Finite Group Databasebeta

← #30609953

Subgroups — #30609953

Order 15616.

1322 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462495C212yesmaximal, complemented
2#21461073C214yesmaximal, complemented
3#21461067C212yesmaximal, complemented
4#21461066C212yesmaximal
5#21461053C214yesmaximal, complemented
6C22×#14519735C211yesmaximal, characteristic
7#16840783C2214yescomplemented
8#16840782—22no—
9#16840779C2214yescomplemented
10#16840778—22no—
11#16840765—22no—
12#16840732C2214yes—
13C4×#11758217—22no—
14#16840607C2218yescomplemented
15#16840606C2218yes—
16C23×C4×D61C411yescharacteristic
17C2×#14519735—22no—
18C2×#14519735—22no—
19C2×#14519735—22no—
20C2×#14519735—22no—
21C2×#14519735C412yes—
22C2×#14519735C412yes—
23C2×#14519735C2212yes—
24C2×#14519735C2212yes—
25C22×C4⋊Dic61C2211yescharacteristic
26C22×#11758198C411yescharacteristic
27C22×#11758198C2211yescharacteristic
28C22×C4×Dic61C411yescharacteristic
29C22×C4×Dic61C2211yescharacteristic
30C22×C4⋊C4×C61C411yescharacteristic
31C2×#11758217—24no—
32#14519844—24no—
33C4⋊#92420—24no—
34C22⋊#92420—24no—
35C22⋊#92420C2318yes—
36#14519832—24no—
37C4×#92420—24no—
38C24×D61C2×C411yescharacteristic
39C23×Dic61C2×C411yescharacteristic
40C23×Dic61C2×C411yescharacteristic
41C23×Dic61C2311yescharacteristic
42C22×C4×D61—21no—
43C22×C4×D61—21no—
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61—28no—
47C22×C4×D61C2×C412yes—
48C22×C4×D61C2×C412yes—
49#14519735—24no—
50#14519735—24no—
51#14519735—24no—
52#14519735—24no—
53#14519735—24no—
54#14519735—24no—
55#14519735—24no—
56#14519735—24no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61C2×C412yes—
60C2×C4⋊Dic61C2312yes—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198C2×C412yes—
66C2×#11758198C2×C412yes—
67C2×#11758198C2×C412yes—
68C2×#11758198C2312yes—
69C2×C4×Dic61—21no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61C2×C412yes—
74C2×C4×Dic61C2×C412yes—
75C2×C4×Dic61C2×C411yescharacteristic
76C2×C4×Dic61C2×C411yescharacteristic
77C2×C4×Dic61D412yes—
78C2×C4×Dic61D412yes—
79C2×C4×Dic61D412yes—
80C2×C4×Dic61D412yes—
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C23×C4×C61C2×C411yescharacteristic
84C23×C4×C61C2×C411yescharacteristic
85C23×C4×C61C2×C411yescharacteristic
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61C2×C412yes—
89C2×C4⋊C4×C61C2×C412yes—
90C24×C61C22×C411yescharacteristic
91C23×D61—21no—
92C23×D61—21no—
93C23×D61—24no—
94C23×D61—42no—
95C23×D61—42no—
96C23×D61C22×C412yes—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—24no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—42no—
103C22×C4×C61—42no—
104C22×C4×C61C22⋊C412yes—
105C22×C4×C61C22⋊C412yes—
106C22×C4×C61C22×C412yes—
107C22×C4×C61C22×C412yes—
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C411yescharacteristic
110C22×Dic61—24no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—42no—
114C22×Dic61—42no—
115C22×Dic61C22⋊C411yescharacteristic
116C22×Dic61C22⋊C411yescharacteristic
117C22×Dic61C22⋊C411yescharacteristic
118C22×Dic61C22⋊C411yescharacteristic
119C22×Dic61C22×C411yescharacteristic
120C22×Dic61C22×C412yes—
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C2×D412yes—
124C22×Dic61C2×D412yes—
125C22×Dic61C2411yescharacteristic
126C2×C4×D61—28no—
127C2×C4×D61—28no—
128C2×C4×D61—28no—
129C2×C4×D61—28no—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—24no—
137C2×C4×D61M1618yescomplemented
138#11758217—48no—
