Finite Group Databasebeta

← #29658158

Subgroups — #29658158

Order 15616.

1056 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460224C211yesmaximal, characteristic
2#21456947C212yesmaximal, complemented
3#21444962C214yesmaximal, complemented
4#21444933C214yesmaximal, complemented
5#21444481C211yesmaximal, characteristic, complemented
6#21444237C212yesmaximal, complemented
7C61×#3152C211yesmaximal, characteristic, complemented
8#16840504—22no—
9#16840478C2212yes—
10#16840477—22no—
11D4×D244—22no—
12#16840397C2212yes—
13#16840328—24no—
14#16840326—22no—
15#16840312C2212yes—
16#16840158—22no—
17#16840110—22no—
18#16840080—24no—
19#16840049C2218yescomplemented
20#16839815C2212yescomplemented
21#16839511C2212yescomplemented
22#16839506C2212yes—
23#16839491C2218yescomplemented
24C2×#137×C61—22no—
25C61×#532C2211yescharacteristic
26C2×C4⋊Q8×C61C2211yescharacteristic, complemented
27C61×#497—22no—
28C61×#494C2212yescomplemented
29C61×#492C2212yes—
30C4×M16×C61C2211yescharacteristic, complemented
31C2×#11758222—22no—
32C22×D244—22no—
33C2×D4×D61—24no—
34#14519777D412yes—
35#14519774D412yes—
36#14519773—24no—
37#14519766—24no—
38#14519765—22no—
39#14519763—28no—
40#14519752—24no—
41#14519751—24no—
42#14519751—22no—
43#14519751D414yes—
44#14519748—24no—
45#14519747—28no—
46#14519745—24no—
47#14519745—24no—
48C4×D244—22no—
49C4×D244C2314yes—
50#14519736—22no—
51#14519735—24no—
52#14519708—28no—
53C4⋊#92418C2314yes—
54#14519697—28no—
55C22⋊#92418—22no—
56C4×#92418—22no—
57C2×#41×C61D412yes—
58C22×Q8×C61—21no—
59#137×C61—48no—
60C2×Q16×C61—24no—
61C2×SD16×C61—24no—
62C2×M16×C61—21no—
63C4⋊Q8×C61—22no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61C2312yes—
66C4⋊Q8×C61C2311yescharacteristic
67#127×C61C2311yescharacteristic
68#124×C61D412yes—
69#121×C61—22no—
70C4×D4×C61—22no—
71C2×C4⋊C4×C61—22no—
72C2×C42×C61C2311yescharacteristic
73#97×C61—21no—
74C4×C8×C61C2312yes—
75C23×D61—24no—
76#41×C61—44no—
77C2×Q8×C61—22no—
78C2×Q8×C61—44no—
79C2×Q8×C61—41no—
80C2×Q8×C61C2×D412yes—
81C2×D4×C61—22no—
82C2×D4×C61—22no—
83C2×D4×C61C2×D412yes—
84C22×C4×C61—22no—
85C22×C4×C61—41no—
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×D411yescharacteristic
88C2×C22⋊D61—28no—
89C2×C22⋊D61—24no—
90D4×D61—88no—
91#11758222—84no—
92C2×D244—22no—
93C2×D244—24no—
94C2×D244—22no—
95C2×D244—84no—
96C2×D244C2×D414yes—
97C2×C4×D61—24no—
98C2×Dic122—22no—
99Q16×C61—48no—
100SD16×C61—48no—
101M16×C61—24no—
102C2×C8×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61—42no—
106C22⋊C4×C61—24no—
107C42×C61C2×D412yes—
108C42×C61C2×D411yescharacteristic
109C42×C61C2411yescharacteristic
110C22⋊Dic61—24no—
111#11758198—28no—
112#11758198—24no—
113#11758198—24no—
114C4⋊Dic61—22no—
115C4⋊Dic61C2×D414yes—
116C4×Dic61—24no—
117C2×#92418—24no—
118C23×C61—22no—
119C23×C61#12011yescharacteristic
120C22×D61—24no—
121C22×D61—84no—
122C22×D61—88no—
123Q8×C61—42no—
124Q8×C61—44no—
125Q8×C61—82no—
126D4×C61—42no—
127D4×C61—44no—
128D4×C61—82no—
129C2×C4×C61—24no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—44no—
133C2×C4×C61—42no—
134C2×C4×C61—82no—
135C2×C4×C61#12012yes—
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61C22×D411yescharacteristic
138C2×C4×C61C22×D412yes—
139C22⋊D61—84no—
140C22⋊D61—88no—
141C2×Dic61—24no—
142D244—28no—
143D244—44no—
144D244—82no—
145D244—84no—
146C4×D61—48no—
147Dic122—82no—
148C8×C61—24no—
149#92418—44no—
150#18715—611nomaximal, Sylow
151C22×C61—22no—
152C22×C61—21no—
153C22×C61—44no—
154C22×C61—42no—
155C22×C61C2×#12011yescharacteristic
156C2×D61—44no—
157C2×D61—44no—
158C2×D61—44no—
159C2×D61—168no—
160C4×C61—22no—
161C4×C61—21no—
162C4×C61—42no—
163C4×C61—82no—
164C4×C61#45312yes—
165C4×C61#57714yes—
166Dic61—44no—
167#3549—611no—
168#3227—612no—
169#3152D6111yescharacteristic, complemented
170#1634—614no—
171#1629—614no—
172#1543—611no—
173#1484—612no—
174C2×C61—41no—
