Finite Group Databasebeta

← #29833336

Subgroups — #29833336

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×#16840197C211yesmaximal, characteristic
2#135×Dic61C211yesmaximal, characteristic, complemented
3C2×#16839844C211yesmaximal, characteristic, complemented
4#21447692C212yesmaximal, complemented
5#21447686C212yesmaximal, complemented
6#21447668C212yesmaximal, complemented
7#21447663C212yesmaximal, complemented
8#21447605C211yesmaximal, characteristic, complemented
9#21447536C211yesmaximal, characteristic, complemented
10M32×Dic61C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13#41×Dic61—21no—
14C2×#14519739—21no—
15#16840197C418yescomplemented
16C2×#14519720C2212yes—
17C2×#14519718C2212yes—
18C2×#14519717C2211yescharacteristic
19Q16×Dic61C2211yescharacteristic, complemented
20Q16×Dic61C2211yescharacteristic, complemented
21SD16×Dic61—21no—
22SD16×Dic61—21no—
23D8×Dic61C2211yescharacteristic, complemented
24D8×Dic61C2211yescharacteristic, complemented
25#16839844—21no—
26#16839844—21no—
27#16839844C2212yes—
28#16839844C2212yes—
29#16839800C2211yescharacteristic
30#16839758—21no—
31C2×#14519697—21no—
32C2×C8×Dic61C2211yescharacteristic
33#16839589C2212yescomplemented
34#16839587C2212yescomplemented
35#16839564C2212yes—
36#16839564C2212yes—
37#16839537C2212yescomplemented
38C16×Dic61C2212yescomplemented
39C2×#135×C61C2211yescharacteristic
40C2×M32×C61C2211yescharacteristic
41C2×#107×C61C2211yescharacteristic
42C61×#358C2212yescomplemented
43C61×#357C2212yescomplemented
44C22×D244—21no—
45C22×C4×D61—41no—
46C2×D488—21no—
47C2×D488—22no—
48C2×D488C2×C414yescomplemented
49C2×D488C2312yes—
50C22×#92418D411yescharacteristic
51Q8×Dic61—21no—
52Q8×Dic61—21no—
53D4×Dic61—21no—
54D4×Dic61—21no—
55D4×Dic61—41no—
56D4×Dic61—41no—
57#14519739—22no—
58#14519739—22no—
59#14519739—41no—
60#14519739—41no—
61#14519736—21no—
62#14519736—41no—
63C2×#11758198—41no—
64C2×C4×Dic61—41no—
65C2×C4×Dic61D411yescharacteristic
66#14519720C2×C418yescomplemented
67#14519718C2×C418yescomplemented
68#14519717C2×C414yescomplemented
69C2×#11758185C2312yes—
70#14519711C2311yescharacteristic
71#14519711C2311yescharacteristic
72#14519709—21no—
73#14519701—21no—
74#14519701—21no—
75#14519698—21no—
76#14519698—21no—
77#14519697—22no—
78#14519697—22no—
79C8×Dic61D411yescharacteristic
80C8×Dic61D411yescharacteristic
81C8×Dic61C2311yescharacteristic
82C8×Dic61C2311yescharacteristic
83C2×#41×C61—21no—
84#135×C61—22no—
85#135×C61C2×C414yescomplemented
86C2×Q16×C61C2311yescharacteristic
87C2×SD16×C61—21no—
88C2×D8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C2×C4⋊C4×C61—21no—
91M32×C61—22no—
92C2×C16×C61C2312yes—
93#107×C61—21no—
94#107×C61C2312yes—
95C23×D61—41no—
96#41×C61—24no—
97#41×C61—42no—
98C2×Q8×C61—21no—
99C2×D4×C61—21no—
100C2×D4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C22×Dic61—41no—
105C22×Dic61—41no—
106C22×Dic61D811yescharacteristic
107C22×Dic61D811yescharacteristic
108C2×D244—24no—
109C2×D244—22no—
110C2×D244—41no—
111C2×D244—42no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×C4×D61—84no—
115Q16×C61—21no—
116Q16×C61C22×C412yescomplemented
117SD16×C61—24no—
118D8×C61—21no—
119D8×C61C22×C412yescomplemented
120C2×C8×C61C22×C412yes—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61#4112yes—
123C2×C8×C61C2411yescharacteristic
