Finite Group Databasebeta

← #29833475

Subgroups — #29833475

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458953C211yesmaximal, characteristic
2#21454735C211yesmaximal, characteristic, complemented
3#21454703C211yesmaximal, characteristic, complemented
4#21447720C214yesmaximal, complemented
5#21447713C214yesmaximal, complemented
6C2×#16839586C211yesmaximal, characteristic, complemented
7C2×#16839573C211yesmaximal, characteristic, complemented
8#21447423C211yesmaximal, characteristic, complemented
9C61×#2148C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11#16840528—21no—
12#16840245—21no—
13#16840203C2212yes—
14#16840202C412yescomplemented
15#16840198C2212yes—
16#16840197C412yescomplemented
17#16840192C414yescomplemented
18C2×#14519717C2211yescharacteristic
19#16839883C2214yes—
20#16839878—22no—
21#16839874C2212yescomplemented
22#16839869C2212yescomplemented
23SD16×Dic61—22no—
24C2×#14519709C2211yescharacteristic
25#16839778—21no—
26#16839763—21no—
27#16839717C2211yescharacteristic
28#16839586C2214yescomplemented
29#16839573C2214yes—
30#16839539C2214yescomplemented
31C2×#135×C61C2211yescharacteristic
32C2×M32×C61C2211yescharacteristic
33C2×#108×C61C2211yescharacteristic
34C61×#362C2212yescomplemented
35C61×#361C2212yescomplemented
36C2×#11758222—21no—
37#14519799—21no—
38#14519799—21no—
39#14519799C2×C412yescomplemented
40#14519799C2312yes—
41C2×Dic244C2×C411yescharacteristic, complemented
42C2×#11758190—21no—
43C2×D488C2×C411yescharacteristic, complemented
44#14519785—24no—
45#14519783—42no—
46C2×#11758194D411yescharacteristic
47M16×D61—42no—
48Q8×Dic61—22no—
49D4×Dic61—22no—
50D4×Dic61—42no—
51#14519736—41no—
52#14519736D411yescharacteristic
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—41no—
55#14519723C2×C414yescomplemented
56#14519721C2×C414yescomplemented
57#14519720C2×C414yescomplemented
58#14519718C2×C414yescomplemented
59#14519717C2×C412yescomplemented
60#14519717C2312yes—
61C2×#11758185C2×C412yescomplemented
62#14519709—21no—
63#14519709C2312yes—
64#14519703—24no—
65#14519701—22no—
66#14519698—22no—
67#14519687C2312yes—
68C8×Dic61D412yes—
69C2×#41×C61—21no—
70#135×C61—21no—
71#135×C61—21no—
72#135×C61C2×C412yes—
73#135×C61C2×C412yes—
74C2×Q16×C61C2311yescharacteristic
75C2×SD16×C61—21no—
76C2×D8×C61C2311yescharacteristic
77C2×M16×C61—21no—
78C22×C8×C61C2311yescharacteristic
79M32×C61—21no—
80M32×C61C2312yes—
81C2×C16×C61—21no—
82#108×C61—21no—
83#108×C61C2312yes—
84#41×C61—22no—
85#41×C61—22no—
86#41×C61—41no—
87#41×C61—41no—
88C2×Q8×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—41no—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C2×C22⋊D61—41no—
94C22×Dic61—42no—
95C22×Dic61—41no—
96#11758222—22no—
97#11758222—22no—
98#11758222—41no—
99#11758222—41no—
100C2×D244—21no—
101C2×C4×D61—41no—
102C2×Dic122—21no—
103Q16×C61—21no—
104Q16×C61C22×C412yes—
105SD16×C61—22no—
106SD16×C61—22no—
107D8×C61—21no—
108D8×C61C22×C412yes—
109M16×C61—22no—
110M16×C61—41no—
111C2×C8×C61—41no—
112C2×C8×C61C22×C411yescharacteristic
113C2×C8×C61C22×C411yescharacteristic
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61#4111yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C2×C8×C61C2411yescharacteristic
118C22⋊Dic61—21no—
119C22⋊Dic61—42no—
120C4⋊Dic61—21no—
121C4⋊Dic61—22no—
122C4⋊Dic61—42no—
123C4⋊Dic61—41no—
