Finite Group Databasebeta

← #30608354

Subgroups — #30608354

Order 15616.

474 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#16840667C212yesmaximal, complemented
2#21460952C212yesmaximal, complemented
3#21460950C212yesmaximal, complemented
4#21456272C211yesmaximal, characteristic
5C2×#14519836—22no—
6#16840667—22no—
7#16840667C2214yes—
8C2×#14519802—22no—
9C2×#14519774C2211yescharacteristic
10#16840152—22no—
11#16840138C411yescharacteristic, complemented
12#16840138C2211yescharacteristic
13#16840136C411yescharacteristic, complemented
14#16840136C2211yescharacteristic
15C4×#11758194C411yescharacteristic, complemented
16C61×#532C411yescharacteristic, complemented
17C2×C22⋊#6923—42no—
18#14519836—24no—
19#14519836—42no—
20C22×D244—41no—
21C22×D244D412yes—
22#14519802—44no—
23#14519802—44no—
24C2×#11758201—22no—
25C2×#11758201—22no—
26C2×#11758199—22no—
27C2×#11758199—22no—
28C2×#11758194—21no—
29#14519774—41no—
30#14519774C2×C411yescharacteristic
31#14519774D412yes—
32#14519774C2311yescharacteristic
33#14519688—21no—
34C4×#92418C2×C411yescharacteristic, complemented
35C4×#92418C2×C411yescharacteristic, complemented
36C2×#41×C61—21no—
37C4⋊Q8×C61C2×C411yescharacteristic, complemented
38#127×C61C2×C411yescharacteristic, complemented
39#124×C61—21no—
40#121×C61—41no—
41C4×D4×C61—41no—
42C2×C42×C61C2×C411yescharacteristic
43C23×D61—22no—
44C22×#6923—82no—
45#41×C61—42no—
46#41×C61—42no—
47C2×Q8×C61—21no—
48C2×D4×C61—21no—
49C2×D4×C61—41no—
50C2×D4×C61—41no—
51C22×C4×C61—21no—
52C22×C4×C61—41no—
53C22×C4×C61C22⋊C411yescharacteristic
54C2×D244—22no—
55C2×D244—42no—
56C2×D244—42no—
57C2×D244—41no—
58C2×D244—42no—
59C2×D244—41no—
60C2×D244—84no—
61C2×D244C22⋊C412yes—
62C2×D244C2×D412yes—
63C22⋊#6923—44no—
64C22⋊#6923—82no—
65C4⋊C4×C61—41no—
66C22⋊C4×C61—42no—
67C42×C61—21no—
68C42×C61C22⋊C411yescharacteristic
69C42×C61C22×C411yescharacteristic
70#11758201—44no—
71#11758201—44no—
72#11758199—44no—
73#11758199—44no—
74#11758194—24no—
75C2×#92418—21no—
76C2×#92418—21no—
77C23×C61—41no—
78C23×C61#9911yescharacteristic
79C22×D61—22no—
80C22×D61—22no—
81C22×D61—42no—
82C22×D61—42no—
83C22×D61—84no—
84C2×#6923—84no—
85C2×#6923—164no—
86Q8×C61—42no—
87D4×C61—42no—
88D4×C61—82no—
89D4×C61—82no—
90C2×C4×C61—22no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—42no—
94C2×C4×C61—41no—
95C2×C4×C61—81no—
96C2×C4×C61—81no—
97C2×C4×C61#9911yescharacteristic
98C2×C4×C61C2×C22⋊C411yescharacteristic
99D244—42no—
100D244—42no—
101D244—82no—
102D244—82no—
103D244—82no—
104D244—84no—
105#92418—22no—
106#92418—22no—
107#13584—611nomaximal, Sylow
108C22×C61—41no—
109C22×C61—82no—
110C22×C61—81no—
111C22×C61#35312yes—
112C22×C61C2×#9911yescharacteristic
113C2×D61—42no—
114C2×D61—42no—
115C2×D61—84no—
116C2×D61—82no—
117C2×D61—82no—
118C2×D61—162no—
119#6923—164no—
120C4×C61—22no—
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—41no—
124C4×C61—81no—
125#3151—611no—
126C2×#353—612no—
127#1409—612no—
128#1408—612no—
129C2×C61—21no—
130C2×C61—22no—
131C2×C61—161no—
