Finite Group Databasebeta

← #30604473

Subgroups — #30604473

Order 15616.

326 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 92 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#21462179C211yesmaximal, characteristic, complemented
2C4×#14519852C211yesmaximal, characteristic, complemented
3#21460786C212yesmaximal, complemented
4D4⋊#11758187C212yesmaximal, complemented
5#21454797C211yesmaximal, characteristic
6C2×#14519852—21no—
7C4⋊#11758187C2212yes—
8#16840657—21no—
9#16840651—21no—
10#16840648—21no—
11C4×#11758187C2212yes—
12#16840294C411yescharacteristic
13#16839892C411yescharacteristic
14#16839886C2211yescharacteristic
15#16839831—21no—
16D4×#92418C2211yescharacteristic
17#16839825—21no—
18#16839820—21no—
19#16839818C411yescharacteristic
20#16839720—21no—
21C4×C8×D61C2211yescharacteristic
22C61×#436C411yescharacteristic
23#14519852—24no—
24C2×#11758187—22no—
25C2×#11758187—22no—
26C22×#92418—21no—
27C2×C8×D61D411yescharacteristic
28C2×C8×D61D411yescharacteristic
29#14519770—21no—
30#14519768C811yescharacteristic
31#14519764C811yescharacteristic
32#14519759—21no—
33#14519757C811yescharacteristic
34#14519748C811yescharacteristic
35C4×Dic122C2×C411yescharacteristic
36C4×C22⋊D61—21no—
37#14519736—21no—
38C42×D61C2×C411yescharacteristic
39C4⋊#92418C2×C411yescharacteristic
40#14519706—21no—
41#14519706—21no—
42C22⋊#92418—21no—
43#14519689—21no—
44#14519687—21no—
45#14519687—21no—
46C4×#92418C2311yescharacteristic
47C8×Dic61—21no—
48C8×Dic61D411yescharacteristic
49C8×Dic61D411yescharacteristic
50C2×M16×C61—21no—
51C4×D4×C61C2×C411yescharacteristic
52C4⋊C8×C61C2×C411yescharacteristic
53C22⋊C8×C61—21no—
54C4×C8×C61C2×C411yescharacteristic
55C2×D4×C61C2×C811yescharacteristic
56C22×C4×C61—21no—
57C2×C22⋊D61—21no—
58C2×C4×D61C22⋊C411yescharacteristic
59C2×C4×D61C2×C811yescharacteristic
60C2×C4×D61M1611yescharacteristic
61C2×Dic122C2×C811yescharacteristic
62M16×C61—42no—
63C2×C8×C61—21no—
64C2×C8×C61C22⋊C411yescharacteristic
65C2×C8×C61C22⋊C411yescharacteristic
66C4⋊C4×C61C2×C811yescharacteristic
67C22⋊C4×C61—21no—
68C42×C61C22×C411yescharacteristic
69C22⋊Dic61—21no—
70C22⋊Dic61—21no—
71C22⋊Dic61—21no—
72#11758198—21no—
73C4⋊Dic61—21no—
74C4⋊Dic61—21no—
75C4⋊Dic61C2×C811yescharacteristic
76#11758196—21no—
77#11758196—21no—
78C4×Dic61—21no—
79C4×Dic61C22⋊C411yescharacteristic
80C4×Dic61C2×C811yescharacteristic
81C4×Dic61M1611yescharacteristic
82C2×#92418—21no—
83C2×#92418—42no—
84C2×#92418C2×D411yescharacteristic
85C2×#92418C2×D411yescharacteristic
86C8×D61—22no—
87C8×D61—22no—
88#11758187—24no—
89#11758187—42no—
90C23×C61—21no—
91C22×D61C22⋊C811yescharacteristic
92D4×C61—22no—
93C2×C4×C61—21no—
94C2×C4×C61—41no—
95C2×C4×C61—41no—
96C2×C4×C61C2×C22⋊C411yescharacteristic
97C2×C4×C61C22×C811yescharacteristic
98C2×C4×C61C2×M1611yescharacteristic
99C22⋊D61—42no—
100C2×Dic61—21no—
101C2×Dic61—21no—
102C2×Dic61C22⋊C811yescharacteristic
103C2×Dic61C22⋊C811yescharacteristic
104C2×Dic61C22⋊C811yescharacteristic
105C4×D61—21no—
106C4×D61—21no—
107C4×D61—22no—
108C4×D61—41no—
109Dic122—22no—
110C8×C61—21no—
111C8×C61—41no—
112C8×C61#10412yes—
113#92418—21no—
114#92418—41no—
115#92418#13512yes—
116#11614—611nomaximal, Sylow
117C22×C61—41no—
118C22×C61—41no—
119C22×C61—41no—
120C22×C61C2×C22⋊C811yescharacteristic
121C2×D61—21no—
122C2×D61—21no—
123C2×D61—21no—
124C4×C61—21no—
125C4×C61—41no—
126C4×C61#41611yescharacteristic
