Finite Group Databasebeta

← #29665219

Subgroups — #29665219

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#21456752C211yesmaximal, characteristic, complemented
2#21455911C211yesmaximal, characteristic, complemented
3#21455815C211yesmaximal, characteristic, complemented
4#21454842C211yesmaximal, characteristic, complemented
5#21445269C212yesmaximal, complemented
6#21445227C212yesmaximal, complemented
7#21445173C212yesmaximal, complemented
8#21445163C212yesmaximal, complemented
9#21444600C211yesmaximal, characteristic, complemented
10#21444479C211yesmaximal, characteristic, complemented
11C61×#1553C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840138C2211yescharacteristic, complemented
15#16840136C2211yescharacteristic, complemented
16#16840112C2211yescharacteristic, complemented
17#16840108C2211yescharacteristic, complemented
18#16840081—21no—
19#16840073—21no—
20#16840053C2212yes—
21#16840039C2212yes—
22#16839974—22no—
23#16839958C2212yes—
24#16839954C2212yes—
25#16839907—22no—
26#16839898—21no—
27#16839895—21no—
28#16839875—22no—
29C4×#11758199—22no—
30C8×D244—22no—
31#16839827C2212yes—
32#16839827C2212yes—
33#16839815C2211yescharacteristic
34#16839815C2211yescharacteristic
35C22⋊C8×D61—22no—
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839510C2212yescomplemented
39#16839505C2212yescomplemented
40C61×#532C2211yescharacteristic
41C61×#432C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C61×#332C2212yescomplemented
44C61×#331C2212yescomplemented
45C2×#11758222—21no—
46C22×D244D411yescharacteristic
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×D488—22no—
50C2×#11758199—22no—
51C2×#11758199—22no—
52C2×C8×D61—22no—
53C2×C8×D61—22no—
54#14519777C2311yescharacteristic
55#14519774C2311yescharacteristic
56#14519765—21no—
57#14519751—21no—
58#14519751—21no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244D412yes—
64C4×D244C2312yes—
65C2×C4⋊Dic61D411yescharacteristic
66C2×#11758198—21no—
67#14519710—22no—
68#14519709—22no—
69C4⋊#92418D412yes—
70C4⋊#92418C2312yes—
71#14519701—22no—
72#14519699—22no—
73#14519699—22no—
74#14519698—22no—
75#14519698—22no—
76#14519697—22no—
77C22⋊#92418—21no—
78C22⋊#92418D412yes—
79#14519689—22no—
80#14519689—22no—
81C4×#92418—21no—
82C4×#92418C2311yescharacteristic
83C4×#92418C2311yescharacteristic
84C2×#41×C61—21no—
85C4⋊Q8×C61C2311yescharacteristic
86#127×C61C2311yescharacteristic
87#124×C61—21no—
88#121×C61—21no—
89#121×C61—21no—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61C2312yes—
94C4⋊C8×C61C2312yes—
95C22⋊C8×C61—21no—
96C22⋊C8×C61—21no—
97#97×C61—21no—
98C4×C8×C61—21no—
99C23×D61—21no—
100#41×C61—82no—
101C2×Q8×C61—21no—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C2×C22⋊D61—41no—
111C22×Dic61—21no—
112#11758222—82no—
113C2×D244—21no—
114C2×D244—24no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244C2×D412yes—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—41no—
122C2×C4×D61—82no—
123C2×Dic122—21no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C22⋊C4×C61—41no—
129C22⋊C4×C61—41no—
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61#4111yescharacteristic
133C42×C61C2411yescharacteristic
134#11758201—82no—
135#11758199—44no—
136#11758199—82no—
137#11758198—22no—
138#11758198—22no—
139#11758198—41no—
140#11758198—41no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61C2×D412yes—
145C2×#92418—22no—
146C2×#92418C2×D412yes—
147C2×#92418C2×D412yes—
148D488—44no—
149C8×D61—82no—
150C8×D61—82no—
151C23×C61—41no—
152C23×C61#12111yescharacteristic
153C22×D61—22no—
154C22×D61—24no—
155C22×D61—41no—
156C22×D61—81no—
157C22×D61—81no—
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—82no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C2×#4111yescharacteristic
174C22⋊D61—82no—
175C2×Dic61—22no—
176C2×Dic61—41no—
177C2×Dic61—81no—
178D244—24no—
179D244—22no—
180D244—42no—
181D244—42no—
182D244—81no—
183D244#13514yes—
184C4×D61—42no—
185C4×D61—42no—
186C4×D61—82no—
187C4×D61—82no—
188Dic122—81no—
189C8×C61—42no—
190C8×C61—42no—
191#92418—22no—
192#92418—22no—
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×C61C2×#12111yescharacteristic
202C2×D61—24no—
203C2×D61—42no—
204C2×D61—42no—
205C2×D61—84no—
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—81no—
214C4×C61C2×#13512yes—
215C4×C61#57612yes—
216Dic61—42no—
217Dic61—81no—
218#3207—611no—
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#1553D6111yescharacteristic, complemented
228#1543—611no—
229C2×C61—41no—
230C2×C61—161no—
231C2×C61#304111yescharacteristic
232C2×C61#305011yescharacteristic
233C2×C61#306811yescharacteristic
234D61—44no—
235D61—161no—
236#532—611no—
237#532C2×D6111yescharacteristic
238C2×C4×D4—611no—
