Finite Group Databasebeta

← #30607405

Subgroups — #30607405

Order 15616.

420 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C22×#14519847C211yesmaximal, characteristic
2C2×#16840686C212yesmaximal, complemented
3#21461695C211yesmaximal, characteristic
4#21460884C212yesmaximal, complemented
5#21447428C211yesmaximal, characteristic
6C22×#11758213—21no—
7C2×#14519847—21no—
8C2×#14519847—24no—
9C2×#14519847C2212yes—
10C2×#14519845C2211yescharacteristic
11C2×#14519845C2211yescharacteristic
12#16840704—21no—
13#16840686—24no—
14#16840636—21no—
15C22×C8×D61C2211yescharacteristic
16C2×#14519716C2212yes—
17C22⋊#11758185C411yescharacteristic, complemented
18#16839541C412yescomplemented
19C61×C22⋊C16C411yescharacteristic, complemented
20C2×#11758213—24no—
21C2×#11758213—22no—
22C2×#11758213—42no—
23C2×#11758213—41no—
24C22×#92420—41no—
25C2×C4⋊#6923—21no—
26C2×C4⋊#6923—21no—
27C2×C22⋊#6923—41no—
28#14519847—24no—
29#14519847—22no—
30#14519847D414yes—
31#14519845—22no—
32#14519845—22no—
33(C2×#6923)⋊C4—41no—
34C22×C4×D61D411yescharacteristic
35C22×#92418Q811yescharacteristic
36C2×C8×D61—22no—
37C2×C8×D61—22no—
38C2×C8×D61C2×C411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61Q811yescharacteristic
41C2×C8×D61C2311yescharacteristic
42#14519716—22no—
43#14519716—22no—
44C2×#11758185C2×C412yes—
45C22×C8×C61C2×C411yescharacteristic
46C2×C16×C61C2×C412yes—
47C23×D61SD1611yescharacteristic
48C22×#6923—41no—
49C22×#6923—41no—
50C22×C4×C61C4⋊C411yescharacteristic
51C22×Dic61SD1611yescharacteristic
52C2×C4×D61—21no—
53C2×C4×D61—21no—
54C2×C4×D61—21no—
55C2×C4×D61—21no—
56C2×C4×D61C4⋊C411yescharacteristic
57C2×C4×D61SD1611yescharacteristic
58C2×C4×D61SD1611yescharacteristic
59C2×C4×D61C2×D411yescharacteristic
60C22⋊#6923—82no—
61C2×#92420—44no—
62C2×#92420—42no—
63C4⋊#6923—41no—
64C4⋊#6923—41no—
65C4⋊#6923—41no—
66C4⋊#6923—41no—
67#11758213—24no—
68#11758213—44no—
69#11758213—42no—
70C2×C8×C61—22no—
71C2×C8×C61C4⋊C411yescharacteristic
72C2×C8×C61C22×C411yescharacteristic
73C2×#92418—22no—
74C2×#92418C4⋊C411yescharacteristic
75C2×#92418C2×Q811yescharacteristic
76C8×D61—21no—
77C8×D61—21no—
78C8×D61—22no—
79C8×D61—22no—
80C8×D61C2×D412yes—
81C8×D61#4112yes—
82C16×C61—22no—
83#11758185—22no—
84C23×C61#10611yescharacteristic
85C22×D61—22no—
86C22×D61—21no—
87C22×D61—21no—
88C22×D61—41no—
89C22×D61#10611yescharacteristic
90C22×D61C2×SD1611yescharacteristic
91C2×#6923—81no—
92C2×#6923—81no—
93C2×#6923—81no—
94C2×#6923—81no—
95C2×#6923—81no—
96C2×#6923—81no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61#10611yescharacteristic
100C2×C4×C61C2×C4⋊C411yescharacteristic
101C2×Dic61—21no—
102C2×Dic61—21no—
103C2×Dic61#10611yescharacteristic
104C2×Dic61C2×SD1611yescharacteristic
105C4×D61—21no—
106C4×D61—21no—
107C4×D61—21no—
108C4×D61—21no—
109C4×D61—21no—
110C4×D61—21no—
111C4×D61#12011yescharacteristic
112C4×D61#12311yescharacteristic
113C4×D61AGL(1,ℤ/8)11yescharacteristic
114C4×D61#13711yescharacteristic
115#92420—44no—
116C8×C61—21no—
117C8×C61C4×D412yes—
118#92418—21no—
119#92418C22⋊Q812yes—
120#12928—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61#36512yescomplemented
124C22×C61C2×#10611yescharacteristic
125C2×D61—21no—
126C2×D61—21no—
127C2×D61—21no—
128C2×D61—21no—
129C2×D61—21no—
130C2×D61—42no—
131C2×D61—42no—
132C2×D61#44811yescharacteristic
133C2×D61#48111yescharacteristic
134#6923—161no—
135#6923—161no—
136C4×C61—21no—
137C4×C61#38511yescharacteristic
138C4×C61#42811yescharacteristic
139Dic61—21no—
140Dic61#45011yescharacteristic
141Dic61#48311yescharacteristic
142C22×#108—611no—
143C2×#365—612no—
