Finite Group Databasebeta

← #30608351

Subgroups — #30608351

Order 15616.

416 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 56 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#21462093C212yesmaximal, complemented
2#21460953C212yesmaximal, complemented
3#21460950C212yesmaximal, complemented
4#21456242C211yesmaximal, characteristic
5#16840743—22no—
6#16840668—22no—
7#16840667C2214yes—
8C2×#14519798—22no—
9#16840313C2211yescharacteristic
10#16840310C411yescharacteristic, complemented
11#16840143—22no—
12#16840141C411yescharacteristic, complemented
13#16840138C2211yescharacteristic
14#16840135C2211yescharacteristic
15C4×C8×D61C411yescharacteristic, complemented
16C61×#492C411yescharacteristic, complemented
17#14519856—42no—
18#14519836—24no—
19#14519836—42no—
20C2×#11758226D412yes—
21C2×#11758224—21no—
22#14519798—44no—
23#14519798—44no—
24C2×#11758202—22no—
25C2×#11758190—22no—
26C2×#11758199—22no—
27C2×C8×D61—21no—
28#14519777C2×C411yescharacteristic
29#14519774C2311yescharacteristic
30#14519766D412yes—
31#14519764—21no—
32#14519752—41no—
33#14519748—41no—
34D4×Dic61—41no—
35#14519739—41no—
36C42×D61C2×C411yescharacteristic
37C4×#92418C2×C411yescharacteristic
38C8×Dic61—21no—
39C2×SD16×C61—22no—
40C4⋊Q8×C61C2×C411yescharacteristic
41#127×C61C2×C411yescharacteristic
42C4×C8×C61C2×C411yescharacteristic
43C2×Q8×C61C22⋊C412yes—
44C2×D4×C61—21no—
45C2×D4×C61—41no—
46C2×C22⋊D61—41no—
47C22×Dic61—41no—
48#11758226—42no—
49#11758226—42no—
50#11758224—42no—
51#11758224—42no—
52C2×D244—22no—
53C2×D244—41no—
54C2×D244C2×D412yes—
55C2×C4×D61—21no—
56C2×C4×D61—22no—
57C2×C4×D61C22⋊C411yescharacteristic
58C2×Dic122—21no—
59C22⋊#6923—44no—
60C4⋊#6923—82no—
61C4×#6923—82no—
62SD16×C61—44no—
63C2×C8×C61—21no—
64C4⋊C4×C61—41no—
65C42×C61C22×C411yescharacteristic
66C22⋊Dic61—42no—
67#11758202—44no—
68#11758199—44no—
69#11758198—42no—
70C4⋊Dic61—41no—
71C4×Dic61—21no—
72C4×Dic61C22⋊C411yescharacteristic
73C2×#92418—21no—
74#11758190—44no—
75C8×D61—22no—
76C8×D61—22no—
77C23×C61—41no—
78C22×D61—22no—
79C22×D61#9911yescharacteristic
80C2×#6923—84no—
81Q8×C61—42no—
82D4×C61—42no—
83D4×C61—82no—
84C2×C4×C61—21no—
85C2×C4×C61—22no—
86C2×C4×C61C2×C22⋊C411yescharacteristic
87C22⋊D61—82no—
88C2×Dic61—21no—
89C2×Dic61—41no—
90C2×Dic61—81no—
91C2×Dic61—81no—
92C2×Dic61#9911yescharacteristic
93D244—42no—
94D244—82no—
95D244—82no—
96C4×D61—21no—
97C4×D61—21no—
98C4×D61—42no—
99C4×D61—42no—
100C4×D61—42no—
101Dic122—42no—
102C8×C61—22no—
103#92418—22no—
104#13583—611nomaximal, Sylow
105C22×C61—82no—
106C22×C61—81no—
107C22×C61C2×#9911yescharacteristic
108C2×D61—21no—
109C2×D61—41no—
110C2×D61—42no—
111C2×D61—82no—
112#6923—164no—
113C4×C61—22no—
114C4×C61—21no—
115C4×C61—42no—
116Dic61—41no—
117Dic61—81no—
118Dic61#35311yescharacteristic
119Dic61#35311yescharacteristic
120#3147—611no—
121#2126—612no—
