Finite Group Databasebeta

← #29833330

Subgroups — #29833330

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840200C211yesmaximal, characteristic
2#21454590C211yesmaximal, characteristic, complemented
3C2×#16839844C211yesmaximal, characteristic, complemented
4#21447694C212yesmaximal, complemented
5#21447682C212yesmaximal, complemented
6#21447670C212yesmaximal, complemented
7#21447665C212yesmaximal, complemented
8#21447534C211yesmaximal, characteristic, complemented
9#21447529C211yesmaximal, characteristic, complemented
10M32×Dic61C211yesmaximal, characteristic, complemented
11C61×#2159C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×D488C2211yescharacteristic
14#16840230—21no—
15C2×#14519739—21no—
16#16840200C418yescomplemented
17C2×#14519722C2212yes—
18C2×D976C2212yes—
19C2×#14519717C2211yescharacteristic
20#16839856C2211yescharacteristic, complemented
21#16839856C2211yescharacteristic, complemented
22#16839849—21no—
23#16839849—21no—
24#16839844—21no—
25#16839844—21no—
26#16839844C2212yes—
27#16839844C2212yes—
28#16839844C2211yescharacteristic, complemented
29#16839844C2211yescharacteristic, complemented
30#16839756—21no—
31C2×#14519697—21no—
32C2×C8×Dic61C2211yescharacteristic
33#16839565C2212yescomplemented
34#16839562C2212yes—
35#16839562C2212yes—
36#16839562C2212yescomplemented
37#16839537C2212yescomplemented
38C16×Dic61C2212yescomplemented
39C2×M32×C61C2211yescharacteristic
40C61×#427C2211yescharacteristic
41C2×#107×C61C2211yescharacteristic
42C61×#367C2212yescomplemented
43C61×#366C2212yescomplemented
44C2×#11758222—21no—
45C22×D244—21no—
46C22×C4×D61—41no—
47#14519799—22no—
48#14519799C2×C414yescomplemented
49C2×Dic244C2311yescharacteristic
50C2×#11758190—21no—
51C2×D488—21no—
52C2×D488—22no—
53C2×D488C2×C414yescomplemented
54C2×D488C2312yes—
55C2×D488C2311yescharacteristic
56C22×#92418D411yescharacteristic
57#14519742—21no—
58#14519742—21no—
59#14519739—22no—
60#14519739—22no—
61#14519739—21no—
62#14519739—21no—
63#14519739—41no—
64#14519739—41no—
65#14519737—41no—
66#14519737—41no—
67#14519734—41no—
68C2×#11758198—41no—
69C2×C4×Dic61D411yescharacteristic
70C42×D61—41no—
71#14519722C2×C418yescomplemented
72D976C2×C418yescomplemented
73#14519717—22no—
74C2×#11758185C2312yes—
75#14519700—21no—
76#14519700—21no—
77#14519697—22no—
78#14519697—22no—
79#14519697—21no—
80#14519697—21no—
81C8×Dic61D411yescharacteristic
82C8×Dic61D411yescharacteristic
83C8×Dic61C2311yescharacteristic
84C8×Dic61C2311yescharacteristic
85C22×C8×C61C2311yescharacteristic
86#117×C61—21no—
87C2×C4⋊C4×C61—21no—
88M32×C61C2×C414yescomplemented
89C2×C16×C61C2312yes—
90#107×C61—21no—
91#107×C61C2311yescharacteristic
92#107×C61C2312yes—
93#107×C61C2311yescharacteristic
94#106×C61—21no—
95C23×D61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C2×C22⋊D61—41no—
99C22×Dic61D811yescharacteristic
100C22×Dic61D811yescharacteristic
101#11758222—24no—
102#11758222—42no—
103C2×D244—24no—
104C2×D244—22no—
105C2×D244—21no—
106C2×D244—41no—
107C2×D244—42no—
108C2×C4×D61—41no—
109C2×C4×D61—42no—
110C2×C4×D61—42no—
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61—84no—
114C2×Dic122—21no—
115C2×C8×C61C22×C412yes—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61#4112yes—
118C2×C8×C61C2411yescharacteristic
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—41no—
123C22⋊C4×C61—41no—
124C42×C61—41no—
125#11758198—42no—
126#11758198—42no—
127#11758198—41no—
128#11758198—41no—
129#11758196—41no—
130#11758196—41no—
131C4×Dic61—41no—
132C4×Dic61—41no—
133C4×Dic61D811yescharacteristic
134C4×Dic61D811yescharacteristic
