Finite Group Databasebeta

← #29596854

Subgroups — #29596854

Order 15616.

460 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448909C211yesmaximal, characteristic
2#21446213C211yesmaximal, characteristic
3#21446036C212yesmaximal
4#21443945C212yesmaximal
5C2×C61×#356C211yesmaximal, characteristic
6#16840240C2211yescharacteristic
7#16839804—22no—
8#16839771—21no—
9C2×#14519696C2211yescharacteristic
10#16839669—21no—
11#16839669C2212yes—
12#16839661—22no—
13C4⋊C4×Dic61—22no—
14C4×C4⋊Dic61—21no—
15#16839523—22no—
16#16839522—21no—
17C2×C4⋊Q8×C61C2211yescharacteristic
18C2×#101×C61C2212yes—
19C61×#356C414yescomplemented
20C2×#11758194—21no—
21Q8×Dic61—41no—
22Q8×Dic61—41no—
23#14519736—21no—
24#14519736—22no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61D412yes—
29C2×C4×Dic61—22no—
30C2×C4×Dic61—21no—
31C2×C4×Dic61D412yes—
32#14519711—42no—
33#14519709—42no—
34#14519701—41no—
35#14519701—41no—
36#14519696—22no—
37#14519684—22no—
38#14519684—22no—
39#14519684—22no—
40C22×Q8×C61D411yescharacteristic
41C22×Q8×C61C2311yescharacteristic
42C2×M16×C61—22no—
43C4⋊Q8×C61—22no—
44C4⋊Q8×C61C2×C412yes—
45C4⋊Q8×C61Q812yes—
46C2×C4⋊C4×C61—22no—
47C2×C42×C61D411yescharacteristic
48#101×C61—22no—
49#101×C61C2×C414yes—
50C2×Q8×C61—21no—
51C2×Q8×C61—41no—
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61—41no—
55C2×Q8×C61C4⋊C412yes—
56C2×Q8×C61C22×C411yescharacteristic
57C2×Q8×C61C2×Q811yescharacteristic
58C2×Q8×C61#4112yes—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D412yes—
63C22×Dic61—22no—
64C22×Dic61—22no—
65M16×C61—24no—
66M16×C61—42no—
67C2×C8×C61—42no—
68C4⋊C4×C61—24no—
69C4⋊C4×C61—24no—
70C42×C61—21no—
71C42×C61C4⋊C412yes—
72C22⋊Dic61—41no—
73C4⋊Dic61—24no—
74C4⋊Dic61—21no—
75C4⋊Dic61—21no—
76C4⋊Dic61—22no—
77C4⋊Dic61—22no—
78C4⋊Dic61—42no—
79C4⋊Dic61—42no—
80C4⋊Dic61—81no—
81C4×Dic61—44no—
82C4×Dic61—41no—
83C4×Dic61—42no—
84C4×Dic61—42no—
85C4×Dic61—81no—
86#11758194—41no—
87#11758194—41no—
88C2×#92418—41no—
89C23×C61#12011yescharacteristic
90Q8×C61—41no—
91Q8×C61—41no—
92Q8×C61—42no—
93Q8×C61—42no—
94Q8×C61—81no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—21no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61C2×C4⋊C411yescharacteristic
106C2×C4×C61C4×D412yes—
107C2×C4×C61C22⋊Q812yes—
108C2×C4×C61#12311yescharacteristic
109C2×Dic61—24no—
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C8×C61—44no—
118#92418—81no—
119#12973—611nomaximal, Sylow
120C22×C61—21no—
121C22×C61—21no—
122C22×C61#38511yescharacteristic
123C22×C61B(4,𝔽2)11yescharacteristic
124C22×C61#45811yescharacteristic
125C4×C61—24no—
126C4×C61—21no—
127C4×C61—22no—
128C4×C61—21no—
129C4×C61—22no—
130C4×C61—42no—
131Dic61—44no—
132Dic61—81no—
133Dic61—82no—
134Dic61—81no—
135#2369—611no—
136C2×#356D6111yescharacteristic
137#1886—611no—
138#1857—612no—
139#1394—612no—
140C2×C61—21no—
141C2×C61—21no—
142C2×C61#186711yescharacteristic
143C2×C61#220411yescharacteristic
