Finite Group Databasebeta

← #29663522

Subgroups — #29663522

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457131C211yesmaximal, characteristic
2#21456746C211yesmaximal, characteristic
3#21456366C211yesmaximal, characteristic, complemented
4#21454562C211yesmaximal, characteristic
5#21445331C212yesmaximal, complemented
6#21445321C212yesmaximal, complemented
7#21445207C212yesmaximal, complemented
8#21445197C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10C2×#16839507C211yesmaximal, characteristic, complemented
11C61×#1542C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840172—22no—
16#16840165C2212yes—
17#16840158C2212yes—
18#16840135—21no—
19#16840113—21no—
20#16840049C2212yes—
21#16840047—21no—
22#16839993—22no—
23#16839968C2212yes—
24#16839968C2212yes—
25#16839964C2212yes—
26#16839962—21no—
27#16839903—22no—
28#16839839C2212yes—
29#16839832—22no—
30#16839827C2212yes—
31#16839805—21no—
32#16839805—21no—
33C2×C4⋊#92418C2211yescharacteristic
34#16839728—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yes—
37#16839512C2212yes—
38#16839507C2212yes—
39#16839507C2212yes—
40C61×#533C2211yescharacteristic
41C61×#431C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C61×#333C2211yescharacteristic
44C61×#333C2211yescharacteristic
45C61×#332C2211yescharacteristic
46C61×#332C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758190—22no—
50C2×#11758190—22no—
51C2×#11758201—22no—
52#14519783—44no—
53C2×#11758194D411yescharacteristic
54C2×#11758194D411yescharacteristic
55C2×#11758189—22no—
56C22×#92418—21no—
57C2×C8×D61—22no—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777D411yescharacteristic
62#14519777C2312yes—
63#14519774D411yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751D412yes—
67C4×D244—21no—
68C4×D244C2312yes—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71#14519710—22no—
72#14519710—22no—
73#14519708—22no—
74#14519708—22no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418C2312yes—
78#14519702—22no—
79#14519702—22no—
80#14519701—22no—
81#14519697—22no—
82#14519689—22no—
83#14519689—22no—
84#14519688D412yes—
85C4×#92418C2312yes—
86C22×C8×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94#117×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2311yescharacteristic
100C4⋊C8×C61C2311yescharacteristic
101C22⋊C8×C61—21no—
102C4×C8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C2×C22⋊D61—22no—
109C22×Dic61—41no—
110#11758222—42no—
111#11758222—42no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244C2×D412yes—
115C2×C4×D61—22no—
116C2×Dic122—21no—
117C2×Dic122—22no—
118C2×Dic122—41no—
119C2×Dic122—81no—
120C2×Dic122—81no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—42no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C4⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C42×C61—41no—
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C42×C61C2411yescharacteristic
138#11758201—82no—
139#11758198—22no—
140#11758198—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146C4⋊Dic61C2×D412yes—
147#11758194—22no—
148#11758194SD1614yescomplemented
149C2×#92418—22no—
150C2×#92418—81no—
151C2×#92418C2×D412yes—
152C2×#92418C2×D412yes—
153#11758190—82no—
154#11758190—82no—
155#11758189—44no—
156C8×D61—44no—
157C23×C61#12111yescharacteristic
158C22×D61—22no—
159Q8×C61—41no—
160Q8×C61—41no—
161Q8×C61—81no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61C2×SD1612yes—
171C2×C4×C61#13712yes—
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C22⋊D61—44no—
176C2×Dic61—22no—
177C2×Dic61—42no—
178C2×Dic61—82no—
179D244—22no—
180D244—22no—
181D244—42no—
182D244—81no—
183C4×D61—42no—
184C4×D61—42no—
185Dic122—42no—
186Dic122—42no—
187Dic122—42no—
188Dic122—82no—
189C8×C61—41no—
190C8×C61—41no—
191C8×C61—41no—
192C8×C61—41no—
193#92418—22no—
194#92418—42no—
195#92418C2×SD1614yes—
196#19232—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61#46112yes—
200C22×C61C2×#12111yescharacteristic
201C2×D61—42no—
202C2×D61—42no—
203C2×D61—42no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—81no—
208C4×C61—81no—
209C4×C61#46112yes—
210C4×C61C22×SD1611yescharacteristic
211C4×C61C2×#13711yescharacteristic
212C4×C61#57611yescharacteristic
213C4×C61#57711yescharacteristic
214Dic61—42no—
215Dic61—82no—
216#3246—611no—
217#3206—611no—
218#3165—611no—
219#2935—611no—
220#1693—612no—
221#1691—612no—
222#1673—612no—
223#1671—612no—
224#1565—611no—
225#1542D6111yescharacteristic, complemented
226C2×#332—611no—
227C2×C61—22no—
228C2×C61C2×#46111yescharacteristic
229C2×C61#306211yescharacteristic
230C2×C61#306911yescharacteristic
