Finite Group Databasebeta

← #29689070

Subgroups — #29689070

Order 15616.

1538 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519757C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic
3#380×D61C211yesmaximal, characteristic
4#21446251C214yesmaximal, complemented
5#21446125C214yesmaximal, complemented
6#21445905C211yesmaximal, characteristic, complemented
7#21445755C211yesmaximal, characteristic, complemented
8#21445753C211yesmaximal, characteristic, complemented
9C61×#1823C211yesmaximal, characteristic, complemented
10C23×Dic122C2211yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C22×#11758189C2212yes—
13C2×#14519757—21no—
14C2×#14519757—21no—
15C2×#14519757—24no—
16C2×#14519757C418yescomplemented
17C2×#14519757C2212yes—
18C2×#14519757C2212yes—
19C22×C4⋊Dic61—21no—
20C2×C4⋊C4×D61C2212yes—
21C22×#11758198—21no—
22C2×C22⋊C4×D61—21no—
23C2×#14519702C2214yescomplemented
24C2×#14519700C2214yescomplemented
25#16839724—24no—
26C2×C22⋊#92418C2211yescharacteristic
27C2×#14519689C2212yes—
28#16839622C2211yescharacteristic
29#16839621—21no—
30#16839620—21no—
31#16839619C2212yes—
32#95×D61—21no—
33#16839524C2214yescomplemented
34C22×C4⋊C4×C61C2211yescharacteristic
35C2×C22⋊C8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C61×#340C2214yescomplemented
38C24×D61D411yescharacteristic
39C22×Dic122—21no—
40C22×Dic122—22no—
41C22×Dic122—44no—
42C22×Dic122C2×C414yescomplemented
43C22×Dic122C2312yes—
44C23×Dic61—21no—
45C23×Dic61D411yescharacteristic
46C22×C4×D61—22no—
47C22×C4×D61—44no—
48C22×C4×D61C2×C414yes—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D414yes—
52C22×C4×D61C2311yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758189—24no—
55C2×#11758189—44no—
56C22×#92418C2312yes—
57#14519757—28no—
58#14519757—28no—
59#14519757—24no—
60#14519757—24no—
61#14519757C2×C418yes—
62#14519757C2×C418yescomplemented
63C2×C4⋊Dic61—28no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61C2312yes—
67C2×#11758196—22no—
68C4⋊C4×D61—44no—
69C2×C22⋊Dic61—21no—
70C2×#11758198—28no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C22⋊C4×D61—44no—
75#14519702—44no—
76#14519700—44no—
77C22⋊#92418—22no—
78#14519689—24no—
79#14519684—21no—
80#14519684—22no—
81C23×C4×C61—21no—
82C23×C4×C61C2311yescharacteristic
83C22×C8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61C2×C414yescomplemented
87C2×C4⋊C4×C61C2312yes—
88C2×C4⋊C4×C61C2312yes—
89C2×C22⋊C4×C61—21no—
90C22⋊C8×C61—22no—
91#95×C61—21no—
92C24×C61C2×D411yescharacteristic
93C23×D61—22no—
94C23×D61—22no—
95C23×D61—24no—
96C23×D61—42no—
97C23×D61C22⋊C414yes—
98C23×D61C2×D411yescharacteristic
99C23×D61C2×D411yescharacteristic
100C22×C4×C61—24no—
101C22×C4×C61—24no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—42no—
105C22×C4×C61—42no—
106C22×C4×C61C22×C412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—24no—
111C22×Dic61—22no—
112C22×Dic61—22no—
113C22×Dic61—42no—
114C22×Dic61—44no—
115C22×Dic61C22⋊C414yes—
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C2×C4×D61—24no—
119C2×C4×D61—24no—
120C2×C4×D61—48no—
121C2×C4×D61—48no—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×C4×D61—44no—
125C2×C4×D61—44no—
126C2×C4×D61—44no—
127C2×C4×D61C22⋊C414yes—
128C2×C4×D61C22⋊C414yes—
129C2×C4×D61C22×C412yes—
130C2×C4×D61C22×C412yes—
131C2×C4×D61C2×D412yes—
132C2×C4×D61C2×D412yes—
133C2×Dic122—24no—
134C2×Dic122—24no—
135C2×Dic122—24no—
136C2×Dic122—22no—
137C2×Dic122—48no—
138C2×Dic122—44no—
139C2×Dic122—48no—
140C2×Dic122C22×C414yes—
141C2×C8×C61—22no—
142C2×C8×C61—24no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61C22×C414yes—
147C22⋊C4×C61—42no—
148C22⋊Dic61—42no—
149#11758198—28no—
150#11758198—28no—
151#11758198—44no—
152C4⋊Dic61—28no—
153C4⋊Dic61—28no—
154C4⋊Dic61—24no—
