Finite Group Databasebeta

← #29593453

Subgroups — #29593453

Order 15616.

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

normal subgroups only

This group has 384 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, complemented
2#21448075C211yesmaximal, characteristic, complemented
3#21447860C211yesmaximal, characteristic, complemented
4#21445174C211yesmaximal, characteristic
5#21443869C211yesmaximal, characteristic, complemented
6#21443784C211yesmaximal, characteristic, complemented
7C61×D4⋊C16C211yesmaximal, characteristic, complemented
8C22×Dic244—21no—
9D4×Dic122—21no—
10C2×#14519722—22no—
11C2×#14519722—21no—
12#16840142C2211yescharacteristic
13#16840139C2211yescharacteristic
14#16840003—21no—
15#16839973—21no—
16#16839957—21no—
17#16839920—22no—
18C4×Dic244C2211yescharacteristic
19#16839564—21no—
20#16839515C2211yescharacteristic
21#16839505—21no—
22C8×D4×C61C2211yescharacteristic
23C61×C4⋊C16C2211yescharacteristic
24C4×C16×C61C2211yescharacteristic
25C22×Dic122—22no—
26C2×Dic244—22no—
27C2×Dic244—21no—
28C2×Dic244—42no—
29C2×Dic244—42no—
30C2×Dic244D411yescharacteristic
31C2×Dic244D412yes—
32C2×D488—21no—
33C2×D488—21no—
34C2×#11758199—41no—
35#14519777—21no—
36#14519776—41no—
37#14519774—21no—
38#14519756—22no—
39#14519754—42no—
40C4×Dic122—21no—
41D4×Dic61—41no—
42#14519722—44no—
43#14519722—42no—
44#14519711D411yescharacteristic
45C4⋊#92418—21no—
46#14519702—22no—
47#14519702—21no—
48#14519697—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×D244—41no—
60C2×D244—41no—
61C2×D244—41no—
62C2×Dic122—21no—
63C2×Dic122—22no—
64C2×Dic122—44no—
65C2×Dic122—42no—
66C2×Dic122—82no—
67C2×C8×C61—21no—
68C2×C8×C61—41no—
69C2×C8×C61—41no—
70C2×C8×C61C2×D411yescharacteristic
71C2×C8×C61C2×D411yescharacteristic
72C4⋊C4×C61D811yescharacteristic
73C22⋊C4×C61—21no—
74C42×C61C2×D411yescharacteristic
75C22⋊Dic61—42no—
76#11758199—82no—
77C4⋊Dic61—21no—
78C4⋊Dic61—41no—
79C4⋊Dic61—41no—
80#11758196—42no—
81#11758196—41no—
82C4×Dic61—41no—
83C2×#92418—41no—
84Dic244—21no—
85Dic244—42no—
86Dic244—44no—
87Dic244—41no—
88Dic244—41no—
89Dic244D812yescomplemented
90D488—42no—
91D488—41no—
92D488—41no—
93C16×C61—22no—
94C16×C61—41no—
95C23×C61—21no—
96C22×D61—81no—
97D4×C61—21no—
98D4×C61SD3212yescomplemented
99C2×C4×C61—21no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61#12011yescharacteristic
103C2×C4×C61C2×D811yescharacteristic
104C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
105C2×Dic61—41no—
106C2×Dic61—41no—
107C2×Dic61—42no—
108C2×Dic61—84no—
109D244—81no—
110D244—81no—
111D244—81no—
112D244—81no—
113D244—161no—
114Dic122—22no—
115Dic122—41no—
116Dic122—42no—
117Dic122—42no—
118Dic122—84no—
119C8×C61—21no—
120C8×C61—41no—
121C8×C61C2×D811yescharacteristic
122C8×C61AGL(1,ℤ/8)11yescharacteristic
123#92418—81no—
124#12318—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—41no—
127C22×C61—41no—
128C22×C61#44711yescharacteristic
129C2×D61—161no—
130C2×D61—161no—
131C2×D61—161no—
132C4×C61—21no—
133C4×C61—41no—
134C4×C61#44711yescharacteristic
135C4×C61#45311yescharacteristic
136C4×C61C2×SD3211yescharacteristic
137C4×C61#51011yescharacteristic
138Dic61—42no—
139Dic61—82no—
140Dic61—81no—
141D4×Q16—611no—
142#2248—611no—
143#2207—611no—
144#1667—611no—
145#1362—611no—
146#1332—611no—
147D4⋊C16D6111yescharacteristic, complemented
148C2×C61—42no—
