Finite Group Databasebeta

← #29575300

Subgroups — #29575300

Order 15616.

654 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16840136C211yesmaximal, characteristic
2#21448909C211yesmaximal, characteristic, complemented
3#21446745C212yesmaximal
4#21443658C212yesmaximal, complemented
5C2×C61×#333C211yesmaximal, characteristic, complemented
6C22×#11758201—22no—
7C2×#14519774C2211yescharacteristic
8#16840136—22no—
9#16840136C414yescomplemented
10C2×#14519699—22no—
11C2×#14519699—22no—
12C2×C4×#92418C2211yescharacteristic
13#16839669—22no—
14#16839661—22no—
15C4⋊C4×Dic61—22no—
16C4×C4⋊Dic61C2212yes—
17C2×C4⋊Q8×C61C2211yescharacteristic
18C2×C4⋊C8×C61C2212yes—
19C61×#333—22no—
20C22×D244—22no—
21C22×D244—21no—
22C2×#11758201—28no—
23C2×#11758201—44no—
24C2×#11758201—44no—
25C22×#92418—21no—
26#14519774—21no—
27#14519774—22no—
28#14519774C2×C412yes—
29C2×C4⋊Dic61—22no—
30C2×C4⋊Dic61D412yes—
31C2×C4⋊Dic61D412yes—
32C2×C4×Dic61—22no—
33C2×C4×Dic61—22no—
34#14519699—44no—
35#14519699—44no—
36C4×#92418—21no—
37C4×#92418C2×C412yes—
38#14519684—24no—
39#14519684—22no—
40C22×Q8×C61D412yes—
41C22×C8×C61—22no—
42C4⋊Q8×C61—22no—
43C4⋊Q8×C61C2×C412yes—
44C4⋊Q8×C61D412yes—
45C2×C4⋊C4×C61—22no—
46C2×C42×C61C2311yescharacteristic
47C4⋊C8×C61—22no—
48C4⋊C8×C61—22no—
49C23×D61—41no—
50C2×Q8×C61—22no—
51C2×Q8×C61—42no—
52C2×Q8×C61—42no—
53C2×Q8×C61C22⋊C414yes—
54C22×C4×C61—22no—
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61C2×D412yes—
57C22×Dic61—22no—
58C22×Dic61—22no—
59C2×D244—22no—
60C2×D244—24no—
61C2×D244—42no—
62C2×D244—42no—
63C2×D244—42no—
64C2×D244—42no—
65C2×D244—42no—
66C2×D244—42no—
67C2×D244—42no—
68C2×D244—41no—
69C2×C8×C61—42no—
70C2×C8×C61—42no—
71C2×C8×C61—42no—
72C4⋊C4×C61—24no—
73C4⋊C4×C61—24no—
74C42×C61C22×C411yescharacteristic
75C42×C61C2×D411yescharacteristic
76C42×C61#4112yes—
77#11758201—416no—
78C4⋊Dic61—22no—
79C4⋊Dic61—22no—
80C4⋊Dic61—22no—
81C4⋊Dic61—82no—
82C4⋊Dic61SD1614yescomplemented
83C4×Dic61—42no—
84C4×Dic61—42no—
85C4×Dic61—82no—
86C2×#92418—22no—
87C2×#92418—22no—
88C2×#92418—22no—
89C23×C61#12011yescharacteristic
90C22×D61—42no—
91C22×D61—82no—
92C22×D61—82no—
93C22×D61—81no—
94C22×D61—81no—
95Q8×C61—44no—
96Q8×C61—82no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—24no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61C2×C22⋊C411yescharacteristic
103C2×C4×C61C4×D412yes—
104C2×C4×C61#12112yes—
105C2×C4×C61#12311yescharacteristic
106C2×C4×C61C2×SD1612yes—
107C2×C4×C61#13512yes—
108C2×C4×C61AGL(1,ℤ/8)12yes—
109C2×C4×C61#13712yes—
110C2×Dic61—24no—
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116D244—44no—
117D244—44no—
118D244—44no—
119D244—82no—
120D244—82no—
121D244—82no—
122C8×C61—82no—
123#92418—24no—
124#9063—611nomaximal, Sylow
125C22×C61#38611yescharacteristic
126C22×C61#44812yes—
127C22×C61#44912yes—
128C22×C61#45311yescharacteristic
129C22×C61#45311yescharacteristic
130C2×D61—84no—
131C2×D61—82no—
132C2×D61—162no—
133C2×D61—161no—
134C2×D61—161no—
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—44no—
138C4×C61C4×SD1612yes—
139C4×C61#44012yes—
140C4×C61#46012yes—
141C4×C61#46412yes—
142Dic61—44no—
143Dic61—82no—
144C2×#492—611no—
145#2369—611no—
146#1982—612no—
147C2×#333D6111yescharacteristic, complemented
148#1279—612no—
149C2×C61#162212yes—
150C2×C61#162311yescharacteristic
151C2×C61#162411yescharacteristic
152C2×C61#188012yes—