135C4×Dic61C2×D411yescharacteristic
136C4×Dic61C2×D411yescharacteristic
137C2×#92418—22no—
138C2×#92418C2×D411yescharacteristic
139C2×#92418C2×D411yescharacteristic
140Dic244—21no—
141Dic244C22×C412yescomplemented
142D488—24no—
143D488—22no—
144D488—21no—
145D488C22×C414yes—
146D488C22×C412yescomplemented
147#11758190—24no—
148C16×C61C22×C414yescomplemented
149#11758185—22no—
150C23×C61C2×D811yescharacteristic
151C22×D61—41no—
152C22×D61—42no—
153C22×D61—44no—
154C22×D61—82no—
155C22×D61—82no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—82no—
160C2×C4×C61C4×D412yes—
161C2×C4×C61C2×D811yescharacteristic
162C2×C4×C61#13512yes—
163C2×C4×C61C22×D411yescharacteristic
164C22⋊D61—44no—
165C2×Dic61—22no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61C2×D811yescharacteristic
169C2×Dic61C2×D811yescharacteristic
170C2×Dic61C2×D811yescharacteristic
171C2×Dic61C2×D811yescharacteristic
172D244—22no—
173D244—24no—
174D244—42no—
175D244—41no—
176D244—44no—
177C4×D61—44no—
178C4×D61—82no—
179C4×D61—84no—
180C4×D61—82no—
181C4×D61—84no—
182Dic122—22no—
183Dic122—41no—
184C8×C61C4×D412yes—
185C8×C61C23×C411yescharacteristic
186C8×C61C2×#4111yescharacteristic
187#92418—21no—
188#92418—21no—
189C4×#509—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61C4×D812yes—
193C22×C61C22×D811yescharacteristic
194C2×D61—42no—
195C2×D61—44no—
196C2×D61—88no—
197C2×D61—82no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—82no—
201C4×C61—82no—
202C4×C61—81no—
203C4×C61—81no—
204C4×C61C4×D812yes—
205C4×C61C2×C4×D411yescharacteristic
206C4×C61C2×#13511yescharacteristic
207Dic61—21no—
208Dic61—21no—
209Dic61—21no—
210Dic61—42no—
211Dic61#50912yescomplemented
212C2×#509—611no—
213C4×#135—611no—
214C2×C4×D8—611no—
215#2177—612no—
216#2175—612no—
217C4×SD32—612no—
218C4×D16—612no—
219#2159D6111yescharacteristic, complemented
220#2140—611no—
221#2139—611no—
222C4×M32—611no—
223C2×C61—22no—
224C2×C61C2×C4×D811yescharacteristic
225C2×C61C2×#50911yescharacteristic
226C2×C61#341411yescharacteristic
227D61—82no—
228D61—84no—
229C2×#135—611no—
230C22×D8—611no—
231C4×#41—1221no—
232C2×C4×D4—1221no—
233#509—618no—
234C2×SD32—612no—
235C2×D16—612no—
236C2×M32—611no—
237C2×M32C2×D6111yescharacteristic
238C4×Q16—611no—
239C4×Q16—611no—
240C4×SD16—1221no—
241C4×SD16—1221no—
242C4×D8—612no—
243C4×D8—612no—
244C4×D8—611no—
245C4×D8—611no—
246C4×D8—1221no—
247C4×D8—1221no—
248#427C2×D6111yescharacteristic
249C2×#107C2×D6111yescharacteristic
250#416—1221no—
251C2×#102—1221no—
252C2×C4×C8—611no—
253#367C2×D6112yescomplemented
254#366C2×D6112yescomplemented
255#358—612no—
256#357—612no—
257#357—612no—
258#357—612no—
259#346—612no—
260C4×C16—612no—
261C61C4×#50911yescharacteristic, complemented, Sylow
262C2×#41—1221no—
263C22×D4—1221no—
264C23×C4—2441no—
265#135—614no—
266#135—1222no—
267C2×Q16—611no—
268C2×SD16—1221no—
269C2×D8—614no—
270C2×D8—612no—
271C2×D8—611no—
272C2×D8—1221no—
273C2×D8—1222no—
274C22×C8—611no—
275C22×C8C22×D6111yescharacteristic
276C4×Q8—1221no—
277C4×Q8—1221no—
278C4×D4—1222no—
279C4×D4—1222no—
280C4×D4—1221no—
281C4×D4—1221no—
282C4×D4—2441no—
283C4×D4—2441no—
284C4×D4—2441no—
285C4×D4—2441no—
286#117—21no—
287#117—2441no—
288C2×C4⋊C4—21no—
289C2×C22⋊C4—2441no—
290C2×C42—611no—
291C2×C42—2441no—
292SD32—618no—
293D16—618no—
294M32—1222no—
295M32C4×D6114yescomplemented
296C2×C16—612no—
297C2×C16C22×D6112yes—
298#107—21no—
299#107C22×D6111yescharacteristic
300#107C22×D6111yescharacteristic
301#107C22×D6112yes—
302#106—21no—
303#103—1221no—
304#103—1221no—
305#102—1222no—
306#102—1222no—
