Finite Group Databasebeta

← #30605436

Subgroups — #30605436

Order 15616.

412 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462047C211yesmaximal, characteristic
2#21461695C211yesmaximal, characteristic
3#21460807C214yesmaximal, complemented
4C2×#16840000C211yesmaximal, characteristic
5C2×#14519852—21no—
6C2×#14519845—21no—
7#16840704—21no—
8#16840657C2212yes—
9C22⋊#11758187—21no—
10#16840636C2212yes—
11C22×C8×D61C2211yescharacteristic
12C2×#14519753—21no—
13#16840000—21no—
14#16840000—21no—
15#16840000C412yescomplemented
16#16840000C2212yes—
17C2×#14519711C411yescharacteristic, complemented
18C2×#14519701—21no—
19C22×Q16×C61C411yescharacteristic, complemented
20C2×C4⋊#6923—22no—
21C2×C22⋊#6923—41no—
22#14519852—42no—
23C2×#11758187—22no—
24#14519845—42no—
25(C2×#6923)⋊C4—41no—
26C22×C4×D61D411yescharacteristic
27C22×#92418D411yescharacteristic
28C2×C8×D61—42no—
29C2×C8×D61C2×C411yescharacteristic
30C2×C8×D61D411yescharacteristic
31C2×C8×D61D411yescharacteristic
32C2×C8×D61C2311yescharacteristic
33#14519753—22no—
34#14519753—22no—
35#14519753—41no—
36#14519753—41no—
37C2×C4⋊Dic61—21no—
38C2×#11758196—41no—
39C2×#11758198—41no—
40#14519711—21no—
41#14519711C2×C412yes—
42#14519701—22no—
43#14519701—22no—
44C22×Q8×C61—21no—
45C2×Q16×C61—21no—
46C2×Q16×C61—42no—
47C2×Q16×C61C2×C412yes—
48C22×C8×C61C2×C411yescharacteristic
49C23×D61D811yescharacteristic
50C22×#6923—42no—
51C2×Q8×C61—22no—
52C2×Q8×C61—42no—
53C2×Q8×C61—42no—
54C2×Q8×C61—41no—
55C22×C4×C61—41no—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×Dic61—41no—
58C22×Dic61SD1611yescharacteristic
59C2×C4×D61—41no—
60C2×C4×D61—41no—
61C2×C4×D61C22⋊C411yescharacteristic
62C2×C4×D61D811yescharacteristic
63C2×C4×D61SD1611yescharacteristic
64C2×C4×D61C2×D411yescharacteristic
65C22⋊#6923—82no—
66C4⋊#6923—42no—
67C4⋊#6923—42no—
68Q16×C61—42no—
69Q16×C61—42no—
70C2×C8×C61—22no—
71C2×C8×C61C22⋊C411yescharacteristic
72C2×C8×C61C22×C411yescharacteristic
73#11758198—42no—
74#11758198—42no—
75C4⋊Dic61—22no—
76C4⋊Dic61—41no—
77#11758196—42no—
78#11758196—42no—
79C2×#92418—41no—
80C2×#92418C22⋊C411yescharacteristic
81C2×#92418C2×D411yescharacteristic
82C8×D61—22no—
83C8×D61—22no—
84C8×D61—41no—
85C8×D61—41no—
86#11758187—42no—
87C23×C61#10211yescharacteristic
88C22×D61—22no—
89C22×D61—22no—
90C22×D61—41no—
91C22×D61#10211yescharacteristic
92C22×D61C2×D811yescharacteristic
93C2×#6923—82no—
94C2×#6923—82no—
95C2×#6923—82no—
96Q8×C61—42no—
97Q8×C61—42no—
98Q8×C61—82no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—42no—
102C2×C4×C61—82no—
103C2×C4×C61#10211yescharacteristic
104C2×C4×C61C2×C22⋊C411yescharacteristic
105C2×Dic61—41no—
106C2×Dic61—42no—
107C2×Dic61—82no—
108C2×Dic61#10211yescharacteristic
109C2×Dic61C2×SD1611yescharacteristic
110C4×D61—21no—
111C4×D61—21no—
112C4×D61—21no—
113C4×D61—21no—
114C4×D61—41no—
115C4×D61—41no—
116C8×C61—21no—
117C8×C61#9912yes—
118#92418—21no—
119#92418—21no—
120#11889—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61#36012yes—
124C22×C61C2×#10211yescharacteristic
125C2×D61—21no—
126C2×D61—21no—
127C2×D61—21no—
128C2×D61—21no—
129C2×D61—21no—
130C2×D61—44no—
131C2×D61#51012yes—
132#6923—162no—
133C4×C61—21no—
134C4×C61—82no—
135C4×C61C2×#9911yescharacteristic
136C4×C61#42111yescharacteristic
137Dic61—21no—
138Dic61—21no—
139Dic61—82no—
140C2×#467—611no—
141#2112—611no—
