Finite Group Databasebeta

← #29573585

Subgroups — #29573585

Order 15616.

498 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 123 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#21448729C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic
3#21443721C212yesmaximal, complemented
4#21443678C212yesmaximal, complemented
5C61×#1460C211yesmaximal, characteristic
6C2×D4×Dic61—21no—
7#16839786C2212yes—
8C2×#14519694—21no—
9#16839661—21no—
10#16839637—22no—
11#16839628C2211yescharacteristic
12#16839622C412yes—
13C4×C4⋊Dic61C2211yescharacteristic
14C2×#14519684C412yes—
15#16839531—21no—
16C2×C4×D4×C61C2211yescharacteristic
17C61×#431C2212yes—
18C61×#323—22no—
19C23×Dic61—22no—
20C2×#11758194—22no—
21D4×Dic61—44no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C4×Dic61—21no—
27C4⋊#92418—22no—
28C4⋊#92418—22no—
29#14519694—42no—
30#14519684—22no—
31#14519684—21no—
32#14519684C2×C411yescharacteristic
33#14519684C2×C412yes—
34#14519684C2×C411yescharacteristic
35C22×D4×C61D411yescharacteristic
36C23×C4×C61C2×C412yes—
37C4×D4×C61—22no—
38C4×D4×C61—22no—
39C2×C4⋊C4×C61D411yescharacteristic
40C2×C22⋊C4×C61D412yes—
41C2×C42×C61C2311yescharacteristic
42C4⋊C8×C61—22no—
43C22⋊C8×C61—22no—
44C4×C8×C61—22no—
45C24×C61C22⋊C412yes—
46C2×D4×C61—22no—
47C2×D4×C61—22no—
48C2×D4×C61SD1612yes—
49C2×D4×C61Q1612yes—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—82no—
53C22×C4×C61C22⋊C412yes—
54C22×C4×C61C22×C411yescharacteristic
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61C2×D411yescharacteristic
57C22×Dic61—21no—
58C22×Dic61—22no—
59C22×Dic61—21no—
60C22×Dic61—44no—
61C22×Dic61—44no—
62C22×Dic61—82no—
63C2×C8×C61—42no—
64C4⋊C4×C61—22no—
65C22⋊C4×C61—24no—
66C42×C61#4112yes—
67C42×C61#4112yes—
68C22⋊Dic61—44no—
69C4⋊Dic61—42no—
70C4⋊Dic61—42no—
71C4⋊Dic61—82no—
72C4×Dic61—42no—
73#11758194—42no—
74#11758194—42no—
75C2×#92418—42no—
76C23×C61—22no—
77C23×C61—22no—
78C23×C61—42no—
79C23×C61—42no—
80C23×C61#10314yes—
81C23×C61C2×C22⋊C411yescharacteristic
82D4×C61—24no—
83D4×C61—24no—
84C2×C4×C61—22no—
85C2×C4×C61—41no—
86C2×C4×C61—41no—
87C2×C4×C61—42no—
88C2×C4×C61—82no—
89C2×C4×C61—82no—
90C2×C4×C61#11711yescharacteristic
91C2×C4×C61#11711yescharacteristic
92C2×C4×C61#12312yes—
93C2×C4×C61#12312yes—
94C2×C4×C61C2×SD1611yescharacteristic
95C2×C4×C61C2×Q1611yescharacteristic
96C2×C4×C61AGL(1,ℤ/8)12yes—
97C2×Dic61—22no—
98C2×Dic61—41no—
99C2×Dic61—41no—
100C2×Dic61—41no—
101C2×Dic61—41no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—41no—
105C2×Dic61—42no—
106C2×Dic61—88no—
107C8×C61—82no—
108#92418—82no—
109#8765—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—21no—
112C22×C61—24no—
113C22×C61—44no—
114C22×C61—44no—
115C22×C61—42no—
116C22×C61—42no—
117C22×C61#38111yescharacteristic
118C22×C61C2×#10311yescharacteristic
119C22×C61#41811yescharacteristic
120C4×C61—22no—
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—42no—
124C4×C61#48112yes—
125C4×C61#48412yes—
126Dic61—42no—
127Dic61—81no—
128Dic61—82no—
129#2349—611no—
130#1800—611no—
131#1460D6111yescharacteristic
132#1303—612no—
133#1286—612no—
