Finite Group Databasebeta

← #29665208

Subgroups — #29665208

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#21456775C211yesmaximal, characteristic, complemented
2#21455919C211yesmaximal, characteristic, complemented
3#21455815C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic, complemented
5#21445255C212yesmaximal, complemented
6#21445241C212yesmaximal, complemented
7#21445183C212yesmaximal, complemented
8#21445159C212yesmaximal, complemented
9#21444600C211yesmaximal, characteristic, complemented
10#21444483C211yesmaximal, characteristic, complemented
11C61×#1553C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840141C2211yescharacteristic, complemented
15#16840134C2211yescharacteristic, complemented
16#16840112C2211yescharacteristic, complemented
17#16840108C2211yescharacteristic, complemented
18#16840098—21no—
19#16840073—21no—
20#16840043C2212yes—
21#16840039C2212yes—
22#16839981—22no—
23#16839968C2212yes—
24#16839954C2212yes—
25#16839924—22no—
26#16839915—21no—
27#16839895—21no—
28#16839864—22no—
29C4×#11758199—22no—
30C8×D244—22no—
31#16839827C2212yes—
32#16839827C2212yes—
33#16839815C2211yescharacteristic
34#16839815C2211yescharacteristic
35C22⋊C8×D61—22no—
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839509C2212yescomplemented
39#16839507C2212yescomplemented
40C61×#532C2211yescharacteristic
41C61×#432C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C61×#332C2212yescomplemented
44C61×#331C2212yescomplemented
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C2×#11758202—22no—
48C2×#11758190—22no—
49C2×#11758200—22no—
50C2×#11758199—22no—
51C2×C8×D61—22no—
52C2×C8×D61—22no—
53#14519777C2311yescharacteristic
54#14519777C2311yescharacteristic
55#14519769—21no—
56#14519756—21no—
57#14519756—21no—
58C4×Dic122—21no—
59C4×Dic122—21no—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244D412yes—
63C4×D244C2312yes—
64#14519736—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×#11758198—21no—
67#14519711—22no—
68#14519710—22no—
69C4⋊#92418D412yes—
70C4⋊#92418C2312yes—
71#14519702—22no—
72#14519701—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
110C22×Dic61—21no—
111C2×D244—21no—
112C2×D244—24no—
113C2×D244C2×D412yes—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—82no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×C8×C61—22no—
120C2×C8×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C22⋊C4×C61—41no—
124C22⋊C4×C61—41no—
125C42×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61#4111yescharacteristic
128C42×C61C2411yescharacteristic
129C22⋊Dic61—41no—
130#11758202—82no—
131#11758200—82no—
132#11758199—44no—
133#11758198—22no—
134#11758198—22no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61C2×D412yes—
141#11758196—41no—
142#11758196—41no—
143C4×Dic61—41no—
144C2×#92418—22no—
145C2×#92418C2×D412yes—
146C2×#92418C2×D412yes—
147#11758190—44no—
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—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—81no—
160D4×C61—82no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—82no—
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C2×Dic61—22no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—81no—
176D244—24no—
177D244—22no—
178D244#13514yes—
179C4×D61—42no—
180C4×D61—42no—
181C4×D61—82no—
182Dic122—42no—
183Dic122—42no—
184C8×C61—42no—
185C8×C61—42no—
186#92418—22no—
187#92418—22no—
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×C61C2×#13512yes—
207C4×C61#57612yes—
208Dic61—42no—
209Dic61—81no—
210Dic61—81no—
211#3210—611no—
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#1553D6111yescharacteristic, complemented
221#1544—611no—
222C2×C61—41no—
223C2×C61—161no—
224C2×C61#304111yescharacteristic
225C2×C61#305011yescharacteristic
226C2×C61#306911yescharacteristic
227D61—44no—
228#533—611no—
229#532C2×D6111yescharacteristic
230C2×C4×D4—611no—
231#494—611no—
232#492—611no—
233#487—611no—
234#486—611no—
235#484—1221no—
236#480—1221no—
237#474—612no—
238#474—612no—
239#464—1222no—
240#461—612no—
241#459—612no—
242#452—1222no—
243#450—1221no—
244#447—1221no—
245#440—1222no—