127C4×C61C2×#10411yescharacteristic
128C4×C61#50412yes—
129Dic61—21no—
130Dic61—21no—
131Dic61—21no—
132Dic61—41no—
133Dic61—41no—
134#2966—611no—
135#2162—611no—
136C4×M32—611no—
137Q8⋊C16—612no—
138D4⋊C16—612no—
139C2×C61—81no—
140C2×C61#147611yescharacteristic
141C2×C61#211311yescharacteristic
142C2×C61#211711yescharacteristic
143D61—41no—
144#529—611no—
145C2×M32—1221no—
146#446—611no—
147C8×Q8—611no—
148#436#692311yescharacteristic
149#435—1221no—
150C8×D4—611no—
151C8×D4—1221no—
152#433—1221no—
153#432—611no—
154#405—1221no—
155C2×C4×C8—611no—
156C4⋊C16—612no—
157C22⋊C16—1221no—
158#346—1221no—
159C4×C16—612no—
160C4×C16—1221no—
161C61#1161411yescharacteristic, complemented, Sylow
162C2×M16—21no—
163C22×C8—611no—
164C22×C8—611no—
165C22×C8—1221no—
166#126—1221no—
167#125—611no—
168#124—611no—
169#123—1221no—
170C22⋊Q8—611no—
171#121—611no—
172C4×Q8—611no—
173C4×D4—1221no—
174C4×D4C2×#692311yescharacteristic
175#117—1221no—
176C2×C42—611no—
177M32—1224no—
178C2×C16—1222no—
179C2×C16—1222no—
180C4⋊C8—611no—
181C4⋊C8—1221no—
182C4⋊C8—1221no—
183C4⋊C8C2×#692311yescharacteristic
184C22⋊C8—21no—
185C22⋊C8—1221no—
186C22⋊C8—1221no—
187#97—1221no—
188#97—1221no—
189C4×C8—611no—
190C4×C8—611no—
191C4×C8—611no—
192C4×C8—1221no—
193C4×C8C2×#692311yescharacteristic
194C2×Q8—611no—
195C2×D4—1221no—
196C2×D4D61⋊C811yescharacteristic
197C22×C4—21no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—611no—
201M16—42no—
202C2×C8—21no—
203C2×C8—611no—
204C2×C8—611no—
205C2×C8—1222no—
206C2×C8—1222no—
207C2×C8—1221no—
208C2×C8—2442no—
209C2×C8C22⋊#692311yescharacteristic
210C2×C8C22⋊#692311yescharacteristic
211C4⋊C4—611no—
212C4⋊C4—1221no—
213C4⋊C4—1221no—
214C4⋊C4—1221no—
215C4⋊C4—1221no—
216C4⋊C4D61⋊C811yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1221no—
219C22⋊C4—1221no—
220C22⋊C4—1221no—
221C22⋊C4—1221no—
222C42—611no—
223C42—611no—
224C42—611no—
225C42—1221no—
226C42C22×#692311yescharacteristic
227C16—1224no—
228C16—2442no—
229C23—21no—
230C23—611no—
231Q8—1222no—
232D4—22no—
233D4—2442no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—611no—
238C2×C4—611no—
239C2×C4—611no—
240C2×C4—1221no—
241C2×C4—1221no—
242C2×C4—1222no—
243C2×C4—1221no—
244C2×C4—1221no—
245C2×C4—2441no—
246C2×C4C2×C22⋊#692311yescharacteristic
247C2×C4C2×D61⋊C811yescharacteristic
248C2×C4D61⋊M1611yescharacteristic
249C8—21no—
250C8—41no—
251C8—612no—
252C8—1221no—
253C8—2441no—
254C8#1451984112yes—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—1221no—
259C22—1221no—
260C22—1221no—
261C22D61⋊(C22⋊C8)11yescharacteristic
262C4—21no—
263C4—41no—
264C4—1221no—
265C4—1221no—
266C4—1221no—
267C4—2441no—
268C4—2441no—
269C4#1684075211yescharacteristic
270C4C2×#1451984111yescharacteristic
271C4#1684082112yes—
272C2—81no—
273C2—2441no—
274C2#2146111111yescharacteristic
275C2#2146205011yescharacteristic
276C2#2146206211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC42C22×#6923
FittingC61×#436C4
socleC22×C61C2×C22⋊C8
prime radical#14519748C8
cosocleC4×#92418C23
hypercenter#436#6923
hypocenterC61#11614
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #21462179C2semidirect
2. C4×#14519852C2semidirect
3. #21460786C2semidirect
4. D4⋊#11758187C2semidirect
161. C61#11614semidirect

Maximal subgroups

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

subgroupindex
#214621792
C4×#145198522
#214607862
D4⋊#117581872
#214547972
#1161461