Finite Group Databasebeta

← #29662808

Subgroups — #29662808

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455823C211yesmaximal, characteristic
2C2×#16840049C211yesmaximal, characteristic
3M16×D244C211yesmaximal, characteristic
4#21445339C212yesmaximal, complemented
5#21445315C212yesmaximal, complemented
6#21445215C212yesmaximal, complemented
7#21445191C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444500C211yesmaximal, characteristic, complemented
11C61×#3165C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13C2×C4×D244C2211yescharacteristic
14#16840090—21no—
15#16840049—21no—
16#16840049—21no—
17#16840049C2212yes—
18#16840049C2211yescharacteristic
19#16840049C2212yes—
20#16840049C2211yescharacteristic
21#16839964C2211yescharacteristic
22#16839964C2211yescharacteristic
23#16839912—21no—
24C4×#11758201—21no—
25C4×#11758201—21no—
26#16839841C2212yes—
27#16839835—22no—
28C8×D244C2212yes—
29#16839815C2211yescharacteristic
30C2×#14519708—21no—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519697—21no—
34#16839725—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839511C2212yescomplemented
38#16839511C2212yescomplemented
39#16839511C2212yescomplemented
40#16839508C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#501—22no—
44C61×#500C2212yescomplemented
45C61×#498C2212yescomplemented
46C61×#428—21no—
47C61×#428—21no—
48C4×M16×C61C2211yescharacteristic
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758201—21no—
52C2×#11758201—21no—
53C2×#11758194—21no—
54C2×#11758189—22no—
55M16×D61—44no—
56C22×#92418—21no—
57C2×C8×D61—22no—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244D411yescharacteristic
61C4×D244D411yescharacteristic
62C4×D244C2311yescharacteristic
63C4×D244C2311yescharacteristic
64C2×C4⋊Dic61D411yescharacteristic
65C2×#11758198—21no—
66#14519708—22no—
67#14519708—22no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75C4⋊#92418C2312yes—
76#14519701—21no—
77#14519701—21no—
78#14519697—22no—
79#14519697—22no—
80#14519697—21no—
81#14519697—21no—
82C22⋊#92418—21no—
83#14519689—22no—
84#14519689—22no—
85C4×#92418—21no—
86C22×Q8×C61—21no—
87C2×M16×C61D411yescharacteristic
88C2×M16×C61D411yescharacteristic
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61C2312yes—
93C4⋊Q8×C61C2311yescharacteristic
94C4⋊Q8×C61C2311yescharacteristic
95#125×C61—21no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100#117×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#107×C61—22no—
105#107×C61—22no—
106#106×C61—22no—
107#106×C61—22no—
108#97×C61D412yes—
109C4×C8×C61C2312yes—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—41no—
115C2×Q8×C61—82no—
116C22×C4×C61—41no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×Dic61—21no—
121C2×D244—22no—
122C2×D244—21no—
123C2×D244—41no—
124C2×D244C2×D411yescharacteristic
125C2×D244C2×D411yescharacteristic
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132M16×C61—22no—
133M16×C61SD1614yescomplemented
134C2×C8×C61C2×D412yes—
135C2×C8×C61C2×D412yes—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C42×C61—41no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61#4111yescharacteristic
151C42×C61C2411yescharacteristic
152#11758201—42no—
153#11758201—42no—
154#11758198—21no—
155#11758198—21no—
156#11758198—21no—
