Finite Group Databasebeta

← #29658298

Subgroups — #29658298

Order 15616.

884 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 179 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#21456757C212yesmaximal
2#21456195C211yesmaximal, characteristic
3#21444915C218yesmaximal, complemented
4#21444484C211yesmaximal, characteristic, complemented
5#21444240C212yesmaximal, complemented
6C61×#3556C211yesmaximal, characteristic, complemented
7C2×#14519774C2211yescharacteristic
8#16840136—22no—
9#16840128—22no—
10#16840110—22no—
11#16840110C2214yes—
12#16839962C2218yes—
13#16839912—24no—
14#16839815C2212yes—
15#16839766—22no—
16C4×#11758194C2211yescharacteristic
17#16839508C2214yescomplemented
18Q8⋊#92418C2218yescomplemented
19C61×#564—22no—
20C61×#557—22no—
21C61×#557—22no—
22C61×#557C2214yescomplemented
23C61×#541—24no—
24C61×#541—22no—
25C61×#533C2212yes—
26C61×#533C2211yescharacteristic
27C22×D244—21no—
28C22×D244—22no—
29C2×#11758201—28no—
30C2×#11758194—21no—
31#14519774—22no—
32#14519774—21no—
33#14519774C2312yes—
34C4⋊#92418C2314yes—
35#14519697—28no—
36#14519697—28no—
37C22⋊#92418—22no—
38#14519688—21no—
39C4×#92418—22no—
40C4×#92418C2312yes—
41C4⋊Q8×C61—22no—
42C4⋊Q8×C61—24no—
43C4⋊Q8×C61D412yes—
44C4⋊Q8×C61D412yes—
45C4⋊Q8×C61C2312yes—
46#126×C61—28no—
47#125×C61—22no—
48#125×C61—28no—
49#125×C61—28no—
50#125×C61—24no—
51#125×C61—28no—
52#125×C61—22no—
53#125×C61D412yes—
54C22⋊Q8×C61—22no—
55C22⋊Q8×C61—22no—
56C22⋊Q8×C61D414yes—
57C4×Q8×C61—24no—
58C4×Q8×C61—24no—
59C4×Q8×C61—24no—
60C4×Q8×C61—22no—
61C4×Q8×C61—22no—
62C4×Q8×C61C2314yes—
63#117×C61—22no—
64#117×C61D412yes—
65C2×C42×C61C2311yescharacteristic
66C23×D61—41no—
67C2×Q8×C61—22no—
68C2×Q8×C61—22no—
69C2×Q8×C61C2×D414yes—
70C22×C4×C61C2×D412yes—
71C22×C4×C61C2×D411yescharacteristic
72C2×D244—22no—
73C2×D244—24no—
74C2×D244—44no—
75C2×D244—42no—
76C2×D244—41no—
77C2×D244—84no—
78C4⋊C4×C61—28no—
79C4⋊C4×C61—28no—
80C4⋊C4×C61—28no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—24no—
85C4⋊C4×C61—24no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61C2×D414yes—
89C4⋊C4×C61C2×D414yes—
90C22⋊C4×C61—24no—
91C22⋊C4×C61—22no—
92C42×C61—28no—
93C42×C61—22no—
94C42×C61—24no—
95C42×C61C2×D412yes—
96C42×C61C2×D411yescharacteristic
97C42×C61C2411yescharacteristic
98#11758201—88no—
99#11758194—42no—
100C2×#92418—24no—
101C2×#92418—22no—
102C23×C61#12011yescharacteristic
103C22×D61—42no—
104C22×D61—82no—
105C22×D61—84no—
106Q8×C61—28no—
107Q8×C61—44no—
108Q8×C61—44no—
109C2×C4×C61—24no—
110C2×C4×C61—22no—
111C2×C4×C61—24no—
112C2×C4×C61—24no—
113C2×C4×C61—24no—
114C2×C4×C61#12012yes—
115C2×C4×C61#12011yescharacteristic
116C2×C4×C61C22×D411yescharacteristic
117C2×C4×C61C22×D412yes—
118D244—44no—
119D244—48no—
120D244—84no—
121D244—82no—
122#92418—42no—
123#92418—44no—
124#18731—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—21no—
127C22×C61C2×#12011yescharacteristic
128C2×D61—84no—
129C2×D61—82no—
130C2×D61—164no—
131C4×C61—22no—
132C4×C61—21no—
133C4×C61—44no—
134C4×C61—44no—
135C4×C61—44no—
136C4×C61#45412yes—
137C4×C61#57714yes—
138#3556D6111yescharacteristic, complemented