144C2×C61#220511yescharacteristic
145C2×C4⋊Q8C2×D6111yescharacteristic
146#519—611no—
147#428—1222no—
148#419—1221no—
149C2×#101—611no—
150C2×#101C2×D6112yes—
151#391—612no—
152#391—1221no—
153#389—1222no—
154C4×C4⋊C4—1222no—
155C4×C4⋊C4—1221no—
156#356Dic6114yescomplemented
157#339—1222no—
158#339—1221no—
159C61#1297311yescharacteristic, complemented, Sylow
160C22×Q8D24411yescharacteristic
161C22×Q8C22×D6111yescharacteristic
162C2×M16—22no—
163C2×M16—1221no—
164C4⋊Q8—22no—
165C4⋊Q8Dic12212yes—
166C4⋊Q8C2×Dic6112yes—
167C4×Q8—2441no—
168C4×Q8—2441no—
169#117—1221no—
170#117—1222no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—612no—
173C2×C4⋊C4—1221no—
174C2×C4⋊C4—1221no—
175C2×C4⋊C4—1221no—
176C2×C42—612no—
177C2×C42—1221no—
178C2×C42—1222no—
179C2×C42D24411yescharacteristic
180#107—2442no—
181#106—2442no—
182#103—2441no—
183#103—2441no—
184#101—22no—
185#101—1222no—
186#101C2×Dic6114yes—
187#95—1222no—
188#95—1222no—
189#95—1222no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C4⋊Dic6112yes—
196C2×Q8C2×Dic12211yescharacteristic
197C2×Q8#1175822412yes—
198C2×Q8C22×Dic6111yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1222no—
202C22×C4—1222no—
203C22×C4C2×D24411yescharacteristic
204C22×C4D4×D6112yes—
205M16—24no—
206M16—42no—
207M16—2441no—
208M16—2441no—
209C2×C8—42no—
210C2×C8—2441no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—1224no—
214C4⋊C4—1221no—
215C4⋊C4—1221no—
216C4⋊C4—1222no—
217C4⋊C4—1222no—
218C4⋊C4—4881no—
219C4⋊C4—2442no—
220C4⋊C4—2442no—
221C22⋊C4—2441no—
222C42—21no—
223C42—4881no—
224C42—2442no—
225C42—2444no—
226C42—2442no—
227C42—2441no—
228C42C4⋊Dic6112yes—
229C23#1451974711yescharacteristic
230Q8—41no—
231Q8—42no—
232Q8—41no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—1224no—
246C2×C4—2442no—
247C2×C4—2442no—
248C2×C4—2441no—
249C2×C4—2441no—
250C2×C4—2442no—
251C2×C4—2442no—
252C2×C4—2442no—
253C2×C4C2×C4⋊Dic6111yescharacteristic
254C2×C4D4×Dic6112yes—
255C2×C4#1451975412yes—
256C2×C4#1451976111yescharacteristic
257C8—44no—
258C8—4881no—
259C22—21no—
260C22—21no—
261C22#1683963711yescharacteristic
262C22#1683995011yescharacteristic
263C22#1683995311yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—42no—
270C4—4881no—
271C4—4882no—
272C4—4881no—
273C4—2444no—
274C2—21no—
275C2—21no—
276C2#2144610211yescharacteristic
277C2#2144784711yescharacteristic
278C2#2144785011yescharacteristic
subgroupG/H
centerC22#16839950
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC2×C61×#356C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
cosocleC22×Q8×C61C23
hypercenterC2×#356D61
hypocenterC61#12973
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
19. C61×#356C4semidirect
156. #356Dic61semidirect
159. C61#12973semidirect

Maximal subgroups

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

subgroupindex
#214489092
#214462132
#214460362
#214439452
C2×C61×#3562
#1297361