Finite Group Databasebeta

← #30609948

Subgroups — #30609948

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#21461070C214yesmaximal, complemented
3#21461067C212yesmaximal, complemented
4#21461064C212yesmaximal
5#21461050C214yesmaximal, complemented
6C22×#14519736C211yesmaximal, characteristic
7#16840783—22no—
8#16840782C2214yescomplemented
9#16840779—22no—
10#16840778C2214yescomplemented
11#16840765—22no—
12#16840732C2214yes—
13C4×#11758217—22no—
14#16840607C2218yescomplemented
15#16840605C2218yes—
16C2×#14519736—22no—
17C2×#14519736—22no—
18C2×#14519736—22no—
19C2×#14519736—22no—
20C2×#14519736C412yes—
21C2×#14519736C412yes—
22C2×#14519736C2212yes—
23C2×#14519736C2212yes—
24C22×C4⋊Dic61C411yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic
26C22×C22⋊Dic61C411yescharacteristic
27C22×C22⋊Dic61C2211yescharacteristic
28C22×C4×Dic61C411yescharacteristic
29C22×C4×Dic61C2211yescharacteristic
30C24×C4×C61C411yescharacteristic
31C2×#11758217—24no—
32#14519844—24no—
33C4⋊#92420—24no—
34C22⋊#92420—24no—
35C22⋊#92420C2318yes—
36#14519832—24no—
37C4×#92420—24no—
38C23×Dic61C2×C411yescharacteristic
39C23×Dic61C2×C411yescharacteristic
40C23×Dic61C2×C411yescharacteristic
41C23×Dic61C2311yescharacteristic
42#14519736—24no—
43#14519736—24no—
44#14519736—24no—
45#14519736—24no—
46#14519736—24no—
47#14519736—24no—
48#14519736—24no—
49#14519736—24no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61C2×C412yes—
55C2×C4⋊Dic61C2×C412yes—
56C2×C4⋊Dic61C2×C412yes—
57C2×C4⋊Dic61C2312yes—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61C2×C412yes—
63C2×C22⋊Dic61C2×C412yes—
64C2×C22⋊Dic61C2×C412yes—
65C2×C22⋊Dic61C2312yes—
66C2×C4×Dic61—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2×C412yes—
71C2×C4×Dic61C2×C412yes—
72C2×C4×Dic61C2×C411yescharacteristic
73C2×C4×Dic61C2×C411yescharacteristic
74C2×C4×Dic61D412yes—
75C2×C4×Dic61D412yes—
76C2×C4×Dic61D412yes—
77C2×C4×Dic61D412yes—
78C2×C4×Dic61C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80C25×C61C2×C411yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61—21no—
83C23×C4×C61—21no—
84C23×C4×C61—21no—
85C23×C4×C61—28no—
86C23×C4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C412yes—
88C23×C4×C61C2×C412yes—
89C23×C4×C61C2×C411yescharacteristic
90C24×C61—21no—
91C24×C61—21no—
92C24×C61—24no—
93C24×C61—42no—
94C24×C61—42no—
95C24×C61C22×C411yescharacteristic
96C24×C61C22×C412yes—
97C22×C4×C61—28no—
98C22×C4×C61—28no—
99C22×C4×C61—28no—
100C22×C4×C61—28no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—24no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—24no—
109C22×C4×C61—24no—
110C22×C4×C61—24no—
111C22×C4×C61—24no—
112C22×C4×C61C22⋊C411yescharacteristic
113C22×C4×C61C22⋊C411yescharacteristic
114C22×C4×C61C22⋊C411yescharacteristic
115C22×C4×C61C22⋊C411yescharacteristic
116C22×C4×C61M1618yescomplemented
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—42no—
125C22×Dic61—42no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61C22⋊C412yes—
129C22×Dic61C22⋊C412yes—
130C22×Dic61C22×C412yes—
131C22×Dic61C22×C412yes—
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C2×D412yes—
136C22×Dic61C2×D412yes—
137C22×Dic61C2411yescharacteristic
138#11758217—48no—
139C2×#92420—28no—
140C22⋊Dic61—24no—
141C22⋊Dic61—24no—
142C22⋊Dic61—24no—
143C22⋊Dic61—24no—
144C4⋊Dic61—24no—
145C4⋊Dic61—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×C61—24no—
164C23×C61—24no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—416no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61C2×C22⋊C411yescharacteristic
175C23×C61C2×C22⋊C411yescharacteristic
176C23×C61C2×M1614yes—
177C23×C61C23×C411yescharacteristic
178C2×C4×C61—28no—
179C2×C4×C61—28no—
180C2×C4×C61—24no—
181C2×C4×C61—24no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—28no—
190C2×C4×C61—28no—
191C2×C4×C61C2×C22⋊C411yescharacteristic
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61C2×M1614yes—
196C2×C4×C61C2×M1614yes—
197C2×Dic61—24no—
198C2×Dic61—24no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61C2×C22⋊C412yes—
214C2×Dic61C2×C22⋊C412yes—
215C2×Dic61C2×C22⋊C412yes—
