Finite Group Databasebeta

← #30610149

Subgroups — #30610149

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#407C212yesmaximal, complemented
2#21461085C218yesmaximal, complemented
3#21461067C212yesmaximal, complemented
4#21461064C212yesmaximal, complemented
5C22×#14519759C211yesmaximal, characteristic
6C2×D61⋊M16—22no—
7#16840734—22no—
8#16840733—22no—
9#16840733C2214yescomplemented
10#16840732C2214yes—
11D61⋊(C22⋊C8)—22no—
12D61⋊(C22⋊C8)C2214yescomplemented
13#16840607C2218yescomplemented
14#16840605C2218yescomplemented
15C23×C22⋊D61C411yescharacteristic
16C23×C4×D61C2211yescharacteristic
17C2×#14519759—22no—
18C2×#14519759—22no—
19C2×#14519759—22no—
20C2×#14519759—22no—
21C2×#14519759C414yes—
22C2×#14519759C2214yes—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×#11758196C411yescharacteristic
25C22×C22⋊Dic61C2211yescharacteristic
26C22×#11758198C411yescharacteristic
27C22×C22⋊C4×C61C411yescharacteristic
28C2×#11758213—24no—
29D61⋊M16—48no—
30C2×#11758217—24no—
31C2×D61⋊C8—24no—
32#14519834—24no—
33#14519834—24no—
34C22⋊#92420—24no—
35C22⋊#92420C2318yes—
36C24×D61C2×C411yescharacteristic
37C22×C22⋊D61—22no—
38C22×C22⋊D61—21no—
39C22×C22⋊D61—21no—
40C22×C22⋊D61—44no—
41C22×C22⋊D61C2×C414yes—
42C23×Dic61C2×C411yescharacteristic
43C23×Dic61C2×C411yescharacteristic
44C23×Dic61C2311yescharacteristic
45C22×C4×D61—44no—
46C22×C4×D61C2×C412yes—
47C22×C4×D61D412yes—
48C22×C4×D61D412yes—
49C22×C4×D61D412yes—
50C22×C4×D61D412yes—
51C22×C4×D61C2312yes—
52#14519759—24no—
53#14519759—24no—
54#14519759—24no—
55#14519759—24no—
56#14519759—24no—
57#14519759—24no—
58#14519759—24no—
59#14519759—24no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61C2×C412yes—
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—22no—
65C2×#11758196—21no—
66C2×#11758196—21no—
67C2×#11758196C2×C414yes—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61C2×C412yes—
71C2×C22⋊Dic61C2312yes—
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198C2×C414yes—
76C25×C61C2×C411yescharacteristic
77C23×C4×C61C2×C411yescharacteristic
78C23×C4×C61C2×C411yescharacteristic
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—21no—
81C2×C22⋊C4×C61—21no—
82C2×C22⋊C4×C61C2×C414yes—
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—42no—
86C24×C61—42no—
87C24×C61—42no—
88C24×C61C22×C411yescharacteristic
89C24×C61C22×C412yes—
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—42no—
100C22×C4×C61—42no—
101C22×C4×C61C22⋊C412yes—
102C22×C4×C61C22⋊C411yescharacteristic
103C22×C4×C61C22⋊C411yescharacteristic
104C22×C4×C61C22×C412yes—
105C22×C4×C61C22×C412yes—
106C2×C22⋊D61—24no—
107C2×C22⋊D61—24no—
108C2×C22⋊D61—48no—
109C2×C22⋊D61—44no—
110C2×C22⋊D61—44no—
111C2×C22⋊D61—48no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—42no—
117C22×Dic61—42no—
118C22×Dic61—42no—
119C22×Dic61C22×C412yes—
120C22×Dic61C22×C412yes—
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C2×D412yes—
123C22×Dic61C2×D412yes—
124C22×Dic61C2411yescharacteristic
125C2×C4×D61—28no—
126C2×C4×D61—44no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61—44no—
130C2×C4×D61—44no—
131C2×C4×D61C22⋊C412yes—
132C2×C4×D61C22⋊C412yes—
133C2×C4×D61C22⋊C412yes—
134C2×C4×D61C22⋊C412yes—
