Finite Group Databasebeta

← #30599190

Subgroups — #30599190

Order 15616.

358 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C4×#14519847C211yesmaximal, characteristic, complemented
2#21461380C211yesmaximal, characteristic, complemented
3#21461286C212yesmaximal
4#21460685C212yesmaximal, complemented
5#21454793C211yesmaximal, characteristic
6#16840784C2212yes—
7C2×#14519847—21no—
8C4×#11758213C2212yes—
9#16840638—21no—
10#16840625—21no—
11#16840623—21no—
12#16840619—22no—
13#16840618—22no—
14#16840299C2211yescharacteristic
15Q8×#92418C411yescharacteristic
16#16839841C411yescharacteristic
17#16839835—21no—
18#16839827—21no—
19#16839827C411yescharacteristic
20#16839822—21no—
21#16839818C2211yescharacteristic
22#16839719—21no—
23C4×C8×D61C2211yescharacteristic
24C61×#446C411yescharacteristic
25C2×#11758213—22no—
26#14519847—24no—
27C4⋊#92420—22no—
28#14519834—22no—
29#14519832—22no—
30#14519832—22no—
31C4×#92420—22no—
32C2×#11758189—21no—
33C2×C8×D61—21no—
34C2×C8×D61D411yescharacteristic
35C2×C8×D61D411yescharacteristic
36#14519772—21no—
37#14519768D411yescharacteristic
38#14519766D411yescharacteristic
39#14519762—21no—
40#14519753Q811yescharacteristic
41#14519752Q811yescharacteristic
42C4×D244—21no—
43C4×D244C2×C411yescharacteristic
44#14519734—21no—
45C42×D61C2311yescharacteristic
46C4⋊#92418—21no—
47C4⋊#92418C2×C411yescharacteristic
48#14519689—21no—
49#14519689—21no—
50#14519689—21no—
51#14519687—21no—
52C4×#92418—21no—
53C4×#92418C2×C411yescharacteristic
54C8×Dic61D411yescharacteristic
55C8×Dic61D411yescharacteristic
56C4×Q8×C61C2×C411yescharacteristic
57C4⋊C8×C61—21no—
58C4⋊C8×C61C2×C411yescharacteristic
59#97×C61—21no—
60C4×C8×C61C2×C411yescharacteristic
61C2×Q8×C61C4⋊C411yescharacteristic
62C2×D244—21no—
63C2×D244C4⋊C411yescharacteristic
64C2×C4×D61—21no—
65C2×C4×D61C2×D411yescharacteristic
66C2×C4×D61C2×Q811yescharacteristic
67C2×C4×D61#4111yescharacteristic
68C2×#92420—42no—
69C2×#92420—42no—
70#11758213—24no—
71#11758213—42no—
72C2×C8×C61—21no—
73C2×C8×C61C22⋊C411yescharacteristic
74C2×C8×C61C22⋊C411yescharacteristic
75C4⋊C4×C61—21no—
76C4⋊C4×C61C4⋊C411yescharacteristic
77C42×C61—21no—
78C42×C61C22×C411yescharacteristic
79#11758198—21no—
80#11758198—21no—
81#11758198—21no—
82#11758198—21no—
83#11758198—21no—
84C4⋊Dic61—21no—
85C4⋊Dic61C4⋊C411yescharacteristic
86#11758196—21no—
87#11758196—21no—
88C4×Dic61C2×D411yescharacteristic
89C4×Dic61C2×D411yescharacteristic
90C4×Dic61#4111yescharacteristic
91C2×#92418—21no—
92C2×#92418C22⋊C411yescharacteristic
93C2×#92418C22⋊C411yescharacteristic
94#11758189—42no—
95C8×D61—22no—
96C8×D61—41no—
97C8×D61—42no—
98C22×D61—21no—
99C22×D61C22⋊Q811yescharacteristic
100Q8×C61—41no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61C2×C22⋊C411yescharacteristic
104C2×C4×C61C2×C4⋊C411yescharacteristic
105C2×C4×C61#11711yescharacteristic
106C2×Dic61—21no—
107C2×Dic61#12111yescharacteristic
108C2×Dic61#12111yescharacteristic
109C2×Dic61C22⋊Q811yescharacteristic
110D244—41no—
111D244—42no—
112C4×D61—21no—
113C4×D61—21no—
114C4×D61—41no—
115C4×D61—41no—
116C4×D61—41no—
117C4×D61—41no—
118#92420—44no—
119#92420—82no—
120C8×C61—21no—
121C8×C61—41no—
122C8×C61#10412yes—
123#92418—21no—
124#92418—21no—
125#92418—41no—
126#8693—611nomaximal, Sylow
127C22×C61#38011yescharacteristic
128C2×D61—21no—
129C2×D61—21no—
130C2×D61—21no—
131C2×D61—41no—
132C2×D61—41no—
133C2×D61—41no—
134C4×C61—21no—
135C4×C61—21no—
136C4×C61—41no—
137C4×C61—41no—
138C4×C61C2×#10411yescharacteristic
139C4×C61#42111yescharacteristic
