Finite Group Databasebeta

← #29834273

Subgroups — #29834273

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840191C212yesmaximal, complemented
2#21456246C211yesmaximal, characteristic, complemented
3C4×#14519796C211yesmaximal, characteristic, complemented
4#21447880C212yesmaximal, complemented
5#21447872C212yesmaximal, complemented
6#21447870C211yesmaximal, characteristic, complemented
7#21447864C211yesmaximal, characteristic, complemented
8#21447856C211yesmaximal, characteristic, complemented
9C4⋊C16×D61C211yesmaximal, characteristic, complemented
10#21447523C212yesmaximal, complemented
11C61×#2206C211yesmaximal, characteristic, complemented
12C2×#14519796—21no—
13C2×#14519796C2212yes—
14C2×#14519796C2211yescharacteristic
15#16840310—21no—
16C4×#11758224—21no—
17#16840191—28no—
18C2×Dic488C2212yes—
19C2×#14519721C2212yes—
20C2×#14519721C2212yes—
21C2×C16×D61C2212yes—
22#16840145—21no—
23#16840145C2212yes—
24#16840142C2211yescharacteristic
25#16840134—21no—
26C4×Dic244C2211yescharacteristic
27#16839856C2211yescharacteristic
28#16839853—21no—
29C4×#11758200—21no—
30D8×Dic61C2211yescharacteristic
31#16839799C2211yescharacteristic
32#16839755—21no—
33C4×C8×D61C2211yescharacteristic
34C4⋊#11758185C2211yescharacteristic
35#16839580C2212yes—
36#16839567C2212yes—
37#16839565C2212yes—
38#16839563C2212yes—
39C2×D16×C61C2212yes—
40C61×#493C2211yescharacteristic
41C4×D8×C61C2211yescharacteristic
42C61×C4⋊C16C2211yescharacteristic
43C61×#357C2212yes—
44C2×#11758224—21no—
45C2×#11758224—22no—
46#14519796—24no—
47#14519796—22no—
48#14519796—24no—
49#14519796—44no—
50#14519796D414yescomplemented
51C2×Dic244—21no—
52C2×Dic244C2312yes—
53C2×Dic244C2311yescharacteristic
54C2×#11758200—21no—
55C2×#11758200—22no—
56C2×#11758200—22no—
57C2×C8×D61D411yescharacteristic
58C2×C8×D61C2311yescharacteristic
59#14519777—21no—
60#14519764—22no—
61#14519748—41no—
62#14519748—41no—
63C4×Dic122—21no—
64#14519742—21no—
65D4×Dic61—21no—
66D4×Dic61—41no—
67D4×Dic61—41no—
68C4×C22⋊D61—41no—
69#14519737—41no—
70#14519735—21no—
71#14519733—41no—
72C2×C4×Dic61—41no—
73C42×D61D411yescharacteristic
74Dic488—24no—
75#14519721—24no—
76#14519721—24no—
77C2×#11758185C2312yes—
78C16×D61—24no—
79#14519711C2311yescharacteristic
80#14519708—21no—
81#14519702—21no—
82#14519700—21no—
83#14519698—21no—
84C4×#92418D411yescharacteristic
85C8×Dic61D411yescharacteristic
86C8×Dic61C2311yescharacteristic
87C2×D8×C61—21no—
88C2×D8×C61C2312yes—
89C2×D8×C61C2311yescharacteristic
90#127×C61—21no—
91C4×D4×C61—21no—
92D16×C61—24no—
93C2×C16×C61C2312yes—
94#107×C61C2311yescharacteristic
95#102×C61—21no—
96C4×C8×C61C2311yescharacteristic
97C2×D4×C61—21no—
98C2×D4×C61—22no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C22×C4×C61—41no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—42no—
104C22×Dic61—41no—
105C22×Dic61—42no—
106C22×Dic61—41no—
107#11758224—24no—
108#11758224—44no—
109#11758224—42no—
110#11758224—44no—
111C2×C4×D61D811yescharacteristic
112C2×C4×D61D811yescharacteristic
113C2×C4×D61C2×D411yescharacteristic
114C2×Dic122—21no—
115C2×Dic122—22no—
116D8×C61—22no—
117D8×C61—21no—
118D8×C61—22no—
119D8×C61—42no—
120D8×C61C2×D412yes—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2411yescharacteristic
123C4⋊C4×C61—21no—
124C22⋊C4×C61—41no—
125C42×C61C2×D411yescharacteristic
126C22⋊Dic61—41no—
127C22⋊Dic61—42no—
128C22⋊Dic61—42no—
129#11758200—24no—
130#11758200—44no—
131#11758200—44no—
