Finite Group Databasebeta

← #29708993

Subgroups — #29708993

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839797C212yesmaximal
2#21448300C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic
4#21447305C212yesmaximal, complemented
5#21447304C212yesmaximal, complemented
6#21447297C211yesmaximal, characteristic, complemented
7#21447291C211yesmaximal, characteristic, complemented
8#21447290C211yesmaximal, characteristic, complemented
9#21445534C212yesmaximal
10#336×D61C211yesmaximal, characteristic
11C61×#2084C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519735C2212yes—
14C2×#14519735C2211yescharacteristic
15C2×#14519735C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839797—28no—
18C2×#14519710C2212yes—
19C2×#14519710C2212yes—
20C2×#14519709C2212yes—
21#16839661C2211yescharacteristic
22#16839660C2211yescharacteristic
23#16839658C2211yescharacteristic
24#16839636—21no—
25#16839633—21no—
26#16839632C2211yescharacteristic
27#16839627C2212yes—
28#16839619—21no—
29#16839619—21no—
30C4⋊C4×Dic61—21no—
31C4⋊C4×Dic61—21no—
32C4⋊C4×Dic61—21no—
33#16839613—21no—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38#16839516C2212yes—
39C2×#106×C61C2212yes—
40C61×#389C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C61×#336C2211yescharacteristic
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C22×#92418C2312yes—
48C2×C8×D61—24no—
49C2×C8×D61—24no—
50C2×C8×D61Q818yes—
51#14519735—22no—
52#14519735—24no—
53#14519735—24no—
54#14519735—24no—
55#14519735D414yes—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758198—21no—
65C2×#11758198—22no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73C42×D61—24no—
74#14519710—24no—
75#14519710—24no—
76#14519709—24no—
77#14519684—21no—
78#14519684—21no—
79#14519684—22no—
80#14519684—22no—
81#14519684—21no—
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C42×C61C2311yescharacteristic
90#106×C61—24no—
91#95×C61—21no—
92C23×D61C2×D411yescharacteristic
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61C2×D412yes—
112C2×C4×D61C2×Q814yes—
113C2×C4×D61#4114yes—
114C2×C4×D61#4112yes—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C2×C8×C61C2×Q814yes—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—42no—
123C4⋊C4×C61—42no—
124C4⋊C4×C61—42no—
125C4⋊C4×C61C2×D412yes—
126C42×C61—22no—
127#11758198—24no—
128#11758198—44no—
129#11758198—42no—
130C4⋊Dic61—21no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—42no—
135C4⋊Dic61—42no—
136C4⋊Dic61—42no—
137C4⋊Dic61C2×D412yes—
138#11758196—42no—
139#11758196—42no—
140C4×Dic61—22no—
141C4×Dic61—24no—
142C4×Dic61—22no—
143C4×Dic61—22no—
144C4×Dic61—44no—
145C4×Dic61—42no—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418C2×Q814yes—
149C8×D61—216no—
150C23×C61C22×D411yescharacteristic
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61#12112yes—
156C22×D61#12312yes—
157C22×D61C4⋊Q812yes—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—22no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61#13512yes—
172C2×C4×C61#13512yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×Q812yes—
175C2×C4×C61C2×#4112yes—
176C2×C4×C61C2×#4111yescharacteristic
177C2×Dic61—22no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—42no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—41no—
191C2×Dic61—42no—
192C2×Dic61#12112yes—
193C2×Dic61#12312yes—
194C2×Dic61C4⋊Q812yes—
195C4×D61—44no—
196C4×D61—44no—
197C4×D61C22⋊Q818yes—
198C4×D61#12518yes—
199C8×C61—24no—
200#92418—24no—
201#23656—611nomaximal, Sylow
202C22×C61C2×#12111yescharacteristic
203C22×C61C2×#12311yescharacteristic
204C22×C61C2×C4⋊Q811yescharacteristic
205C22×C61C2×#13511yescharacteristic
206C22×C61C2×#13511yescharacteristic
207C22×C61#57511yescharacteristic
208C22×C61#57511yescharacteristic
209C2×D61—22no—
210C2×D61—22no—
211C2×D61—22no—
212C2×D61—22no—
213C2×D61—22no—
214C2×D61—22no—
215C2×D61#39814yes—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—42no—
219C4×C61—81no—
220C4×C61—82no—
221C4×C61C2×C22⋊Q812yes—
222C4×C61C2×#12512yes—
223Dic61—41no—
224Dic61—41no—
225Dic61—42no—
226Dic61—81no—
227Dic61—82no—
228Dic61#39814yes—
229C2×#427—612no—
230#2296—611no—
231#2285—611no—
232#2085—612no—
233#2085—612no—
234#2084—611no—
235#2084D6111yescharacteristic, complemented
236#2083—611no—
237#2083—611no—
238#1735—612no—
239C2×#336—611no—
240C2×C61C2×#39811yescharacteristic
241C2×C61#303511yescharacteristic
242C2×C61#303511yescharacteristic
243C2×C61#309111yescharacteristic
244C2×C61#309111yescharacteristic
245C2×C61#316011yescharacteristic
246C2×C61#316211yescharacteristic
247D61—24no—
