Finite Group Databasebeta

← #29646103

Subgroups — #29646103

Order 15616.

1386 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519755C211yesmaximal, characteristic
2#407×D61C212yesmaximal
3#21444181C218yesmaximal, complemented
4#21444149C211yesmaximal, characteristic, complemented
5#21444148C212yesmaximal, complemented
6C61×#1463C211yesmaximal, characteristic, complemented
7C23×Dic122C411yescharacteristic
8C23×C4×D61C2211yescharacteristic
9C2×M16×D61—22no—
10C2×#14519755—22no—
11C2×#14519755—22no—
12C2×#14519755—22no—
13C2×#14519755—22no—
14C2×#14519755C414yes—
15C2×#14519755C2214yes—
16C22×#11758196C411yescharacteristic
17C22×#11758196C2211yescharacteristic
18C22×#11758198C411yescharacteristic
19C22×#11758198C411yescharacteristic
20#16839726C2212yes—
21#16839725—22no—
22#16839724—22no—
23#16839724C2214yes—
24C22⋊C8×D61—22no—
25C22⋊C8×D61C2214yes—
26#16839488C2214yescomplemented
27#16839487C2218yescomplemented
28C22×C4⋊C4×C61C2211yescharacteristic
29C61×#407C2212yes—
30C61×#324C2214yescomplemented
31C24×D61C2×C411yescharacteristic
32C22×Dic122—21no—
33C22×Dic122—22no—
34C22×Dic122—21no—
35C22×Dic122—44no—
36C22×Dic122C2×C414yes—
37C23×Dic61C2×C411yescharacteristic
38C23×Dic61C2×C411yescharacteristic
39C23×Dic61C2×C411yescharacteristic
40C22×C4×D61—44no—
41C22×C4×D61C2×C412yes—
42C22×C4×D61D412yes—
43C22×C4×D61D412yes—
44C22×C4×D61D412yes—
45C22×C4×D61D412yes—
46C22×C4×D61C2312yes—
47C2×#11758194—22no—
48C2×#11758189—24no—
49M16×D61—48no—
50C2×C8×D61—24no—
51#14519755—24no—
52#14519755—24no—
53#14519755—24no—
54#14519755—24no—
55#14519755—24no—
56#14519755—24no—
57#14519755—24no—
58#14519755—24no—
59C2×#11758196—22no—
60C2×#11758196—22no—
61C2×#11758196—21no—
62C2×#11758196—22no—
63C2×#11758196—21no—
64C2×#11758196C2×C414yes—
65C2×#11758196C2×C412yes—
66C2×#11758196C2312yes—
67C2×#11758198—22no—
68C2×#11758198—21no—
69C2×#11758198—21no—
70C2×#11758198—22no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198C2×C414yes—
74C2×#11758198C2×C414yes—
75C22⋊#92418—22no—
76C22⋊#92418C2314yes—
77#14519689—24no—
78#14519689—24no—
79C23×C4×C61C2×C411yescharacteristic
80C23×C4×C61C2×C411yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C2×M16×C61—22no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61C2×C412yes—
86C2×C4⋊C4×C61C2312yes—
87C22⋊C8×C61—22no—
88C22⋊C8×C61C2314yes—
89C24×C61C22×C411yescharacteristic
90C23×D61—24no—
91C23×D61—42no—
92C23×D61—42no—
93C23×D61C22⋊C412yes—
94C23×D61C22⋊C411yescharacteristic
95C23×D61C22⋊C411yescharacteristic
96C23×D61C22×C412yes—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—42no—
102C22×C4×C61—42no—
103C22×C4×C61—42no—
104C22×C4×C61C22×C412yes—
105C22×C4×C61C22×C412yes—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—24no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—42no—
116C22×Dic61—42no—
117C22×Dic61C22⋊C411yescharacteristic
118C22×Dic61C22⋊C412yes—
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61C22×C412yes—
121C22×Dic61C22×C412yes—
122C22×Dic61C22×C412yes—
123C2×C4×D61—28no—
124C2×C4×D61—44no—
125C2×C4×D61—44no—
126C2×C4×D61—44no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61C22⋊C412yes—
130C2×C4×D61C22⋊C412yes—
131C2×C4×D61C22⋊C412yes—
132C2×C4×D61C22⋊C412yes—
133C2×C4×D61C2×D412yes—
134C2×C4×D61C2×D412yes—
135C2×C4×D61C2×D412yes—
136C2×C4×D61C2×D412yes—
137C2×Dic122—24no—
138C2×Dic122—24no—
139C2×Dic122—44no—
140C2×Dic122—48no—
141C2×Dic122—44no—
142C2×Dic122—48no—