139#3208—612no—
140#3138—611no—
141#1626—618no—
142#1544—611no—
143#1485—612no—
144C2×C61—41no—
145C2×C61#301012yes—
146C2×C61C2×#45411yescharacteristic
147D61—162no—
148#564—22no—
149#557—22no—
150#557—22no—
151#557C2×D6114yescomplemented
152#541—24no—
153#541—22no—
154#533C2×D6112yes—
155#533C2×D6111yescharacteristic
156C2×#127—611no—
157#492—1222no—
158#490—1222no—
159#486—614no—
160#486—1222no—
161#460—618no—
162#450—1224no—
163#431—612no—
164#418—1222no—
165C4×M16—611no—
166#332—614no—
167Q8⋊C8—618no—
168C61#1873111yescharacteristic, complemented, Sylow
169C22×D4—1222no—
170C22×D4—1221no—
171C2×SD16—1228no—
172C2×M16—1221no—
173C4⋊Q8—22no—
174C4⋊Q8—24no—
175C4⋊Q8C22⋊D6112yes—
176C4⋊Q8C22⋊D6112yes—
177C4⋊Q8C22×D6112yes—
178#127—612no—
179#127—1222no—
180#127—1221no—
181#126—28no—
182#125—22no—
183#125—28no—
184#125—28no—
185#125—24no—
186#125—28no—
187#125—22no—
188#125C22⋊D6112yes—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8C22⋊D6114yes—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8C22×D6114yes—
198#117—22no—
199#117C22⋊D6112yes—
200C2×C42C22×D6111yescharacteristic
201C4⋊C8—614no—
202#102—1228no—
203#102—1228no—
204C22⋊C8—1222no—
205#97—1221no—
206C4×C8—612no—
207C4×C8—1222no—
208C24—2441no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8C2×C22⋊D6114yes—
212C2×D4—1222no—
213C2×D4—1224no—
214C2×D4—4884no—
215C2×D4—2444no—
216C2×D4—2442no—
217C2×D4—2441no—
218C22×C4C2×C22⋊D6112yes—
219C22×C4C2×C22⋊D6111yescharacteristic
220SD16—4888no—
221M16—2442no—
222C2×C8—1222no—
223C2×C8—1224no—
224C4⋊C4—28no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4—22no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4C2×C22⋊D6114yes—
235C4⋊C4C2×C22⋊D6114yes—
236C22⋊C4—24no—
237C22⋊C4—22no—
238C42—28no—
239C42—24no—
240C42—22no—
241C42C2×C22⋊D6112yes—
242C42C2×C22⋊D6111yescharacteristic
243C42C23×D6111yescharacteristic
244C23—4884no—
245C23—4882no—
246C23—2442no—
247C23#1451974611yescharacteristic
248Q8—28no—
249Q8—44no—
250Q8—44no—
251D4—4884no—
252D4—4882no—
253D4—2444no—
254D4—2448no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4#1451974612yes—
261C2×C4#1451974611yescharacteristic
262C2×C4C22×C22⋊D6111yescharacteristic
263C2×C4C22×C22⋊D6112yes—
264C8—2442no—
265C8—2444no—
266C22—22no—
267C22—21no—
268C22—9764no—
269C22—4884no—
270C22—4882no—
271C22C2×#1451974611yescharacteristic
272C4—22no—
273C4—21no—
274C4—44no—
275C4—44no—
276C4—44no—
277C4#1683993512yes—
278C4#1684057014yes—
279C2—41no—
280C2—9762no—
281C2#2145510512yes—
282C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3556C2
socleC22×C61C2×#120
prime radicalC2×#14519774C22
hypercenter#3556D61
hypocenterC61#18731
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
3. #21444915C2semidirect
4. #21444484C2semidirect
5. #21444240C2semidirect
6. C61×#3556C2semidirect
17. #16839508C22semidirect
18. Q8⋊#92418C22semidirect
22. C61×#557C22semidirect
138. #3556D61semidirect
151. #557C2×D61semidirect
168. C61#18731semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214561952
#214449152
#214444842
#214442402
C61×#35562
#1873161