155#11758196—44no—
156C2×#92418—22no—
157C2×#92418—24no—
158#11758189—48no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61C2×C22⋊C412yes—
162C23×C61AGL(1,ℤ/8)12yes—
163C23×C61#13712yes—
164C23×C61C22×D411yescharacteristic
165C22×D61—22no—
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—22no—
169C22×D61—21no—
170C22×D61—22no—
171C22×D61—21no—
172C22×D61—28no—
173C22×D61—28no—
174C22×D61—44no—
175C22×D61—48no—
176C22×D61—48no—
177C22×D61C2×C22⋊C412yes—
178C22×D61C2×C22⋊C412yes—
179C22×D61#13714yes—
180C2×C4×C61—22no—
181C2×C4×C61—24no—
182C2×C4×C61—24no—
183C2×C4×C61—24no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—48no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61C2×C22⋊C412yes—
191C2×C4×C61C23×C411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61H(5,𝔽2)14yes—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—24no—
197C2×Dic61—44no—
198C2×Dic61—48no—
199C2×Dic61—44no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61C2×C22⋊C412yes—
204C2×Dic61C2×C22⋊C412yes—
205C4×D61—24no—
206C4×D61—24no—
207C4×D61—24no—
208C4×D61—24no—
209C4×D61—44no—
210C4×D61—44no—
211C4×D61—88no—
212Dic122—24no—
213Dic122—24no—
214Dic122—48no—
215Dic122—44no—
216C8×C61—42no—
217#92418—42no—
218#21765—611nomaximal, Sylow
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—24no—
222C22×C61—24no—
223C22×C61#41714yes—
224C22×C61C22×C22⋊C411yescharacteristic
225C22×C61#53412yes—
226C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C22×C61C2×#13712yes—
228C22×C61C2×#13711yescharacteristic
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—24no—
232C2×D61—24no—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—416no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61#41914yes—
240C4×C61—22no—
241C4×C61—44no—
242C4×C61—82no—
243C4×C61#51812yes—
244C4×C61#53412yes—
245Dic61—22no—
246Dic61—22no—
247Dic61—44no—
248Dic61—82no—
249C22×C22⋊Q8—611no—
250C2×#407—611no—
251C2×#380—611no—
252#1894—614no—
253#1872—614no—
254#1823—611no—
255#1823D6111yescharacteristic, complemented
256#1787—611no—
257#1787—611no—
258C2×C61—24no—
259C2×C61#230811yescharacteristic
260C2×C61C2×#41911yescharacteristic
261C2×C61C2×#41711yescharacteristic
262C2×C61#319312yes—
263C2×C61#319512yes—
264D61—24no—
265D61—42no—
266C23×Q8—611no—
267C24×C4—611no—
268C22×M16—612no—
269C2×C22⋊Q8—618no—
270C2×C22⋊Q8—612no—
271C2×C22⋊Q8—612no—
272C2×C22⋊Q8—1221no—
273C2×C22⋊Q8—1221no—
274C2×C22⋊Q8—1224no—
275C22×C4⋊C4—612no—
276C22×C4⋊C4—1221no—
277C22×C4⋊C4C2×D6111yescharacteristic
278C22×C22⋊C4—1221no—
279C22×C22⋊C4—1221no—
280C2×#103—614no—
281C2×#103—614no—
282#407—1224no—
283C2×C22⋊C8—611no—
284C2×C22⋊C8—612no—
285C2×C22⋊C8C2×D6111yescharacteristic
286#380—611no—
287#380—612no—
288#380—1221no—
289#380—1221no—
290#380C2×D6111yescharacteristic
291C2×#95—1221no—
292#340—614no—
293#340C2×D6114yescomplemented
294C61#2176511yescharacteristic, complemented, Sylow
295C25—611no—
296C22×Q8—614no—
297C22×Q8—612no—
298C22×Q8—1221no—
299C22×Q8—1222no—
300C22×Q8—2444no—
301C23×C4—21no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—611no—
305C23×C4—614no—
306C23×C4—611no—
307C23×C4—611no—
308C23×C4—614no—
309C23×C4—1222no—
310C23×C4—1221no—
311C23×C4—2444no—
312C23×C4C22×D6111yescharacteristic
313C2×M16—1224no—
314C2×M16—2444no—
315C22×C8—612no—
316C22×C8C22×D6112yes—
317C22⋊Q8—618no—
318C22⋊Q8—618no—
319C22⋊Q8—1224no—
320C22⋊Q8—1224no—
321C22⋊Q8—1228no—
322C22⋊Q8—1228no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—612no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4—1228no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—2444no—
331C2×C4⋊C4C4×D6114yescomplemented
332C2×C4⋊C4C22×D6112yes—
333C2×C4⋊C4C22×D6112yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1228no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—2444no—