157#11758198—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61C2×D411yescharacteristic
160C4⋊Dic61C2×D411yescharacteristic
161#11758194—42no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—42no—
167#11758189—44no—
168C8×D61—44no—
169C23×C61#12111yescharacteristic
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—81no—
175Q8×C61—41no—
176Q8×C61—41no—
177Q8×C61—82no—
178Q8×C61—81no—
179Q8×C61—82no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—82no—
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61C2×SD1612yes—
191C2×C4×C61#13712yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—42no—
198D244—21no—
199D244—21no—
200D244—21no—
201D244—42no—
202D244#13712yes—
203C4×D61—41no—
204C4×D61—41no—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—42no—
208C4×D61—42no—
209C8×C61—22no—
210C8×C61C2×SD1614yes—
211#92418—42no—
212#92418—41no—
213#92418—42no—
214#92418—41no—
215#19165—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61#46112yes—
219C22×C61C2×#12111yescharacteristic
220C2×D61—22no—
221C2×D61—41no—
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—84no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—81no—
231C4×C61—82no—
232C4×C61—81no—
233C4×C61#46112yes—
234C4×C61C22×SD1611yescharacteristic
235C4×C61C2×#13711yescharacteristic
236C4×C61C2×#13711yescharacteristic
237C4×C61#57511yescharacteristic
238Dic61—41no—
239Dic61—41no—
240#3165D6111yescharacteristic, complemented
241#3101—611no—
242C2×#476—611no—
243D4×M16—611no—
244#1695—612no—
245#1691—612no—
246#1675—612no—
247#1671—612no—
248#1565—611no—
249#1550—611no—
250#1548—611no—
251C2×C61—22no—
252C2×C61C2×#46111yescharacteristic
253C2×C61#305611yescharacteristic
254C2×C61#306811yescharacteristic
255D61—42no—
256D61—81no—
257C22×M16—1221no—
258#533C2×D6111yescharacteristic
259C2×C4⋊Q8C2×D6111yescharacteristic
260C2×C4×D4—611no—
261#501—22no—
262#500C2×D6112yescomplemented
263#498C2×D6112yescomplemented
264#483—1221no—
265#476—611no—
266#476—611no—
267#476—612no—
268#476—612no—
269#476—1221no—
270#476—1221no—
271#461—611no—
272#461—611no—
273#450—1221no—
274C4×SD16—1221no—
275C4×SD16—1221no—
276#436—612no—
277#435—1222no—
278C8×D4—612no—
279#431—611no—
280#428—21no—
281#428—21no—
282C2×#106—1221no—
283#423—611no—
284C2×C4⋊C8—611no—
285C2×#102—1221no—
286#407—1221no—
287C4×M16C2×D6111yescharacteristic
288#335—612no—
289#334—612no—
290#333—612no—
291#333—612no—
292#333—612no—
293#332—612no—
294C61#1916511yescharacteristic, complemented, Sylow
295C22×Q8—21no—
296C22×D4—611no—
297C23×C4—1221no—
298C2×SD16—1221no—
299C2×SD16—1221no—
300C2×M16—1221no—
301C2×M16—1222no—
302C2×M16—2444no—
303C2×M16C22⋊D6111yescharacteristic
304C2×M16C22⋊D6111yescharacteristic
305C22×C8—1221no—
306C22×C8—1222no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8C22×D6112yes—
311C4⋊Q8C22×D6111yescharacteristic
312C4⋊Q8C22×D6111yescharacteristic
313#125—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×D4—611no—
319C4×D4—611no—
320C4×D4—611no—
321C4×D4—611no—
322C4×D4—1221no—
323C4×D4—1221no—
324#117—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—611no—
328C2×C22⋊C4—1221no—
329C2×C42C22×D6111yescharacteristic
330#107—22no—
331#107—22no—
332#106—22no—
333#106—22no—
334#106—1222no—
335#106—1222no—
336#106—1221no—
337#106—1221no—
338C4⋊C8—612no—
339C4⋊C8—612no—
340C4⋊C8—611no—
341C4⋊C8—611no—
342C4⋊C8—1221no—