140Dic61—21no—
141Dic61—21no—
142Dic61—21no—
143Dic61—41no—
144#2966—611no—
145C4×#108—611no—
146#1568—611no—
147#1529—612no—
148#1291—612no—
149C2×C61#180611yescharacteristic
150C2×C61#181311yescharacteristic
151C2×C61#181411yescharacteristic
152D61—41no—
153D61—81no—
154#529—611no—
155#446#692311yescharacteristic
156C8×Q8—611no—
157#436—611no—
158#435—1221no—
159C8×D4—611no—
160C8×D4—1221no—
161#433—1221no—
162#432—611no—
163#432—612no—
164C2×#108—1221no—
165#405—1221no—
166C4×M16—612no—
167C2×C4×C8—611no—
168#341—1221no—
169#335—1221no—
170#334—1221no—
171#330—1222no—
172#329—1222no—
173C61#869311yescharacteristic, complemented, Sylow
174C2×M16—1221no—
175C2×M16—1222no—
176C22×C8—611no—
177C22×C8—611no—
178C22×C8—1221no—
179#126—1221no—
180#125—611no—
181#124—611no—
182#123—1221no—
183C22⋊Q8—611no—
184#121—611no—
185C4×Q8C2×#692311yescharacteristic
186C4×D4—611no—
187C4×D4—1221no—
188#117—1221no—
189C2×C42—611no—
190#108—1224no—
191C4⋊C8—21no—
192C4⋊C8—611no—
193C4⋊C8—1221no—
194C4⋊C8—1222no—
195C4⋊C8C2×#692311yescharacteristic
196C22⋊C8—1221no—
197C22⋊C8—1221no—
198C22⋊C8—1221no—
199C22⋊C8—1222no—
200#97—21no—
201#97—1221no—
202#97—1222no—
203#97—1222no—
204C4×C8—611no—
205C4×C8—611no—
206C4×C8—611no—
207C4×C8—1221no—
208C4×C8—1222no—
209C4×C8C2×#692311yescharacteristic
210C2×Q8C4⋊#692311yescharacteristic
211C2×D4—611no—
212C2×D4—1221no—
213C22×C4—611no—
214C22×C4—611no—
215C22×C4—611no—
216C22×C4—1221no—
217M16—1224no—
218M16—2442no—
219M16—2442no—
220C2×C8—21no—
221C2×C8—611no—
222C2×C8—611no—
223C2×C8—1222no—
224C2×C8—1221no—
225C2×C8—2441no—
226C2×C8—2442no—
227C2×C8—2442no—
228C2×C8—2442no—
229C2×C8C22⋊#692311yescharacteristic
230C2×C8C22⋊#692311yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—611no—
233C4⋊C4—1221no—
234C4⋊C4—1221no—
235C4⋊C4—1221no—
236C4⋊C4C4⋊#692311yescharacteristic
237C22⋊C4—1221no—
238C22⋊C4—1221no—
239C22⋊C4—1221no—
240C22⋊C4—1221no—
241C22⋊C4—1221no—
242C42—21no—
243C42—611no—
244C42—611no—
245C42—611no—
246C42C22×#692311yescharacteristic
247C23—611no—
248C23—1221no—
249Q8—41no—
250D4—2441no—
251D4—2442no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—611no—
255C2×C4—611no—
256C2×C4—611no—
257C2×C4—1221no—
258C2×C4—1221no—
259C2×C4—1221no—
260C2×C4—2441no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4—2441no—
264C2×C4C2×C22⋊#692311yescharacteristic
265C2×C4C2×C4⋊#692311yescharacteristic
266C2×C4#1451985611yescharacteristic
267C8—21no—
268C8—41no—
269C8—1221no—
270C8—1221no—
271C8—4882no—
272C8—2444no—
273C8—2441no—
274C8#1451984212yes—
275C22—1221no—
276C22—1221no—
277C22—1221no—
278C22—2441no—
279C22—2441no—
280C22—2441no—
281C22#1684070411yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—1221no—
287C4—1221no—
288C4—1221no—
289C4—2441no—
290C4C2×#1451984211yescharacteristic
291C4#1684075911yescharacteristic
292C2—4881no—
293C2—2441no—
294C2#2146169211yescharacteristic
295C2#2146170411yescharacteristic
296C2#2146170511yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#446C4
socleC22×C61#380
prime radical#14519752Q8
cosocleC42×D61C23
hypercenter#446#6923
hypocenterC61#8693
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #446

Complements

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

NH ≅ G/Ntype
1. C4×#14519847C2semidirect
2. #21461380C2semidirect
4. #21460685C2semidirect
173. C61#8693semidirect

Maximal subgroups

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

subgroupindex
C4×#145198472
#214613802
#214612862
#214606852
#214547932
#869361