Finite Group Databasebeta

← #29662839

Subgroups — #29662839

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456201C211yesmaximal, characteristic, complemented
2#21455823C211yesmaximal, characteristic
3C2×#16840043C211yesmaximal, characteristic
4#21454554C211yesmaximal, characteristic
5#21445397C212yesmaximal, complemented
6#21445369C212yesmaximal, complemented
7#21445273C212yesmaximal, complemented
8#21445245C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444492C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840132—22no—
16#16840114C2212yescomplemented
17#16840114C2212yescomplemented
18#16840090—21no—
19#16840049C2211yescharacteristic
20#16840049C2211yescharacteristic
21#16840043—21no—
22#16840043—21no—
23#16840043C2212yes—
24#16840043C2212yes—
25#16839964C2211yescharacteristic
26#16839964C2211yescharacteristic
27#16839912—21no—
28C4×#11758201—21no—
29C4×#11758201—21no—
30#16839839C2212yes—
31#16839832—22no—
32#16839827C2212yes—
33#16839815C2211yescharacteristic
34C2×#14519711—21no—
35#16839770—21no—
36#16839770—21no—
37C2×#14519699—21no—
38#16839725—21no—
39C4×#11758194C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839509C2212yescomplemented
43#16839509C2212yescomplemented
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yescomplemented
48C61×#332C2212yescomplemented
49C22×Dic122—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52C2×#11758201—21no—
53C2×#11758201—21no—
54C2×#11758194D411yescharacteristic
55C2×#11758194D411yescharacteristic
56C2×#11758189—22no—
57M16×D61—44no—
58C2×C8×D61—22no—
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777C2312yes—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244D411yescharacteristic
66C4×D244D411yescharacteristic
67C4×D244C2311yescharacteristic
68C4×D244C2311yescharacteristic
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×#11758198—21no—
73#14519711—22no—
74#14519711—22no—
75#14519708—21no—
76#14519708—21no—
77C4⋊#92418C2311yescharacteristic
78C4⋊#92418C2311yescharacteristic
79#14519701—21no—
80#14519701—21no—
81#14519700—22no—
82#14519700—22no—
83#14519700—22no—
84#14519700—22no—
85#14519699—22no—
86#14519699—22no—
87#14519697—21no—
88#14519697—21no—
89C22⋊#92418—21no—
90#14519689—22no—
91#14519689—22no—
92#14519688D412yes—
93C4×#92418—21no—
94C4×#92418C2312yes—
95C2×M16×C61—21no—
96C22×C8×C61—21no—
97C4⋊Q8×C61C2311yescharacteristic
98C4⋊Q8×C61C2311yescharacteristic
99#125×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61—21no—
104#117×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C4⋊C8×C61C2312yes—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×Dic61—21no—
117C22×Dic61—41no—
118C2×D244—22no—
119C2×D244—21no—
120C2×D244—41no—
121C2×D244C2×D411yescharacteristic
122C2×D244C2×D411yescharacteristic
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×Dic122—22no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×Dic122—41no—
133M16×C61—42no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—42no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C42×C61—41no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61#4111yescharacteristic
148C42×C61C2411yescharacteristic
149#11758201—42no—
150#11758201—42no—
151#11758198—21no—
152#11758198—21no—
153#11758198—21no—
154#11758198—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—41no—
159C4⋊Dic61—41no—
160C4⋊Dic61C2×D411yescharacteristic
161C4⋊Dic61C2×D411yescharacteristic
162#11758194—22no—
163#11758194—22no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418C2×D412yes—
167C2×#92418C2×D412yes—
168#11758189—44no—
169C8×D61—44no—
170C23×C61#12111yescharacteristic
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—81no—
176Q8×C61—41no—
177Q8×C61—41no—
178Q8×C61—81no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#13512yes—
188C2×C4×C61AGL(1,ℤ/8)12yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—82no—
197D244—21no—
198D244—21no—
199D244—21no—
200D244—42no—
201D244#13712yes—
202C4×D61—41no—
203C4×D61—41no—
204C4×D61—41no—
205C4×D61—41no—
206C4×D61—42no—
207C4×D61—42no—
208Dic122—42no—
209Dic122—42no—
210Dic122—82no—
211C8×C61—42no—
212C8×C61—42no—
213#92418—22no—
214#92418—22no—
215#92418—41no—
216#92418—41no—
217#19167—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61#46312yes—
221C22×C61C2×#12111yescharacteristic
222C2×D61—22no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61—84no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—81no—
233C4×C61—81no—
234C4×C61#46312yes—
235C4×C61C2×#13511yescharacteristic
236C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C61C2×#13711yescharacteristic
238C4×C61#57511yescharacteristic
239Dic61—41no—
240Dic61—41no—
241Dic61—82no—
242#3139—611no—
243#3101—611no—
244C2×#474—611no—
245D4×M16—611no—
246#1703—612no—
247#1699—612no—
248#1683—612no—
249#1679—612no—
250#1565—611no—
251#1550D6111yescharacteristic, complemented
252#1546—611no—
253C2×C61—22no—
254C2×C61C2×#46311yescharacteristic
255C2×C61#305611yescharacteristic
256C2×C61#306911yescharacteristic
257D61—42no—
