Finite Group Databasebeta

← #29569113

Subgroups — #29569113

Order 15616.

478 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 87 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#21452191C211yesmaximal, characteristic
2#21447347C211yesmaximal, characteristic
3#21447292C212yesmaximal, complemented
4#21443643C212yesmaximal, complemented
5C61×#1270C211yesmaximal, characteristic
6C2×#14519754—22no—
7C2×#14519711—22no—
8C2×#14519710—21no—
9C2×#14519698—21no—
10#16839685—22no—
11#16839682—21no—
12#16839660C2212yes—
13#16839642—21no—
14#16839632C2211yescharacteristic
15#16839525—22no—
16#16839517C2211yescharacteristic
17C61×#390C2211yescharacteristic
18C61×#336C2212yes—
19C22×Dic122—21no—
20C23×Dic61—21no—
21C22×#92418—21no—
22#14519754—44no—
23#14519754—44no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—22no—
26C2×C4⋊Dic61D412yes—
27C2×C4⋊Dic61D412yes—
28C2×#11758196—22no—
29C2×#11758196—22no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—21no—
32#14519711—44no—
33#14519710—42no—
34#14519698—42no—
35#14519684—21no—
36#14519684—22no—
37C22×D4×C61D411yescharacteristic
38C22×C8×C61—22no—
39C2×C4⋊C4×C61D411yescharacteristic
40C2×C22⋊C4×C61—22no—
41C2×C42×C61C2311yescharacteristic
42C24×C61—21no—
43C2×D4×C61—21no—
44C2×D4×C61—41no—
45C2×D4×C61—41no—
46C2×D4×C61Q1612yes—
47C22×C4×C61—21no—
48C22×C4×C61C2×D412yes—
49C22×C4×C61C2×D411yescharacteristic
50C22×Dic61—22no—
51C22×Dic61—22no—
52C22×Dic61—44no—
53C22×Dic61—41no—
54C22×Dic61—81no—
55C2×Dic122—44no—
56C2×Dic122—81no—
57C2×C8×C61—24no—
58C2×C8×C61—42no—
59C2×C8×C61—42no—
60C4⋊C4×C61—41no—
61C22⋊C4×C61—42no—
62C22⋊C4×C61—42no—
63C42×C61—41no—
64C22⋊Dic61—44no—
65C22⋊Dic61—81no—
66C4⋊Dic61—22no—
67C4⋊Dic61—42no—
68C4⋊Dic61—41no—
69C4⋊Dic61—41no—
70C4⋊Dic61—42no—
71C4⋊Dic61—42no—
72C4⋊Dic61D814yes—
73#11758196—44no—
74#11758196—42no—
75#11758196—42no—
76C2×#92418—22no—
77C2×#92418—41no—
78C2×#92418—41no—
79C23×C61—22no—
80C23×C61—42no—
81C23×C61—41no—
82C23×C61—42no—
83C23×C61—41no—
84C23×C61#12011yescharacteristic
85D4×C61—42no—
86D4×C61—81no—
87C2×C4×C61—21no—
88C2×C4×C61—22no—
89C2×C4×C61—21no—
90C2×C4×C61—22no—
91C2×C4×C61—42no—
92C2×C4×C61—41no—
93C2×C4×C61C2×D812yes—
94C2×C4×C61C2×Q1611yescharacteristic
95C2×C4×C61AGL(1,ℤ/8)12yes—
96C2×C4×C61#13711yescharacteristic
97C2×Dic61—24no—
98C2×Dic61—42no—
99C2×Dic61—42no—
100C2×Dic61—42no—
101C2×Dic61—42no—
102C2×Dic61—42no—
103C2×Dic61—84no—
104C2×Dic61—82no—
105C2×Dic61—81no—
106Dic122—84no—
107C8×C61—44no—
108#92418—42no—
109#7797—611nomaximal, Sylow
110C22×C61—42no—
111C22×C61—44no—
112C22×C61—82no—
113C22×C61—81no—
114C22×C61—81no—
115C22×C61#44712yes—
116C22×C61#45111yescharacteristic
117C22×C61#45311yescharacteristic
118C22×C61#45612yes—
119C22×C61#45811yescharacteristic
120C4×C61—22no—
121C4×C61—22no—
122C4×C61—21no—
123C4×C61—21no—
124C4×C61—81no—
125Dic61—44no—
126Dic61—82no—
127Dic61—161no—
128#2683—611no—
129#2093—611no—
130#2083—612no—
