Finite Group Databasebeta

← #29589505

Subgroups — #29589505

Order 15616.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#21444157C212yesmaximal
2#21443765C212yesmaximal, complemented
3#21443753C212yesmaximal, complemented
4C61×#2326C211yesmaximal, characteristic
5#16839726—22no—
6#16839488C2214yes—
7#16839486—22no—
8C2×#124×C61C411yescharacteristic
9C2×C4×Q8×C61C411yescharacteristic
10C2×C4×D4×C61C2211yescharacteristic
11C61×#391C2211yescharacteristic
12C61×#390C411yescharacteristic
13C61×#388—22no—
14C61×#385—22no—
15C61×#384C2211yescharacteristic
16C61×#376C411yescharacteristic
17C2×#11758194—42no—
18C22⋊#92418—24no—
19C22⋊#92418—42no—
20C22×Q8×C61C2×C411yescharacteristic
21C22×D4×C61C2×C411yescharacteristic
22C23×C4×C61D412yes—
23#124×C61—22no—
24#124×C61—22no—
25C4×Q8×C61—42no—
26C4×D4×C61—42no—
27C2×C4⋊C4×C61—21no—
28C2×C4⋊C4×C61—22no—
29C2×C4⋊C4×C61C2×C411yescharacteristic
30C2×C4⋊C4×C61C2311yescharacteristic
31C2×C22⋊C4×C61—22no—
32C2×C22⋊C4×C61—22no—
33C2×C22⋊C4×C61D412yes—
34C2×C42×C61—21no—
35C2×C42×C61C2×C411yescharacteristic
36C2×C42×C61C2×C411yescharacteristic
37C2×C42×C61C2×C411yescharacteristic
38#95×C61—21no—
39#95×C61—21no—
40#95×C61—22no—
41C24×C61C22⋊C412yes—
42C2×Q8×C61—21no—
43C2×Q8×C61—41no—
44C2×Q8×C61—41no—
45C2×Q8×C61M1612yescomplemented
46C2×D4×C61—21no—
47C2×D4×C61—21no—
48C2×D4×C61—41no—
49C2×D4×C61—41no—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—82no—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D412yes—
61C4⋊C4×C61—41no—
62C4⋊C4×C61—41no—
63C4⋊C4×C61—42no—
64C4⋊C4×C61—82no—
65C22⋊C4×C61—28no—
66C22⋊C4×C61—42no—
67C22⋊C4×C61—82no—
68C42×C61—22no—
69C42×C61—21no—
70C42×C61—42no—
71C42×C61—41no—
72C42×C61M1612yescomplemented
73#11758194—84no—
74C2×#92418—44no—
75C23×C61—22no—
76C23×C61—22no—
77C23×C61—22no—
78C23×C61—42no—
79C23×C61—42no—
80C23×C61C2×C22⋊C411yescharacteristic
81Q8×C61—42no—
82Q8×C61—42no—
83D4×C61—42no—
84D4×C61—42no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61—22no—
90C2×C4×C61—21no—
91C2×C4×C61—41no—
92C2×C4×C61—44no—
93C2×C4×C61—41no—
94C2×C4×C61—41no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61—42no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—42no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—82no—
105C2×C4×C61#9912yes—
106C2×C4×C61C2×M1611yescharacteristic
107C2×C4×C61C2×M1611yescharacteristic
108#92418—84no—
109#11535—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—21no—
112C22×C61—44no—
113C22×C61—42no—
114C22×C61—42no—
115C22×C61—42no—
116C22×C61—82no—
117C22×C61#40711yescharacteristic
118C22×C61C2×#9911yescharacteristic
119C22×C61#41311yescharacteristic
120C4×C61—22no—
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—42no—
124C4×C61—42no—
125C4×C61—41no—
126C4×C61—42no—
127C4×C61—81no—
128C4×C61—81no—
129#2326D6111yescharacteristic
130#1465—612no—
131#1325—612no—
132#1319—612no—
133C2×C61—21no—
134C2×C61—21no—
135C2×C61—82no—
136C2×C61#146511yescharacteristic
137C2×C61#212711yescharacteristic
138C2×C61#213511yescharacteristic
139C2×#124Dic6111yescharacteristic
140C2×C4×Q8Dic6111yescharacteristic
141C2×C4×D4C2×D6111yescharacteristic
142#407—1222no—
143#391C2×D6111yescharacteristic
144#390Dic6111yescharacteristic
145#388—22no—
146#385—22no—
147#384C2×D6111yescharacteristic
148#376Dic6111yescharacteristic
149#324—614no—
150#323—1222no—
151C61#1153511yescharacteristic, complemented, Sylow
152C22×Q8C2×Dic6111yescharacteristic
153C22×D4C2×Dic6111yescharacteristic
154C23×C4C22⋊D6112yes—
155C2×M16—2442no—
156#124—22no—
157#124—22no—
158C4×Q8—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×Dic6111yescharacteristic
163C2×C4⋊C4C22×D6111yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D6112yes—
167C2×C42—21no—
168C2×C42C2×Dic6111yescharacteristic
169C2×C42C2×Dic6111yescharacteristic
170C2×C42C2×Dic6111yescharacteristic
171C22⋊C8—1224no—
172C22⋊C8—2442no—
173#95—21no—
174#95—21no—
175#95—22no—
176C24C22⋊Dic6112yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#1175819412yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊Dic6111yescharacteristic
193C22×C4C22⋊Dic6111yescharacteristic
194C22×C4C22×Dic6111yescharacteristic
195C22×C4C2×C22⋊D6112yes—
196M16—4884no—
197C2×C8—2444no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42#1175819412yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic6111yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#1451969212yes—
241C2×C4C2×#1175819411yescharacteristic
242C2×C4C2×#1175819411yescharacteristic
243C8—4884no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#1683972611yescharacteristic
252C22C2×#1451969211yescharacteristic
253C22#1683974611yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2144415711yescharacteristic
267C2#2144748811yescharacteristic
268C2#2144751411yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2326C2
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC2×C4⋊C4×C61C23
hypercenter#2326D61
hypocenterC61#11535
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #21443765C2semidirect
3. #21443753C2semidirect
45. C2×Q8×C61M16semidirect
72. C42×C61M16semidirect
151. C61#11535semidirect
180. C2×Q8#11758194semidirect
209. C42#11758194semidirect

Maximal subgroups

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

subgroupindex
#214441572
#214437652
#214437532
C61×#23262
#1153561