Finite Group Databasebeta

← #29597058

Subgroups — #29597058

Order 15616.

548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16840015C211yesmaximal, characteristic
2C2×#16839574C212yesmaximal, complemented
3C2×#16839544C211yesmaximal, characteristic, complemented
4#21443919C212yesmaximal, complemented
5C2×C61×#430C211yesmaximal, characteristic, complemented
6C22×D488C2211yescharacteristic
7C2×#14519751—21no—
8C2×#14519717C2212yes—
9#16840015—22no—
10#16840015C414yes—
11C2×#14519709C2211yescharacteristic
12C2×#14519699—21no—
13#16839574—24no—
14#16839544—22no—
15C22×M16×C61—21no—
16C22×M16×C61C2211yescharacteristic
17C61×#430—24no—
18C2×#108×C61—21no—
19C2×#108×C61C2212yes—
20C22×D244—21no—
21C22×C22⋊D61—41no—
22C2×D488—22no—
23C2×D488—22no—
24C2×D488—21no—
25C2×D488C2×C412yes—
26#14519751—24no—
27#14519751—42no—
28C2×C4⋊Dic61—21no—
29C2×#11758198—41no—
30#14519717—22no—
31#14519717—22no—
32C2×#11758185—22no—
33#14519709—21no—
34#14519709C2×C412yes—
35#14519699—24no—
36C23×C4×C61D411yescharacteristic
37C2×M16×C61—22no—
38C2×M16×C61—22no—
39C2×M16×C61—24no—
40C2×M16×C61—22no—
41C2×M16×C61—41no—
42C2×M16×C61—42no—
43C2×M16×C61C2×C412yes—
44C2×M16×C61D412yes—
45C22×C8×C61—41no—
46C22×C8×C61D411yescharacteristic
47C22×C8×C61C2311yescharacteristic
48#108×C61—24no—
49#108×C61—22no—
50#108×C61D414yescomplemented
51C24×C61SD1611yescharacteristic
52C23×D61—41no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61D812yes—
59C22×C4×C61SD1611yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C2×C22⋊D61—44no—
62C2×C22⋊D61—84no—
63C22×Dic61—41no—
64C2×D244—22no—
65C2×D244—41no—
66C2×D244—41no—
67C2×D244—41no—
68C2×D244—41no—
69C2×D244—41no—
70M16×C61—24no—
71M16×C61—24no—
72M16×C61—24no—
73M16×C61—44no—
74M16×C61—42no—
75C2×C8×C61—21no—
76C2×C8×C61—21no—
77C2×C8×C61—44no—
78C2×C8×C61—42no—
79C2×C8×C61C22⋊C412yes—
80C2×C8×C61C22×C411yescharacteristic
81C2×C8×C61C2×D412yes—
82C2×C8×C61C2×D411yescharacteristic
83C2×C8×C61#4112yes—
84#11758198—44no—
85C4⋊Dic61—22no—
86C4⋊Dic61—41no—
87D488—24no—
88D488—42no—
89#11758185—42no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—42no—
93C23×C61#10212yes—
94C23×C61C2×SD1611yescharacteristic
95C22×D61—42no—
96C22×D61—82no—
97C22×D61—82no—
98C22×D61—81no—
99C22×D61—81no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61#10212yes—
107C2×C4×C61C2×C22⋊C411yescharacteristic
108C2×C4×C61#12011yescharacteristic
109C2×C4×C61#12311yescharacteristic
110C2×C4×C61C2×D811yescharacteristic
111C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
112C2×C4×C61#13711yescharacteristic
113C22⋊D61—88no—
114C2×Dic61—42no—
115C2×Dic61—82no—
116D244—42no—
117D244—42no—
118D244—81no—
119D244—81no—
120C8×C61—22no—
121C8×C61—44no—
122C8×C61C4×D412yes—
123C8×C61#12112yes—
124#13010—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—22no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—81no—
130C22×C61C2×#10211yescharacteristic
131C22×C61#44811yescharacteristic
132C22×C61#48111yescharacteristic
133C2×D61—82no—
134C2×D61—82no—
135C2×D61—82no—
136C2×D61—162no—
137C2×D61—162no—
138C4×C61—22no—
139C4×C61#38611yescharacteristic
140C4×C61#41711yescharacteristic
141C4×C61#44711yescharacteristic
142C4×C61#48011yescharacteristic
143Dic61—82no—
144C2×#469—611no—
145C2×#430D6111yescharacteristic, complemented
146C2×#361—612no—
147C2×#349—611no—
148#1383—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61—42no—
152C2×C61#187411yescharacteristic