132C2×C61C2×#35311yescharacteristic
133D61—82no—
134D61—82no—
135C2×AGL(1,ℤ/8)—1222no—
136#532#692311yescharacteristic, complemented
137C2×#127—611no—
138#496—1222no—
139#493—611no—
140#493—611no—
141#492—611no—
142#492—611no—
143C2×#99—1222no—
144C4×M16—611no—
145#353—614no—
146#353—1222no—
147C61#1358411yescharacteristic, complemented, Sylow
148C2×#41—21no—
149C22×D4—612no—
150C22×D4—2441no—
151AGL(1,ℤ/8)—2444no—
152AGL(1,ℤ/8)—2444no—
153C2×SD16—1222no—
154C2×SD16—1222no—
155C2×D8—1222no—
156C2×D8—1222no—
157C2×M16—1221no—
158C4⋊Q8C2×#692311yescharacteristic, complemented
159#127—612no—
160#127—611no—
161#127—611no—
162#127—2441no—
163#127C2×#692311yescharacteristic, complemented
164#124—21no—
165#121—41no—
166C4×D4—41no—
167C2×C22⋊C4—2442no—
168C2×C42C2×#692311yescharacteristic
169#99—1224no—
170#99—2442no—
171#97—1221no—
172C4×C8—611no—
173C4×C8—611no—
174C24—1222no—
175#41—42no—
176#41—42no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—612no—
182C2×D4—612no—
183C2×D4—1222no—
184C2×D4—4884no—
185C2×D4—2441no—
186C2×D4—2441no—
187C2×D4—2442no—
188C2×D4—2442no—
189C2×D4—2442no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—4882no—
193C22×C4C22⋊#692311yescharacteristic
194SD16—2444no—
195SD16—2444no—
196D8—2444no—
197D8—2444no—
198M16—1224no—
199C2×C8—1221no—
200C2×C8—1221no—
201C4⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—4882no—
204C22⋊C4—2444no—
205C42—21no—
206C42C22⋊#692311yescharacteristic
207C42C22×#692311yescharacteristic
208C23—41no—
209C23—1222no—
210C23—1222no—
211C23—4884no—
212C23—2442no—
213C23—2442no—
214C23#1451983511yescharacteristic
215Q8—42no—
216D4—42no—
217D4—82no—
218D4—82no—
219D4—4882no—
220D4—4882no—
221D4—4882no—
222D4—4884no—
223D4—2442no—
224D4—2442no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—9764no—
233C2×C4—4884no—
234C2×C4#1451983511yescharacteristic
235C2×C4C2×C22⋊#692311yescharacteristic
236C8—1222no—
237C8—1222no—
238C22—41no—
239C22—82no—
240C22—81no—
241C22—9762no—
242C22—4882no—
243C22—4882no—
244C22—4884no—
245C22—2442no—
246C22—2442no—
247C22#1684066612yes—
248C22C2×#1451983511yescharacteristic
249C4—22no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—9764no—
255C2—21no—
256C2—22no—
257C2—161no—
258C2—4882no—
259C2—4882no—
260C2C2×#1684066611yescharacteristic
subgroupG/H
centerC2C2×#16840666
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#532C4
socleC2×C61C2×#353
prime radical#21456272C2
cosocle#14519774C23
hypercenter#532#6923
hypocenterC61#13584
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#16840667C2semidirect
2. #21460952C2semidirect
3. #21460950C2semidirect
11. #16840138C4semidirect
13. #16840136C4semidirect
15. C4×#11758194C4semidirect
16. C61×#532C4semidirect
34. C4×#92418C2×C4semidirect
35. C4×#92418C2×C4semidirect
37. C4⋊Q8×C61C2×C4semidirect
38. #127×C61C2×C4semidirect
136. #532#6923semidirect
147. C61#13584semidirect
158. C4⋊Q8C2×#6923semidirect
163. #127C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406672
#214609522
#214609502
#214562722
#1358461