Finite Group Databasebeta

← #29679811

Subgroups — #29679811

Order 15616.

1774 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D61C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic
3#21448201C211yesmaximal, characteristic
4#21445910C211yesmaximal, characteristic, complemented
5#21445909C211yesmaximal, characteristic, complemented
6#21445907C211yesmaximal, characteristic, complemented
7#21445904C214yesmaximal, complemented
8#21445537C214yesmaximal, complemented
9C61×#1824C211yesmaximal, characteristic, complemented
10C23×C4×D61—21no—
11C23×C4×D61C2211yescharacteristic
12C22×#11758189C2212yes—
13C2×#14519735C2212yes—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×#11758196—21no—
16C2×C4⋊C4×D61—24no—
17C2×C4⋊C4×D61—21no—
18C2×C4⋊C4×D61—21no—
19C2×C4⋊C4×D61C418yescomplemented
20C2×C4⋊C4×D61C2212yes—
21C2×C4⋊C4×D61C2212yes—
22#16839724—24no—
23C2×C22⋊#92418C2211yescharacteristic
24C2×#14519689C2212yes—
25#16839624—21no—
26#16839622C2211yescharacteristic
27#16839619C2212yes—
28#16839617—21no—
29C22⋊C4×Dic61—21no—
30#16839609—21no—
31#16839525C2214yescomplemented
32#16839524C2214yescomplemented
33#16839524C2214yescomplemented
34C22×C4⋊C4×C61C2211yescharacteristic
35C2×C22⋊C8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C61×#340C2214yescomplemented
38C24×D61D411yescharacteristic
39C23×Dic61—21no—
40C23×Dic61D411yescharacteristic
41C22×C4×D61—28no—
42C22×C4×D61—28no—
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—48no—
46C22×C4×D61C2×C414yes—
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D414yes—
50C22×C4×D61C2311yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C2×#11758189—24no—
53C2×#11758189—28no—
54C22×#92418C2312yes—
55#14519735—28no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61C2×C414yes—
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2312yes—
61C2×#11758196—28no—
62C2×#11758196—22no—
63C2×#11758196—22no—
64C4⋊C4×D61—28no—
65C4⋊C4×D61—28no—
66C4⋊C4×D61—24no—
67C4⋊C4×D61—24no—
68C4⋊C4×D61C2×C418yes—
69C4⋊C4×D61C2×C418yes—
70C2×C22⋊Dic61—21no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×C4×Dic61—22no—
74C22⋊#92418—22no—
75#14519689—24no—
76#14519684—21no—
77#14519684—22no—
78C23×C4×C61—21no—
79C23×C4×C61C2311yescharacteristic
80C22×C8×C61C2312yes—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61C2×C414yes—
84C2×C4⋊C4×C61C2312yes—
85C2×C4⋊C4×C61C2312yes—
86C2×C22⋊C4×C61—21no—
87C22⋊C8×C61—22no—
88#95×C61—21no—
89C24×C61C2×D411yescharacteristic
90C23×D61—22no—
91C23×D61—22no—
92C23×D61—24no—
93C23×D61—24no—
94C23×D61C22⋊C414yes—
95C23×D61C2×D411yescharacteristic
96C23×D61C2×D411yescharacteristic
97C22×C4×C61—24no—
98C22×C4×C61—24no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—44no—
102C22×C4×C61C22×C412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—24no—
107C22×Dic61—24no—
108C22×Dic61—22no—
109C22×Dic61—22no—
110C22×Dic61—44no—
111C22×Dic61C22⋊C414yes—
112C22×Dic61C2×D411yescharacteristic
113C22×Dic61C2×D411yescharacteristic
114C2×C4×D61—24no—
115C2×C4×D61—24no—
116C2×C4×D61—24no—
117C2×C4×D61—216no—
118C2×C4×D61—216no—
119C2×C4×D61—28no—
120C2×C4×D61—28no—
121C2×C4×D61—48no—
122C2×C4×D61—432no—
123C2×C4×D61—48no—
124C2×C4×D61C22⋊C414yes—
125C2×C4×D61C22⋊C414yes—
126C2×C4×D61C22×C412yes—
127C2×C4×D61C22×C412yes—
128C2×C4×D61C2×D412yes—
129C2×C4×D61C2×D412yes—
130C2×C4×D61#4118yes—
131C2×C8×C61—22no—
132C2×C8×C61—24no—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61C22×C414yes—
137C22⋊C4×C61—42no—
138C22⋊Dic61—42no—
139#11758198—44no—
140#11758198—44no—
141C4⋊Dic61—24no—
142C4⋊Dic61—22no—
143C4⋊Dic61—24no—