239#493—611no—
240#492—611no—
241#487—611no—
242#486—611no—
243#481—1221no—
244#480—1221no—
245#476—612no—
246#474—612no—
247#463—1222no—
248#459—612no—
249#459—612no—
250#449—1222no—
251#447—1221no—
252#447—1221no—
253#442—1222no—
254C4×D8—1222no—
255C8×D4—612no—
256C8×D4—612no—
257C8×D4—1222no—
258#432C2×D6111yescharacteristic
259#431—611no—
260#431—611no—
261#431C2×D6111yescharacteristic
262C2×C22⋊C8—1222no—
263#335—612no—
264#334—612no—
265#333—612no—
266#332C2×D6112yescomplemented
267#331—612no—
268#331C2×D6112yescomplemented
269C61#1938911yescharacteristic, complemented, Sylow
270C2×#41—21no—
271C2×#41—1221no—
272C22×D4—611no—
273C23×C4—1221no—
274C2×SD16—1222no—
275C2×D8—1222no—
276C2×D8—1222no—
277C2×D8—1222no—
278C22×C8—1222no—
279C22×C8—1222no—
280C4⋊Q8—611no—
281C4⋊Q8C22×D6111yescharacteristic
282#127—611no—
283#127C22×D6111yescharacteristic
284#124—21no—
285#124—1221no—
286#121—21no—
287#121—21no—
288#121—1221no—
289#121—1221no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—612no—
293C4×D4—612no—
294C4×D4—1221no—
295C4×D4—1221no—
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⋊C8C22×D6112yes—
306C4⋊C8C22×D6112yes—
307#103—1222no—
308#102—1222no—
309#102—1222no—
310#102—1222no—
311#102—1222no—
312#102—1222no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—612no—
316C22⋊C8—1222no—
317C22⋊C8—1222no—
318C22⋊C8—1221no—
319#97—21no—
320C4×C8—21no—
321C4×C8—611no—
322C4×C8—611no—
323C4×C8—1221no—
324C24—1221no—
325#41—82no—
326#41—4882no—
327C2×Q8—21no—
328C2×Q8—1221no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—612no—
334C2×D4—1224no—
335C2×D4—1221no—
336C2×D4—1221no—
337C2×D4—1221no—
338C2×D4—1221no—
339C2×D4—2441no—
340C22×C4—41no—
341C22×C4—1221no—
342C22×C4—1222no—
343C22×C4—1222no—
344C22×C4—4882no—
345C22×C4—2441no—
346C22×C4#1175822211yescharacteristic
347C22×C4D4×D6111yescharacteristic
348C22×C4D4×D6111yescharacteristic
349SD16—4882no—
350D8—4882no—
351D8—2444no—
352D8—2444no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—612no—
356C2×C8—612no—
357C2×C8—1222no—
358C2×C8—4882no—
359C2×C8—4882no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—612no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—1222no—
369C22⋊C4—1222no—
370C22⋊C4—2441no—
371C22⋊C4—2441no—
372C42#1175822211yescharacteristic
373C42D4×D6111yescharacteristic
374C42D4×D6111yescharacteristic
375C42C23×D6111yescharacteristic
376C23—41no—
377C23—1224no—
378C23—1222no—
379C23—4881no—
380C23—4881no—
381C23—2441no—
382C23#1451975011yescharacteristic
383Q8—42no—
384Q8—4881no—
385D4—42no—
386D4—42no—
387D4—81no—
388D4—82no—
389D4—614no—
390D4—1224no—
391D4—1222no—
392D4—4881no—
393D4—4882no—
394D4—2442no—
395D4—2442no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—82no—
401C2×C4—1222no—
402C2×C4—4882no—
403C2×C4—4882no—
404C2×C4—4881no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4#1451975011yescharacteristic
409C2×C4#1451975011yescharacteristic
410C2×C4#1451975011yescharacteristic
411C2×C4C2×D4×D6111yescharacteristic
412C2×C4C2×D4×D6111yescharacteristic
413C2×C4C2×#1175822211yescharacteristic
414C8—42no—
415C8—42no—
416C8—1222no—
417C8—1222no—
418C8—2442no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—1224no—
426C22—4884no—
427C22—4881no—
428C22—4881no—
429C22—4881no—
430C22—2442no—
431C22—2442no—
432C22C2×#1451975011yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—81no—
438C4—4881no—
439C4—2442no—
440C4#135×D6112yes—
441C4#1684056612yes—
442C2—41no—
443C2—161no—
444C2—9761no—
445C2—2444no—
446C2#2145532111yescharacteristic
447C2#2145539911yescharacteristic
448C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1553C2
socleC22×C61C2×#121
hypercenter#1553D61
hypocenterC61#19389
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1553

Complements

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

NH ≅ G/Ntype
1. #21456752C2semidirect
2. #21455911C2semidirect
3. #21455815C2semidirect
4. #21454842C2semidirect
5. #21445269C2semidirect
6. #21445227C2semidirect
7. #21445173C2semidirect
8. #21445163C2semidirect
9. #21444600C2semidirect
10. #21444479C2semidirect
11. C61×#1553C2semidirect
14. #16840138C22semidirect
15. #16840136C22semidirect
16. #16840112C22semidirect
17. #16840108C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
38. #16839510C22semidirect
39. #16839505C22semidirect
43. C61×#332C22semidirect
44. C61×#331C22semidirect
227. #1553D61semidirect
266. #332C2×D61semidirect
268. #331C2×D61semidirect
269. C61#19389semidirect

Maximal subgroups

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

subgroupindex
#214567522
#214559112
#214558152
#214548422
#214452692
#214452272
#214451732
#214451632
#214446002
#214444792
C61×#15532
#1938961