139C2×#92420—28no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—24no—
142#11758198—24no—
143#11758198—24no—
144#11758198—24no—
145#11758198—24no—
146C4⋊Dic61—24no—
147C4⋊Dic61—24no—
148C4×Dic61—22no—
149C4×Dic61—22no—
150C4×Dic61—22no—
151C4×Dic61—22no—
152C4×Dic61C22⋊C412yes—
153C4×Dic61C22⋊C412yes—
154C4×Dic61C22⋊C412yes—
155C4×Dic61C22⋊C412yes—
156C4×Dic61C2×D412yes—
157C4×Dic61C2×D412yes—
158C4×Dic61C2×D412yes—
159C4×Dic61C2×D412yes—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61C2×C22⋊C411yescharacteristic
164C23×C61C2×C22⋊C411yescharacteristic
165C23×C61C23×C411yescharacteristic
166C22×D61—24no—
167C22×D61—24no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—416no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61C2×M1614yes—
178C2×C4×C61—24no—
179C2×C4×C61—24no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—24no—
183C2×C4×C61—22no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61C2×C22⋊C412yes—
191C2×C4×C61C2×C22⋊C412yes—
192C2×C4×C61C2×M1614yes—
193C2×Dic61—22no—
194C2×Dic61—24no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
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×Dic61C2×C22⋊C411yescharacteristic
207C2×Dic61C2×C22⋊C411yescharacteristic
208C2×Dic61C2×C22⋊C412yes—
209C2×Dic61C2×C22⋊C411yescharacteristic
210C2×Dic61C2×C22⋊C411yescharacteristic
211C2×Dic61C2×M1614yes—
212C2×Dic61C22×D412yes—
213C4×D61—28no—
214C4×D61—28no—
215C4×D61—28no—
216C4×D61—28no—
217#92420—48no—
218#17357—611nomaximal, Sylow
219C22×C61—28no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61#41012yes—
227C22×C61#41312yes—
228C22×C61C22×C22⋊C411yescharacteristic
229C22×C61C22×M1612yes—
230C2×D61—24no—
231C2×D61—44no—
232C2×D61—48no—
233C2×D61—48no—
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—44no—
237C4×C61—24no—
238C4×C61—22no—
239C4×C61—22no—
240C4×C61—42no—
241Dic61—21no—
242Dic61—21no—
243Dic61—22no—
244Dic61—22no—
245Dic61—42no—
246Dic61#40712yes—
247Dic61#40712yes—
248C22×#117—611no—
249#2920—612no—
250#1464—614no—
251#1463—612no—
252#1462—612no—
253#1458—614no—
254C2×C61—22no—
255C2×C61—22no—
256C2×C61—22no—
257C2×C61C2×#40711yescharacteristic
258C2×C61C2×#41011yescharacteristic
259C2×C61C2×#41311yescharacteristic
260D61—44no—
261C24×C4—611no—
262C2×#117—612no—
263C2×#117—612no—
264C2×#117—612no—
265C2×#117—612no—
266C2×#117—1222no—
267C2×#117—1222no—
268C2×#117—1222no—
269C2×#117—1222no—
270C22×C4⋊C4—611no—
271C22×C4⋊C4#692311yescharacteristic
272C22×C22⋊C4—611no—
273C22×C22⋊C4—611no—
274C22×C42—611no—
275C22×C42—611no—
276#432—614no—
277#432—1222no—
278#431—614no—
279#431—1222no—
280#423—1222no—
281#407—614no—
282C4×M16—1222no—
283#324—618no—
284#323—618no—
285C61#1735711yescharacteristic, complemented, Sylow
286C25—611no—
287C23×C4—611no—
288C23×C4—611no—
289C23×C4—612no—
290C23×C4—611no—
291C23×C4—612no—
292C23×C4—1221no—
293C23×C4—1221no—
294C23×C4—1221no—
295C23×C4—1221no—
296C23×C4—1228no—
297C23×C4C2×#692311yescharacteristic
298C23×C4C2×#692311yescharacteristic
299C23×C4C2×#692311yescharacteristic
300C2×M16—1224no—
301#117—1224no—
302#117—1224no—
303#117—1224no—
304#117—1224no—
305#117—1224no—
306#117—1224no—
307#117—1224no—
308#117—1224no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—612no—
312C2×C4⋊C4—612no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4C2×#692312yes—
316C2×C4⋊C4C2×#692312yes—
317C2×C22⋊C4—612no—
318C2×C22⋊C4—612no—
319C2×C22⋊C4—612no—
320C2×C22⋊C4—612no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C42—612no—