144C4⋊Dic61C22×C414yes—
145#11758196—28no—
146#11758196—28no—
147C4×Dic61—44no—
148C2×#92418—22no—
149C2×#92418—24no—
150#11758189—48no—
151C23×C61—22no—
152C23×C61—22no—
153C23×C61C2×C22⋊C412yes—
154C23×C61#13712yes—
155C23×C61#13712yes—
156C23×C61C22×D411yescharacteristic
157C22×D61—216no—
158C22×D61—216no—
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—28no—
166C22×D61—28no—
167C22×D61—28no—
168C22×D61C2×C22⋊C412yes—
169C22×D61C2×C22⋊C412yes—
170C22×D61#12314yes—
171C22×D61AGL(1,ℤ/8)14yes—
172C2×C4×C61—22no—
173C2×C4×C61—24no—
174C2×C4×C61—24no—
175C2×C4×C61—24no—
176C2×C4×C61—48no—
177C2×C4×C61—44no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61C2×C22⋊C412yes—
182C2×C4×C61C23×C411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4114yes—
185C2×Dic61—22no—
186C2×Dic61—28no—
187C2×Dic61—24no—
188C2×Dic61—48no—
189C2×Dic61—44no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61C2×C22⋊C412yes—
194C2×Dic61C2×C22⋊C412yes—
195C2×Dic61#12314yes—
196C4×D61—28no—
197C4×D61—24no—
198C4×D61—24no—
199C4×D61—28no—
200C4×D61—416no—
201C4×D61—416no—
202C4×D61#11718yes—
203C4×D61#12318yes—
204C8×C61—42no—
205#92418—42no—
206#20700—611nomaximal, Sylow
207C22×C61—24no—
208C22×C61—24no—
209C22×C61—22no—
210C22×C61—22no—
211C22×C61#41914yes—
212C22×C61C22×C22⋊C411yescharacteristic
213C22×C61C2×#12312yes—
214C22×C61C2×AGL(1,ℤ/8)12yes—
215C22×C61C2×#13711yescharacteristic
216C22×C61C2×#13711yescharacteristic
217C2×D61—232no—
218C2×D61—24no—
219C2×D61—24no—
220C2×D61—24no—
221C2×D61—24no—
222C2×D61—22no—
223C2×D61—22no—
224C2×D61—22no—
225C2×D61—22no—
226C2×D61#38114yes—
227C2×D61#41814yes—
228C4×C61—22no—
229C4×C61—44no—
230C4×C61—82no—
231C4×C61C2×#11712yes—
232C4×C61C2×#12312yes—
233Dic61—22no—
234Dic61—44no—
235Dic61—82no—
236Dic61#38114yes—
237C23×C4⋊C4—611no—
238C2×#407—611no—
239#2280—611no—
240#1824—611no—
241#1824—611no—
242#1824—611no—
243#1824D6111yescharacteristic, complemented
244#1823—614no—
245#1736—614no—
246C2×C61—24no—
247C2×C61C2×#38111yescharacteristic
248C2×C61C2×#41811yescharacteristic
249C2×C61C2×#41911yescharacteristic
250C2×C61#309212yes—
251C2×C61#309212yes—
252D61—24no—
253D61—24no—
254C24×C4—611no—
255C24×C4—1221no—
256C22×M16—612no—
257C2×#117—612no—
258C22×C4⋊C4—611no—
259C22×C4⋊C4—618no—
260C22×C4⋊C4—612no—
261C22×C4⋊C4—612no—
262C22×C4⋊C4—1221no—
263C22×C4⋊C4—1221no—
264C22×C4⋊C4—1224no—
265C22×C4⋊C4—1221no—
266C22×C4⋊C4C2×D6111yescharacteristic
267#407—1224no—
268C2×C22⋊C8—611no—
269C2×C22⋊C8—612no—
270C2×C22⋊C8C2×D6111yescharacteristic
271#381—1221no—
272#380—611no—
273#380—612no—
274#380C2×D6111yescharacteristic
275#379—1221no—
276C4×C22⋊C4—1221no—
277C4×C22⋊C4—1221no—
278#340—614no—
279#340—614no—
280#340—614no—
281#340C2×D6114yescomplemented
282C61#2070011yescharacteristic, complemented, Sylow
283C25—611no—
284C23×C4—21no—
285C23×C4—611no—
286C23×C4—611no—
287C23×C4—614no—
288C23×C4—611no—
289C23×C4—611no—
290C23×C4—614no—
291C23×C4—611no—
292C23×C4—1228no—
293C23×C4—1222no—
294C23×C4—1222no—
295C23×C4—1228no—
296C23×C4—1221no—
297C23×C4—2448no—
298C23×C4C22×D6111yescharacteristic
299C2×M16—1224no—
300C2×M16—1228no—
301C22×C8—612no—
302C22×C8C22×D6112yes—
303#117—1228no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—614no—
307C2×C4⋊C4—612no—
308C2×C4⋊C4—618no—
309C2×C4⋊C4—618no—
310C2×C4⋊C4—612no—
311C2×C4⋊C4—1224no—
312C2×C4⋊C4—1224no—
313C2×C4⋊C4—1228no—
314C2×C4⋊C4—1228no—
315C2×C4⋊C4—1222no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1228no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4C4×D6114yes—
321C2×C4⋊C4C22×D6112yes—
