Finite Group Databasebeta

← #29574392

Subgroups — #29574392

Order 15616.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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#21446636C211yesmaximal, characteristic
2#21445952C212yesmaximal, complemented
3C2×#16839502C211yesmaximal, characteristic
4#21443700C212yesmaximal, complemented
5C61×#2379C211yesmaximal, characteristic
6C2×#14519701—21no—
7C2×#14519698—21no—
8C2×#14519688C2211yescharacteristic
9C2×#14519688C2212yes—
10#16839632C2211yescharacteristic
11#16839524—22no—
12#16839524—22no—
13#16839502C414yescomplemented
14C2×#124×C61C2211yescharacteristic
15C61×#391—21no—
16C61×#388—22no—
17C61×#386—21no—
18C61×#385—22no—
19C61×#382C2212yes—
20C22×#92418—21no—
21C22×#92418—22no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—22no—
24#14519701—42no—
25#14519698—42no—
26#14519688—22no—
27#14519688—22no—
28#14519688C2×C414yescomplemented
29C22×Q8×C61D411yescharacteristic
30C22×D4×C61D411yescharacteristic
31C23×C4×C61—21no—
32#124×C61—22no—
33#124×C61C2×C412yes—
34#124×C61Q812yes—
35C2×C4⋊C4×C61—21no—
36C2×C4⋊C4×C61D412yes—
37C2×C4⋊C4×C61D412yes—
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—21no—
41C2×C42×C61—21no—
42C2×C42×C61C2311yescharacteristic
43#95×C61—22no—
44#95×C61—22no—
45#95×C61—22no—
46C24×C61—21no—
47C2×Q8×C61—21no—
48C2×Q8×C61—41no—
49C2×Q8×C61—41no—
50C2×Q8×C61C4⋊C412yes—
51C2×D4×C61—21no—
52C2×D4×C61—41no—
53C2×D4×C61—41no—
54C2×D4×C61C4⋊C412yes—
55C22×C4×C61—21no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61—81no—
59C22×C4×C61—82no—
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D412yes—
62C22×Dic61—41no—
63C4⋊C4×C61—42no—
64C4⋊C4×C61—42no—
65C4⋊C4×C61—81no—
66C22⋊C4×C61—24no—
67C22⋊C4×C61—24no—
68C22⋊C4×C61—81no—
69C22⋊C4×C61—82no—
70C42×C61—81no—
71C42×C61C22×C411yescharacteristic
72C42×C61C2×Q811yescharacteristic
73C42×C61#4112yes—
74C4⋊Dic61—41no—
75C4⋊Dic61—41no—
76C4⋊Dic61—44no—
77C2×#92418—22no—
78C2×#92418—24no—
79C2×#92418—42no—
80C2×#92418—42no—
81C2×#92418—41no—
82C2×#92418—41no—
83C23×C61—22no—
84C23×C61—41no—
85C23×C61—42no—
86C23×C61—42no—
87C23×C61—41no—
88C23×C61#12011yescharacteristic
89Q8×C61—42no—
90Q8×C61—81no—
91D4×C61—42no—
92D4×C61—81no—
93C2×C4×C61—21no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—21no—
97C2×C4×C61—22no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—84no—
107C2×C4×C61C2×C4⋊C411yescharacteristic
108C2×C4×C61C4×D412yes—
109C2×C4×C61C22⋊Q812yes—
110C2×C4×C61#12311yescharacteristic
111C2×Dic61—81no—
112C2×Dic61—82no—
113#92418—42no—
114#92418—44no—
115#8914—611nomaximal, Sylow
116C22×C61—42no—
117C22×C61—44no—
118C22×C61—81no—
119C22×C61—81no—
120C22×C61—82no—
121C22×C61#38511yescharacteristic
122C22×C61#45311yescharacteristic
123C22×C61#45412yes—
124C22×C61#45511yescharacteristic
125C22×C61#45512yes—
126C4×C61—21no—
127C4×C61—21no—
128C4×C61—22no—
129C4×C61—22no—
130C4×C61—42no—
131C4×C61—82no—
132C4×C61—82no—
133Dic61—161no—
134#2379D6111yescharacteristic
135#1959—611no—
136#1836—612no—
137C2×#329—611no—
138#1295—612no—
139C2×C61—82no—
140C2×C61#165711yescharacteristic
141C2×C61#165911yescharacteristic
142C2×C61#166212yes—
143C2×C61#185611yescharacteristic
144C2×C61#186012yes—
145C2×#124C2×D6111yescharacteristic
146C2×#103—1221no—
147C2×#102—1221no—
148C2×#97—612no—
149C2×#97—611no—
150#391—21no—
151#388—22no—
152#386—21no—
153#385—22no—
154#384—611no—
155#382C2×D6112yes—
156#340—1222no—
157#340—1222no—
158#329—614no—
159C61#891411yescharacteristic, complemented, Sylow
160C22×Q8C22⋊D6111yescharacteristic
161C22×D4C22⋊D6111yescharacteristic
162C23×C4—21no—
163C22×C8—1222no—
164C22×C8—1221no—
165#124—22no—
166#124Dic12212yes—
167#124C4×D6112yes—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—1222no—
170C2×C4⋊C4—1221no—
171C2×C4⋊C4C22⋊D6112yes—
172C2×C4⋊C4C22⋊D6112yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C42—21no—
177C2×C42C22×D6111yescharacteristic
178#103—2442no—
179#102—2442no—
180#97—614no—
181#97—1222no—
182#97—1222no—
183#95—22no—
184#95—22no—
185#95—22no—
186C24—21no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8#1175819612yes—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4#1175819612yes—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—82no—
199C22×C4—81no—
200C22×C4—2441no—
201C22×C4C2×C22⋊D6111yescharacteristic
202C22×C4C2×C22⋊D6112yes—
203C2×C8—1222no—
204C2×C8—1224no—
205C2×C8—2442no—
206C2×C8—2441no—
207C2×C8—2442no—
208C2×C8—2441no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—2441no—
213C4⋊C4—2441no—
214C4⋊C4—2444no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—81no—
218C22⋊C4—82no—
219C42—81no—
220C42C2×Dic12211yescharacteristic
221C42C2×C4×D6111yescharacteristic
222C42#1175822212yes—
223C23—22no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23#1451974611yescharacteristic
229Q8—42no—
230Q8—81no—
231D4—42no—
232D4—81no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—4881no—
248C2×C4—4882no—
249C2×C4C2×#1175819611yescharacteristic
250C2×C4C4×C22⋊D6112yes—
251C2×C4#1451975612yes—
252C2×C4#1451976011yescharacteristic
253C8—2442no—
254C8—2444no—
255C22—42no—
256C22—44no—
257C22—81no—
258C22—82no—
259C22—81no—
260C22#1683963511yescharacteristic
261C22#1683993011yescharacteristic
262C22#1683993512yes—
263C22#1683994012yes—
264C22#1683994011yescharacteristic
265C4—21no—
266C4—21no—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—82no—
271C4—82no—
272C4—9761no—
273C2—82no—
274C2#2144511711yescharacteristic
275C2#2144512511yescharacteristic
276C2#2144514212yes—
277C2#2144602911yescharacteristic
278C2#2144605212yes—
subgroupG/H
centerC23#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2379C2
socleC23×C61#120
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#2379D61
hypocenterC61#8914
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
2. #21445952C2semidirect
4. #21443700C2semidirect
13. #16839502C4semidirect
28. #14519688C2×C4semidirect
159. C61#8914semidirect

Maximal subgroups

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

subgroupindex
#214466362
#214459522
C2×#168395022
#214437002
C61×#23792
#891461