Finite Group Databasebeta

← #29611221

Subgroups — #29611221

Order 15616.

6494 conjugacy classes of subgroups; only the 675 normal subgroups are shown.

normal subgroups only

This group has 6494 conjugacy classes of subgroups — too many to draw. Only its 675 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
1C25×Dic61C211yesmaximal, characteristic
2C2×#16839634C216yesmaximal, complemented
3C2×#16839618C211yesmaximal, characteristic
4#21444041C216yesmaximal, complemented
5C2×C61×#379C211yesmaximal, characteristic
6C24×Dic61—24no—
7C24×Dic61—224no—
8C24×Dic61C2216yes—
9C22×#11758196C2214yescomplemented
10C22×C22⋊Dic61C2216yes—
11#16839634—224no—
12#16839618C418yescomplemented
13C2×#14519684C22112yescomplemented
14C26×C61C2211yescharacteristic
15C22×C22⋊C4×C61C2213yes—
16C22×C22⋊C4×C61C2213yes—
17C61×#379—24no—
18C23×Dic61—296no—
19C23×Dic61—2128no—
20C23×Dic61—212no—
21C23×Dic61—212no—
22C23×Dic61—248no—
23C23×Dic61D4124yes—
24C23×Dic61C2314yes—
25C23×Dic61C2314yes—
26C2×#11758196—224no—
27C2×C22⋊Dic61—212no—
28C2×C22⋊Dic61—212no—
29C2×C22⋊Dic61C2×C4124yescomplemented
30#14519684—224no—
31C25×C61—212no—
32C25×C61—212no—
33C25×C61C2×C414yes—
34C25×C61D414yes—
35C25×C61Q814yes—
36C25×C61C2313yes—
37C23×C4×C61C2311yescharacteristic
38C23×C4×C61C2313yes—
39C2×C22⋊C4×C61—26no—
40C2×C22⋊C4×C61—26no—
41C2×C22⋊C4×C61—212no—
42C2×C22⋊C4×C61—212no—
43C24×C61—232no—
44C24×C61—296no—
45C24×C61—248no—
46C24×C61—224no—
47C24×C61—248no—
48C24×C61—224no—
49C24×C61—212no—
50C24×C61—212no—
51C24×C61C4⋊C4116yescomplemented
52C24×C61C22×C416yes—
53C24×C61C2×D416yes—
54C24×C61C2×D4112yes—
55C24×C61C2×Q816yes—
56C24×C61#41112yes—
57C24×C61C2411yescharacteristic
58C22×C4×C61—21no—
59C22×C4×C61—23no—
60C22×C4×C61—26no—
61C22×C4×C61—26no—
62C22×C4×C61—212no—
63C22×Dic61—2256no—
64C22×Dic61—296no—
65C22×Dic61—296no—
66C22×Dic61—248no—
67C22×Dic61—24no—
68C22×Dic61—212no—
69C22×Dic61—296no—
70C22×Dic61—212no—
71C22×Dic61—212no—
72C22×Dic61C22×C418yescomplemented
73C22×Dic61C2×D4124yes—
74C22⋊C4×C61—412no—
75C22⋊C4×C61—412no—
76C22⋊Dic61—248no—
77#11758196—416no—
78C23×C61—232no—
79C23×C61—2128no—
80C23×C61—296no—
81C23×C61—2128no—
82C23×C61—224no—
83C23×C61—224no—
84C23×C61—212no—
85C23×C61—224no—
86C23×C61—212no—
87C23×C61—224no—
88C23×C61—248no—
89C23×C61—296no—
90C23×C61C2×C4⋊C4112yes—
91C23×C61C4×D4124yes—
92C23×C61#120112yes—
93C23×C61C22⋊Q8124yes—
94C23×C61#123112yes—
95C23×C61C23×C411yescharacteristic
96C23×C61C22×D411yescharacteristic
97C23×C61C22×D416yes—
98C23×C61C22×Q811yescharacteristic
99C23×C61C2×#4116yes—
100C2×C4×C61—44no—
101C2×C4×C61—43no—
102C2×C4×C61—43no—
103C2×C4×C61—46no—
104C2×C4×C61—412no—
105C2×Dic61—2192no—
106C2×Dic61—224no—
107C2×Dic61—224no—
108C2×Dic61—224no—
109C2×Dic61—44no—
110C2×Dic61—412no—
111C2×Dic61#120116yes—
112#14693—611nomaximal, Sylow
113C22×C61—2128no—
114C22×C61—212no—
115C22×C61—248no—
116C22×C61—212no—
117C22×C61—224no—
118C22×C61—224no—
119C22×C61—248no—
120C22×C61#385124yescomplemented
121C22×C61C22×C4⋊C411yescharacteristic
122C22×C61C2×C4×D416yes—
123C22×C61C2×#12014yes—
124C22×C61C2×#12013yes—
125C22×C61C2×C22⋊Q816yes—
126C22×C61C2×#12313yes—
127C22×C61#525112yes—
128C4×C61—81no—
129C4×C61—83no—
130Dic61—216no—
131Dic61—48no—
132C25×C4—611no—
133C2×#385—616no—
134C2×#379—611no—
135C2×#379D6111yescharacteristic
136C22⋊#95—616no—
137C2×C61—224no—
138C2×C61C2×#38513yes—
139C2×C61C4×#12014yes—