124C4⋊C4×C61—22no—
125C4⋊C4×C61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129#11758198—42no—
130#11758198—42no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61—41no—
134C4⋊Dic61—41no—
135C4×Dic61—41no—
136C4×Dic61—41no—
137C4×Dic61—41no—
138C4×Dic61D811yescharacteristic
139C4×Dic61D811yescharacteristic
140C4×Dic61C2×D411yescharacteristic
141C4×Dic61C2×D411yescharacteristic
142C2×#92418—22no—
143C2×#92418C2×D411yescharacteristic
144C2×#92418C2×D411yescharacteristic
145D488—24no—
146D488—22no—
147D488C22×C414yes—
148C16×C61—22no—
149#11758185C22×C414yescomplemented
150C23×C61—41no—
151C23×C61C2×D811yescharacteristic
152C22×D61—42no—
153C22×D61—44no—
154C22×D61—82no—
155C22×D61—82no—
156Q8×C61—22no—
157Q8×C61—41no—
158D4×C61—22no—
159D4×C61—44no—
160D4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—44no—
163C2×C4×C61—42no—
164C2×C4×C61—82no—
165C2×C4×C61C4×D412yes—
166C2×C4×C61C2×D811yescharacteristic
167C2×C4×C61#13512yes—
168C2×C4×C61C22×D411yescharacteristic
169C2×Dic61—22no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—82no—
174C2×Dic61—82no—
175C2×Dic61C2×D811yescharacteristic
176C2×Dic61C2×D811yescharacteristic
177C2×Dic61C2×D811yescharacteristic
178C2×Dic61C2×D811yescharacteristic
179D244—24no—
180D244—42no—
181D244—44no—
182C4×D61—84no—
183C4×D61—84no—
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×C61—42no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61C4×D812yes—
196C22×C61C22×D811yescharacteristic
197C2×D61—44no—
198C2×D61—88no—
199C2×D61—82no—
200C2×D61—82no—
201C4×C61—42no—
202C4×C61—82no—
203C4×C61C4×D812yes—
204C4×C61C2×C4×D411yescharacteristic
205C4×C61C2×#13511yescharacteristic
206Dic61—21no—
207Dic61—21no—
208Dic61—21no—
209Dic61—81no—
210Dic61—81no—
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#2159—611no—
220#2140—611no—
221#2139D6111yescharacteristic, complemented
222C4×M32—611no—
223C2×C61—22no—
224C2×C61—82no—
225C2×C61C2×C4×D811yescharacteristic
226C2×C61C2×#50911yescharacteristic
227C2×C61#341411yescharacteristic
228D61—84no—
229C2×#135C2×D6111yescharacteristic
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—611no—
243C4×D8—611no—
244C4×D8—612no—
245C4×D8—612no—
246C4×D8—1221no—
247C4×D8—1221no—
248#427—611no—
249C2×#107C2×D6111yescharacteristic
250#416—1221no—
251C2×#102—1221no—
252C2×C4×C8—611no—
253#367—612no—
254#366—612no—
255#358C2×D6112yescomplemented
256#357—612no—
257#357—612no—
258#357C2×D6112yescomplemented
259#346—612no—
260C4×C16—612no—
261C61C4×#50911yescharacteristic, complemented, Sylow
262C2×#41—21no—
263C22×D4—1221no—
264C23×C4—2441no—
265#135—22no—
266#135C4×D6114yescomplemented
267C2×Q16C22×D6111yescharacteristic
268C2×SD16—21no—
269C2×D8—614no—
270C2×D8—612no—
271C2×D8—1221no—
272C2×D8—1222no—
273C2×D8C22×D6111yescharacteristic
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—1221no—
287#117—2441no—
288C2×C4⋊C4—21no—
289C2×C22⋊C4—2441no—
290C2×C42—611no—
291C2×C42—2441no—
292SD32—618no—
293D16—618no—
294M32—22no—
295M32—614no—
296C2×C16—612no—
297C2×C16C22×D6112yes—
298#107—21no—
299#107—611no—
300#107—611no—
301#107C22×D6112yes—
302#106—1221no—
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—24no—
315#41—42no—
316C2×Q8—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—1222no—
320C2×D4—1224no—
321C2×D4—2441no—
