D6111yescharacteristic
268C22×SD16C2×D6111yescharacteristic
269C22×SD16C2×D6111yescharacteristic
270C23×C8—611no—
271C2×#123—1221no—
272C2×#123—1221no—
273C2×#121C2×D6111yescharacteristic
274C2×C4×Q8—612no—
275C22×C4⋊C4—611no—
276#448—24no—
277C2×C4⋊C8—612no—
278C2×#103—612no—
279C2×#103—612no—
280C2×#103—1224no—
281C2×#103C2×D6111yescharacteristic
282C2×#103C2×D6111yescharacteristic
283C2×#102—612no—
284C2×C22⋊C8C2×D6111yescharacteristic
285#399—1222no—
286#391—612no—
287#391—1222no—
288#390—612no—
289#384—612no—
290#380—611no—
291#380—1221no—
292C4×C22⋊C4—611no—
293C4×C22⋊C4—1221no—
294#340—612no—
295#340—612no—
296#340—612no—
297#336—612no—
298C61#2452811yescharacteristic, complemented, Sylow
299C22×Q8—21no—
300C22×Q8—22no—
301C22×Q8—24no—
302C22×Q8—42no—
303C22×Q8C22⋊D6114yes—
304C22×Q8C22×D6111yescharacteristic
305C22×Q8C22×D6111yescharacteristic
306C22×D4—21no—
307C22×D4C22×D6111yescharacteristic
308C23×C4—21no—
309C23×C4—1221no—
310C23×C4C22×D6111yescharacteristic
311C2×SD16—24no—
312C2×SD16—22no—
313C2×SD16—22no—
314C2×SD16—42no—
315C22×C8—612no—
316C22×C8—1222no—
317C22×C8—2442no—
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6111yescharacteristic
320#123—2444no—
321#123—2444no—
322#121—22no—
323#121—22no—
324C4×Q8—1224no—
325C4×Q8—2442no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4C22×D6111yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—611no—
337C2×C22⋊C4—611no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1221no—
341C2×C42—612no—
342C2×C42—1222no—
343C4⋊C8—2442no—
344#103—22no—
345#103—22no—
346#103—1224no—
347#103—1224no—
348#102—2442no—
349C22⋊C8—22no—
350#95—612no—
351#95—1222no—
352#95—1222no—
353#95—1221no—
354C24—21no—
355C24D4×D6111yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—28no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8#1175822414yes—
367C2×Q8C2×C22⋊D6112yes—
368C2×Q8C2×C22⋊D6112yes—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—44no—
383C22×C4—612no—
384C22×C4—612no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4D4×D6111yescharacteristic
391C22×C4#1175822412yes—
392C22×C4C2×C22⋊D6112yes—
393C22×C4C23×D6111yescharacteristic
394SD16—44no—
395SD16—44no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—2444no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—2444no—
411C4⋊C4—2444no—
412C4⋊C4—2444no—
413C4⋊C4—2442no—
414C22⋊C4—42no—
415C22⋊C4—614no—
416C22⋊C4—1222no—
417C22⋊C4—2444no—
418C22⋊C4—2444no—
419C22⋊C4—2442no—
420C42—1222no—
421C42—1222no—
422C42—2444no—
423C23—22no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23#1451974912yes—
431C23SD16×D6112yes—
432C23#1451980412yes—
433C23C2×D4×D6111yescharacteristic
434Q8—22no—
435Q8—24no—
436Q8—22no—
437Q8—22no—
438Q8—48no—
439Q8—42no—
440Q8#1451975814yescomplemented
441D4—41no—
442D4—41no—
443D4—42no—
444D4—84no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—48no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—614no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—4884no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4#1451974912yes—
471C2×C4C22×C22⋊D6111yescharacteristic
472C2×C4C2×#1175822412yes—
473C2×C4C2×#1175822411yescharacteristic
474C2×C4#1451982212yes—
475C8—41no—
476C8—41no—
477C8—2442no—
478C22—22no—
479C22—24no—
480C22—24no—
481C22—22no—
482C22—81no—
483C22—82no—
484C22—81no—
485C22—82no—
486C22—81no—
487C22#1683996014yes—
488C22C2×#1451974911yescharacteristic
489C22#1684039912yes—
490C22C2×SD16×D6111yescharacteristic
491C22C2×#1451980411yescharacteristic
492C22#1684057812yes—
493C4—22no—
494C4—24no—
495C4—42no—
496C4—81no—
497C4—1224no—
498C4—4882no—
499C4—4882no—
500C4—2442no—
501C4C2×#1451975811yescharacteristic
502C4#1684032111yescharacteristic
503C4#1684044312yes—
504C2—24no—
505C2—161no—
506C2#2144921611yescharacteristic
507C2C2×#1683996011yescharacteristic
508C2#2145535111yescharacteristic
509C2#2145767712yes—
510C2#2145781412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#448C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenterC2×#448D61
hypocenterC61#24528
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C22×#14519701C2semidirect
3. Q8×C22⋊Dic61C2semidirect
4. #21447230C2semidirect
5. #21447132C2semidirect
7. #21446176C2semidirect
8. #21445996C2semidirect
9. #21445906C2semidirect
10. #21445822C2semidirect
151. C22⋊Dic61SD16semidirect
184. Q8×C61#123semidirect
298. C61#24528semidirect
440. Q8#14519758semidirect

Maximal subgroups

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

subgroupindex
C22×#145197012
#214502912
Q8×C22⋊Dic612
#214472302
#214471322
#214462482
#214461762
#214459962
#214459062
#214458222
C2×C61×#4482
#2452861