132#11758198—41no—
133C4⋊Dic61—21no—
134C4⋊Dic61—41no—
135C4⋊Dic61—41no—
136#11758196—41no—
137#11758196—41no—
138C4×Dic61—41no—
139C4×Dic61—41no—
140C4×Dic61—41no—
141C4×Dic61D811yescharacteristic
142C4×Dic61D811yescharacteristic
143C4×Dic61C2×D411yescharacteristic
144C2×#92418C2×D411yescharacteristic
145C2×#92418C2×D411yescharacteristic
146Dic244—22no—
147Dic244—21no—
148Dic244—22no—
149Dic244—42no—
150Dic244C2×D412yes—
151C8×D61—22no—
152C8×D61C2×D412yes—
153C8×D61#4112yes—
154C16×C61—22no—
155#11758185—22no—
156C23×C61—41no—
157C23×C61—42no—
158C22×D61C2×D811yescharacteristic
159D4×C61—22no—
160D4×C61—42no—
161D4×C61—41no—
162D4×C61—42no—
163D4×C61—82no—
164D4×C61—82no—
165C2×C4×C61—41no—
166C2×C4×C61—82no—
167C2×C4×C61C2×D811yescharacteristic
168C2×C4×C61C2×D811yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C22⋊D61—44no—
171C22⋊D61—84no—
172C2×Dic61—41no—
173C2×Dic61—44no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—82no—
177C2×Dic61—84no—
178C2×Dic61C2×D811yescharacteristic
179C2×Dic61C2×D811yescharacteristic
180C2×Dic61C2×D811yescharacteristic
181C4×D61—22no—
182C4×D61—41no—
183C4×D61#12112yes—
184C4×D61AGL(1,ℤ/8)12yes—
185Dic122—22no—
186Dic122—41no—
187Dic122—42no—
188Dic122—42no—
189C8×C61—21no—
190C8×C61C22×D411yescharacteristic
191C8×C61C2×#4111yescharacteristic
192#92418—21no—
193#92418#12112yes—
194#33579—611nomaximal, Sylow
195C22×C61—42no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61—82no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61C22×D811yescharacteristic
202C2×D61—21no—
203C2×D61—21no—
204C2×D61#45912yes—
205C4×C61—21no—
206C4×C61—81no—
207C4×C61#50812yescomplemented
208C4×C61C2×#12111yescharacteristic
209C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
210Dic61—21no—
211Dic61—21no—
212Dic61—42no—
213Dic61—81no—
214Dic61—82no—
215Dic61#45912yes—
216C2×#508—612no—
217#3147—611no—
218C4×#135—611no—
219#2211—612no—
220#2210—612no—
221#2209—611no—
222#2208—611no—
223#2207—611no—
224#2206D6111yescharacteristic, complemented
225C2×C4⋊C16—611no—
226#2138—612no—
227C2×C61—82no—
228C2×C61—162no—
229C2×C61C2×#45911yescharacteristic
230C2×C61C2×#50811yescharacteristic
231C2×C61#341411yescharacteristic
232D61—22no—
233C2×#135—611no—
234C2×#135—612no—
235C2×#135—1221no—
236#532—1221no—
237C4×#41—1221no—
238#508—1228no—
239C2×Q32—612no—
240C2×SD32—612no—
241C2×SD32—612no—
242C2×D16C2×D6112yes—
243C22×C16—612no—
244#495—612no—
245#495—1221no—
246#494—611no—
247#493C2×D6111yescharacteristic
248#492—1221no—
249C4×Q16—611no—
250C4×Q16—611no—
251C4×SD16—1221no—
252C4×SD16—1221no—
253C4×D8—611no—
254C4×D8C2×D6111yescharacteristic
255#427—611no—
256#416—1221no—
257C2×C4×C8—611no—
258C4⋊C16—612no—
259C4⋊C16—611no—
260C4⋊C16C2×D6111yescharacteristic
261#358—612no—
262#358—612no—
263#357—612no—
264#357C2×D6112yes—
265C61#3357911yescharacteristic, complemented, Sylow
266C2×#41—1221no—
267C2×#41—1222no—
268#135—614no—
269#135—1222no—
270#135—1224no—
271#135—1224no—
272#135—2444no—
273C2×Q16—611no—
274C2×Q16—612no—
275C2×Q16—1221no—
276C2×SD16—1221no—
277C2×SD16—1222no—
278C2×SD16—1222no—
279C2×D8—21no—
280C2×D8C22×D6112yes—
281C2×D8C22×D6111yescharacteristic
282C22×C8—611no—
283C22×C8—611no—
284C4⋊Q8—1221no—
285#127—21no—
286#124—1222no—
287#121—2441no—
288#121—2441no—
289C4×Q8—1221no—
290C4×Q8—1221no—
291C4×D4—21no—
292C4×D4—1221no—
293C4×D4—2441no—
294C4×D4—2441no—
295C4×D4—2441no—
296C4×D4—2441no—
297#117—1221no—
298#117—2441no—
