Finite Group Databasebeta

← #29687208

Subgroups — #29687208

Order 15616.

964 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840090C211yesmaximal, characteristic
2C22⋊Q8×Dic61C211yesmaximal, characteristic
3C2×C4⋊C4×Dic61C211yesmaximal, characteristic
4#21446463C212yesmaximal, complemented
5#21446400C212yesmaximal, complemented
6#21446007C211yesmaximal, characteristic
7#21445906C211yesmaximal, characteristic
8#21445715C212yesmaximal, complemented
9#21445651C212yesmaximal, complemented
10#21445583C211yesmaximal, characteristic
11C61×#1872C211yesmaximal, characteristic
12C2×Q8×Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C4×Dic61—21no—
15C22×C4×Dic61C2211yescharacteristic
16#16840090C418yescomplemented
17C2×#14519710C2212yes—
18C2×#14519708C2212yes—
19C2×#14519701C2212yes—
20C2×C22⋊#92418C2211yescharacteristic
21C2×#14519687C2212yes—
22#16839669C2211yescharacteristic
23#16839661C2211yescharacteristic
24#16839648—21no—
25#16839637—21no—
26#16839622C2211yescharacteristic
27C4⋊C4×Dic61—21no—
28C4⋊C4×Dic61—21no—
29C4⋊C4×Dic61—21no—
30C4⋊C4×Dic61C2212yes—
31C4⋊C4×Dic61C2212yes—
32C4⋊C4×Dic61C2211yescharacteristic
33C22⋊C4×Dic61—21no—
34C2×#14519684—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839516C2212yes—
39C2×C22⋊Q8×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#340C2212yes—
44C23×Dic61—21no—
45C23×Dic61D411yescharacteristic
46C23×Dic61D411yescharacteristic
47C22×#92418C2312yes—
48Q8×Dic61—24no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2×C414yes—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2312yes—
57C2×C22⋊Dic61—21no—
58C2×C4×Dic61—24no—
59C2×C4×Dic61—22no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—44no—
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C2×C4×Dic61C2311yescharacteristic
68#14519710—24no—
69#14519708C2×C418yescomplemented
70#14519701C2×C418yescomplemented
71C22⋊#92418C2×C414yes—
72#14519687—24no—
73#14519684—21no—
74#14519684—22no—
75#14519684—22no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684—21no—
80C22×Q8×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61C2311yescharacteristic
83C22×C8×C61C2312yes—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61C2×C414yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—24no—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2312yes—
91C2×C22⋊C4×C61—21no—
92#103×C61—24no—
93C22⋊C8×C61—22no—
94C24×C61C2×D411yescharacteristic
95C2×Q8×C61—21no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61C22×C412yes—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—24no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—42no—
106C22×C4×C61—42no—
107C22×C4×C61C22×C412yes—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4112yes—
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—22no—
112C22×Dic61—22no—
113C22×Dic61—22no—
114C22×Dic61—24no—
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—41no—
121C22×Dic61—42no—
122C22×Dic61—42no—
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C2×C8×C61—22no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61C22×C412yes—
135C4⋊C4×C61#4114yes—
136C22⋊C4×C61—24no—
137C22⋊Dic61—42no—
138C4⋊Dic61—24no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—42no—
143C4⋊Dic61—42no—
144C4⋊Dic61C22×C414yes—
145C4×Dic61—22no—
146C4×Dic61—22no—
147C4×Dic61—22no—
148C4×Dic61—42no—
149C4×Dic61—44no—
150C4×Dic61—44no—
151C4×Dic61—42no—
152C4×Dic61#4114yes—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418C22×C414yes—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—21no—
159C23×C61—21no—
160C23×C61C4×D412yes—
161C23×C61AGL(1,ℤ/8)12yes—
162C23×C61#13712yes—
163C23×C61C22×D411yescharacteristic
164Q8×C61—24no—
165Q8×C61—42no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—24no—
169C2×C4×C61—24no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—24no—
173C2×C4×C61—48no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61C4×D412yes—
181C2×C4×C61C23×C411yescharacteristic
182C2×C4×C61C2×#4112yes—
183C2×C4×C61C2×#4112yes—
184C2×C4×C61C2×#4111yescharacteristic
185C2×Dic61—28no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—24no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—44no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—48no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61#12312yes—
207C2×Dic61#12312yes—
208C2×Dic61#13712yes—
209C8×C61—42no—
210#92418—24no—
211#21579—611nomaximal, Sylow
212C22×C61—22no—
213C22×C61—22no—
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—22no—
217C22×C61—22no—
218C22×C61#44014yes—
219C22×C61C2×C4×D411yescharacteristic
220C22×C61C2×#12311yescharacteristic
221C22×C61#52511yescharacteristic
222C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C61C2×#13711yescharacteristic
224C22×C61C2×#13711yescharacteristic
225C22×C61#57511yescharacteristic
226C4×C61—22no—
227C4×C61—44no—
228C4×C61—42no—
229C4×C61—42no—
230C4×C61C4×#4112yes—
231C4×C61#52912yes—
232Dic61—22no—
233Dic61—22no—
234Dic61—22no—
235Dic61—41no—
236Dic61—44no—
237Dic61—81no—
238Dic61—81no—
239C2×#483—611no—
240C4×C22⋊Q8—611no—