326C2×C42—612no—
327C2×C42—612no—
328C2×C42—612no—
329C2×C42—611no—
330C2×C42—611no—
331C2×C42—611no—
332C2×C42—611no—
333C2×C42—612no—
334C2×C42—612no—
335C2×C42—1221no—
336C2×C42—1221no—
337C2×C42—1221no—
338C2×C42—1221no—
339C4⋊C8—1224no—
340C4⋊C8—1224no—
341C22⋊C8—618no—
342C22⋊C8—1224no—
343#97—1224no—
344C4×C8—1224no—
345C24—612no—
346C24—1224no—
347C24—1221no—
348C24—1221no—
349C24—2442no—
350C24—2442no—
351C24C22×#692311yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—611no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—611no—
364C22×C4—612no—
365C22×C4—611no—
366C22×C4—612no—
367C22×C4—611no—
368C22×C4—611no—
369C22×C4—618no—
370C22×C4—612no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1224no—
376C22×C4—1224no—
377C22×C4—1228no—
378C22×C4—1228no—
379C22×C4—1228no—
380C22×C4—1228no—
381C22×C4—1224no—
382C22×C4—1224no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4C22⋊#692312yes—
388C22×C4C22⋊#692312yes—
389C22×C4C22×#692312yes—
390C22×C4C22×#692312yes—
391C22×C4C22×#692311yescharacteristic
392C22×C4C22×#692311yescharacteristic
393M16—2448no—
394C2×C8—1228no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—1224no—
398C4⋊C4—1224no—
399C22⋊C4—1224no—
400C22⋊C4—1224no—
401C22⋊C4—1224no—
402C22⋊C4—1224no—
403C42—612no—
404C42—612no—
405C42—612no—
406C42—612no—
407C42—612no—
408C42—612no—
409C42—612no—
410C42—612no—
411C42—1222no—
412C42—1222no—
413C42—1222no—
414C42—1222no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—614no—
419C23—1224no—
420C23—1224no—
421C23—1222no—
422C23—1222no—
423C23—2442no—
424C23—2442no—
425C23—24416no—
426C23—2442no—
427C23—2442no—
428C23—2442no—
429C23—2442no—
430C23C2×C22⋊#692311yescharacteristic
431C23C2×C22⋊#692311yescharacteristic
432C23C23×#692311yescharacteristic
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—614no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—612no—
451C2×C4—612no—
452C2×C4—1228no—
453C2×C4—1228no—
454C2×C4—1228no—
455C2×C4—1228no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1224no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—2441no—
464C2×C4—2442no—
465C2×C4—2441no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4C2×C22⋊#692312yes—
470C2×C4C2×C22⋊#692312yes—
471C2×C4D61⋊M1614yes—
472C8—2448no—
473C22—28no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—1224no—
481C22—2444no—
482C22—2448no—
483C22—2448no—
484C22—2444no—
485C22—2442no—
486C22—2442no—
487C22#1684074312yes—
488C22#1684074912yes—
489C22C22×C22⋊#692311yescharacteristic
490C22C2×D61⋊M1612yes—
491C4—24no—
492C4—22no—
493C4—22no—
494C4—42no—
495C4—612no—
496C4—612no—
497C4—1221no—
498C4—1221no—
499C4—1222no—
500C4—1222no—
501C4—2442no—
502C2—22no—
503C2—22no—
504C2—22no—
505C2—2444no—
506C2D61⋊#40711yescharacteristic
507C2C2×#1684074311yescharacteristic
508C2C2×#1684074911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×D61C2×C4
cosocleC22×Dic61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61#17357
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #21462495C2semidirect
2. #21461073C2semidirect
3. #21461067C2semidirect
5. #21461053C2semidirect
7. #16840783C22semidirect
9. #16840779C22semidirect
14. #16840607C22semidirect
137. C2×C4×D61M16semidirect
285. C61#17357semidirect

Maximal subgroups

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

subgroupindex
#214624952
#214610732
#214610672
#214610662
#214610532
C22×#145197352
#1735761