153C2×C61#188711yescharacteristic
154D61—162no—
155C22×SD16—1222no—
156C2×C4⋊Q8C2×D6111yescharacteristic
157C2×#127—611no—
158#492—614no—
159#492—1222no—
160C2×C4⋊C8C2×D6112yes—
161C2×#102—1222no—
162C2×#102—1222no—
163C2×C4×C8—611no—
164#391—1222no—
165#389—1222no—
166C4×C4⋊C4—612no—
167C4×C4⋊C4—1222no—
168#333—22no—
169C61#906311yescharacteristic, complemented, Sylow
170C22×Q8D24412yes—
171C22×D4—1221no—
172C22×D4—1222no—
173C2×SD16—1228no—
174C2×SD16—2444no—
175C2×SD16—2444no—
176C22×C8—22no—
177C22×C8—1221no—
178C4⋊Q8—22no—
179C4⋊Q8C4×D6112yes—
180C4⋊Q8C22⋊D6112yes—
181#127—612no—
182#127—1222no—
183#127—1221no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—612no—
186C2×C4⋊C4—612no—
187C2×C4⋊C4—1222no—
188C2×C42—1222no—
189C2×C42—1222no—
190C2×C42C22×D6111yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193#102—2444no—
194#102—2444no—
195C4×C8—612no—
196C4×C8—1221no—
197#95—1224no—
198#95—1222no—
199C24—2441no—
200C2×Q8—22no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8#1175819814yes—
204C2×D4—1222no—
205C2×D4—1224no—
206C2×D4—2442no—
207C2×D4—2442no—
208C2×D4—2442no—
209C2×D4—2442no—
210C2×D4—2441no—
211C2×D4—2442no—
212C2×D4—2442no—
213C2×D4—2442no—
214C22×C4—22no—
215C22×C4—1222no—
216C22×C4—1222no—
217C22×C4C2×D24411yescharacteristic
218C22×C4D4×D6112yes—
219SD16—24416no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—1222no—
224C2×C8—1222no—
225C2×C8—1222no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—614no—
229C4⋊C4—1222no—
230C4⋊C4—1222no—
231C4⋊C4—1222no—
232C4⋊C4—4882no—
233C42—4882no—
234C42—2442no—
235C42—2442no—
236C42C2×C4×D6111yescharacteristic
237C42#1175822412yes—
238C42C2×C22⋊D6111yescharacteristic
239C23—4882no—
240C23—4882no—
241C23—4881no—
242C23—4881no—
243C23—2442no—
244C23#1451974711yescharacteristic
245Q8—44no—
246Q8—82no—
247D4—4882no—
248D4—4882no—
249D4—4882no—
250D4—2444no—
251D4—2444no—
252D4—2444no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—1224no—
259C2×C4—2442no—
260C2×C4—2442no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4—2442no—
264C2×C4C2×#1175819811yescharacteristic
265C2×C4#1451973712yes—
266C2×C4#1451974912yes—
267C2×C4#1451976111yescharacteristic
268C2×C4SD16×D6112yes—
269C2×C4#1451980012yes—
270C2×C4#1451980412yes—
271C2×C4#1451980912yes—
272C8—82no—
273C8—1224no—
274C22—9762no—
275C22—9761no—
276C22—9761no—
277C22—4884no—
278C22—4882no—
279C22#1683963911yescharacteristic
280C22#1683990212yes—
281C22#1683991012yes—
282C22#1683993211yescharacteristic
283C22#1683993211yescharacteristic
284C4—21no—
285C4—21no—
286C4—44no—
287C4—4882no—
288C4—2444no—
289C4#1683984712yes—
290C4#1683986512yes—
291C4#1683996012yes—
292C4#1683998612yes—
293C2—9762no—
294C2#2144489212yes—
295C2#2144489811yescharacteristic
296C2#2144490311yescharacteristic
297C2#2144617212yes—
298C2#2144621511yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#333C2
socleC23×C61#120
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenterC2×#333D61
hypocenterC61#9063
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
2. #21448909C2semidirect
4. #21443658C2semidirect
5. C2×C61×#333C2semidirect
9. #16840136C4semidirect
82. C4⋊Dic61SD16semidirect
147. C2×#333D61semidirect
169. C61#9063semidirect

Maximal subgroups

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

subgroupindex
C2×#168401362
#214489092
#214467452
#214436582
C2×C61×#3332
#906361