175C2×C61—82no—
176C2×C61#300912yes—
177C2×C61C2×#45311yescharacteristic
178D61—84no—
179C2×#137—22no—
180H(3,𝔽4)—1222no—
181#555—612no—
182#555—1222no—
183D42—612no—
184D42—1222no—
185#535—1224no—
186#535—1222no—
187#532—612no—
188#532C2×D6111yescharacteristic
189C2×C4⋊Q8C2×D6111yescharacteristic, complemented
190#498—1222no—
191#497—22no—
192#494C2×D6112yescomplemented
193#492C2×D6112yes—
194#486—1222no—
195#481—1224no—
196#476—618no—
197#431—612no—
198C4×M16C2×D6111yescharacteristic, complemented
199#333—612no—
200#331—612no—
201D4⋊C8—618no—
202C61#1871511yescharacteristic, complemented, Sylow
203C2×#41—1222no—
204C2×#41C22⋊D6112yes—
205C22×Q8—21no—
206C22×D4—1222no—
207C22×D4—1224no—
208#137—48no—
209C2×Q16—24no—
210C2×SD16—24no—
211C2×M16—21no—
212C4⋊Q8—22no—
213C4⋊Q8—21no—
214C4⋊Q8—612no—
215C4⋊Q8C22×D6112yes—
216C4⋊Q8C22×D6111yescharacteristic
217#127—612no—
218#127—1224no—
219#127C22×D6111yescharacteristic
220#124—1228no—
221#124—1224no—
222#124—1222no—
223#124C22⋊D6112yes—
224#121—22no—
225#121—614no—
226#121—1224no—
227#121—1224no—
228#121—1224no—
229#121—1222no—
230#120—1228no—
231#120—1224no—
232#120—1224no—
233C4×D4—22no—
234C4×D4—614no—
235C4×D4—1222no—
236#117—1224no—
237#117—1222no—
238C2×C4⋊C4—22no—
239C2×C42C22×D6111yescharacteristic
240#106—1228no—
241C4⋊C8—614no—
242#102—1228no—
243C22⋊C8—1222no—
244#97—21no—
245C4×C8—1222no—
246C4×C8C22×D6112yes—
247C24—1224no—
248#41—44no—
249#41—4884no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—1222no—
254C2×Q8C2×C22⋊D6112yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—614no—
258C2×D4—1224no—
259C2×D4—1228no—
260C2×D4—1224no—
261C2×D4—1222no—
262C2×D4—1222no—
263C2×D4—4884no—
264C2×D4—4888no—
265C2×D4C2×C22⋊D6112yes—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—1224no—
269C22×C4D4×D6112yes—
270C22×C4C2×C22⋊D6111yescharacteristic
271Q16—48no—
272SD16—48no—
273M16—24no—
274C2×C8—22no—
275C2×C8—1224no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C4⋊C4—614no—
280C4⋊C4—1222no—
281C22⋊C4—24no—
282C22⋊C4—1228no—
283C22⋊C4—1224no—
284C22⋊C4—1224no—
285C22⋊C4—1224no—
286C42—1224no—
287C42D4×D6112yes—
288C42C2×C22⋊D6111yescharacteristic
289C42C23×D6111yescharacteristic
290C23—22no—
291C23—1224no—
292C23—4884no—
293C23—4888no—
294C23#1451974511yescharacteristic
295Q8—42no—
296Q8—44no—
297Q8—82no—
298Q8—4882no—
299D4—42no—
300D4—44no—
301D4—82no—
302D4—1228no—
303D4—4882no—
304D4—4884no—
305D4—4884no—
306D4—4888no—
307D4—2444no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—1224no—
315C2×C4—2448no—
316C2×C4#1451974512yes—
317C2×C4#1451974511yescharacteristic
318C2×C4C2×D4×D6112yes—
319C2×C4C22×C22⋊D6111yescharacteristic
320C8—24no—
321C8—2444no—
322C22—21no—
323C22—22no—
324C22—42no—
325C22—44no—
326C22—9768no—
327C22—2444no—
328C22—2444no—
329C22—2444no—
330C22C2×#1451974511yescharacteristic
331C4—21no—
332C4—22no—
333C4—42no—
334C4—82no—
335C4—2444no—
336C4#1683992912yes—
337C4#1684057114yes—
338C2—41no—
339C2—82no—
340C2—4884no—
341C2#2145509512yes—
342C2C2×#1683992911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3152C2
socleC22×C61C2×#120
prime radical#21460224C2
hypercenter#3152D61
hypocenterC61#18715
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. #21456947C2semidirect
3. #21444962C2semidirect
4. #21444933C2semidirect
5. #21444481C2semidirect
6. #21444237C2semidirect
7. C61×#3152C2semidirect
19. #16840049C22semidirect
20. #16839815C22semidirect
21. #16839511C22semidirect
23. #16839491C22semidirect
26. C2×C4⋊Q8×C61C22semidirect
28. C61×#494C22semidirect
30. C4×M16×C61C22semidirect
169. #3152D61semidirect
189. C2×C4⋊Q8C2×D61semidirect
192. #494C2×D61semidirect
198. C4×M16C2×D61semidirect
202. C61#18715semidirect

Maximal subgroups

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

subgroupindex
#214602242
#214569472
#214449622
#214449332
#214444812
#214442372
C61×#31522
#1871561