153C2×C61#223812yes—
154D61—162no—
155C22×D8—611no—
156C22×M16—21no—
157C22×M16C2×D6111yescharacteristic
158C2×#121—1221no—
159C2×M32—612no—
160#469—614no—
161#469—1222no—
162#430—24no—
163C2×#108—21no—
164C2×#108C2×D6112yes—
165C2×#106—611no—
166C2×#102—1221no—
167#361—1224no—
168#349—1222no—
169C61#1301011yescharacteristic, complemented, Sylow
170C22×D4—1221no—
171C22×D4—2441no—
172C23×C4C22⋊D6111yescharacteristic
173C2×D8—612no—
174C2×D8—1221no—
175C2×D8—1222no—
176C2×D8—1222no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—24no—
180C2×M16—22no—
181C2×M16—41no—
182C2×M16—42no—
183C2×M16C4×D6112yes—
184C2×M16D24412yes—
185C22×C8—41no—
186C22×C8C22⋊D6111yescharacteristic
187C22×C8C22×D6111yescharacteristic
188#121—1224no—
189#121—2442no—
190C2×C4⋊C4—1221no—
191C2×C22⋊C4—2441no—
192M32—1222no—
193M32—1222no—
194C2×C16—1222no—
195#108—24no—
196#108—22no—
197#108C22⋊D6114yescomplemented
198#106—612no—
199#106—1221no—
200#102—1224no—
201C24—2441no—
202C24#1175820111yescharacteristic
203C2×D4—1222no—
204C2×D4—4884no—
205C2×D4—2444no—
206C2×D4—2441no—
207C2×D4—2441no—
208C2×D4—2441no—
209C2×D4—2441no—
210C2×D4—2441no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2441no—
216C22×C4#1175819812yes—
217C22×C4#1175819912yes—
218C22×C4#1175820111yescharacteristic
219C22×C4C2×C22⋊D6111yescharacteristic
220D8—1224no—
221D8—2442no—
222M16—24no—
223M16—24no—
224M16—24no—
225M16—44no—
226M16—42no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—44no—
230C2×C8—42no—
231C2×C8#1175819812yes—
232C2×C8C2×C4×D6111yescharacteristic
233C2×C8C2×D24411yescharacteristic
234C2×C8#1175822212yes—
235C2×C8C2×C22⋊D6112yes—
236C4⋊C4—1222no—
237C4⋊C4—2441no—
238C22⋊C4—2444no—
239C16—2442no—
240C23—21no—
241C23—21no—
242C23—42no—
243C23—4882no—
244C23—4882no—
245C23—4881no—
246C23—4881no—
247C23—2442no—
248C23#1451969712yes—
249C23C2×#1175820111yescharacteristic
250D4—4888no—
251D4—4881no—
252D4—4881no—
253D4—2442no—
254D4—2442no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—4882no—
262C2×C4—2442no—
263C2×C4#1451969712yes—
264C2×C4C2×#1175819811yescharacteristic
265C2×C4#1451974611yescharacteristic
266C2×C4#1451976011yescharacteristic
267C2×C4C2×#1175819911yescharacteristic
268C2×C4#1451980311yescharacteristic
269C2×C4#1451980811yescharacteristic
270C8—22no—
271C8—44no—
272C8C4×C22⋊D6112yes—
273C8#1451975112yes—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—81no—
279C22—9762no—
280C22—9762no—
281C22—4882no—
282C22—4882no—
283C22—4882no—
284C22C2×#1451969711yescharacteristic
285C22#1683990111yescharacteristic
286C22#1684008011yescharacteristic
287C4—22no—
288C4—4882no—
289C4#1683964111yescharacteristic
290C4#1683976111yescharacteristic
291C4#1683989611yescharacteristic
292C4#1684007511yescharacteristic
293C2—21no—
294C2—21no—
295C2—42no—
296C2—9762no—
297C2#2144613611yescharacteristic
298C2#2144803312yes—
subgroupG/H
centerC22#16840080
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#430C2
socleC22×C61#481
prime radicalC2×#16840015C2
cosocleC22×C8×C61C23
hypercenterC2×#430D61
hypocenterC61#13010
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
2. C2×#16839574C2semidirect
3. C2×#16839544C2semidirect
4. #21443919C2semidirect
5. C2×C61×#430C2semidirect
50. #108×C61D4semidirect
145. C2×#430D61semidirect
169. C61#13010semidirect
197. #108C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400152
C2×#168395742
C2×#168395442
#214439192
C2×C61×#4302
#1301061