131#1271—612no—
132#1270D6111yescharacteristic
133C2×C61—82no—
134C2×C61#161012yes—
135C2×C61#161411yescharacteristic
136C2×C61#161911yescharacteristic
137C2×C61#164912yes—
138C2×C61#166011yescharacteristic
139C2×C22⋊Q8—1222no—
140C2×#107—1222no—
141C2×#107—1221no—
142C2×#102—1221no—
143#396—1222no—
144#395—1221no—
145#390C2×D6111yescharacteristic
146#389—612no—
147#386—1221no—
148#384—611no—
149#340—1222no—
150#336—611no—
151#336C2×D6112yes—
152C61#779711yescharacteristic, complemented, Sylow
153C22×Q8—1221no—
154C22×D4D24411yescharacteristic
155C23×C4—1221no—
156C22×C8—22no—
157C22×C8—1221no—
158C22⋊Q8—2444no—
159C22⋊Q8—2444no—
160C2×C4⋊C4—612no—
161C2×C4⋊C4—612no—
162C2×C4⋊C4—1222no—
163C2×C4⋊C4—1222no—
164C2×C4⋊C4—1221no—
165C2×C4⋊C4—1222no—
166C2×C4⋊C4D24411yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—1221no—
169C2×C22⋊C4—1222no—
170C2×C42C22×D6111yescharacteristic
171#107—2442no—
172#107—2444no—
173#102—2442no—
174#95—1221no—
175#95—1222no—
176C24—21no—
177C2×Q8—4881no—
178C2×Q8—2444no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4Dic24412yes—
183C22×C4—21no—
184C22×C4—1222no—
185C22×C4—1222no—
186C22×C4—4881no—
187C22×C4—2444no—
188C22×C4—2441no—
189C22×C4C2×D24411yescharacteristic
190C22×C4D4×D6112yes—
191C2×C8—24no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—1222no—
195C2×C8—2441no—
196C2×C8—2441no—
197C4⋊C4—41no—
198C4⋊C4—614no—
199C4⋊C4—1222no—
200C4⋊C4—2441no—
201C4⋊C4—2441no—
202C4⋊C4—2442no—
203C4⋊C4—2444no—
204C4⋊C4—2442no—
205C4⋊C4—2442no—
206C4⋊C4—2442no—
207C4⋊C4—2442no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—4881no—
211C22⋊C4—2444no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1451974711yescharacteristic
219Q8—4884no—
220D4—42no—
221D4—81no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—1224no—
229C2×C4—4882no—
230C2×C4—4881no—
231C2×C4—4884no—
232C2×C4—2442no—
233C2×C4—2442no—
234C2×C4—2442no—
235C2×C4—2442no—
236C2×C4—2442no—
237C2×C4D8×D6112yes—
238C2×C4C2×Dic24411yescharacteristic
239C2×C4#1451980412yes—
240C2×C4#1451981011yescharacteristic
241C8—44no—
242C8—2442no—
243C22—44no—
244C22—42no—
245C22—81no—
246C22—81no—
247C22—82no—
248C22#1683989712yes—
249C22#1683992011yescharacteristic
250C22#1683993211yescharacteristic
251C22#1683994612yes—
252C22#1683995311yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—81no—
258C4—9761no—
259C4—4882no—
260C4—2444no—
261C2—82no—
262C2#2144482312yes—
263C2#2144484611yescharacteristic
264C2#2144487411yescharacteristic
265C2#2144507112yes—
266C2#2144513111yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61#120
prime radicalC22×D4×C61D4
hypercenter#1270D61
hypocenterC61#7797
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
3. #21447292C2semidirect
4. #21443643C2semidirect
152. C61#7797semidirect

Maximal subgroups

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

subgroupindex
#214521912
#214473472
#214472922
#214436432
C61×#12702
#779761