343C4⋊C8—1221no—
344#103—1221no—
345#103—1221no—
346#102—1222no—
347#102—1222no—
348#102—1221no—
349#102—1221no—
350C22⋊C8—1221no—
351C22⋊C8—1222no—
352C22⋊C8—1222no—
353#97C22⋊D6112yes—
354C4×C8—1221no—
355C4×C8C22×D6112yes—
356C24—1221no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—82no—
362C2×D4—611no—
363C2×D4—611no—
364C2×D4—1222no—
365C2×D4—1221no—
366C2×D4—2441no—
367C22×C4—41no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—2442no—
374C22×C4—2442no—
375C22×C4D4×D6111yescharacteristic
376C22×C4#1175822411yescharacteristic
377C22×C4C2×C22⋊D6111yescharacteristic
378SD16—2442no—
379SD16—2442no—
380M16—22no—
381M16—2442no—
382M16—2444no—
383M16#1175820114yescomplemented
384C2×C8—1221no—
385C2×C8—1221no—
386C2×C8—1222no—
387C2×C8—1222no—
388C2×C8—2442no—
389C2×C8—2444no—
390C2×C8C2×C22⋊D6112yes—
391C2×C8C2×C22⋊D6112yes—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—611no—
403C4⋊C4—611no—
404C4⋊C4—1221no—
405C22⋊C4—41no—
406C22⋊C4—1221no—
407C22⋊C4—1221no—
408C22⋊C4—1221no—
409C22⋊C4—1221no—
410C42—41no—
411C42D4×D6111yescharacteristic
412C42#1175822411yescharacteristic
413C42C2×C22⋊D6111yescharacteristic
414C42C23×D6111yescharacteristic
415C23—1221no—
416C23—1221no—
417C23—4881no—
418C23—2442no—
419C23—2442no—
420C23#1451974811yescharacteristic
421Q8—41no—
422Q8—41no—
423Q8—81no—
424Q8—82no—
425Q8—82no—
426D4—612no—
427D4—1221no—
428D4—1221no—
429D4—1221no—
430D4—2442no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4#1451974811yescharacteristic
448C2×C4#1451974811yescharacteristic
449C2×C4#1451974811yescharacteristic
450C2×C4C2×#1175820112yes—
451C2×C4#1451980812yes—
452C2×C4C2×D4×D6111yescharacteristic
453C2×C4C22×C22⋊D6111yescharacteristic
454C2×C4C2×#1175822411yescharacteristic
455C8—22no—
456C8—2442no—
457C8—2441no—
458C8—2442no—
459C8—2441no—
460C8C2×#1175820114yes—
461C22—21no—
462C22—21no—
463C22—1222no—
464C22—4884no—
465C22—2441no—
466C22—2441no—
467C22—2441no—
468C22—2441no—
469C22—2441no—
470C22#1683996412yes—
471C22C2×#1451974811yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—81no—
476C4—82no—
477C4—81no—
478C4—2441no—
479C4—2441no—
480C4#1683996412yes—
481C4C22×#1175820111yescharacteristic
482C4#137×D6111yescharacteristic
483C4C2×#1451980811yescharacteristic
484C4#1684056111yescharacteristic
485C2—22no—
486C2—4881no—
487C2—2442no—
488C2C2×#1683996411yescharacteristic
489C2#2145544311yescharacteristic
490C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3165C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#3165D61
hypocenterC61#19165
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #21445339C2semidirect
5. #21445315C2semidirect
6. #21445215C2semidirect
7. #21445191C2semidirect
8. #21444571C2semidirect
9. #21444509C2semidirect
10. #21444500C2semidirect
11. C61×#3165C2semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839511C22semidirect
38. #16839511C22semidirect
39. #16839511C22semidirect
40. #16839508C22semidirect
44. C61×#500C22semidirect
45. C61×#498C22semidirect
133. M16×C61SD16semidirect
240. #3165D61semidirect
262. #500C2×D61semidirect
263. #498C2×D61semidirect
294. C61#19165semidirect
383. M16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214558232
C2×#168400492
M16×D2442
#214453392
#214453152
#214452152
#214451912
#214445712
#214445092
#214445002
C61×#31652
#1916561