Finite Group Databasebeta

← #29836676

Subgroups — #29836676

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840191C211yesmaximal, characteristic, complemented
2#21457504C212yesmaximal, complemented
3#21448077C211yesmaximal, characteristic, complemented
4#21447910C211yesmaximal, characteristic, complemented
5#21447902C211yesmaximal, characteristic, complemented
6#21447872C212yesmaximal, complemented
7#21447748C212yesmaximal, complemented
8#21447739C212yesmaximal, complemented
9D16×Dic61C211yesmaximal, characteristic, complemented
10C16×C22⋊D61C211yesmaximal, characteristic, complemented
11C22×D16×C61C211yesmaximal, characteristic, complemented
12C2×#14519796C2212yescomplemented
13#16840406—22no—
14#16840191—24no—
15C2×Dic488C2211yescharacteristic, complemented
16C2×#14519721C2212yescomplemented
17C2×#14519721C2212yescomplemented
18C2×C16×D61C2211yescharacteristic, complemented
19#16840145C2212yescomplemented
20#16840003C2212yes—
21#16839995C2212yescomplemented
22#16839975—22no—
23#16839913—22no—
24#16839904—22no—
25D8×Dic61C2212yescomplemented
26C8×C22⋊D61C2211yescharacteristic
27#16839588C2211yescharacteristic, complemented
28#16839580C2211yescharacteristic, complemented
29#16839567C2212yescomplemented
30#16839563C2212yescomplemented
31#16839563C2212yescomplemented
32C22⋊#11758185C2211yescharacteristic
33#16839541C2211yescharacteristic, complemented
34C16×Dic61C2211yescharacteristic, complemented
35C22×D8×C61C2212yes—
36C2×D16×C61—24no—
37C2×D16×C61C2212yes—
38C2×D16×C61C2211yescharacteristic
39C2×D16×C61C2211yescharacteristic
40C22×C16×C61C2211yescharacteristic
41C2×#11758224—22no—
42#14519796—24no—
43#14519796—42no—
44C2×Dic244C2312yes—
45C2×#11758200—22no—
46C2×#11758200—22no—
47C2×C8×D61C2311yescharacteristic
48#14519764—22no—
49#14519758—42no—
50#14519756—22no—
51#14519749—42no—
52#14519748—22no—
53#14519746—42no—
54D4×Dic61—22no—
55C4×C22⋊D61D411yescharacteristic
56C2×C22⋊Dic61—42no—
57Dic488—22no—
58#14519721—24no—
59#14519721—42no—
60C2×#11758185C2311yescharacteristic
61C2×#11758185C2311yescharacteristic
62C16×D61—22no—
63#14519711C2312yes—
64#14519706D411yescharacteristic
65#14519702—22no—
66#14519698—22no—
67#14519698—22no—
68C22⋊#92418D411yescharacteristic
69#14519689D411yescharacteristic
70C8×Dic61C2311yescharacteristic
71C22×D4×C61—22no—
72C2×D8×C61—24no—
73C2×D8×C61—22no—
74C2×D8×C61—42no—
75C2×D8×C61C2312yes—
76C2×D8×C61C2312yes—
77C22×C8×C61C2311yescharacteristic
78D16×C61—24no—
79D16×C61—22no—
80D16×C61D414yescomplemented
81C2×C16×C61—22no—
82C2×C16×C61C2311yescharacteristic
83C2×C16×C61C2311yescharacteristic
84C24×C61—42no—
85C2×D4×C61—22no—
86C2×D4×C61—22no—
87C2×D4×C61—44no—
88C2×D4×C61—42no—
89C2×D4×C61—82no—
90C22×C4×C61C2×D411yescharacteristic
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C2×C22⋊D61D811yescharacteristic
94C22×Dic61—42no—
95C22×Dic61—42no—
96#11758224—44no—
97#11758224—82no—
98C2×C4×D61C2×D411yescharacteristic
99C2×Dic122—22no—
100D8×C61—22no—
101D8×C61—22no—
102D8×C61—22no—
103D8×C61—44no—
104D8×C61C2×D414yes—
105C2×C8×C61—21no—
106C2×C8×C61—21no—
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2411yescharacteristic
109C22⋊Dic61—42no—
110C22⋊Dic61—42no—
111C22⋊Dic61—42no—
112C22⋊Dic61—42no—
113C22⋊Dic61—42no—
114C22⋊Dic61D811yescharacteristic
115#11758200—44no—
116#11758200—82no—
117#11758198D811yescharacteristic
118C4⋊Dic61—22no—
119#11758196—42no—
120#11758196D811yescharacteristic
121C4×Dic61C2×D411yescharacteristic
122C2×#92418C2×D411yescharacteristic
123C2×#92418C2×D411yescharacteristic
124Dic244—22no—
125Dic244—22no—
126C8×D61—21no—
127C8×D61—21no—
128C16×C61—21no—
129C16×C61C2×D412yes—
130#11758185—21no—
131#11758185—21no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—84no—
135C23×C61—84no—
136C23×C61—162no—
137C23×C61C2×D811yescharacteristic
138C22×D61C2×D811yescharacteristic
139D4×C61—24no—
140D4×C61—42no—
141D4×C61—42no—
142D4×C61—42no—
143D4×C61—84no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61C2×D811yescharacteristic
147C2×C4×C61C22×D411yescharacteristic
148C22⋊D61—22no—
149C22⋊D61—84no—
150C22⋊D61—162no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—84no—
154C2×Dic61—162no—
155C2×Dic61C2×D811yescharacteristic
156C2×Dic61C2×D811yescharacteristic
157C2×Dic61C2×D811yescharacteristic
158C4×D61—21no—
159C4×D61—21no—
160Dic122—42no—
161Dic122—42no—
162C8×C61—21no—
163C8×C61C22×D411yescharacteristic
164C8×C61H(5,𝔽2)11yescharacteristic