216C2×Dic61C22×D412yes—
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—24no—
227C22×C61—44no—
228C22×C61—44no—
229C22×C61—48no—
230C22×C61—48no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61#41012yes—
234C22×C61#41312yes—
235C22×C61C22×C22⋊C411yescharacteristic
236C22×C61C22×M1612yes—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—24no—
240C4×C61#40712yes—
241C4×C61#40712yes—
242Dic61—22no—
243Dic61—22no—
244Dic61—22no—
245Dic61—22no—
246Dic61—42no—
247Dic61—42no—
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×C61—44no—
258C2×C61C2×#40711yescharacteristic
259C2×C61C2×#41011yescharacteristic
260C2×C61C2×#41311yescharacteristic
261C24×C4#692311yescharacteristic
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—611no—
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
286C25C2×#692311yescharacteristic
287C23×C4—28no—
288C23×C4—21no—
289C23×C4—21no—
290C23×C4—21no—
291C23×C4—21no—
292C23×C4—611no—
293C23×C4—611no—
294C23×C4—611no—
295C23×C4—611no—
296C23×C4C2×#692311yescharacteristic
297C23×C4C2×#692312yes—
298C23×C4C2×#692312yes—
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—612no—
310C2×C4⋊C4—612no—
311C2×C4⋊C4—612no—
312C2×C4⋊C4—612no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1222no—
316C2×C4⋊C4—1222no—
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—21no—
346C24—21no—
347C24—24no—
348C24—42no—
349C24—42no—
350C24C22×#692311yescharacteristic
351C24C22×#692312yes—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—612no—
368C22×C4—612no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—611no—
374C22×C4—611no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—2442no—
382C22×C4—2442no—
383C22×C4—2442no—
384C22×C4—2442no—
385C22×C4#1175821718yescomplemented
386C22×C4C22⋊#692311yescharacteristic
387C22×C4C22⋊#692311yescharacteristic
388C22×C4C22⋊#692311yescharacteristic
389C22×C4C22⋊#692311yescharacteristic
390C22×C4C22×#692311yescharacteristic
391C22×C4C22×#692312yes—
392C22×C4C22×#692311yescharacteristic
393M16—2448no—
394C2×C8—1228no—
395C4⋊C4—1224no—
396C4⋊C4—1224no—
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—24no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—416no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C2×C22⋊#692311yescharacteristic
430C23C2×C22⋊#692311yescharacteristic
431C23C2×#1175821714yes—
432C23C23×#692311yescharacteristic
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—612no—
447C2×C4—612no—
448C2×C4—612no—
449C2×C4—612no—
450C2×C4—1224no—
451C2×C4—1224no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—2441no—
455C2×C4—2442no—
456C2×C4—2441no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4C2×C22⋊#692311yescharacteristic
467C2×C4C2×C22⋊#692311yescharacteristic
468C2×C4C2×C22⋊#692311yescharacteristic
469C2×C4C2×C22⋊#692311yescharacteristic
470C2×C4C2×#1175821714yes—
471C2×C4C2×#1175821714yes—
472C8—2448no—
473C22—28no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—24no—
481C22—44no—
482C22—48no—
483C22—48no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22#1684074112yes—
488C22#1684074812yes—
489C22C22×C22⋊#692311yescharacteristic
490C22C22×#1175821712yes—
491C4—21no—
492C4—21no—
493C4—24no—
494C4—1222no—
495C4—1222no—
496C4—1222no—
497C4—1222no—
498C4—2442no—
499C4—2442no—
500C4#1684073212yes—
501C4#1684073212yes—
502C2—22no—
503C2—22no—
504C2—22no—
505C2—44no—
506C2C2×#1684073211yescharacteristic
507C2C2×#1684074111yescharacteristic
508C2C2×#1684074811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC24×C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC22×Dic61C24
hypercenterC24×C4#6923
hypocenterC61#17357
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #21462495C2semidirect
2. #21461070C2semidirect
3. #21461067C2semidirect
5. #21461050C2semidirect
8. #16840782C22semidirect
10. #16840778C22semidirect
14. #16840607C22semidirect
116. C22×C4×C61M16semidirect
285. C61#17357semidirect
385. C22×C4#11758217semidirect

Maximal subgroups

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

subgroupindex
#214624952
#214610702
#214610672
#214610642
#214610502
C22×#145197362
#1735761