322C2×C4⋊C4C22×D6112yes—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1222no—
326C2×C22⋊C4—1221no—
327C2×C42—1222no—
328C22⋊C8—22no—
329C22⋊C8—1224no—
330C22⋊C8—1222no—
331#95—21no—
332#95—1222no—
333#95—1221no—
334C24—611no—
335C24—611no—
336C24—614no—
337C24—1224no—
338C24—1224no—
339C24—1222no—
340C24—1222no—
341C24D4×D6111yescharacteristic
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—44no—
347C22×C4—611no—
348C22×C4—612no—
349C22×C4—612no—
350C22×C4—618no—
351C22×C4—611no—
352C22×C4—614no—
353C22×C4—612no—
354C22×C4—612no—
355C22×C4—614no—
356C22×C4—614no—
357C22×C4—1224no—
358C22×C4—12216no—
359C22×C4—12216no—
360C22×C4—1224no—
361C22×C4—1228no—
362C22×C4—1228no—
363C22×C4—1224no—
364C22×C4—1224no—
365C22×C4—1222no—
366C22×C4—1224no—
367C22×C4—1222no—
368C22×C4—2448no—
369C22×C4—24432no—
370C22×C4—2448no—
371C22×C4—2444no—
372C22×C4C2×C4×D6112yes—
373C22×C4D4×D6112yes—
374C22×C4D4×D6111yescharacteristic
375C22×C4C23×D6111yescharacteristic
376M16—2448no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—1222no—
380C2×C8—1224no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—614no—
385C4⋊C4—1228no—
386C4⋊C4—1228no—
387C4⋊C4—1222no—
388C4⋊C4—1224no—
389C4⋊C4—1224no—
390C4⋊C4C2×C4×D6114yes—
391C22⋊C4—42no—
392C22⋊C4—2444no—
393C22⋊C4—2444no—
394C22⋊C4—2442no—
395C42—2444no—
396C23—22no—
397C23—22no—
398C23—612no—
399C23—612no—
400C23—614no—
401C23—614no—
402C23—1228no—
403C23—1228no—
404C23—12216no—
405C23—12216no—
406C23—1222no—
407C23—1228no—
408C23—1222no—
409C23—1221no—
410C23—1221no—
411C23—1221no—
412C23—1221no—
413C23C22⋊C4×D6112yes—
414C23#1451980612yes—
415C23#1451980912yes—
416C23C2×D4×D6111yescharacteristic
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—612no—
427C2×C4—614no—
428C2×C4—618no—
429C2×C4—618no—
430C2×C4—612no—
431C2×C4—1228no—
432C2×C4—1228no—
433C2×C4—1228no—
434C2×C4—1222no—
435C2×C4—1224no—
436C2×C4—1224no—
437C2×C4—1224no—
438C2×C4—24416no—
439C2×C4—24416no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2448no—
444C2×C4—2444no—
445C2×C4C22⋊C4×D6112yes—
446C2×C4C22×C4×D6111yescharacteristic
447C2×C4C2×D4×D6111yescharacteristic
448C2×C4#41×D6114yes—
449C8—42no—
450C8—2442no—
451C22—22no—
452C22—24no—
453C22—24no—
454C22—22no—
455C22—614no—
456C22—614no—
457C22—12232no—
458C22—1222no—
459C22—1224no—
460C22—1224no—
461C22—1224no—
462C22—1224no—
463C22—1222no—
464C22—1222no—
465C22—1222no—
466C22#1683976914yes—
467C22C2×C22⋊C4×D6111yescharacteristic
468C22#123×D6112yes—
469C22AGL(1,ℤ/8)×D6112yes—
470C22C2×#1451980611yescharacteristic
471C22C2×#1451980911yescharacteristic
472C4—22no—
473C4—44no—
474C4—82no—
475C4—614no—
476C4—1222no—
477C4—4882no—
478C4—2444no—
479C4#117×D6112yes—
480C4#123×D6112yes—
481C2—24no—
482C2—1224no—
483C2—1224no—
484C2#381×D6111yescharacteristic
485C2#418×D6111yescharacteristic
486C2C2×#1683976911yescharacteristic
487C2#2145574112yes—
488C2#2145574612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1824C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1824D61
hypocenterC61#20700
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. #21445910C2semidirect
5. #21445909C2semidirect
6. #21445907C2semidirect
7. #21445904C2semidirect
8. #21445537C2semidirect
9. C61×#1824C2semidirect
19. C2×C4⋊C4×D61C4semidirect
31. #16839525C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
37. C61×#340C22semidirect
243. #1824D61semidirect
281. #340C2×D61semidirect
282. C61#20700semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D612
C2×#168397242
#214482012
#214459102
#214459092
#214459072
#214459042
#214455372
C61×#18242
#2070061