322C2×D4—2442no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—4884no—
328C22×C4—2442no—
329C22×C4—2442no—
330C22×C4—2441no—
331C22×C4—2441no—
332C22×C4D4×D6111yescharacteristic
333Q16—21no—
334Q16C2×C4×D6112yescomplemented
335SD16—24no—
336D8—21no—
337D8—614no—
338D8—1222no—
339D8—1224no—
340D8C2×C4×D6112yescomplemented
341C2×C8—611no—
342C2×C8—611no—
343C2×C8—1222no—
344C2×C8C2×C4×D6112yes—
345C2×C8D4×D6111yescharacteristic
346C2×C8#1175822412yes—
347C2×C8C23×D6111yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—41no—
350C4⋊C4—1221no—
351C4⋊C4—1221no—
352C4⋊C4—2441no—
353C4⋊C4—2441no—
354C22⋊C4—2442no—
355C22⋊C4—2442no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C22⋊C4—2441no—
359C42—611no—
360C42—611no—
361C42—611no—
362C42—611no—
363C42—2441no—
364C42—2441no—
365C42—2441no—
366C16—22no—
367C16—614no—
368C23—41no—
369C23—4882no—
370C23—4882no—
371C23—2442no—
372C23—2444no—
373C23D8×D6111yescharacteristic
374Q8—22no—
375Q8—41no—
376D4—22no—
377D4—44no—
378D4—41no—
379D4—1224no—
380D4—2442no—
381D4—2444no—
382C2×C4—41no—
383C2×C4—44no—
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—4884no—
393C2×C4—4882no—
394C2×C4—4882no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4#1451973712yes—
399C2×C4D8×D6111yescharacteristic
400C2×C4#1451979812yes—
401C2×C4C2×D4×D6111yescharacteristic
402C8—1221no—
403C8—1221no—
404C8#1451973712yes—
405C8C22×C4×D6111yescharacteristic
406C8C2×#1175822411yescharacteristic
407C22—21no—
408C22—21no—
409C22—42no—
410C22—81no—
411C22—81no—
412C22—4888no—
413C22—4882no—
414C22—4882no—
415C22—2444no—
416C22#1683984212yes—
417C22C2×D8×D6111yescharacteristic
418C4—42no—
419C4—82no—
420C4—612no—
421C4—1221no—
422C4—1221no—
423C4—1221no—
424C4—4881no—
425C4—4881no—
426C4#1683984212yes—
427C4C2×#1451973711yescharacteristic
428C4C2×#1451979811yescharacteristic
429C2—22no—
430C2—82no—
431C2—4884no—
432C2C2×#1683984211yescharacteristic
433C2#509×D6111yescharacteristic
434C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
prime radicalC2×#16840197C2
cosocleC2×C8×C61C24
hypercenter#2139D61
hypocenterC61C4×#509
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
2. #135×Dic61C2semidirect
3. C2×#16839844C2semidirect
4. #21447692C2semidirect
5. #21447686C2semidirect
6. #21447668C2semidirect
7. #21447663C2semidirect
8. #21447605C2semidirect
9. #21447536C2semidirect
10. M32×Dic61C2semidirect
11. C61×#2139C2semidirect
15. #16840197C4semidirect
19. Q16×Dic61C22semidirect
20. Q16×Dic61C22semidirect
23. D8×Dic61C22semidirect
24. D8×Dic61C22semidirect
33. #16839589C22semidirect
34. #16839587C22semidirect
37. #16839537C22semidirect
38. C16×Dic61C22semidirect
42. C61×#358C22semidirect
43. C61×#357C22semidirect
48. C2×D488C2×C4semidirect
66. #14519720C2×C4semidirect
67. #14519718C2×C4semidirect
68. #14519717C2×C4semidirect
85. #135×C61C2×C4semidirect
116. Q16×C61C22×C4semidirect
119. D8×C61C22×C4semidirect
149. #11758185C22×C4semidirect
211. Dic61#509semidirect
221. #2139D61semidirect
255. #358C2×D61semidirect
258. #357C2×D61semidirect
261. C61C4×#509semidirect
266. #135C4×D61semidirect
334. Q16C2×C4×D61semidirect
340. D8C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
#135×Dic612
C2×#168398442
#214476922
#214476862
#214476682
#214476632
#214476052
#214475362
M32×Dic612
C61×#21392
C4×#50961