122#1409—612no—
123#1408—612no—
124C2×C61—21no—
125C2×C61—161no—
126C2×C61C2×#35311yescharacteristic
127D61—41no—
128D61—82no—
129C2×#135—1222no—
130#532—611no—
131#532—611no—
132#495—1222no—
133#494—611no—
134#493—611no—
135#492—611no—
136#492#692311yescharacteristic, complemented
137#410—1222no—
138C2×C4×C8—611no—
139#354—1222no—
140#353—614no—
141C61#1358311yescharacteristic, complemented, Sylow
142C2×#41—612no—
143C2×#41—1221no—
144#135—2444no—
145#135—2444no—
146C2×Q16—1222no—
147C2×SD16—22no—
148C2×SD16—1222no—
149C2×D8—1222no—
150C22×C8—1221no—
151C4⋊Q8—611no—
152C4⋊Q8C2×#692311yescharacteristic
153#127—611no—
154#127C2×#692311yescharacteristic
155#124—612no—
156#124—1221no—
157#121—2441no—
158#121—2441no—
159C4×D4—2441no—
160C4×D4—2441no—
161#117—2442no—
162C2×C42—611no—
163#99—1224no—
164#99—2442no—
165C4×C8—611no—
166C4×C8—1221no—
167C4×C8C2×#692311yescharacteristic
168#41—2442no—
169#41—2442no—
170#41—2442no—
171#41—2442no—
172C2×Q8—1221no—
173C2×Q8C22⋊#692312yes—
174C2×D4—21no—
175C2×D4—41no—
176C2×D4—612no—
177C2×D4—1222no—
178C2×D4—2441no—
179C2×D4—2441no—
180C22×C4—611no—
181C22×C4—1222no—
182C22×C4—1221no—
183C22×C4—2441no—
184Q16—2444no—
185SD16—44no—
186SD16—2444no—
187D8—2444no—
188C2×C8—21no—
189C2×C8—1222no—
190C2×C8—1222no—
191C2×C8—1221no—
192C4⋊C4—41no—
193C4⋊C4—4882no—
194C4⋊C4—2441no—
195C22⋊C4—2442no—
196C22⋊C4—2442no—
197C22⋊C4—2444no—
198C42—611no—
199C42—1221no—
200C42—4882no—
201C42C22×#692311yescharacteristic
202C23—41no—
203C23—611no—
204C23—1222no—
205Q8—42no—
206Q8—2442no—
207D4—42no—
208D4—82no—
209D4—4882no—
210D4—4882no—
211D4—4882no—
212D4—2442no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—611no—
216C2×C4—1221no—
217C2×C4—1221no—
218C2×C4—1221no—
219C2×C4—4881no—
220C2×C4—4881no—
221C2×C4—4884no—
222C2×C4—2442no—
223C2×C4—2442no—
224C2×C4—2442no—
225C2×C4—2441no—
226C2×C4C2×C22⋊#692311yescharacteristic
227C8—22no—
228C8—1222no—
229C22—82no—
230C22—81no—
231C22—1221no—
232C22—4882no—
233C22—2441no—
234C22—2442no—
235C22D61⋊#9911yescharacteristic
236C4—22no—
237C4—21no—
238C4—42no—
239C4—611no—
240C4—611no—
241C4—9764no—
242C4—4881no—
243C4—2441no—
244C2—21no—
245C2—161no—
246C2—4882no—
247C2—2441no—
248C2D61⋊#35311yescharacteristic
subgroupG/H
centerC2D61⋊#353
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#492C4
socleC2×C61C2×#353
prime radical#21456242C2
cosocle#14519774C23
hypercenter#492#6923
hypocenterC61#13583
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #21462093C2semidirect
2. #21460953C2semidirect
3. #21460950C2semidirect
10. #16840310C4semidirect
12. #16840141C4semidirect
15. C4×C8×D61C4semidirect
16. C61×#492C4semidirect
136. #492#6923semidirect
141. C61#13583semidirect

Maximal subgroups

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

subgroupindex
#214620932
#214609532
#214609502
#214562422
#1358361