134C2×C61—21no—
135C2×C61—21no—
136C2×C61—44no—
137C2×C61#182411yescharacteristic
138C2×C61#182711yescharacteristic
139C2×C61#182911yescharacteristic
140C2×C4×D4—1221no—
141C2×C4×D4C2×D6111yescharacteristic
142#431C2×D6112yes—
143#423—612no—
144C2×#100—1221no—
145#389—1221no—
146#385—1222no—
147#382—611no—
148#380—612no—
149C4×C4⋊C4—611no—
150C2×#95—612no—
151#343—1221no—
152#323—22no—
153C61#876511yescharacteristic, complemented, Sylow
154C22×D4D24411yescharacteristic
155C23×C4—1222no—
156C23×C4C4×D6112yes—
157C2×M16—1222no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—2444no—
161C2×C4⋊C4—1222no—
162C2×C4⋊C4—1221no—
163C2×C4⋊C4—1221no—
164C2×C4⋊C4D24411yescharacteristic
165C2×C22⋊C4—1222no—
166C2×C22⋊C4C22⋊D6112yes—
167C2×C42—1221no—
168C2×C42C22×D6111yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—1222no—
171C4⋊C8—1222no—
172#100—2442no—
173C22⋊C8—22no—
174C4×C8—22no—
175#95—611no—
176#95—612no—
177#95—611no—
178#95—1222no—
179#95—1221no—
180C24#1175819812yes—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4#1175819012yes—
184C2×D4Dic24412yes—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—82no—
188C22×C4—1221no—
189C22×C4—1222no—
190C22×C4—1221no—
191C22×C4—4882no—
192C22×C4—2444no—
193C22×C4—2444no—
194C22×C4#1175819812yes—
195C22×C4C2×C4×D6111yescharacteristic
196C22×C4C2×D24411yescharacteristic
197C22×C4C2×C22⋊D6111yescharacteristic
198M16—2442no—
199M16—2442no—
200C2×C8—42no—
201C2×C8—2442no—
202C4⋊C4—22no—
203C4⋊C4—4882no—
204C4⋊C4—2442no—
205C4⋊C4—2442no—
206C22⋊C4—24no—
207C22⋊C4—2444no—
208C42—2442no—
209C42#1175822412yes—
210C42#1175822412yes—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23#1451970214yes—
216C23C2×#1175819811yescharacteristic
217D4—24no—
218D4—24no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—1222no—
226C2×C4—4888no—
227C2×C4—2441no—
228C2×C4—2442no—
229C2×C4—2441no—
230C2×C4—2442no—
231C2×C4—2441no—
232C2×C4—2442no—
233C2×C4—2441no—
234C2×C4—2441no—
235C2×C4#1451973311yescharacteristic
236C2×C4#1451973311yescharacteristic
237C2×C4#1451975812yes—
238C2×C4#1451976112yes—
239C2×C4C2×#1175819011yescharacteristic
240C2×C4C2×Dic24411yescharacteristic
241C2×C4#1451980512yes—
242C8—82no—
243C8—4882no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22#1683962311yescharacteristic
252C22C2×#1451970211yescharacteristic
253C22#1683976811yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—4882no—
259C4—4881no—
260C4—2442no—
261C4#1684008112yes—
262C4#1684009812yes—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#2144591011yescharacteristic
267C2#2144592011yescharacteristic
268C2#2144592811yescharacteristic
subgroupG/H
centerC22#16839623
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1460C2
socleC22×C61#381
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1460D61
hypocenterC61#8765
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #21443721C2semidirect
4. #21443678C2semidirect
153. C61#8765semidirect

Maximal subgroups

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

subgroupindex
#214487292
#214458112
#214437212
#214436782
C61×#14602
#876561