241C2×C4×C4⋊C4—611no—
242#1928—612no—
243#1919—612no—
244#1872D6111yescharacteristic
245#1852—611no—
246#1823—611no—
247#1776—612no—
248#1763—612no—
249#1749—611no—
250C2×C61—24no—
251C2×C61C4×#12311yescharacteristic
252C2×C61C2×#44011yescharacteristic
253C2×C61C4×#13711yescharacteristic
254C2×C61#309211yescharacteristic
255C2×C61#309511yescharacteristic
256C2×C61#311311yescharacteristic
257C2×C61#311411yescharacteristic
258C2×C22⋊Q8C2×D6111yescharacteristic
259C2×C4×Q8—611no—
260C22×C4⋊C4—611no—
261C22×C4⋊C4C2×D6111yescharacteristic
262C22×C42—611no—
263C22×C42—1221no—
264#483—618no—
265C2×#107—612no—
266C2×#106—612no—
267C2×#103—612no—
268C2×#103C2×D6112yes—
269C2×C22⋊C8—611no—
270C2×C22⋊C8C2×D6111yescharacteristic
271C2×#97—612no—
272#391—611no—
273#389—611no—
274#387—1221no—
275#385—1221no—
276#380—611no—
277C4×C4⋊C4—612no—
278C4×C4⋊C4—612no—
279C4×C4⋊C4—611no—
280C4×C4⋊C4—1221no—
281C4×C4⋊C4—1221no—
282C4×C4⋊C4—1221no—
283C4×C22⋊C4—1221no—
284C2×#95—1221no—
285#340—612no—
286#340—612no—
287#340—612no—
288#340C2×D6112yes—
289#336—612no—
290C61#2157911yescharacteristic, complemented, Sylow
291C22×Q8C22×D6111yescharacteristic
292C23×C4—21no—
293C23×C4—611no—
294C23×C4—611no—
295C23×C4—1221no—
296C23×C4C22×D6111yescharacteristic
297C22×C8—612no—
298C22×C8C22×D6112yes—
299C22⋊Q8—22no—
300C22⋊Q8C4×D6114yes—
301C4×Q8—1224no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—614no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1222no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4C22×D6111yescharacteristic
314C2×C4⋊C4C22×D6112yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—1221no—
317C2×C42—611no—
318C2×C42—611no—
319C2×C42—611no—
320C2×C42—611no—
321C2×C42—1224no—
322C2×C42—1221no—
323C2×C42—1221no—
324C2×C42—1222no—
325C2×C42—1222no—
326C2×C42—2444no—
327#107—1224no—
328#106—618no—
329#103—24no—
330#103—618no—
331C22⋊C8—22no—
332C22⋊C8—614no—
333#97—1224no—
334#95—1222no—
335#95—1222no—
336#95—1221no—
337#95—1221no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341C24D4×D6111yescharacteristic
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8C2×C4×D6112yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—611no—
355C22×C4—611no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—1224no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—2441no—
368C22×C4—2442no—
369C22×C4—2442no—
370C22×C4C2×C4×D6112yes—
371C22×C4D4×D6111yescharacteristic
372C22×C4#1175822412yes—
373C22×C4C23×D6111yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—614no—
378C2×C8—1222no—
379C2×C8—1222no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—24no—
384C4⋊C4—614no—
385C4⋊C4—1222no—
386C4⋊C4—1224no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—2442no—
390C4⋊C4—2442no—
391C4⋊C4C2×C4×D6112yes—
392C4⋊C4#1175822414yes—
393C22⋊C4—24no—
394C22⋊C4—2442no—
395C42—614no—
396C42—1222no—
397C42—1222no—
398C42—1222no—
399C42—2442no—
400C42—2444no—
401C42—2444no—
402C42—2442no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23#1451973712yes—
408C23#1451980412yes—
409C23#1451980912yes—
410C23C2×D4×D6111yescharacteristic
411Q8—24no—
412Q8—42no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—48no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—612no—
428C2×C4—612no—
429C2×C4—612no—
430C2×C4—1228no—
431C2×C4—1221no—
432C2×C4—1221no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1221no—
436C2×C4—1221no—
437C2×C4—1224no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—2444no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4—2448no—
451C2×C4#1451973712yes—
452C2×C4C22×C4×D6111yescharacteristic
453C2×C4#41×D6112yes—
454C2×C4C2×#1175822412yes—
455C2×C4C2×#1175822411yescharacteristic
456C8—42no—
457C8—1224no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22#1683986514yes—
465C22C2×#1451973711yescharacteristic
466C22#123×D6111yescharacteristic
467C22#1684026811yescharacteristic
468C22C2×#1451980411yescharacteristic
469C22#137×D6111yescharacteristic
470C22C2×#1451980911yescharacteristic
471C22#1684055811yescharacteristic
472C4—22no—
473C4—42no—
474C4—44no—
475C4—42no—
476C4—1222no—
477C4—1222no—
478C4—1222no—
479C4—4881no—
480C4—4881no—
481C4—2444no—
482C4—2441no—
483C4C4×#1175822412yes—
484C4#1684029312yes—
485C2—24no—
486C2#2144834411yescharacteristic
487C2C2×#1683986511yescharacteristic
488C2#2145464311yescharacteristic
489C2#2145574311yescharacteristic
490C2#2145576411yescharacteristic
491C2#2145594411yescharacteristic
492C2#2145595811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1872D61
hypocenterC61#21579
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #21446463C2semidirect
5. #21446400C2semidirect
8. #21445715C2semidirect
9. #21445651C2semidirect
16. #16840090C4semidirect
69. #14519708C2×C4semidirect
70. #14519701C2×C4semidirect
290. C61#21579semidirect

Maximal subgroups

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

subgroupindex
C2×#168400902
C22⋊Q8×Dic612
C2×C4⋊C4×Dic612
#214464632
#214464002
#214460072
#214459062
#214457152
#214456512
#214455832
C61×#18722
#2157961