Finite Group Databasebeta

← #29593481

Subgroups — #29593481

Order 15616.

368 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 73 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
1D4×Dic244C211yesmaximal, characteristic
2#21448072C211yesmaximal, characteristic, complemented
3#21447870C211yesmaximal, characteristic, complemented
4#21445184C211yesmaximal, characteristic
5#21443869C211yesmaximal, characteristic, complemented
6#21443784C211yesmaximal, characteristic, complemented
7C61×D4⋊C16C211yesmaximal, characteristic
8C22×Dic244—21no—
9D4×Dic122—21no—
10C2×Dic488—22no—
11C2×Dic488—21no—
12#16840142C2211yescharacteristic
13#16840142C2211yescharacteristic
14#16840042—21no—
15#16840003—21no—
16#16839973—21no—
17#16839920—22no—
18C4×Dic244C2211yescharacteristic
19#16839567—21no—
20#16839515C2211yescharacteristic
21#16839509—21no—
22C8×D4×C61C2211yescharacteristic
23C61×C4⋊C16C2211yescharacteristic
24C4×C16×C61C2211yescharacteristic
25C22×Dic122—22no—
26C2×Dic244—22no—
27C2×Dic244—21no—
28C2×Dic244—21no—
29C2×Dic244—21no—
30C2×Dic244—42no—
31C2×Dic244—42no—
32C2×Dic244D411yescharacteristic
33C2×Dic244D412yes—
34#14519777—21no—
35#14519777—21no—
36#14519776—41no—
37#14519756—22no—
38#14519754—42no—
39C4×Dic122—21no—
40D4×Dic61—41no—
41Dic488—44no—
42Dic488—42no—
43#14519711D411yescharacteristic
44#14519710—41no—
45C4⋊#92418—21no—
46#14519702—22no—
47#14519702—21no—
48#14519698—41no—
49C22×C8×C61—21no—
50C4×D4×C61D411yescharacteristic
51C2×C16×C61—21no—
52C2×C16×C61—21no—
53C4⋊C8×C61D411yescharacteristic
54C22⋊C8×C61—21no—
55C4×C8×C61C2311yescharacteristic
56C2×D4×C61D811yescharacteristic
57C22×C4×C61—21no—
58C22×Dic61—42no—
59C2×Dic122—21no—
60C2×Dic122—22no—
61C2×Dic122—44no—
62C2×Dic122—42no—
63C2×Dic122—41no—
64C2×Dic122—82no—
65C2×C8×C61—21no—
66C2×C8×C61—41no—
67C2×C8×C61—41no—
68C2×C8×C61C2×D411yescharacteristic
69C2×C8×C61C2×D411yescharacteristic
70C4⋊C4×C61D811yescharacteristic
71C22⋊C4×C61—21no—
72C42×C61C2×D411yescharacteristic
73C22⋊Dic61—42no—
74C4⋊Dic61—21no—
75C4⋊Dic61—41no—
76C4⋊Dic61—41no—
77C4⋊Dic61—41no—
78C4⋊Dic61—41no—
79#11758196—42no—
80#11758196—41no—
81C4×Dic61—41no—
82C2×#92418—41no—
83Dic244—21no—
84Dic244—42no—
85Dic244—42no—
86Dic244—44no—
87Dic244—41no—
88Dic244—41no—
89Dic244—41no—
90Dic244—41no—
91Dic244D812yes—
92C16×C61—22no—
93C16×C61—41no—
94C23×C61—21no—
95D4×C61—21no—
96D4×C61Q3212yescomplemented
97C2×C4×C61—21no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61#12011yescharacteristic
101C2×C4×C61C2×D811yescharacteristic
102C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
103C2×Dic61—41no—
104C2×Dic61—41no—
105C2×Dic61—42no—
106C2×Dic61—84no—
107C2×Dic61—81no—
108Dic122—22no—
109Dic122—41no—
110Dic122—42no—
111Dic122—42no—
112Dic122—84no—
113Dic122—81no—
114Dic122—81no—
115C8×C61—21no—
116C8×C61—41no—
117C8×C61C2×D811yescharacteristic
118C8×C61AGL(1,ℤ/8)11yescharacteristic
119#92418—81no—
120#12322—611nomaximal, Sylow
121C22×C61—22no—
122C22×C61—41no—
123C22×C61—41no—
124C22×C61#44711yescharacteristic
125C4×C61—21no—
126C4×C61—41no—
127C4×C61#44711yescharacteristic
128C4×C61#45311yescharacteristic
129C4×C61C2×Q3211yescharacteristic
130C4×C61#51011yescharacteristic
131Dic61—42no—
132Dic61—82no—
133Dic61—81no—
134Dic61—161no—
135D4×Q16—611no—
136#2247—611no—
137#2209—611no—
138#1669—611no—