231D61—82no—
232C2×#131—1221no—
233#533C2×D6111yescharacteristic
234#532—611no—
235C2×C4⋊Q8—611no—
236#501—1222no—
237#500—612no—
238#498—612no—
239#492—1221no—
240#487—1221no—
241#476—612no—
242#475—1221no—
243#465—1222no—
244#461—612no—
245#461—612no—
246#461—612no—
247#460—1221no—
248#448—1222no—
249#436—612no—
250#435—1222no—
251C8×D4—612no—
252#431C2×D6111yescharacteristic
253#428—1221no—
254#428—1221no—
255C2×C4⋊C8—611no—
256C2×C4⋊C8C2×D6111yescharacteristic
257#408—1221no—
258C4×M16—611no—
259#335—612no—
260#334—612no—
261#333C2×D6111yescharacteristic
262#333C2×D6111yescharacteristic
263#332—612no—
264#332—612no—
265#332C2×D6111yescharacteristic
266#332C2×D6111yescharacteristic
267C61#1923211yescharacteristic, complemented, Sylow
268C2×#41—1221no—
269C22×Q8—1221no—
270C2×SD16—1222no—
271C2×SD16—1222no—
272C2×SD16—1222no—
273#131—2444no—
274C2×M16—611no—
275C2×M16—611no—
276C2×M16—1222no—
277C22×C8—21no—
278C22×C8—1222no—
279C22×C8—1221no—
280C4⋊Q8—611no—
281C4⋊Q8—612no—
282C4⋊Q8—1221no—
283C4⋊Q8—1221no—
284C4⋊Q8—1221no—
285C4⋊Q8C22×D6111yescharacteristic
286C4⋊Q8C22×D6111yescharacteristic
287#127—611no—
288#125—21no—
289#124—1221no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292#121—612no—
293#121—1221no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×D4—612no—
297C4×D4—1221no—
298#117—21no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4—1221no—
301C2×C42C22×D6111yescharacteristic
302#107—1222no—
303#107—1222no—
304#106—1222no—
305#106—1222no—
306#106—1222no—
307C4⋊C8—21no—
308C4⋊C8—612no—
309C4⋊C8—1221no—
310C4⋊C8C22×D6111yescharacteristic
311C4⋊C8C22×D6111yescharacteristic
312C4⋊C8C22×D6111yescharacteristic
313C4⋊C8C22×D6111yescharacteristic
314#103—1222no—
315#103—1222no—
316#103—1222no—
317#102—1222no—
318C22⋊C8—21no—
319C22⋊C8—1222no—
320C22⋊C8—1222no—
321#97—612no—
322C4×C8—21no—
323C4×C8—612no—
324#41—2442no—
325#41—2442no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—1221no—
329C2×Q8—1222no—
330C2×Q8—4881no—
331C2×Q8—4881no—
332C2×Q8—2441no—
333C2×D4—612no—
334C2×D4—1222no—
335C2×D4—1221no—
336C2×D4—1221no—
337C22×C4—1222no—
338C22×C4—2441no—
339C22×C4#1175822211yescharacteristic
340C22×C4D4×D6111yescharacteristic
341C22×C4D4×D6111yescharacteristic
342SD16—4882no—
343SD16—4882no—
344SD16—4882no—
345M16—614no—
346M16—1222no—
347M16—2444no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—42no—
353C2×C8—612no—
354C2×C8—612no—
355C2×C8—1222no—
356C2×C8—4881no—
357C2×C8—2444no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—612no—
365C4⋊C4—1221no—
366C4⋊C4—1222no—
367C4⋊C4—1222no—
368C4⋊C4—2441no—
369C4⋊C4—2441no—
370C22⋊C4—41no—
371C22⋊C4—1222no—
372C22⋊C4—1222no—
373C42—41no—
374C42#1175822211yescharacteristic
375C42D4×D6111yescharacteristic
376C42D4×D6111yescharacteristic
377C42C23×D6111yescharacteristic
378C23—1222no—
379C23#1451975011yescharacteristic
380Q8—41no—
381Q8—41no—
382Q8—81no—
383Q8—4882no—
384Q8—2442no—
385Q8—2442no—
386Q8—2442no—
387D4—1222no—
388D4—1222no—
389D4—4881no—
390D4—2444no—
391D4—2442no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—1222no—
398C2×C4—4882no—
399C2×C4—2442no—
400C2×C4—2442no—
401C2×C4—2442no—
402C2×C4#1451975011yescharacteristic
403C2×C4#1451975011yescharacteristic
404C2×C4#1451975011yescharacteristic
405C2×C4SD16×D6112yes—
406C2×C4#1451980912yes—
407C2×C4C2×D4×D6111yescharacteristic
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4C2×#1175822211yescharacteristic
410C8—41no—
411C8—41no—
412C8—41no—
413C8—41no—
414C8—614no—
415C8—1222no—
416C8—2442no—
417C22—21no—
418C22—21no—
419C22—2442no—
420C22—2442no—
421C22—2442no—
422C22#1683996612yes—
423C22C2×#1451975011yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—81no—
428C4—81no—
429C4—4882no—
430C4—2442no—
431C4#1683996612yes—
432C4C2×SD16×D6111yescharacteristic
433C4C2×#1451980911yescharacteristic
434C4#1684056411yescharacteristic
435C4#1684056911yescharacteristic
436C2—22no—
437C2—4882no—
438C2C2×#1683996611yescharacteristic
439C2#2145550211yescharacteristic
440C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1542D61
hypocenterC61#19232
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
3. #21456366C2semidirect
5. #21445331C2semidirect
6. #21445321C2semidirect
7. #21445207C2semidirect
8. #21445197C2semidirect
9. #21444572C2semidirect
10. C2×#16839507C2semidirect
11. C61×#1542C2semidirect
148. #11758194SD16semidirect
225. #1542D61semidirect
267. C61#19232semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214567462
#214563662
#214545622
#214453312
#214453212
#214452072
#214451972
#214445722
C2×#168395072
C61×#15422
#1923261