299C2×C42—611no—
300C2×C42—2441no—
301Q32—1224no—
302SD32—1224no—
303SD32—1224no—
304D16—24no—
305C2×C16—612no—
306C2×C16—1224no—
307C2×C16C22×D6112yes—
308#107—611no—
309#107C22×D6111yescharacteristic
310#106—1221no—
311#103—1221no—
312#103—1221no—
313#102—21no—
314#102—1221no—
315C4×C8—611no—
316C4×C8—611no—
317C4×C8—611no—
318C4×C8C22×D6111yescharacteristic
319#41—1224no—
320#41—2444no—
321#41—2444no—
322#41—2442no—
323C2×Q8—1221no—
324C2×Q8—1222no—
325C2×D4—21no—
326C2×D4—22no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—2442no—
330C2×D4—2441no—
331C22×C4—41no—
332C22×C4—611no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—2441no—
336C22×C4—2442no—
337C22×C4—2441no—
338Q16—612no—
339Q16—1222no—
340Q16—1221no—
341Q16—1222no—
342Q16—2442no—
343SD16—1224no—
344SD16—2444no—
345SD16—2444no—
346D8—22no—
347D8—21no—
348D8—22no—
349D8—42no—
350D8D4×D6112yes—
351C2×C8—611no—
352C2×C8—612no—
353C2×C8—612no—
354C2×C8—611no—
355C2×C8—1222no—
356C2×C8D4×D6111yescharacteristic
357C2×C8C23×D6111yescharacteristic
358C4⋊C4—21no—
359C4⋊C4—1221no—
360C4⋊C4—2441no—
361C4⋊C4—2441no—
362C4⋊C4—2441no—
363C4⋊C4—2441no—
364C22⋊C4—41no—
365C22⋊C4—2441no—
366C22⋊C4—2441no—
367C22⋊C4—2442no—
368C22⋊C4—2442no—
369C42—611no—
370C42—611no—
371C42—611no—
372C42—2441no—
373C42—2441no—
374C42—2441no—
375C42D4×D6111yescharacteristic
376C16—22no—
377C16—1222no—
378C23—41no—
379C23—42no—
380C23—611no—
381Q8—1222no—
382Q8—2442no—
383Q8—2442no—
384Q8—2441no—
385D4—22no—
386D4—42no—
387D4—41no—
388D4—42no—
389D4—82no—
390D4—82no—
391D4—4884no—
392D4—2444no—
393C2×C4—41no—
394C2×C4—82no—
395C2×C4—611no—
396C2×C4—612no—
397C2×C4—612no—
398C2×C4—611no—
399C2×C4—611no—
400C2×C4—1222no—
401C2×C4—4884no—
402C2×C4—4882no—
403C2×C4—2441no—
404C2×C4—2444no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4—2441no—
408C2×C4D8×D6111yescharacteristic
409C2×C4D8×D6111yescharacteristic
410C2×C4C2×D4×D6111yescharacteristic
411C8—21no—
412C8—612no—
413C8—1221no—
414C8C2×D4×D6111yescharacteristic
415C8#41×D6111yescharacteristic
416C22—42no—
417C22—81no—
418C22—81no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—612no—
423C22—1221no—
424C22—1221no—
425C22C2×D8×D6111yescharacteristic
426C4—21no—
427C4—81no—
428C4—612no—
429C4—1221no—
430C4—1221no—
431C4—4881no—
432C4—4882no—
433C4—2442no—
434C4#1684019112yescomplemented
435C4#121×D6111yescharacteristic
436C4AGL(1,ℤ/8)×D6111yescharacteristic
437C2—82no—
438C2—162no—
439C2—1222no—
440C2#459×D6111yescharacteristic
441C2C2×#1684019111yescharacteristic
442C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2206C2
socleC22×C61C22×D8
hypercenter#2206D61
hypocenterC61#33579
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2206

Complements

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

NH ≅ G/Ntype
1. C2×#16840191C2semidirect
2. #21456246C2semidirect
3. C4×#14519796C2semidirect
4. #21447880C2semidirect
5. #21447872C2semidirect
6. #21447870C2semidirect
7. #21447864C2semidirect
8. #21447856C2semidirect
9. C4⋊C16×D61C2semidirect
10. #21447523C2semidirect
11. C61×#2206C2semidirect
50. #14519796D4semidirect
207. C4×C61#508semidirect
224. #2206D61semidirect
265. C61#33579semidirect
434. C4#16840191semidirect

Maximal subgroups

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

subgroupindex
C2×#168401912
#214562462
C4×#145197962
#214478802
#214478722
#214478702
#214478642
#214478562
C4⋊C16×D612
#214475232
C61×#22062
#3357961