248C23×C8—612no—
249C2×#117—611no—
250C2×#117—612no—
251C2×#117—611no—
252C22×C42—611no—
253#427—1228no—
254C2×#107—612no—
255C2×#107—612no—
256C2×#106—612no—
257C2×#106C2×D6112yes—
258#389—611no—
259#389—611no—
260#389—611no—
261#389C2×D6111yescharacteristic
262#385—1221no—
263#385—1221no—
264#384—611no—
265#384C2×D6111yescharacteristic
266#382—612no—
267#380—1221no—
268#380—1221no—
269C4×C4⋊C4—1221no—
270C4×C4⋊C4—1221no—
271C4×C4⋊C4—1221no—
272C4×C4⋊C4—1221no—
273#340—612no—
274#340—612no—
275#340—612no—
276#340C2×D6112yes—
277#336—612no—
278#336—611no—
279#336C2×D6111yescharacteristic
280C61#2365611yescharacteristic, complemented, Sylow
281C23×C4—611no—
282C23×C4—611no—
283C23×C4—611no—
284C22×C8—612no—
285C22×C8—618no—
286C22×C8—1224no—
287C22×C8—1224no—
288C22×C8C22×D6112yes—
289#117—614no—
290#117—1222no—
291#117—1224no—
292#117—1224no—
293#117—1224no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—612no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1221no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4C22×D6111yescharacteristic
306C2×C4⋊C4C22×D6112yes—
307C2×C4⋊C4C22×D6111yescharacteristic
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1222no—
310C2×C22⋊C4—1221no—
311C2×C42—611no—
312C2×C42—611no—
313C2×C42—611no—
314C2×C42—1224no—
315C2×C42—1221no—
316C2×C42—1221no—
317C2×C42—1222no—
318C2×C42C22×D6111yescharacteristic
319#107—1224no—
320#107—1224no—
321#106—24no—
322#106—1224no—
323#95—21no—
324#95—1222no—
325#95—1222no—
326#95—1221no—
327#95—1221no—
328#95—1221no—
329C24—611no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—611no—
334C22×C4—612no—
335C22×C4—614no—
336C22×C4—614no—
337C22×C4—612no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—1222no—
341C22×C4—1222no—
342C22×C4—1221no—
343C22×C4—1222no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—1221no—
347C22×C4—2441no—
348C22×C4—2441no—
349C22×C4D4×D6111yescharacteristic
350C22×C4D4×D6111yescharacteristic
351C22×C4C23×D6111yescharacteristic
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—614no—
355C2×C8—12216no—
356C2×C8—1222no—
357C2×C8—1222no—
358C2×C8Q8×D6114yes—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—612no—
367C4⋊C4—1221no—
368C4⋊C4—1222no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—2442no—
372C4⋊C4—2442no—
373C4⋊C4—2442no—
374C4⋊C4—2442no—
375C4⋊C4—2442no—
376C4⋊C4D4×D6112yes—
377C22⋊C4—1224no—
378C22⋊C4—2444no—
379C22⋊C4—2442no—
380C42—22no—
381C42—1222no—
382C42—1222no—
383C42—1224no—
384C42—1222no—
385C42—2444no—
386C42—2442no—
387C23—612no—
388C23—612no—
389C23—612no—
390C23—1221no—
391C23—1221no—
392C23—1221no—
393C23—1221no—
394C23C2×D4×D6111yescharacteristic
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—612no—
409C2×C4—612no—
410C2×C4—612no—
411C2×C4—618no—
412C2×C4—618no—
413C2×C4—1222no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—2444no—
421C2×C4—2444no—
422C2×C4—2442no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4—2442no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4#1451979812yes—
431C2×C4#1451979812yes—
432C2×C4C2×D4×D6111yescharacteristic
433C2×C4C2×Q8×D6112yes—
434C2×C4#41×D6112yes—
435C2×C4#41×D6111yescharacteristic
436C8—24no—
437C8—1224no—
438C22—614no—
439C22—1222no—
440C22—1222no—
441C22—1222no—
442C22—1222no—
443C22—1222no—
444C22—1222no—
445C22#121×D6111yescharacteristic
446C22#123×D6111yescharacteristic
447C22C4⋊Q8×D6111yescharacteristic
448C22C2×#1451979811yescharacteristic
449C22C2×#1451979811yescharacteristic
450C22#1684055811yescharacteristic
451C22#1684056111yescharacteristic
452C4—22no—
453C4—22no—
454C4—42no—
455C4—81no—
456C4—82no—
457C4—614no—
458C4—4881no—
459C4—4882no—
460C4—2441no—
461C4—2441no—
462C4—2442no—
463C4C22⋊Q8×D6112yes—
464C4#125×D6112yes—
465C2—1224no—
466C2#398×D6111yescharacteristic
467C2#2145526911yescharacteristic
468C2#2145527511yescharacteristic
469C2#2145572911yescharacteristic
470C2#2145573311yescharacteristic
471C2#2145632811yescharacteristic
472C2#2145634011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2084C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2084D61
hypocenterC61#23656
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
4. #21447305C2semidirect
5. #21447304C2semidirect
6. #21447297C2semidirect
7. #21447291C2semidirect
8. #21447290C2semidirect
11. C61×#2084C2semidirect
235. #2084D61semidirect
280. C61#23656semidirect

Maximal subgroups

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

subgroupindex
C2×#168397972
#214483002
#214482242
#214473052
#214473042
#214472972
#214472912
#214472902
#214455342
#336×D612
C61×#20842
#2365661