149C2×C61#161911yescharacteristic
150C2×C61#218511yescharacteristic
151C2×C61#219311yescharacteristic
152D61—321no—
153C22×Q16—1221no—
154D4×Q8—1221no—
155C2×SD32—1221no—
156C2×SD32—1222no—
157#494—611no—
158#493—611no—
159#467—1221no—
160#462—1221no—
161#459—1221no—
162#451—1222no—
163C4×Q16—611no—
164C8×D4C2×D6111yescharacteristic
165C4⋊C16C2×D6111yescharacteristic
166#357—1221no—
167C4×C16C2×D6111yescharacteristic
168#335—611no—
169#331—1221no—
170C61#1231811yescharacteristic, complemented, Sylow
171C22×Q8—1222no—
172C2×Q16—611no—
173C2×Q16—612no—
174C2×Q16—1222no—
175C2×Q16—1221no—
176C2×Q16—2442no—
177C2×Q16—2442no—
178C2×D8—1221no—
179C2×D8—1221no—
180C2×D8—2441no—
181C22×C8—21no—
182C4⋊Q8—1221no—
183C4⋊Q8—2441no—
184#127—1221no—
185C22⋊Q8—1222no—
186C22⋊Q8—2442no—
187C4×Q8—1221no—
188C4×D4—2441no—
189C4×D4D24411yescharacteristic
190SD32—2444no—
191SD32—2442no—
192C2×C16—21no—
193C2×C16—21no—
194#107—611no—
195C4⋊C8—1221no—
196C4⋊C8D24411yescharacteristic
197#103—1222no—
198#103—1221no—
199#102—2441no—
200C22⋊C8—21no—
201C4×C8C22×D6111yescharacteristic
202C2×Q8—1221no—
203C2×Q8—1222no—
204C2×Q8—4882no—
205C2×Q8—2442no—
206C2×Q8—2444no—
207C2×D4—2441no—
208C2×D4—2441no—
209C2×D4—2441no—
210C2×D4D48811yescharacteristic
211C22×C4—21no—
212C22×C4—2442no—
213Q16—612no—
214Q16—1221no—
215Q16—2442no—
216Q16—2444no—
217Q16—2441no—
218Q16—2441no—
219D8—4882no—
220D8—2442no—
221D8—2441no—
222D8—2441no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—2441no—
227C2×C8D4×D6111yescharacteristic
228C2×C8D4×D6111yescharacteristic
229C4⋊C4—1221no—
230C4⋊C4—2442no—
231C4⋊C4—2441no—
232C4⋊C4—2441no—
233C4⋊C4—2441no—
234C4⋊C4D48811yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—2442no—
237C42—2441no—
238C42C2×D24411yescharacteristic
239C16—22no—
240C16—41no—
241C23—21no—
242C23—4881no—
243Q8—1222no—
244Q8—4884no—
245Q8—2442no—
246Q8—2442no—
247Q8—2441no—
248D4—21no—
249D4—9761no—
250D4—4881no—
251D4—4881no—
252D4—4881no—
253D4—4881no—
254D4#1451972212yescomplemented
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—4884no—
259C2×C4—2441no—
260C2×C4—2442no—
261C2×C4—2441no—
262C2×C4#1451974711yescharacteristic
263C2×C4C2×D48811yescharacteristic
264C2×C4#1451980511yescharacteristic
265C8—21no—
266C8—41no—
267C8—4881no—
268C8D8×D6111yescharacteristic
269C8#1451980411yescharacteristic
270C22—22no—
271C22—41no—
272C22—41no—
273C22—9761no—
274C22—9761no—
275C22—9761no—
276C22#1683989811yescharacteristic
277C4—21no—
278C4—41no—
279C4—4882no—
280C4—4881no—
281C4—2442no—
282C4#1683989711yescharacteristic
283C4#1683993111yescharacteristic
284C4C2×#1451972211yescharacteristic
285C4#1684020511yescharacteristic
286C2—42no—
287C2—19521no—
288C2#2144487511yescharacteristic
289C2#2144773811yescharacteristic
290C2#2144778511yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×D4⋊C16C2
socleC22×C61#447
prime radical#21445174C2
hypercenterD4⋊C16D61
hypocenterC61#12318
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
1. D4×Dic244C2semidirect
2. #21448075C2semidirect
3. #21447860C2semidirect
5. #21443869C2semidirect
6. #21443784C2semidirect
7. C61×D4⋊C16C2semidirect
89. Dic244D8semidirect
98. D4×C61SD32semidirect
147. D4⋊C16D61semidirect
170. C61#12318semidirect
254. D4#14519722semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2442
#214480752
#214478602
#214451742
#214438692
#214437842
C61×D4⋊C162
#1231861