140C2×C61#284611yescharacteristic
141C2×C61#284814yes—
142C2×C61#284913yes—
143C26C2×D6111yescharacteristic
144C24×C4—616no—
145C24×C4—12224no—
146C24×C4—1224no—
147C22×C4⋊C4—614no—
148C22×C22⋊C4—616no—
149C22×C22⋊C4C2×D6113yes—
150C22×C22⋊C4C2×D6113yes—
151#385—12224no—
152#379—24no—
153#379—618no—
154C2×#95—6112no—
155C61#1469311yescharacteristic, complemented, Sylow
156C25—212no—
157C25—212no—
158C25Dic12214yes—
159C25C4×D6114yes—
160C25C22⋊D6114yes—
161C25C22×D6113yes—
162C23×C4—614no—
163C23×C4—6124no—
164C23×C4—614no—
165C23×C4—12248no—
166C23×C4—122128no—
167C23×C4—12296no—
168C23×C4—12212no—
169C23×C4—12212no—
170C23×C4C22×D6111yescharacteristic
171C23×C4C22×D6113yes—
172C2×C4⋊C4—12224no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—26no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—6124no—
178C2×C22⋊C4—12212no—
179C2×C22⋊C4—12212no—
180#95—12224no—
181C24—232no—
182C24—296no—
183C24—212no—
184C24—212no—
185C24—248no—
186C24—224no—
187C24—248no—
188C24—224no—
189C24#11758196116yescomplemented
190C24C2×Dic12216yes—
191C24C2×C4×D6116yes—
192C24D4×D61112yes—
193C24#11758224112yes—
194C24C2×C22⋊D6116yes—
195C24C23×D6111yescharacteristic
196C22×C4—26no—
197C22×C4—26no—
198C22×C4—21no—
199C22×C4—23no—
200C22×C4—212no—
201C22×C4—6124no—
202C22×C4—618no—
203C22×C4—12296no—
204C22×C4—122256no—
205C22×C4—12296no—
206C22×C4—12296no—
207C22×C4—12212no—
208C22×C4—1224no—
209C22×C4—12248no—
210C22×C4—12212no—
211C22×C4—12212no—
212C4⋊C4—24416no—
213C22⋊C4—412no—
214C22⋊C4—412no—
215C22⋊C4—12248no—
216C23—232no—
217C23—2128no—
218C23—296no—
219C23—2128no—
220C23—224no—
221C23—224no—
222C23—212no—
223C23—224no—
224C23—212no—
225C23—224no—
226C23—248no—
227C23—296no—
228C23C2×#11758196112yes—
229C23#14519737124yes—
230C23#14519745112yes—
231C23#14519754124yes—
232C23#14519758112yes—
233C23C22×C4×D6111yescharacteristic
234C23C2×D4×D6116yes—
235C23C22×C22⋊D6111yescharacteristic
236C23C22×Dic12211yescharacteristic
237C23C2×#1175822416yes—
238C2×C4—44no—
239C2×C4—43no—
240C2×C4—43no—
241C2×C4—46no—
242C2×C4—412no—
243C2×C4—6116no—
244C2×C4—122192no—
245C2×C4—12224no—
246C2×C4—12224no—
247C2×C4—12224no—
248C2×C4—2444no—
249C2×C4—24412no—
250C22—2128no—
251C22—212no—
252C22—248no—
253C22—212no—
254C22—224no—
255C22—224no—
256C22—248no—
257C22#16839634124yescomplemented
258C22C22×#1175819611yescharacteristic
259C22C2×#1451973716yes—
260C22#120×D6114yes—
261C22C2×#1451974513yes—
262C22C2×#1451975416yes—
263C22C2×#1451975813yes—
264C22#16840268112yes—
265C4—81no—
266C4—83no—
267C4—12216no—
268C4—2448no—
269C2—224no—
270C2C2×#1683963413yes—
271C2#2144831814yes—
272C2#2145409511yescharacteristic
273C2#2145410414yes—
274C2#2145410913yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#379C2
socle, cosocleC24×C61C24
prime radicalC26×C61C22
hypercenterC2×#379D61
hypocenterC61#14693
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
2. C2×#16839634C2semidirect
4. #21444041C2semidirect
9. C22×#11758196C22semidirect
12. #16839618C4semidirect
13. C2×#14519684C22semidirect
29. C2×C22⋊Dic61C2×C4semidirect
51. C24×C61C4⋊C4semidirect
72. C22×Dic61C22×C4semidirect
120. C22×C61#385semidirect
155. C61#14693semidirect
189. C24#11758196semidirect
257. C22#16839634semidirect

Maximal subgroups

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

subgroupindex
C25×Dic612
C2×#168396342
C2×#168396182
#214440412
C2×C61×#3792
#1469361