Finite Group Databasebeta

← #29665216

Subgroups — #29665216

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455906C211yesmaximal, characteristic, complemented
2#21455815C211yesmaximal, characteristic, complemented
3#21454842C211yesmaximal, characteristic, complemented
4#21445267C212yesmaximal, complemented
5#21445225C212yesmaximal, complemented
6#21445171C212yesmaximal, complemented
7#21445157C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444524C211yesmaximal, characteristic, complemented
10#21444481C211yesmaximal, characteristic, complemented
11C61×#3207C211yesmaximal, characteristic, complemented
12C2×C4×D244C2211yescharacteristic
13#16840078—21no—
14#16840073—21no—
15#16840049C2212yes—
16#16840039C2212yes—
17#16839954C2212yes—
18#16839954C2212yes—
19#16839895—21no—
20#16839895—21no—
21#16839872—22no—
22C4×#11758199—22no—
23C8×D244C2212yes—
24C8×D244C2212yes—
25#16839827—22no—
26#16839819C2211yescharacteristic
27#16839815C2211yescharacteristic
28#16839815C2211yescharacteristic
29C22⋊C8×D61—22no—
30#16839515C2212yescomplemented
31#16839513C2212yescomplemented
32#16839511C2212yescomplemented
33#16839508C2212yescomplemented
34#16839506C2212yescomplemented
35#16839506C2212yescomplemented
36C61×#532C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#493C2211yescharacteristic, complemented
39C61×#492C2211yescharacteristic, complemented
40C61×#487C2211yescharacteristic, complemented
41C61×#486C2211yescharacteristic, complemented
42C61×#463—22no—
43C61×#449—22no—
44C61×#431C2211yescharacteristic
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C2×#11758199—22no—
48C2×#11758199—22no—
49C2×C8×D61—22no—
50C2×C8×D61—22no—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244D412yes—
54C4×D244C2312yes—
55C2×C4⋊Dic61D411yescharacteristic
56C2×#11758198—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519698—22no—
63#14519698—22no—
64#14519697—22no—
65#14519697—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69#14519689—22no—
70#14519689—22no—
71#14519688—21no—
72C4×#92418—21no—
73C4×#92418—21no—
74C2×#41×C61—21no—
75C2×#41×C61—21no—
76C2×SD16×C61—22no—
77C2×D8×C61—22no—
78C4⋊Q8×C61C2311yescharacteristic
79C4⋊Q8×C61C2311yescharacteristic
80#127×C61C2311yescharacteristic
81#127×C61C2311yescharacteristic
82#124×C61—21no—
83#124×C61—21no—
84#121×C61—21no—
85#121×C61—21no—
86#121×C61—21no—
87#121×C61—21no—
88C4×D4×C61—21no—
89C4×D4×C61—21no—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61D412yes—
94#103×C61—22no—
95#102×C61—22no—
96C22⋊C8×C61D412yes—
97C4×C8×C61C2311yescharacteristic
98C4×C8×C61C2311yescharacteristic
99C23×D61—21no—
100#41×C61—82no—
101#41×C61—82no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C22×C4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C22×Dic61—21no—
118C2×D244—21no—
119C2×D244—24no—
120C2×D244C2×D412yes—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—82no—
124SD16×C61—82no—
125D8×C61—82no—
126C2×C8×C61C2×D412yes—
127C2×C8×C61C2×D412yes—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C22⋊C4×C61—41no—
133C22⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61#4111yescharacteristic
139C42×C61C2411yescharacteristic
140#11758199—44no—
141#11758199—44no—
142#11758198—22no—
143#11758198—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61C2×D412yes—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418—22no—
149C8×D61—82no—
150C8×D61—82no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61#12111yescharacteristic
154C22×D61—22no—
155C22×D61—24no—
156C22×D61—81no—
157C22×D61—81no—
158Q8×C61—42no—
159Q8×C61—81no—
160D4×C61—42no—
161D4×C61—42no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—81no—
165D4×C61—81no—
166D4×C61—82no—
167D4×C61—82no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—82no—
174C2×C4×C61—82no—
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×Dic61—22no—
182C2×Dic61—81no—
183D244—24no—
184D244—22no—
185D244#13514yes—
186C4×D61—42no—
187C4×D61—42no—
188C4×D61—82no—
189C8×C61—22no—
190C8×C61—22no—
191#92418—42no—
192#92418—42no—
193#92418—42no—
194#19389—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61C2×#12111yescharacteristic
205C2×D61—24no—
206C2×D61—42no—
207C2×D61—42no—
208C2×D61—84no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—81no—
214C4×C61—81no—
215C4×C61C2×#13512yes—
216C4×C61#57612yes—
217Dic61—42no—
218#3207D6111yescharacteristic, complemented
219#3109—611no—
220#3100—611no—
221#2972—611no—
222#1683—612no—
223#1677—612no—
224#1667—612no—
225#1665—612no—
226#1572—611no—
227#1553—611no—
228#1543—611no—
229C2×C61—41no—
230C2×C61—161no—
231C2×C61—161no—
232C2×C61#304111yescharacteristic
233C2×C61#305011yescharacteristic
234C2×C61#306811yescharacteristic
235D61—44no—
236#532C2×D6111yescharacteristic
237#532C2×D6111yescharacteristic
238C2×C4×D4—611no—