142#1807—611no—
143#1348—614no—
144C2×C61—22no—
145C2×C61#151911yescharacteristic
146C2×C61#214111yescharacteristic
147C2×C61C2×#36011yescharacteristic
148D61—22no—
149D61—41no—
150C22×Q16#692311yescharacteristic, complemented
151C23×C8—611no—
152C2×C22⋊Q8—1221no—
153C2×M32—1221no—
154#467—612no—
155#467—612no—
156#467—1221no—
157#467—1221no—
158C2×#107—611no—
159C2×#106—1221no—
160C2×#103—1221no—
161#380—1221no—
162C22⋊C16—612no—
163C22⋊C16—1221no—
164#340—612no—
165C61#1188911yescharacteristic, complemented, Sylow
166C22×Q8—21no—
167C23×C4—611no—
168C2×Q16—21no—
169C2×Q16—42no—
170C2×Q16C2×#692312yes—
171C22×C8—611no—
172C22×C8—611no—
173C22×C8—611no—
174C22×C8—611no—
175C22×C8—611no—
176C22×C8—2442no—
177C22×C8C2×#692311yescharacteristic
178C22⋊Q8—1222no—
179C22⋊Q8—1222no—
180C22⋊Q8—2441no—
181C22⋊Q8—2441no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4—1222no—
184C2×C4⋊C4—2441no—
185C2×C22⋊C4—2441no—
186C2×C22⋊C4—2441no—
187M32—2442no—
188C2×C16—1222no—
189#107—612no—
190#107—1221no—
191#106—2442no—
192#103—1222no—
193#103—1222no—
194#95—2441no—
195C24—611no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—41no—
200C22×C4—41no—
201C22×C4—611no—
202C22×C4—611no—
203C22×C4—611no—
204C22×C4—611no—
205C22×C4—611no—
206C22×C4—2441no—
207C22×C4—2441no—
208C22×C4—2442no—
209C22×C4—2441no—
210C22×C4C22⋊#692311yescharacteristic
211Q16—42no—
212Q16—42no—
213C2×C8—22no—
214C2×C8—611no—
215C2×C8—611no—
216C2×C8—1222no—
217C2×C8—1222no—
218C2×C8—2441no—
219C2×C8—2441no—
220C2×C8—2441no—
221C2×C8C22⋊#692311yescharacteristic
222C2×C8C22×#692311yescharacteristic
223C4⋊C4—1222no—
224C4⋊C4—2442no—
225C4⋊C4—2442no—
226C4⋊C4—2442no—
227C4⋊C4—2442no—
228C4⋊C4—2441no—
229C22⋊C4—4882no—
230C22⋊C4—2442no—
231C22⋊C4—2442no—
232C16—2442no—
233C23—611no—
234C23—611no—
235C23—1222no—
236C23—1222no—
237C23—2441no—
238C23#1451983911yescharacteristic
239Q8—42no—
240Q8—42no—
241Q8—82no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—611no—
247C2×C4—611no—
248C2×C4—1221no—
249C2×C4—1221no—
250C2×C4—1221no—
251C2×C4—1221no—
252C2×C4—4882no—
253C2×C4—4882no—
254C2×C4—4882no—
255C2×C4—4882no—
256C2×C4—2441no—
257C2×C4—2441no—
258C2×C4—2441no—
259C2×C4—2442no—
260C2×C4#1451983911yescharacteristic
261C2×C4C2×C22⋊#692311yescharacteristic
262C8—21no—
263C8—1221no—
264C8—1221no—
265C8#1451983512yes—
266C22—21no—
267C22—21no—
268C22—612no—
269C22—1221no—
270C22—1221no—
271C22—1221no—
272C22—1221no—
273C22—1221no—
274C22—2444no—
275C22#1684067912yes—
276C22C2×#1451983911yescharacteristic
277C4—21no—
278C4—82no—
279C4—1221no—
280C4—1221no—
281C4—9762no—
282C4—4882no—
283C4C2×#1451983511yescharacteristic
284C4#1684076011yescharacteristic
285C2—22no—
286C2—1222no—
287C2—2441no—
288C2#2146125511yescharacteristic
289C2#2146212911yescharacteristic
290C2C2×#1684067911yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC22×Q16×C61C4
socleC22×C61C2×#102
prime radicalC23×D61D8
cosocleC2×C8×D61C23
hypercenterC22×Q16#6923
hypocenterC61#11889
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #21460807C2semidirect
15. #16840000C4semidirect
17. C2×#14519711C4semidirect
19. C22×Q16×C61C4semidirect
150. C22×Q16#6923semidirect
165. C61#11889semidirect

Maximal subgroups

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

subgroupindex
#214620472
#214616952
#214608072
C2×#168400002
#1188961