139#1362—611no—
140#1332—611no—
141D4⋊C16D6111yescharacteristic
142C2×C61—42no—
143C2×C61#161911yescharacteristic
144C2×C61#218811yescharacteristic
145C2×C61#219511yescharacteristic
146C22×Q16—1221no—
147D4×Q8—1221no—
148C2×Q32—1221no—
149C2×Q32—1222no—
150#494—611no—
151#494—611no—
152#474—1221no—
153#467—1221no—
154#462—1221no—
155#451—1222no—
156C4×Q16—611no—
157C8×D4C2×D6111yescharacteristic
158C4⋊C16C2×D6111yescharacteristic
159#358—1221no—
160C4×C16C2×D6111yescharacteristic
161#335—611no—
162#332—1221no—
163C61#1232211yescharacteristic, complemented, Sylow
164C22×Q8—1222no—
165C2×Q16—611no—
166C2×Q16—612no—
167C2×Q16—1222no—
168C2×Q16—1221no—
169C2×Q16—1221no—
170C2×Q16—1221no—
171C2×Q16—2442no—
172C2×Q16—2442no—
173C22×C8—21no—
174C4⋊Q8—1221no—
175C4⋊Q8—1221no—
176C4⋊Q8—2441no—
177C22⋊Q8—1222no—
178C22⋊Q8—2442no—
179C4×Q8—1221no—
180C4×D4—2441no—
181C4×D4D24411yescharacteristic
182Q32—2444no—
183Q32—2442no—
184C2×C16—21no—
185C2×C16—21no—
186#107—611no—
187#107—2441no—
188C4⋊C8—1221no—
189C4⋊C8D24411yescharacteristic
190#103—1222no—
191#103—1221no—
192#102—2441no—
193C22⋊C8—21no—
194C4×C8C22×D6111yescharacteristic
195C2×Q8—1221no—
196C2×Q8—1222no—
197C2×Q8—4882no—
198C2×Q8—2442no—
199C2×Q8—2444no—
200C2×Q8—2441no—
201C2×D4D48811yescharacteristic
202C22×C4—21no—
203C22×C4—2442no—
204Q16—612no—
205Q16—1221no—
206Q16—2442no—
207Q16—2442no—
208Q16—2444no—
209Q16—2441no—
210Q16—2441no—
211Q16—2441no—
212Q16—2441no—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—2441no—
217C2×C8D4×D6111yescharacteristic
218C2×C8D4×D6111yescharacteristic
219C4⋊C4—1221no—
220C4⋊C4—2442no—
221C4⋊C4—2441no—
222C4⋊C4—2441no—
223C4⋊C4—2441no—
224C4⋊C4—2441no—
225C4⋊C4—2441no—
226C4⋊C4D48811yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—2442no—
229C42—2441no—
230C42C2×D24411yescharacteristic
231C16—22no—
232C16—41no—
233C23—21no—
234Q8—1222no—
235Q8—4884no—
236Q8—4881no—
237Q8—4881no—
238Q8—2442no—
239Q8—2442no—
240Q8—2441no—
241D4—21no—
242D4Dic48812yescomplemented
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—4881no—
247C2×C4—4884no—
248C2×C4—2441no—
249C2×C4—2442no—
250C2×C4—2441no—
251C2×C4#1451974711yescharacteristic
252C2×C4C2×D48811yescharacteristic
253C2×C4#1451980511yescharacteristic
254C8—21no—
255C8—41no—
256C8—4881no—
257C8D8×D6111yescharacteristic
258C8#1451980411yescharacteristic
259C22—22no—
260C22—41no—
261C22—41no—
262C22#1683989811yescharacteristic
263C4—21no—
264C4—41no—
265C4—9761no—
266C4—4882no—
267C4—4881no—
268C4—2442no—
269C4#1683989711yescharacteristic
270C4#1683993111yescharacteristic
271C4C2×Dic48811yescharacteristic
272C4#1684020511yescharacteristic
273C2—42no—
274C2#2144487511yescharacteristic
275C2#2144775511yescharacteristic
276C2#2144779911yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×D4⋊C16C2
socleC22×C61#447
prime radicalC2×D4×C61D8
hypercenterD4⋊C16D61
hypocenterC61#12322
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ D4⋊C16

Complements

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

NH ≅ G/Ntype
2. #21448072C2semidirect
3. #21447870C2semidirect
5. #21443869C2semidirect
6. #21443784C2semidirect
96. D4×C61Q32semidirect
163. C61#12322semidirect
242. D4Dic488semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2442
#214480722
#214478702
#214451842
#214438692
#214437842
C61×D4⋊C162
#1232261