341#103—2444no—
342#103—2444no—
343C22⋊C8—22no—
344C22⋊C8—1224no—
345C22⋊C8—1222no—
346#95—21no—
347#95—1222no—
348#95—1221no—
349C24—611no—
350C24—611no—
351C24—614no—
352C24—1224no—
353C24—1222no—
354C24—1222no—
355C24—2442no—
356C24D4×D6111yescharacteristic
357C2×Q8—614no—
358C2×Q8—1222no—
359C2×Q8—1224no—
360C2×Q8—1224no—
361C2×Q8—1224no—
362C2×Q8—2448no—
363C2×Q8—2444no—
364C2×Q8—2448no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—614no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—612no—
378C22×C4—614no—
379C22×C4—614no—
380C22×C4—1224no—
381C22×C4—1224no—
382C22×C4—1222no—
383C22×C4—1224no—
384C22×C4—1222no—
385C22×C4—2448no—
386C22×C4—2448no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2444no—
390C22×C4—2444no—
391C22×C4—2444no—
392C22×C4—2442no—
393C22×C4—2444no—
394C22×C4C2×C4×D6112yes—
395C22×C4D4×D6112yes—
396C22×C4D4×D6111yescharacteristic
397C22×C4C23×D6111yescharacteristic
398M16—2448no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—1222no—
402C2×C8—1224no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—1228no—
407C4⋊C4—1228no—
408C4⋊C4—1224no—
409C4⋊C4—2444no—
410C4⋊C4C2×C4×D6114yes—
411C22⋊C4—42no—
412C22⋊C4—1228no—
413C22⋊C4—1228no—
414C22⋊C4—2444no—
415C22⋊C4—2442no—
416C23—22no—
417C23—22no—
418C23—612no—
419C23—614no—
420C23—612no—
421C23—1228no—
422C23—1221no—
423C23—1221no—
424C23—1222no—
425C23—1228no—
426C23—1221no—
427C23—1222no—
428C23—1221no—
429C23—1222no—
430C23—2444no—
431C23—2448no—
432C23—2448no—
433C23C22⋊C4×D6112yes—
434C23#1451980112yes—
435C23#1451980612yes—
436C23C2×D4×D6111yescharacteristic
437Q8—1224no—
438Q8—1224no—
439Q8—2448no—
440Q8—2444no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—612no—
452C2×C4—612no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1224no—
456C2×C4—1224no—
457C2×C4—1224no—
458C2×C4—1224no—
459C2×C4—1224no—
460C2×C4—4888no—
461C2×C4—2444no—
462C2×C4—2444no—
463C2×C4—2444no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2448no—
468C2×C4—2444no—
469C2×C4C22⋊C4×D6112yes—
470C2×C4C22×C4×D6111yescharacteristic
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4#1451982314yes—
473C8—42no—
474C8—2442no—
475C22—24no—
476C22—22no—
477C22—22no—
478C22—24no—
479C22—614no—
480C22—1224no—
481C22—1222no—
482C22—1224no—
483C22—1222no—
484C22—1222no—
485C22—1222no—
486C22—1222no—
487C22—24416no—
488C22—2442no—
489C22—2442no—
490C22#1683976014yes—
491C22C2×C22⋊C4×D6111yescharacteristic
492C22#1684032312yes—
493C22C2×#1451980111yescharacteristic
494C22#137×D6112yes—
495C22C2×#1451980611yescharacteristic
496C4—22no—
497C4—44no—
498C4—82no—
499C4—1222no—
500C4—1222no—
501C4—4882no—
502C4—2444no—
503C4#1684023612yes—
504C4#1684032312yes—
505C2—24no—
506C2—1224no—
507C2—2442no—
508C2#2144839611yescharacteristic
509C2#419×D6111yescharacteristic
510C2C2×#1683976011yescharacteristic
511C2#2145662712yes—
512C2#2145664212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1823C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1823D61
hypocenterC61#21765
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #21446251C2semidirect
5. #21446125C2semidirect
6. #21445905C2semidirect
7. #21445755C2semidirect
8. #21445753C2semidirect
9. C61×#1823C2semidirect
16. C2×#14519757C4semidirect
23. C2×#14519702C22semidirect
24. C2×#14519700C22semidirect
33. #16839524C22semidirect
37. C61×#340C22semidirect
42. C22×Dic122C2×C4semidirect
62. #14519757C2×C4semidirect
86. C2×C4⋊C4×C61C2×C4semidirect
255. #1823D61semidirect
293. #340C2×D61semidirect
294. C61#21765semidirect
331. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197572
C2×#168397242
#380×D612
#214462512
#214461252
#214459052
#214457552
#214457532
C61×#18232
#2176561