258D61—81no—
259C22×M16—1221no—
260#533C2×D6111yescharacteristic
261C2×C4⋊Q8—611no—
262C2×C4×D4—611no—
263#491—1222no—
264#487—612no—
265#487—612no—
266#483—1221no—
267#476—611no—
268#476—611no—
269#474—612no—
270#474—612no—
271#474—1221no—
272#474—1221no—
273#461—611no—
274#461—611no—
275#450—1221no—
276C4×SD16—1221no—
277C4×SD16—1221no—
278#436—612no—
279#435—1222no—
280C8×D4—612no—
281#431—611no—
282C2×#107—1221no—
283#423C2×D6111yescharacteristic
284C2×C4⋊C8C2×D6111yescharacteristic
285#419—1221no—
286#419—1221no—
287C2×#102—1221no—
288#407—1221no—
289C4×M16—611no—
290#335—612no—
291#334—612no—
292#333C2×D6112yescomplemented
293#332—612no—
294#332—612no—
295#332C2×D6112yescomplemented
296C61#1916711yescharacteristic, complemented, Sylow
297C22×Q8—1221no—
298C22×D4—611no—
299C23×C4—1221no—
300C2×SD16—1221no—
301C2×SD16—1221no—
302C2×M16—21no—
303C2×M16—611no—
304C2×M16—611no—
305C2×M16—1222no—
306C2×M16—2444no—
307C22×C8—21no—
308C22×C8—1222no—
309C4⋊Q8—612no—
310C4⋊Q8—1221no—
311C4⋊Q8—1221no—
312C4⋊Q8—1221no—
313C4⋊Q8C22×D6111yescharacteristic
314C4⋊Q8C22×D6111yescharacteristic
315#125—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×D4—611no—
321C4×D4—611no—
322C4×D4—611no—
323C4×D4—611no—
324C4×D4—1221no—
325C4×D4—1221no—
326#117—21no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C22⋊C4—1221no—
331C2×C42C22×D6111yescharacteristic
332#107—1222no—
333#107—1222no—
334#106—1221no—
335#106—1221no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—611no—
339C4⋊C8—611no—
340C4⋊C8C22×D6112yes—
341C4⋊C8C22×D6112yes—
342#103—1222no—
343#103—1222no—
344#103—1222no—
345#103—1222no—
346#103—1221no—
347#103—1221no—
348#102—1221no—
349#102—1221no—
350#102—1222no—
351#102—1222no—
352C22⋊C8—1222no—
353C22⋊C8—1222no—
354C22⋊C8—1221no—
355#97—612no—
356C4×C8—612no—
357C4×C8—1221no—
358C24—1221no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—1222no—
362C2×Q8—2442no—
363C2×Q8—2442no—
364C2×Q8—2441no—
365C2×D4—611no—
366C2×D4—611no—
367C2×D4—1222no—
368C2×D4—1221no—
369C2×D4—2441no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—2441no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4#1175822211yescharacteristic
379C22×C4D4×D6111yescharacteristic
380C22×C4D4×D6111yescharacteristic
381SD16—2442no—
382SD16—2442no—
383M16—42no—
384M16—1222no—
385M16—1222no—
386M16—2444no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—42no—
390C2×C8—612no—
391C2×C8—612no—
392C2×C8—1221no—
393C2×C8—1221no—
394C2×C8—2444no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—611no—
402C4⋊C4—611no—
403C4⋊C4—1221no—
404C4⋊C4—1222no—
405C4⋊C4—1222no—
406C4⋊C4—2441no—
407C4⋊C4—2441no—
408C22⋊C4—41no—
409C22⋊C4—1221no—
410C22⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—1221no—
413C42—41no—
414C42#1175822211yescharacteristic
415C42D4×D6111yescharacteristic
416C42D4×D6111yescharacteristic
417C42C23×D6111yescharacteristic
418C23—1221no—
419C23—1221no—
420C23—4881no—
421C23—2442no—
422C23—2442no—
423C23#1451975011yescharacteristic
424Q8—41no—
425Q8—41no—
426Q8—81no—
427Q8—4882no—
428Q8—2442no—
429Q8—2442no—
430D4—612no—
431D4—1221no—
432D4—1221no—
433D4—1221no—
434D4—2442no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—4882no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4#1451975011yescharacteristic
452C2×C4#1451975011yescharacteristic
453C2×C4#1451975011yescharacteristic
454C2×C4#1451980012yes—
455C2×C4#1451980412yes—
456C2×C4C2×D4×D6111yescharacteristic
457C2×C4C2×D4×D6111yescharacteristic
458C2×C4C2×#1175822211yescharacteristic
459C8—42no—
460C8—42no—
461C8—1222no—
462C8—1222no—
463C8—2441no—
464C8—2441no—
465C22—21no—
466C22—21no—
467C22—1222no—
468C22—4884no—
469C22—2441no—
470C22—2441no—
471C22—2441no—
472C22—2441no—
473C22—2441no—
474C22#1683998012yes—
475C22C2×#1451975011yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—81no—
480C4—81no—
481C4—4882no—
482C4—2441no—
483C4—2441no—
484C4#1683998012yes—
485C4C2×#1451980011yescharacteristic
486C4C2×#1451980411yescharacteristic
487C4#137×D6111yescharacteristic
488C4#1684055811yescharacteristic
489C2—22no—
490C2—4881no—
491C2—2442no—
492C2C2×#1683998011yescharacteristic
493C2#2145545211yescharacteristic
494C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1550D61
hypocenterC61#19167
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21456201C2semidirect
5. #21445397C2semidirect
6. #21445369C2semidirect
7. #21445273C2semidirect
8. #21445245C2semidirect
9. #21444569C2semidirect
10. #21444492C2semidirect
11. C61×#1550C2semidirect
16. #16840114C22semidirect
17. #16840114C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839509C22semidirect
43. #16839509C22semidirect
47. C61×#333C22semidirect
48. C61×#332C22semidirect
251. #1550D61semidirect
292. #333C2×D61semidirect
295. #332C2×D61semidirect
296. C61#19167semidirect

Maximal subgroups

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

subgroupindex
#214562012
#214558232
C2×#168400432
#214545542
#214453972
#214453692
#214452732
#214452452
#214445692
#214444922
C61×#15502
#1916761