246C4×D8—1222no—
247C8×D4—612no—
248C8×D4—612no—
249C8×D4—1222no—
250#432C2×D6111yescharacteristic
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#332C2×D6112yescomplemented
260#331C2×D6112yescomplemented
261C61#1938811yescharacteristic, complemented, Sylow
262C2×#41—21no—
263C22×D4—611no—
264C23×C4—1221no—
265C2×Q16—1222no—
266C2×SD16—1222no—
267C2×SD16—1222no—
268C2×D8—1222no—
269C22×C8—1222no—
270C22×C8—1222no—
271C4⋊Q8—611no—
272C4⋊Q8—611no—
273C4⋊Q8C22×D6111yescharacteristic
274#127C22×D6111yescharacteristic
275#125—1221no—
276#124—21no—
277C22⋊Q8—1221no—
278C22⋊Q8—1221no—
279#121—21no—
280#121—21no—
281C4×Q8—1221no—
282C4×Q8—1221no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—612no—
286C4×D4—612no—
287C4×D4—1221no—
288C4×D4—1221no—
289#117—1221no—
290C2×C4⋊C4—611no—
291C2×C22⋊C4—1221no—
292C2×C42C22×D6111yescharacteristic
293#107—1222no—
294#107—1222no—
295C4⋊C8—612no—
296C4⋊C8—612no—
297C4⋊C8C22×D6112yes—
298C4⋊C8C22×D6112yes—
299#103—1222no—
300#103—1222no—
301#102—1222no—
302#102—1222no—
303#102—1222no—
304#102—1222no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—612no—
308C22⋊C8—1222no—
309C22⋊C8—1222no—
310C22⋊C8—1221no—
311#97—21no—
312C4×C8—21no—
313C4×C8—611no—
314C4×C8—611no—
315C4×C8—1221no—
316C24—1221no—
317#41—82no—
318C2×Q8—21no—
319C2×Q8—1221no—
320C2×Q8—1221no—
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×C4#1175822211yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4D4×D6111yescharacteristic
336Q16—4882no—
337SD16—4882no—
338SD16—2444no—
339D8—2444no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—612no—
343C2×C8—612no—
344C2×C8—1222no—
345C2×C8—4882no—
346C2×C8—4882no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—612no—
350C4⋊C4—1221no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—1221no—
355C4⋊C4—2441no—
356C4⋊C4—2441no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—1222no—
360C22⋊C4—1222no—
361C22⋊C4—2441no—
362C42—2441no—
363C42#1175822211yescharacteristic
364C42D4×D6111yescharacteristic
365C42D4×D6111yescharacteristic
366C42C23×D6111yescharacteristic
367C23—41no—
368C23—1224no—
369C23—1222no—
370C23—4881no—
371C23—4881no—
372C23#1451975011yescharacteristic
373Q8—42no—
374Q8—2442no—
375Q8—2442no—
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—82no—
388C2×C4—1222no—
389C2×C4—4882no—
390C2×C4—4881no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4#1451975011yescharacteristic
396C2×C4#1451975011yescharacteristic
397C2×C4#1451975011yescharacteristic
398C2×C4C2×D4×D6111yescharacteristic
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C2×#1175822211yescharacteristic
401C8—42no—
402C8—42no—
403C8—1222no—
404C8—1222no—
405C8—2442no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—1224no—
413C22—4884no—
414C22—2442no—
415C22—2442no—
416C22C2×#1451975011yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—81no—
422C4—4881no—
423C4—4881no—
424C4—2442no—
425C4#135×D6112yes—
426C4#1684056612yes—
427C2—41no—
428C2—161no—
429C2—2444no—
430C2#2145532111yescharacteristic
431C2#2145539911yescharacteristic
432C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1553C2
socleC22×C61C2×#121
prime radical#21455815C2
hypercenter#1553D61
hypocenterC61#19388
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. #21456775C2semidirect
2. #21455919C2semidirect
4. #21454842C2semidirect
5. #21445255C2semidirect
6. #21445241C2semidirect
7. #21445183C2semidirect
8. #21445159C2semidirect
9. #21444600C2semidirect
10. #21444483C2semidirect
11. C61×#1553C2semidirect
14. #16840141C22semidirect
15. #16840134C22semidirect
16. #16840112C22semidirect
17. #16840108C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
38. #16839509C22semidirect
39. #16839507C22semidirect
43. C61×#332C22semidirect
44. C61×#331C22semidirect
220. #1553D61semidirect
259. #332C2×D61semidirect
260. #331C2×D61semidirect
261. C61#19388semidirect

Maximal subgroups

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

subgroupindex
#214567752
#214559192
#214558152
#214548422
#214452552
#214452412
#214451832
#214451592
#214446002
#214444832
C61×#15532
#1938861