Finite Group Databasebeta

← #29665205

Subgroups — #29665205

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455914C211yesmaximal, characteristic, complemented
2#21455815C211yesmaximal, characteristic
3#21454842C211yesmaximal, characteristic, complemented
4#21445253C212yesmaximal, complemented
5#21445239C212yesmaximal, complemented
6#21445181C212yesmaximal, complemented
7#21445157C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444524C211yesmaximal, characteristic, complemented
10#21444484C211yesmaximal, characteristic, complemented
11C61×#3210C211yesmaximal, characteristic, complemented
12C2×C4×D244C2211yescharacteristic
13#16840095—21no—
14#16840073—21no—
15#16840039C2212yes—
16#16840039C2212yes—
17#16839964C2212yes—
18#16839954C2212yes—
19#16839912—21no—
20#16839895—21no—
21#16839860—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#16839508C2212yescomplemented
33#16839508C2212yescomplemented
34#16839508C2212yescomplemented
35#16839506C2212yescomplemented
36C61×#533C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#494C2211yescharacteristic, complemented
39C61×#492C2211yescharacteristic, complemented
40C61×#487C2211yescharacteristic, complemented
41C61×#486C2211yescharacteristic, complemented
42C61×#464—22no—
43C61×#452—22no—
44C61×#431C2211yescharacteristic
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C2×#11758201—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#14519710—22no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—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×Q16×C61—22no—
76C2×SD16×C61—22no—
77C4⋊Q8×C61C2311yescharacteristic
78C4⋊Q8×C61C2311yescharacteristic
79C4⋊Q8×C61C2311yescharacteristic
80#127×C61C2311yescharacteristic
81#125×C61—21no—
82#124×C61—21no—
83C22⋊Q8×C61—21no—
84C22⋊Q8×C61—21no—
85#121×C61—21no—
86#121×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61—21no—
89C4×D4×C61—21no—
90C4×D4×C61—21no—
91#117×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—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×Dic61—21no—
113C2×D244—21no—
114C2×D244—24no—
115C2×D244C2×D412yes—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—82no—
119Q16×C61—82no—
120SD16×C61—82no—
121C2×C8×C61C2×D412yes—
122C2×C8×C61C2×D412yes—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C22⋊C4×C61—41no—
134C42×C61—41no—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61#4111yescharacteristic
138C42×C61C2411yescharacteristic
139#11758201—44no—
140#11758199—44no—
141#11758198—22no—
142#11758198—22no—
143C4⋊Dic61—21no—
144C4⋊Dic61C2×D412yes—
145C2×#92418—22no—
146C2×#92418—22no—
147C2×#92418—22no—
148C8×D61—82no—
149C8×D61—82no—
150C23×C61—41no—
151C23×C61#12111yescharacteristic
152C22×D61—22no—
153C22×D61—24no—
154C22×D61—81no—
155C22×D61—81no—
156Q8×C61—42no—
157Q8×C61—42no—
158Q8×C61—42no—
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—81no—
162D4×C61—82no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—82no—
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C2×Dic61—22no—
177C2×Dic61—81no—
178D244—24no—
179D244—22no—
180D244#13514yes—
181C4×D61—42no—
182C4×D61—42no—
183C4×D61—82no—
184C8×C61—22no—
185C8×C61—22no—
186#92418—42no—
187#92418—42no—
188#92418—42no—
189#19388—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61C2×#12111yescharacteristic
197C2×D61—24no—
198C2×D61—42no—
199C2×D61—42no—
200C2×D61—84no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—81no—
206C4×C61—81no—
207C4×C61—81no—
208C4×C61C2×#13512yes—
209C4×C61#57612yes—
210Dic61—42no—
211#3210D6111yescharacteristic, complemented
212#3110—611no—
213#3100—611no—
214#2972—611no—
215#1681—612no—
216#1679—612no—
217#1669—612no—
218#1665—612no—
219#1572—611no—
220#1553—611no—
221#1544—611no—
222C2×C61—41no—
223C2×C61—161no—
224C2×C61#304111yescharacteristic
225C2×C61#305011yescharacteristic
226C2×C61#306911yescharacteristic
227D61—44no—
228#533C2×D6111yescharacteristic
229#532C2×D6111yescharacteristic
230C2×C4×D4—611no—
231#494C2×D6111yescharacteristic, complemented
232#492C2×D6111yescharacteristic, complemented
233#487C2×D6111yescharacteristic, complemented
234#486C2×D6111yescharacteristic, complemented
235#484—1221no—
236#480—1221no—
237#474—612no—
238#474—612no—
239#464—22no—
240#461—612no—
241#459—612no—
242#452—22no—
243#450—1221no—
244#447—1221no—
245#440—1222no—
246C4×D8—1222no—
247C8×D4—612no—
248C8×D4—612no—