307#102—1221no—
308#102—1221no—
309C4×C8—611no—
310C4×C8—611no—
311C4×C8—611no—
312C4×C8—611no—
313C24—2441no—
314#41—1224no—
315#41—2442no—
316C2×Q8—1221no—
317C2×D4—1222no—
318C2×D4—1221no—
319C2×D4—1224no—
320C2×D4—2441no—
321C2×D4—2441no—
322C2×D4—2442no—
323C22×C4—41no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—4884no—
327C22×C4—2441no—
328C22×C4—2441no—
329C22×C4—2442no—
330C22×C4—2442no—
331C22×C4—2441no—
332C22×C4D4×D6111yescharacteristic
333Q16—612no—
334Q16—1221no—
335SD16—1224no—
336D8—612no—
337D8—614no—
338D8—1222no—
339D8—1221no—
340D8—1224no—
341C2×C8—611no—
342C2×C8—611no—
343C2×C8—1222no—
344C2×C8C2×C4×D6112yes—
345C2×C8D4×D6111yescharacteristic
346C2×C8#1175822612yes—
347C2×C8C23×D6111yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—2441no—
353C4⋊C4—2441no—
354C22⋊C4—41no—
355C22⋊C4—2442no—
356C22⋊C4—2442no—
357C22⋊C4—2441no—
358C22⋊C4—2441no—
359C42—41no—
360C42—611no—
361C42—611no—
362C42—611no—
363C42—611no—
364C42—2441no—
365C42—2441no—
366C16—1222no—
367C16C2×C4×D6114yescomplemented
368C23—4882no—
369C23—4882no—
370C23—2441no—
371C23—2442no—
372C23—2444no—
373C23D8×D6111yescharacteristic
374Q8—1222no—
375Q8—2441no—
376D4—1224no—
377D4—1222no—
378D4—2444no—
379D4—2442no—
380D4—2441no—
381D4—2444no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—611no—
387C2×C4—611no—
388C2×C4—611no—
389C2×C4—611no—
390C2×C4—1222no—
391C2×C4—4884no—
392C2×C4—4882no—
393C2×C4—4884no—
394C2×C4—4882no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2444no—
398C2×C4#1451973912yes—
399C2×C4D8×D6111yescharacteristic
400C2×C4#1451980012yes—
401C2×C4C2×D4×D6111yescharacteristic
402C8—1221no—
403C8—1221no—
404C8#1451973912yes—
405C8C22×C4×D6111yescharacteristic
406C8C2×#1175822611yescharacteristic
407C22—21no—
408C22—21no—
409C22—4888no—
410C22—4881no—
411C22—4882no—
412C22—4882no—
413C22—4881no—
414C22—2442no—
415C22—2444no—
416C22#1683984412yes—
417C22C2×D8×D6111yescharacteristic
418C4—81no—
419C4—82no—
420C4—81no—
421C4—612no—
422C4—1221no—
423C4—1221no—
424C4—1221no—
425C4—2442no—
426C4#1683984412yes—
427C4C2×#1451973911yescharacteristic
428C4C2×#1451980011yescharacteristic
429C2—22no—
430C2—4884no—
431C2—4882no—
432C2C2×#1683984411yescharacteristic
433C2#509×D6111yescharacteristic
434C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
prime radicalC2×#16840200C2
cosocleC2×C8×C61C24
hypercenter#2159D61
hypocenterC61C4×#509
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
2. #21454590C2semidirect
3. C2×#16839844C2semidirect
4. #21447694C2semidirect
5. #21447682C2semidirect
6. #21447670C2semidirect
7. #21447665C2semidirect
8. #21447534C2semidirect
9. #21447529C2semidirect
10. M32×Dic61C2semidirect
11. C61×#2159C2semidirect
16. #16840200C4semidirect
20. #16839856C22semidirect
21. #16839856C22semidirect
28. #16839844C22semidirect
29. #16839844C22semidirect
33. #16839565C22semidirect
36. #16839562C22semidirect
37. #16839537C22semidirect
38. C16×Dic61C22semidirect
42. C61×#367C22semidirect
43. C61×#366C22semidirect
48. #14519799C2×C4semidirect
53. C2×D488C2×C4semidirect
71. #14519722C2×C4semidirect
72. D976C2×C4semidirect
88. M32×C61C2×C4semidirect
141. Dic244C22×C4semidirect
146. D488C22×C4semidirect
148. C16×C61C22×C4semidirect
211. Dic61#509semidirect
219. #2159D61semidirect
253. #367C2×D61semidirect
254. #366C2×D61semidirect
261. C61C4×#509semidirect
295. M32C4×D61semidirect
367. C16C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
#214545902
C2×#168398442
#214476942
#214476822
#214476702
#214476652
#214475342
#214475292
M32×Dic612
C61×#21592
C4×#50961