165#92418—21no—
166#92418—21no—
167#33846—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—44no—
171C22×C61—82no—
172C22×C61—82no—
173C22×C61—82no—
174C22×C61—82no—
175C22×C61—82no—
176C22×C61—168no—
177C22×C61#50812yescomplemented
178C22×C61C22×D811yescharacteristic
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—21no—
182C4×C61—21no—
183C4×C61D4211yescharacteristic
184C4×C61#57611yescharacteristic
185Dic61—21no—
186Dic61—21no—
187Dic61—21no—
188Dic61—82no—
189Dic61—82no—
190C2×#508—611no—
191C22×D16D6111yescharacteristic, complemented
192#3280—612no—
193#2249—611no—
194#2216—611no—
195#2215—611no—
196#2210—612no—
197#2187—612no—
198#2186—612no—
199C4×D16—611no—
200D4×C16—611no—
201C2×C61—22no—
202C2×C61—84no—
203C2×C61—162no—
204C2×C61D4×D811yescharacteristic
205C2×C61C2×#50811yescharacteristic
206C2×C61#341511yescharacteristic
207D61—41no—
208C2×#135—612no—
209C22×D8C2×D6112yes—
210#546—1222no—
211#508—1224no—
212C2×Q32—611no—
213C2×SD32—612no—
214C2×SD32—612no—
215C2×D16—24no—
216C2×D16C2×D6112yes—
217C2×D16C2×D6111yescharacteristic
218C2×D16C2×D6111yescharacteristic
219C22×C16—611no—
220C22×C16C2×D6111yescharacteristic
221#495—612no—
222#467—612no—
223#466—612no—
224#463—1222no—
225#450—1222no—
226#449—1222no—
227C4×D8—612no—
228C8×D4—611no—
229#366—611no—
230C4⋊C16—611no—
231#358—612no—
232#357—612no—
233#357—612no—
234C22⋊C16—611no—
235C22⋊C16—611no—
236C4×C16—611no—
237C61#3384611yescharacteristic, complemented, Sylow
238C2×#41—1222no—
239C22×D4—22no—
240#135—1224no—
241#135—2442no—
242C2×Q16—612no—
243C2×SD16—1222no—
244C2×SD16—1222no—
245C2×D8—24no—
246C2×D8—22no—
247C2×D8—42no—
248C2×D8C22×D6112yes—
249C2×D8C22×D6112yes—
250C22×C8—611no—
251C22×C8C22×D6111yescharacteristic
252#124—1222no—
253#123—2442no—
254C22⋊Q8—1222no—
255#121—1222no—
256#121—2442no—
257#120—2442no—
258C4×D4—611no—
259C4×D4—1222no—
260C2×C22⋊C4—2442no—
261Q32—1222no—
262SD32—1224no—
263SD32—2442no—
264D16—24no—
265D16—22no—
266D16C22⋊D6114yescomplemented
267C2×C16—22no—
268C2×C16—611no—
269C2×C16—611no—
270C2×C16—1222no—
271C2×C16C22×D6111yescharacteristic
272C2×C16C22×D6111yescharacteristic
273#107—612no—
274C4⋊C8—611no—
275#103—1222no—
276#102—1222no—
277#102—1222no—
278C22⋊C8—611no—
279C22⋊C8—611no—
280C4×C8—611no—
281C24—42no—
282#41—4882no—
283#41—2444no—
284C2×Q8—1222no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—44no—
288C2×D4—42no—
289C2×D4—82no—
290C2×D4—611no—
291C2×D4—2442no—
292C2×D4—2442no—
293C22×C4—611no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4D4×D6111yescharacteristic
297Q16—1222no—
298Q16—1222no—
299SD16—4882no—
300SD16—2444no—
301D8—22no—
302D8—22no—
303D8—22no—
304D8—44no—
305D8C2×C22⋊D6114yes—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—611no—
309C2×C8—611no—
310C2×C8—1221no—
311C2×C8—1221no—
312C2×C8D4×D6111yescharacteristic
313C2×C8C23×D6111yescharacteristic
314C4⋊C4—611no—
315C4⋊C4—1222no—
316C4⋊C4—2442no—
317C22⋊C4—611no—
318C22⋊C4—611no—
319C22⋊C4—2442no—
320C22⋊C4—2442no—
321C22⋊C4—2442no—
322C22⋊C4—2442no—
323C22⋊C4—2442no—
324C42—611no—
325C16—21no—
326C16—1221no—
327C16—1221no—
328C16C2×C22⋊D6112yes—
329C23—42no—
330C23—42no—
331C23—84no—
332C23—84no—
333C23—162no—
334C23—611no—
335C23D8×D6111yescharacteristic
336Q8—2442no—
337Q8—2442no—
338D4—24no—
339D4—42no—
340D4—42no—
341D4—42no—
342D4—84no—
343D4—1222no—
344D4—9762no—
345D4—4884no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—611no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—9762no—
354C2×C4—4884no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4D8×D6111yescharacteristic
358C2×C4C2×D4×D6111yescharacteristic
359C8—21no—
360C8—1221no—
361C8—1221no—
362C8C22×C22⋊D6111yescharacteristic
363C8#1451982211yescharacteristic
364C22—21no—
365C22—21no—
366C22—44no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—168no—
373C22—1221no—
374C22—1221no—
375C22—1221no—
376C22#1684019112yescomplemented
377C22C2×D8×D6111yescharacteristic
378C4—21no—
379C4—1221no—
380C4—1221no—
381C4—1221no—
382C4—4882no—
383C4—4882no—
384C4D4×C22⋊D6111yescharacteristic
385C4#1684056311yescharacteristic
386C2—22no—
387C2—84no—
388C2—162no—
389C2—2441no—
390C2D8×C22⋊D6111yescharacteristic
391C2C2×#1684019111yescharacteristic
392C2#2145895611yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC22×D16×C61C2
socleC22×C61C22×D8
hypercenterC22×D16D61
hypocenterC61#33846
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×D16