249C8×D4—1222no—
250#432—611no—
251#431—611no—
252#431—611no—
253#431C2×D6111yescharacteristic
254C2×C22⋊C8—1222no—
255#335—612no—
256#334—612no—
257#332—612no—
258#332—612no—
259#332—612no—
260#331—612no—
261C61#1938811yescharacteristic, complemented, Sylow
262C2×#41—21no—
263C22×D4—611no—
264C23×C4—1221no—
265C2×Q16—22no—
266C2×SD16—22no—
267C2×SD16—1222no—
268C2×D8—1222no—
269C22×C8—1222no—
270C22×C8—1222no—
271C4⋊Q8C22×D6111yescharacteristic
272C4⋊Q8C22×D6111yescharacteristic
273C4⋊Q8C22×D6111yescharacteristic
274#127C22×D6111yescharacteristic
275#125—21no—
276#124—21no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279#121—21no—
280#121—21no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—612no—
286C4×D4—612no—
287C4×D4—1221no—
288C4×D4—1221no—
289#117—21no—
290C2×C4⋊C4—611no—
291C2×C22⋊C4—1221no—
292C2×C42C22×D6111yescharacteristic
293#107—1222no—
294#107—1222no—
295C4⋊C8—612no—
296C4⋊C8—612no—
297C4⋊C8—612no—
298C4⋊C8C22⋊D6112yes—
299#103—22no—
300#103—1222no—
301#102—22no—
302#102—1222no—
303#102—1222no—
304#102—1222no—
305C22⋊C8—1221no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308C22⋊C8—1222no—
309C22⋊C8—1222no—
310C22⋊C8C22⋊D6112yes—
311#97—1221no—
312C4×C8—1221no—
313C4×C8—1221no—
314C4×C8C22×D6111yescharacteristic
315C4×C8C22×D6111yescharacteristic
316C24—1221no—
317#41—82no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—41no—
325C2×D4—612no—
326C2×D4—1224no—
327C2×D4—1221no—
328C22×C4—41no—
329C22×C4—1221no—
330C22×C4—1222no—
331C22×C4—1222no—
332C22×C4—4882no—
333C22×C4D4×D6111yescharacteristic
334C22×C4#1175822411yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336Q16—82no—
337SD16—82no—
338SD16—2444no—
339D8—2444no—
340C2×C8—1222no—
341C2×C8—1222no—
342C2×C8—1222no—
343C2×C8—4882no—
344C2×C8—4882no—
345C2×C8C2×C22⋊D6112yes—
346C2×C8C2×C22⋊D6112yes—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—612no—
356C4⋊C4—1221no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—1222no—
361C22⋊C4—1222no—
362C42—41no—
363C42D4×D6111yescharacteristic
364C42#1175822411yescharacteristic
365C42C2×C22⋊D6111yescharacteristic
366C42C23×D6111yescharacteristic
367C23—41no—
368C23—1224no—
369C23—1222no—
370C23—4881no—
371C23—4881no—
372C23#1451974811yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—42no—
376D4—42no—
377D4—42no—
378D4—81no—
379D4—82no—
380D4—614no—
381D4—1224no—
382D4—1222no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—82no—
390C2×C4—1222no—
391C2×C4—4882no—
392C2×C4—4881no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4#1451974811yescharacteristic
396C2×C4#1451974811yescharacteristic
397C2×C4#1451974811yescharacteristic
398C2×C4C2×D4×D6111yescharacteristic
399C2×C4C22×C22⋊D6111yescharacteristic
400C2×C4C2×#1175822411yescharacteristic
401C8—22no—
402C8—22no—
403C8—2442no—
404C8—2442no—
405C8—2442no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—1224no—
413C22—4884no—
414C22—2442no—
415C22—2442no—
416C22C2×#1451974811yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—2442no—
425C4#135×D6112yes—
426C4#1684056512yes—
427C2—41no—
428C2—161no—
429C2—2444no—
430C2#2145531811yescharacteristic
431C2#2145539611yescharacteristic
432C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3210C2
socleC22×C61C2×#121
prime radical#21455815C2
hypercenter#3210D61
hypocenterC61#19388
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3210

Complements

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

NH ≅ G/Ntype
1. #21455914C2semidirect
3. #21454842C2semidirect
4. #21445253C2semidirect
5. #21445239C2semidirect
6. #21445181C2semidirect
7. #21445157C2semidirect
8. #21444599C2semidirect
9. #21444524C2semidirect
10. #21444484C2semidirect
11. C61×#3210C2semidirect
30. #16839515C22semidirect
31. #16839513C22semidirect
32. #16839508C22semidirect
33. #16839508C22semidirect
34. #16839508C22semidirect
35. #16839506C22semidirect
38. C61×#494C22semidirect
39. C61×#492C22semidirect
40. C61×#487C22semidirect
41. C61×#486C22semidirect
211. #3210D61semidirect
231. #494C2×D61semidirect
232. #492C2×D61semidirect
233. #487C2×D61semidirect
234. #486C2×D61semidirect
261. C61#19388semidirect

Maximal subgroups

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

subgroupindex
#214559142
#214558152
#214548422
#214452532
#214452392
#214451812
#214451572
#214445992
#214445242
#214444842
C61×#32102
#1938861