239#493C2×D6111yescharacteristic, complemented
240#492C2×D6111yescharacteristic, complemented
241#487C2×D6111yescharacteristic, complemented
242#486C2×D6111yescharacteristic, complemented
243#481—1221no—
244#480—1221no—
245#476—612no—
246#474—612no—
247#463—22no—
248#459—612no—
249#459—612no—
250#449—22no—
251#447—1221no—
252#447—1221no—
253#442—1222no—
254C4×D8—1222no—
255C8×D4—612no—
256C8×D4—612no—
257C8×D4—1222no—
258#432—611no—
259#431—611no—
260#431—611no—
261#431C2×D6111yescharacteristic
262C2×C22⋊C8—1222no—
263#335—612no—
264#334—612no—
265#333—612no—
266#332—612no—
267#331—612no—
268#331—612no—
269C61#1938911yescharacteristic, complemented, Sylow
270C2×#41—21no—
271C2×#41—21no—
272C22×D4—611no—
273C23×C4—1221no—
274C2×SD16—22no—
275C2×D8—22no—
276C2×D8—1222no—
277C2×D8—1222no—
278C22×C8—1222no—
279C22×C8—1222no—
280C4⋊Q8C22×D6111yescharacteristic
281C4⋊Q8C22×D6111yescharacteristic
282#127C22×D6111yescharacteristic
283#127C22×D6111yescharacteristic
284#124—21no—
285#124—21no—
286#121—21no—
287#121—21no—
288#121—21no—
289#121—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—612no—
295C4×D4—612no—
296C4×D4—1221no—
297C4×D4—1221no—
298C2×C4⋊C4—611no—
299C2×C22⋊C4—1221no—
300C2×C42C22×D6111yescharacteristic
301#107—1222no—
302#106—1222no—
303C4⋊C8—612no—
304C4⋊C8—612no—
305C4⋊C8—612no—
306C4⋊C8C22⋊D6112yes—
307#103—22no—
308#102—22no—
309#102—1222no—
310#102—1222no—
311#102—1222no—
312#102—1222no—
313C22⋊C8—1221no—
314C22⋊C8—1221no—
315C22⋊C8—1221no—
316C22⋊C8—1222no—
317C22⋊C8—1222no—
318C22⋊C8C22⋊D6112yes—
319#97—1221no—
320C4×C8—1221no—
321C4×C8—1221no—
322C4×C8C22×D6111yescharacteristic
323C4×C8C22×D6111yescharacteristic
324C24—1221no—
325#41—82no—
326#41—82no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—612no—
338C2×D4—1224no—
339C2×D4—1221no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—1221no—
343C22×C4—1222no—
344C22×C4—1222no—
345C22×C4—4882no—
346C22×C4D4×D6111yescharacteristic
347C22×C4#1175822411yescharacteristic
348C22×C4C2×C22⋊D6111yescharacteristic
349SD16—82no—
350D8—82no—
351D8—2444no—
352D8—2444no—
353C2×C8—1222no—
354C2×C8—1222no—
355C2×C8—1222no—
356C2×C8—4882no—
357C2×C8—4882no—
358C2×C8C2×C22⋊D6112yes—
359C2×C8C2×C22⋊D6112yes—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—612no—
365C4⋊C4—1221no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—1222no—
371C22⋊C4—1222no—
372C42D4×D6111yescharacteristic
373C42#1175822411yescharacteristic
374C42C2×C22⋊D6111yescharacteristic
375C42C23×D6111yescharacteristic
376C23—41no—
377C23—41no—
378C23—1224no—
379C23—1222no—
380C23—4881no—
381C23—4881no—
382C23#1451974811yescharacteristic
383Q8—42no—
384Q8—81no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—81no—
390D4—81no—
391D4—82no—
392D4—82no—
393D4—614no—
394D4—1224no—
395D4—1222no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—1222no—
404C2×C4—4882no—
405C2×C4—4881no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4#1451974811yescharacteristic
409C2×C4#1451974811yescharacteristic
410C2×C4#1451974811yescharacteristic
411C2×C4C2×D4×D6111yescharacteristic
412C2×C4C22×C22⋊D6111yescharacteristic
413C2×C4C2×#1175822411yescharacteristic
414C8—22no—
415C8—22no—
416C8—2442no—
417C8—2442no—
418C8—2442no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—1224no—
429C22—4884no—
430C22—2442no—
431C22—2442no—
432C22C2×#1451974811yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—81no—
438C4—81no—
439C4—2442no—
440C4#135×D6112yes—
441C4#1684056512yes—
442C2—41no—
443C2—161no—
444C2—161no—
445C2—2444no—
446C2#2145531811yescharacteristic
447C2#2145539611yescharacteristic
448C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3207C2
socleC22×C61C2×#121
hypercenter#3207D61
hypocenterC61#19389
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3207

Complements

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

NH ≅ G/Ntype
1. #21455906C2semidirect
2. #21455815C2semidirect
3. #21454842C2semidirect
4. #21445267C2semidirect
5. #21445225C2semidirect
6. #21445171C2semidirect
7. #21445157C2semidirect
8. #21444599C2semidirect
9. #21444524C2semidirect
10. #21444481C2semidirect
11. C61×#3207C2semidirect
30. #16839515C22semidirect
31. #16839513C22semidirect
32. #16839511C22semidirect
33. #16839508C22semidirect
34. #16839506C22semidirect
35. #16839506C22semidirect
38. C61×#493C22semidirect
39. C61×#492C22semidirect
40. C61×#487C22semidirect
41. C61×#486C22semidirect
218. #3207D61semidirect
239. #493C2×D61semidirect
240. #492C2×D61semidirect
241. #487C2×D61semidirect
242. #486C2×D61semidirect
269. C61#19389semidirect

Maximal subgroups

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

subgroupindex
#214559062
#214558152
#214548422
#214452672
#214452252
#214451712
#214451572
#214445992
#214445242
#214444812
C61×#32072
#1938961