144#2112—611no—
145#1807—611no—
146#1383—612no—
147C2×C61—22no—
148C2×C61#185311yescharacteristic
149C2×C61C2×#36511yescharacteristic
150C2×C61#215511yescharacteristic
151D61—21no—
152D61—21no—
153D61—41no—
154C22×M16—1221no—
155C23×C8—611no—
156C2×M32—612no—
157C2×#108—612no—
158C2×#108—1224no—
159C2×#108—1221no—
160C2×#106—611no—
161C2×#106—611no—
162#380—1221no—
163#365—1224no—
164C22⋊C16—611no—
165C22⋊C16—612no—
166C22⋊C16#692311yescharacteristic, complemented
167#340—1221no—
168C61#1292811yescharacteristic, complemented, Sylow
169C23×C4—611no—
170C2×M16—1222no—
171C2×M16—1224no—
172C2×M16—2441no—
173C2×M16—2442no—
174C22×C8—611no—
175C22×C8—611no—
176C22×C8—611no—
177C22×C8—611no—
178C22×C8—611no—
179C22×C8—1222no—
180C22×C8—1222no—
181C22×C8—2441no—
182C22×C8C2×#692311yescharacteristic
183C2×C4⋊C4—1221no—
184C2×C4⋊C4—1221no—
185C2×C22⋊C4—2441no—
186M32—1222no—
187M32—1222no—
188C2×C16—612no—
189C2×C16C2×#692312yes—
190#108—614no—
191#108—1222no—
192#108—1224no—
193#106—1222no—
194#106—1222no—
195#95—2441no—
196C24—611no—
197C22×C4—611no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—611no—
201C22×C4—611no—
202C22×C4—1221no—
203C22×C4—1221no—
204C22×C4—1221no—
205C22×C4—1221no—
206C22×C4—2441no—
207C22×C4—2441no—
208C22×C4C4⋊#692311yescharacteristic
209M16—1224no—
210M16—2442no—
211M16—2444no—
212C2×C8—22no—
213C2×C8—611no—
214C2×C8—612no—
215C2×C8—612no—
216C2×C8—611no—
217C2×C8—1222no—
218C2×C8—1221no—
219C2×C8—1221no—
220C2×C8—1222no—
221C2×C8—1222no—
222C2×C8—2442no—
223C2×C8—2444no—
224C2×C8C4⋊#692311yescharacteristic
225C2×C8C22×#692311yescharacteristic
226C4⋊C4—2441no—
227C4⋊C4—2441no—
228C4⋊C4—2441no—
229C4⋊C4—2441no—
230C22⋊C4—4882no—
231C16—22no—
232C16—1222no—
233C23—611no—
234C23—611no—
235C23—1221no—
236C23—1221no—
237C23—1222no—
238C23—2441no—
239C23#1451984511yescharacteristic
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—611no—
243C2×C4—611no—
244C2×C4—611no—
245C2×C4—611no—
246C2×C4—611no—
247C2×C4—611no—
248C2×C4—1221no—
249C2×C4—1221no—
250C2×C4—1221no—
251C2×C4—1221no—
252C2×C4—1221no—
253C2×C4—1221no—
254C2×C4—1221no—
255C2×C4—1221no—
256C2×C4—4881no—
257C2×C4—4881no—
258C2×C4—4881no—
259C2×C4—4881no—
260C2×C4—4881no—
261C2×C4—4881no—
262C2×C4#1451984511yescharacteristic
263C2×C4C2×C4⋊#692311yescharacteristic
264C8—21no—
265C8—612no—
266C8—1221no—
267C8—2444no—
268C8D4×#692312yes—
269C22—21no—
270C22—21no—
271C22—611no—
272C22—611no—
273C22—1221no—
274C22—1221no—
275C22—1221no—
276C22—1221no—
277C22—1221no—
278C22—2442no—
279C22—2442no—
280C22#1684068612yescomplemented
281C22C2×#1451984511yescharacteristic
282C4—21no—
283C4—611no—
284C4—611no—
285C4—1221no—
286C4—9761no—
287C4—9761no—
288C4#1684070911yescharacteristic
289C4#1684077311yescharacteristic
290C2—22no—
291C2—1221no—
292C2—1221no—
293C2—2441no—
294C2#2146176911yescharacteristic
295C2C2×#1684068611yescharacteristic
296C2#2146216711yescharacteristic
subgroupG/H
centerC22C2×#14519845
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×C22⋊C16C4
socleC22×C61C2×#106
prime radicalC23×D61SD16
cosocleC2×C8×D61C23
hypercenterC22⋊C16#6923
hypocenterC61#12928
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#16840686C2semidirect
4. #21460884C2semidirect
17. C22⋊#11758185C4semidirect
18. #16839541C4semidirect
19. C61×C22⋊C16C4semidirect
123. C22×C61#365semidirect
166. C22⋊C16#6923semidirect
168. C61#12928semidirect
280. C22#16840686semidirect

Maximal subgroups

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

subgroupindex
C22×#145198472
C2×#168406862
#214616952
#214608842
#214474282
#1292861