124C4×Dic61—42no—
125C4×Dic61C2×D412yes—
126#11758194—21no—
127#11758194—21no—
128C2×#92418C2×D412yes—
129Dic244—21no—
130Dic244C22×C412yes—
131D488—21no—
132D488C22×C412yes—
133#11758190—22no—
134#11758190—22no—
135#11758189—44no—
136C8×D61—44no—
137C16×C61—22no—
138#11758185C22×C414yes—
139C23×C61—41no—
140C23×C61AGL(1,ℤ/8)11yescharacteristic
141C22×D61—41no—
142Q8×C61—22no—
143Q8×C61—41no—
144D4×C61—22no—
145D4×C61—42no—
146D4×C61—42no—
147D4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C22⋊D61—42no—
158C22⋊D61—42no—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—42no—
162C2×Dic61—162no—
163C2×Dic61—161no—
164D244—22no—
165D244—41no—
166C4×D61—42no—
167C4×D61—42no—
168Dic122—22no—
169Dic122—41no—
170C8×C61—42no—
171C8×C61C4×D411yescharacteristic
172C8×C61C4×D411yescharacteristic
173C8×C61C23×C411yescharacteristic
174C8×C61C2×#4111yescharacteristic
175#92418—22no—
176#33469—611nomaximal, Sylow
177C22×C61—41no—
178C22×C61—42no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61#44211yescharacteristic
182C22×C61#44211yescharacteristic
183C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
184C2×D61—42no—
185C2×D61—161no—
186C4×C61—42no—
187C4×C61#44211yescharacteristic
188C4×C61#44211yescharacteristic
189C4×C61C2×C4×D411yescharacteristic
190C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
191Dic61—42no—
192Dic61—42no—
193Dic61—82no—
194#3414—611no—
195#2957—611no—
196#2952—611no—
197AGL(1,ℤ/16)—614no—
198#2180—614no—
199C2×#365—611no—
200#2148D6111yescharacteristic, complemented
201C2×#360—611no—
202#2110—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61—82no—
207C2×C61C2×#44211yescharacteristic
208D61—82no—
209C2×#135—611no—
210C2×#135C2×D6111yescharacteristic
211#568—1221no—
212#520—1221no—
213#510—612no—
214#510—612no—
215#509—612no—
216#509—612no—
217#508—614no—
218C2×M32—611no—
219C2×M32C2×D6111yescharacteristic
220#444—614no—
221#443—1222no—
222#442—612no—
223#441—612no—
224C4×SD16—1222no—
225C2×#108C2×D6111yescharacteristic
226C2×#106—611no—
227#421—1221no—
228#417—1221no—
229#405—611no—
230#365—614no—
231#362C2×D6112yescomplemented
232#361C2×D6112yescomplemented
233#360—614no—
234#347—614no—
235C61#3346911yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—1221no—
238#135—21no—
239#135—21no—
240#135—612no—
241#135—612no—
242#135—1221no—
243#135—1221no—
244#135C4×D6112yes—
245#135C4×D6112yes—
246C2×Q16—611no—
247C2×Q16C22×D6111yescharacteristic
248C2×SD16—21no—
249C2×SD16—1221no—
250C2×D8—611no—
251C2×D8C22×D6111yescharacteristic
252#131—1224no—
253#131—2442no—
254C2×M16—21no—
255C2×M16—611no—
256C2×M16—2442no—
257C22×C8C22×D6111yescharacteristic
258C4×Q8—1222no—
259C4×D4—1222no—
260C4×D4—2442no—
261#117—611no—
262#117—2441no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—2441no—
265Q32—614no—
266SD32—614no—
267SD32—614no—
268D16—614no—
269M32—21no—
270M32—612no—
271M32—612no—
272M32C22×D6112yes—
273C2×C16—21no—
274C2×C16—612no—
275#108—21no—
276#108C22×D6112yes—
277#106—612no—
278#106—1221no—
279#104—1224no—
280#103—1222no—
281#102—1222no—
282#97—612no—
283C4×C8—612no—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—1222no—
289#41—1222no—
290#41—2441no—
291#41—2441no—
292C2×Q8—21no—
293C2×Q8—1221no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—1221no—
297C2×D4—2441no—
298C22×C4—41no—