143M16×C61—44no—
144C2×C8×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147#11758198—24no—
148#11758198—24no—
149#11758198—24no—
150#11758198—24no—
151#11758196—24no—
152#11758196—24no—
153#11758196—24no—
154#11758196—24no—
155#11758194—44no—
156C2×#92418—24no—
157#11758189—48no—
158C8×D61—48no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61C2×C22⋊C411yescharacteristic
163C23×C61C2×C22⋊C411yescharacteristic
164C23×C61C23×C411yescharacteristic
165C22×D61—24no—
166C22×D61—24no—
167C22×D61—22no—
168C22×D61—24no—
169C22×D61—24no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—42no—
173C22×D61—41no—
174C22×D61—42no—
175C22×D61—41no—
176C22×D61—416no—
177C22×D61#10114yes—
178C22×D61C2×C22⋊C412yes—
179C22×D61C2×C22⋊C412yes—
180C2×C4×C61—24no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—24no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61C2×C22⋊C412yes—
193C2×C4×C61C22×D412yes—
194C2×Dic61—22no—
195C2×Dic61—24no—
196C2×Dic61—24no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—24no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61#9914yes—
206C2×Dic61C2×C22⋊C412yes—
207C2×Dic61C2×C22⋊C412yes—
208C4×D61—24no—
209C4×D61—24no—
210C4×D61—24no—
211C4×D61—24no—
212C4×D61—42no—
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—42no—
216Dic122—48no—
217Dic122—48no—
218C8×C61—44no—
219#92418—44no—
220#17397—611nomaximal, Sylow
221C22×C61—28no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61C2×#9912yes—
229C22×C61C2×#10112yes—
230C22×C61C22×C22⋊C411yescharacteristic
231C22×C61#56812yes—
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—22no—
237C2×D61—44no—
238C2×D61—48no—
239C2×D61—48no—
240C2×D61—44no—
241C2×D61C2×#10114yes—
242C4×C61—22no—
243C4×C61—22no—
244C4×C61—42no—
245C4×C61—42no—
246Dic61—22no—
247Dic61—42no—
248Dic61—42no—
249Dic61C2×#9914yes—
250C22×C22⋊Q8—611no—
251C2×#407—612no—
252#1472—618no—
253#1463—612no—
254#1463—611no—
255#1463D6111yescharacteristic, complemented
256C2×C61—22no—
257C2×C61—22no—
258C2×C61—22no—
259C2×C61#287911yescharacteristic
260C2×C61C22×#9911yescharacteristic
261C2×C61C22×#10111yescharacteristic
262D61—24no—
263D61—42no—
264C23×Q8—611no—
265C24×C4—611no—
266C22×M16—1222no—
267C2×C22⋊Q8—614no—
268C2×C22⋊Q8—614no—
269C2×C22⋊Q8—1222no—
270C2×C22⋊Q8—1222no—
271C2×C22⋊Q8—1222no—
272C2×C22⋊Q8—1222no—
273C22×C4⋊C4—611no—
274C22×C4⋊C4—611no—
275C22×C4⋊C4C2×D6111yescharacteristic
276C22×C22⋊C4—611no—
277C22×C22⋊C4—611no—
278#407—612no—
279#407—614no—
280#407—1222no—
281#407—1222no—
282#407C2×D6112yes—
283C2×C22⋊C8—614no—
284C2×C22⋊C8—1222no—
285#324—614no—
286#324—618no—
287#324C2×D6114yescomplemented
288C61#1739711yescharacteristic, complemented, Sylow
289C25—611no—
290C22×Q8—614no—
291C22×Q8—1221no—
292C22×Q8—1222no—
293C22×Q8—1221no—
294C22×Q8—2444no—
295C23×C4—611no—
296C23×C4—612no—
297C23×C4—612no—
298C23×C4—612no—
299C23×C4—612no—
300C23×C4—612no—
301C23×C4—612no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—2444no—
305C23×C4C4×D6111yescharacteristic
306C23×C4C4×D6111yescharacteristic
307C23×C4C22×D6111yescharacteristic
308C2×M16—22no—
309C2×M16—1224no—
310C2×M16—1222no—
311C2×M16—2448no—
312C22×C8—1224no—
313C22⋊Q8—1224no—
314C22⋊Q8—1224no—
315C22⋊Q8—1224no—
316C22⋊Q8—1224no—
317C22⋊Q8—1224no—
318C22⋊Q8—1224no—
319C22⋊Q8—1224no—
320C22⋊Q8—1224no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—614no—
324C2×C4⋊C4—612no—