Complements

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

NH ≅ G/Ntype
1. C2×#16840191C2semidirect
2. #21457504C2semidirect
3. #21448077C2semidirect
4. #21447910C2semidirect
5. #21447902C2semidirect
6. #21447872C2semidirect
7. #21447748C2semidirect
8. #21447739C2semidirect
9. D16×Dic61C2semidirect
10. C16×C22⋊D61C2semidirect
11. C22×D16×C61C2semidirect
12. C2×#14519796C22semidirect
15. C2×Dic488C22semidirect
16. C2×#14519721C22semidirect
17. C2×#14519721C22semidirect
18. C2×C16×D61C22semidirect
19. #16840145C22semidirect
21. #16839995C22semidirect
25. D8×Dic61C22semidirect
27. #16839588C22semidirect
28. #16839580C22semidirect
29. #16839567C22semidirect
30. #16839563C22semidirect
31. #16839563C22semidirect
33. #16839541C22semidirect
34. C16×Dic61C22semidirect
80. D16×C61D4semidirect
177. C22×C61#508semidirect
191. C22×D16D61semidirect
237. C61#33846semidirect
266. D16C22⋊D61semidirect
376. C22#16840191semidirect

Maximal subgroups

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

subgroupindex
C2×#168401912
#214575042
#214480772
#214479102
#214479022
#214478722
#214477482
#214477392
D16×Dic612
C16×C22⋊D612
C22×D16×C612
#3384661