135C2×C4×D61C2×D412yes—
136C2×C4×D61C2×D412yes—
137C2×C4×D61C2×D412yes—
138C2×C4×D61C2×D412yes—
139#11758217—48no—
140C2×#92420—28no—
141#11758213—48no—
142D61⋊C8—48no—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—24no—
145C22⋊Dic61—24no—
146C22⋊Dic61—24no—
147#11758198—24no—
148#11758198—24no—
149C4⋊Dic61—24no—
150C4⋊Dic61—24no—
151#11758196—24no—
152#11758196—24no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61—22no—
157C23×C61—24no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—42no—
166C23×C61—88no—
167C23×C61C2×C22⋊C411yescharacteristic
168C23×C61C2×C22⋊C411yescharacteristic
169C23×C61C23×C411yescharacteristic
170C22×D61—24no—
171C22×D61—22no—
172C22×D61—24no—
173C22×D61—24no—
174C22×D61—24no—
175C22×D61—41no—
176C22×D61—42no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—42no—
181C22×D61—416no—
182C22×D61#9914yes—
183C22×D61C2×C22⋊C412yes—
184C22×D61C2×C22⋊C412yes—
185C2×C4×C61—24no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61C2×C22⋊C412yes—
194C2×C4×C61C2×C22⋊C412yes—
195C22⋊D61—48no—
196C22⋊D61—48no—
197C2×Dic61—24no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61C2×C22⋊C412yes—
210C2×Dic61C22×D412yes—
211C4×D61—24no—
212C4×D61—24no—
213C4×D61—24no—
214C4×D61—24no—
215C4×D61—42no—
216C4×D61—42no—
217C4×D61—42no—
218C4×D61—42no—
219#92420—48no—
220#17392—611nomaximal, Sylow
221C22×C61—28no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—84no—
232C22×C61—84no—
233C22×C61—84no—
234C22×C61C2×#9912yes—
235C22×C61#41312yes—
236C22×C61C22×C22⋊C411yescharacteristic
237C22×C61#56812yes—
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—22no—
243C2×D61—44no—
244C2×D61—48no—
245C2×D61—48no—
246C2×D61—44no—
247C2×D61C2×#9914yes—
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—42no—
251Dic61—22no—
252Dic61—22no—
253Dic61—42no—
254Dic61—42no—
255C22×#123—611no—
256C2×#407—612no—
257#1468—618no—
258#1463—612no—
259#1462—612no—
260C2×C61—22no—
261C2×C61—22no—
262C2×C61—22no—
263C2×C61—82no—
264C2×C61#287911yescharacteristic
265C2×C61C22×#9911yescharacteristic
266C2×C61C2×#41311yescharacteristic
267D61—24no—
268D61—42no—
269C23×D4—611no—
270C24×C4—611no—
271C22×M16—1222no—
272C2×#123—614no—
273C2×#123—614no—
274C2×#123—1222no—
275C2×#123—1222no—
276C2×#123—1222no—
277C2×#123—1222no—
278C22×C4⋊C4—611no—
279C22×C4⋊C4—611no—
280C22×C22⋊C4—611no—
281C22×C22⋊C4—611no—
282C22×C22⋊C4#692311yescharacteristic
283#407—614no—
284#407—614no—
285#407—1222no—
286#407—1222no—
287C2×C22⋊C8—614no—
288C2×C22⋊C8—1222no—
289#324—618no—
290#323—618no—
291C61#1739211yescharacteristic, complemented, Sylow
292C25—611no—
293C25C2×#692311yescharacteristic
294C22×D4—614no—
295C22×D4—1221no—
296C22×D4—1222no—
297C22×D4—1221no—
298C22×D4—2444no—
299C23×C4—611no—
300C23×C4—612no—
301C23×C4—612no—
302C23×C4—612no—
303C23×C4—612no—
304C23×C4—612no—
305C23×C4—612no—
306C23×C4—611no—
307C23×C4—611no—
308C23×C4—2444no—
309C23×C4C2×#692311yescharacteristic
310C23×C4C2×#692311yescharacteristic
311C2×M16—1224no—
312C2×M16—1224no—
313C2×M16—2448no—
314C22×C8—1224no—
315#123—1224no—
316#123—1224no—
317#123—1224no—
318#123—1224no—
319#123—1224no—
320#123—1224no—
321#123—1224no—
322#123—1224no—
323C2×C4⋊C4—612no—
324C2×C4⋊C4—614no—