325C2×C4⋊C4—612no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4C4×D6112yes—
332C2×C4⋊C4C22×D6112yes—
333C2×C22⋊C4—614no—
334C2×C22⋊C4—614no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1221no—
341C22⋊C8—22no—
342C22⋊C8—614no—
343C22⋊C8—1224no—
344C22⋊C8—1224no—
345C22⋊C8—1222no—
346C22⋊C8C22×D6114yes—
347C24—611no—
348C24—612no—
349C24—612no—
350C24—611no—
351C24—1224no—
352C24—2442no—
353C24—2442no—
354C24C2×C4×D6111yescharacteristic
355C2×Q8—1224no—
356C2×Q8—1224no—
357C2×Q8—2444no—
358C2×Q8—2448no—
359C2×Q8—2448no—
360C2×Q8—2444no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—611no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—611no—
372C22×C4—612no—
373C22×C4—612no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—612no—
378C22×C4—612no—
379C22×C4—612no—
380C22×C4—612no—
381C22×C4—612no—
382C22×C4—1228no—
383C22×C4—1224no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—2444no—
389C22×C4—2444no—
390C22×C4—2444no—
391C22×C4—2444no—
392C22×C4—2444no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4C2×C4×D6112yes—
396C22×C4C2×C4×D6112yes—
397C22×C4C2×C4×D6111yescharacteristic
398C22×C4D4×D6112yes—
399C22×C4D4×D6112yes—
400C22×C4C23×D6111yescharacteristic
401M16—44no—
402M16—2448no—
403M16—2444no—
404C2×C8—24no—
405C2×C8—1224no—
406C2×C8—2448no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C4⋊C4—1224no—
412C4⋊C4—1224no—
413C22⋊C4—1224no—
414C22⋊C4—1224no—
415C22⋊C4—1224no—
416C22⋊C4—1224no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—612no—
421C23—614no—
422C23—612no—
423C23—1224no—
424C23—1224no—
425C23—1224no—
426C23—1222no—
427C23—1224no—
428C23—2442no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23—2442no—
433C23—2441no—
434C23—24416no—
435C23C22⋊C4×D6111yescharacteristic
436C23C22⋊C4×D6111yescharacteristic
437C23C22×C4×D6111yescharacteristic
438Q8—2448no—
439Q8—2448no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—612no—
453C2×C4—614no—
454C2×C4—612no—
455C2×C4—1224no—
456C2×C4—1224no—
457C2×C4—1224no—
458C2×C4—1224no—
459C2×C4—1224no—
460C2×C4—1224no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1224no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4C22⋊C4×D6112yes—
475C2×C4C2×D4×D6112yes—
476C8—44no—
477C8—2444no—
478C22—28no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—614no—
486C22—1222no—
487C22—1222no—
488C22—1222no—
489C22—1222no—
490C22—1222no—
491C22—2444no—
492C22—2448no—
493C22—2448no—
494C22—2444no—
495C22#99×D6112yes—
496C22#101×D6112yes—
497C22C2×C22⋊C4×D6111yescharacteristic
498C22#1684052812yes—
499C4—22no—
500C4—22no—
501C4—42no—
502C4—42no—
503C4—614no—
504C4—1222no—
505C4—2442no—
506C4—2442no—
507C2—22no—
508C2—22no—
509C2—22no—
510C2—1224no—
511C2—2442no—
512C2#2145426211yescharacteristic
513C2C2×#99×D6111yescharacteristic
514C2C2×#101×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1463C2
socleC22×C61C22×C22⋊C4
prime radicalC24×D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1463D61
hypocenterC61#17397
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #21444181C2semidirect
4. #21444149C2semidirect
5. #21444148C2semidirect
6. C61×#1463C2semidirect
26. #16839488C22semidirect
27. #16839487C22semidirect
30. C61×#324C22semidirect
255. #1463D61semidirect
287. #324C2×D61semidirect
288. C61#17397semidirect

Maximal subgroups

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

subgroupindex
C22×#145197552
#407×D612
#214441812
#214441492
#214441482
C61×#14632
#1739761