299C22×C4—2442no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4D4×D6111yescharacteristic
303Q16—21no—
304Q16—612no—
305Q16—1221no—
306Q16C2×C4×D6112yes—
307SD16—22no—
308SD16—22no—
309SD16—1222no—
310SD16—1222no—
311D8—21no—
312D8—612no—
313D8—1221no—
314D8C2×C4×D6112yes—
315M16—22no—
316M16—41no—
317M16—1221no—
318M16—1221no—
319M16—2444no—
320C2×C8—41no—
321C2×C8—612no—
322C2×C8—2444no—
323C2×C8C2×C4×D6111yescharacteristic
324C2×C8C2×C4×D6111yescharacteristic
325C2×C8D4×D6111yescharacteristic
326C2×C8#1175822411yescharacteristic
327C2×C8#1175822411yescharacteristic
328C2×C8C23×D6111yescharacteristic
329C4⋊C4—1221no—
330C4⋊C4—1222no—
331C4⋊C4—2442no—
332C4⋊C4—2441no—
333C22⋊C4—1221no—
334C22⋊C4—2442no—
335C42—612no—
336C42—2442no—
337C16—22no—
338C16—614no—
339C23—41no—
340C23—2441no—
341C23#1451980411yescharacteristic
342Q8—22no—
343Q8—41no—
344Q8—1222no—
345Q8—2441no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—41no—
350D4—1222no—
351D4—2442no—
352D4—2442no—
353D4—2441no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—1222no—
358C2×C4—9761no—
359C2×C4—9762no—
360C2×C4—2442no—
361C2×C4—2441no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4#1451973711yescharacteristic
365C2×C4#1451973711yescharacteristic
366C2×C4#1451980111yescharacteristic
367C2×C4#1451980111yescharacteristic
368C2×C4#1451980411yescharacteristic
369C2×C4C2×D4×D6111yescharacteristic
370C8—42no—
371C8—1222no—
372C8#1451973711yescharacteristic
373C8#1451973711yescharacteristic
374C8C22×C4×D6111yescharacteristic
375C8C2×#1175822411yescharacteristic
376C22—41no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—9761no—
381C22—2442no—
382C22#1683987111yescharacteristic
383C22#1683987111yescharacteristic
384C22C2×#1451980411yescharacteristic
385C4—42no—
386C4—4882no—
387C4—2442no—
388C4—2442no—
389C4#1683987111yescharacteristic
390C4#1683987111yescharacteristic
391C4C2×#1451973711yescharacteristic
392C4C2×#1451980111yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—4882no—
398C2C2×#1683987111yescharacteristic
subgroupG/H
centerC2C2×#16839871
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2148C2
socleC2×C61C2×#442
prime radical#21458953C2
cosocleC2×C8×C61C24
hypercenter#2148D61
hypocenterC61#33469
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. #21454735C2semidirect
3. #21454703C2semidirect
4. #21447720C2semidirect
5. #21447713C2semidirect
6. C2×#16839586C2semidirect
7. C2×#16839573C2semidirect
8. #21447423C2semidirect
9. C61×#2148C2semidirect
14. #16840202C4semidirect
16. #16840197C4semidirect
17. #16840192C4semidirect
21. #16839874C22semidirect
22. #16839869C22semidirect
28. #16839586C22semidirect
30. #16839539C22semidirect
34. C61×#362C22semidirect
35. C61×#361C22semidirect
39. #14519799C2×C4semidirect
41. C2×Dic244C2×C4semidirect
43. C2×D488C2×C4semidirect
55. #14519723C2×C4semidirect
56. #14519721C2×C4semidirect
57. #14519720C2×C4semidirect
58. #14519718C2×C4semidirect
59. #14519717C2×C4semidirect
61. C2×#11758185C2×C4semidirect
200. #2148D61semidirect
231. #362C2×D61semidirect
232. #361C2×D61semidirect
235. C61#33469semidirect

Maximal subgroups

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

subgroupindex
#214589532
#214547352
#214547032
#214477202
#214477132
C2×#168395862
C2×#168395732
#214474232
C61×#21482
#3346961