325C2×C4⋊C4—612no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—612no—
335C2×C22⋊C4—612no—
336C2×C22⋊C4—614no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4C2×#692314yes—
343C22⋊C8—618no—
344C22⋊C8—1224no—
345C22⋊C8—1224no—
346C22⋊C8—1224no—
347C24—21no—
348C24—21no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—611no—
353C24—612no—
354C24—612no—
355C24—611no—
356C24—1224no—
357C24—2442no—
358C24—2442no—
359C24C22×#692311yescharacteristic
360C24C22×#692312yes—
361C2×D4—1224no—
362C2×D4—1224no—
363C2×D4—2444no—
364C2×D4—2448no—
365C2×D4—2444no—
366C2×D4—2448no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—612no—
372C22×C4—612no—
373C22×C4—611no—
374C22×C4—611no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—612no—
378C22×C4—612no—
379C22×C4—612no—
380C22×C4—612no—
381C22×C4—612no—
382C22×C4—612no—
383C22×C4—612no—
384C22×C4—612no—
385C22×C4—1228no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—2444no—
391C22×C4—2444no—
392C22×C4—2444no—
393C22×C4—2444no—
394C22×C4—2444no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4C22⋊#692312yes—
399C22×C4C22⋊#692311yescharacteristic
400C22×C4C22⋊#692311yescharacteristic
401C22×C4C22×#692312yes—
402C22×C4C22×#692312yes—
403M16—2448no—
404M16—2448no—
405C2×C8—1228no—
406C2×C8—2448no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—1224no—
414C22⋊C4—1224no—
415C22⋊C4—1224no—
416C22⋊C4—1224no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—88no—
431C23—612no—
432C23—614no—
433C23—612no—
434C23—1224no—
435C23—1224no—
436C23—1222no—
437C23—1224no—
438C23—1224no—
439C23—2441no—
440C23—2442no—
441C23—2441no—
442C23—2441no—
443C23—2441no—
444C23—2442no—
445C23—24416no—
446C23C2×C22⋊#692311yescharacteristic
447C23C2×C22⋊#692311yescharacteristic
448C23C23×#692311yescharacteristic
449D4—2448no—
450D4—2448no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—612no—
460C2×C4—612no—
461C2×C4—1224no—
462C2×C4—1224no—
463C2×C4—1224no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4C2×C22⋊#692312yes—
482C2×C4C2×C22⋊#692312yes—
483C8—2448no—
484C22—28no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22—614no—
498C22—1222no—
499C22—1222no—
500C22—1222no—
501C22—1222no—
502C22—1222no—
503C22—2444no—
504C22—2448no—
505C22—2448no—
506C22—2444no—
507C22D61⋊#9912yes—
508C22#1684074812yes—
509C22C22×C22⋊#692311yescharacteristic
510C22#1684084012yes—
511C4—22no—
512C4—22no—
513C4—42no—
514C4—1222no—
515C4—1222no—
516C4—2442no—
517C4—2442no—
518C2—22no—
519C2—22no—
520C2—22no—
521C2—82no—
522C2—1224no—
523C2—2442no—
524C2#2146240511yescharacteristic
525C2C2×D61⋊#9911yescharacteristic
526C2C2×#1684074811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C22⋊C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC22×Dic61C24
hypercenterC22×C22⋊C4#6923
hypocenterC61#17392
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. D61⋊#407C2semidirect
2. #21461085C2semidirect
3. #21461067C2semidirect
4. #21461064C2semidirect
9. #16840733C22semidirect
12. D61⋊(C22⋊C8)C22semidirect
13. #16840607C22semidirect
14. #16840605C22semidirect
291. C61#17392semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4072
#214610852
#214610672
#214610642
C22×#145197592
#1739261