Finite Group Databasebeta

← #29596849

Subgroups — #29596849

Order 15616.

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

normal subgroups only

This group has 492 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#21448931C211yesmaximal, characteristic
2#21446223C211yesmaximal, characteristic
3#21446036C212yesmaximal, complemented
4#21443945C212yesmaximal, complemented
5C2×C61×#355C211yesmaximal, characteristic
6#16840240C2211yescharacteristic
7#16839804—22no—
8#16839767—21no—
9C2×#14519696C2211yescharacteristic
10#16839669C2212yes—
11#16839655—22no—
12#16839642—21no—
13C22⋊C4×Dic61—22no—
14#16839608—21no—
15#16839523—22no—
16#16839522—21no—
17C2×#124×C61C2211yescharacteristic
18C2×#101×C61C2212yes—
19C61×#355C414yescomplemented
20C23×Dic61—21no—
21C2×#11758194—21no—
22Q8×Dic61—42no—
23#14519736—21no—
24#14519736—22no—
25C2×C4⋊Dic61D412yes—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—21no—
28C2×C4×Dic61—22no—
29C2×C4×Dic61D412yes—
30#14519711—42no—
31#14519709—42no—
32#14519698—42no—
33#14519696—22no—
34#14519684—21no—
35#14519684—21no—
36#14519684—22no—
37#14519684—22no—
38C22×Q8×C61C2311yescharacteristic
39C22×D4×C61D411yescharacteristic
40C2×M16×C61—22no—
41#124×C61—22no—
42#124×C61C2×C412yes—
43#124×C61Q812yes—
44C2×C22⋊C4×C61—22no—
45C2×C42×C61D411yescharacteristic
46#101×C61—22no—
47#101×C61C2×C414yescomplemented
48C24×C61—21no—
49C2×Q8×C61—41no—
50C2×Q8×C61—41no—
51C2×Q8×C61C22×C411yescharacteristic
52C2×Q8×C61C2×Q811yescharacteristic
53C2×Q8×C61#4112yes—
54C2×D4×C61—21no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61C4⋊C412yes—
58C22×C4×C61—21no—
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D412yes—
61C22×Dic61—22no—
62C22×Dic61—22no—
63C22×Dic61—44no—
64C22×Dic61—81no—
65M16×C61—24no—
66M16×C61—42no—
67C2×C8×C61—42no—
68C22⋊C4×C61—24no—
69C22⋊C4×C61—24no—
70C42×C61—21no—
71C42×C61C4⋊C412yes—
72C22⋊Dic61—44no—
73C22⋊Dic61—41no—
74C22⋊Dic61—81no—
75C4⋊Dic61—22no—
76C4⋊Dic61—22no—
77C4⋊Dic61—22no—
78C4⋊Dic61—42no—
79C4×Dic61—44no—
80C4×Dic61—41no—
81C4×Dic61—42no—
82C4×Dic61—42no—
83#11758194—41no—
84#11758194—41no—
85C2×#92418—41no—
86C23×C61—22no—
87C23×C61—42no—
88C23×C61—42no—
89C23×C61—41no—
90C23×C61—41no—
91C23×C61#12011yescharacteristic
92Q8×C61—41no—
93Q8×C61—41no—
94Q8×C61—42no—
95D4×C61—42no—
96D4×C61—81no—
97C2×C4×C61—21no—
98C2×C4×C61—22no—
99C2×C4×C61—21no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—42no—
103C2×C4×C61C2×C4⋊C411yescharacteristic
104C2×C4×C61C4×D412yes—
105C2×C4×C61C22⋊Q812yes—
106C2×C4×C61#12311yescharacteristic
107C2×Dic61—24no—
108C2×Dic61—42no—
109C2×Dic61—42no—
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—84no—
115C8×C61—44no—
116#92418—81no—
117#12972—611nomaximal, Sylow
118C22×C61—22no—
119C22×C61—44no—
120C22×C61—42no—
121C22×C61—81no—
122C22×C61—81no—
123C22×C61—82no—
124C22×C61#38511yescharacteristic
125C22×C61B(4,𝔽2)11yescharacteristic
126C22×C61#45811yescharacteristic
127C4×C61—21no—
128C4×C61—22no—
129C4×C61—21no—
130C4×C61—22no—
131C4×C61—42no—
132Dic61—44no—
133Dic61—82no—
134Dic61—82no—
135#2372—611no—
136C2×#355D6111yescharacteristic
137#1888—611no—
138#1857—612no—
139#1394—612no—
140C2×C61—21no—
141C2×C61—21no—
142C2×C61—82no—
143C2×C61#186711yescharacteristic
144C2×C61#220412yes—
145C2×#124C2×D6111yescharacteristic
146#519—611no—
147#428—1222no—
148#418—1221no—
149C2×#101—611no—
150C2×#101C2×D6112yes—
151#391—612no—
152#388—1222no—
153#386—1221no—
154C4×C22⋊C4—1222no—
155#376—1221no—
156#355Dic6114yescomplemented
157#339—1222no—
158#339—1221no—
159C61#1297211yescharacteristic, complemented, Sylow
160C22×Q8C22×D6111yescharacteristic
161C22×D4D24411yescharacteristic
162C23×C4—1221no—
163C2×M16—22no—
164C2×M16—1221no—
165#124—22no—
166#124Dic12212yes—
167#124C2×Dic6112yes—
168C4×Q8—2442no—
169#117—1221no—
170#117—1222no—
171C2×C4⋊C4—612no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—1221no—
174C2×C22⋊C4—1222no—
175C2×C42—612no—
176C2×C42—1222no—
177C2×C42D24411yescharacteristic
178#107—2442no—
179#106—2442no—
180#102—2442no—
181#101—22no—
182#101—1222no—
183#101C2×Dic6114yescomplemented
184#95—1221no—
185#95—1221no—
186#95—1222no—
187#95—1222no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×Dic12211yescharacteristic
192C2×Q8#1175822412yes—
193C2×Q8C22×Dic6111yescharacteristic
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4C4⋊Dic6112yes—
198C22×C4—21no—
199C22×C4—1222no—
200C22×C4—1222no—
201C22×C4—4881no—
202C22×C4—2444no—
203C22×C4C2×D24411yescharacteristic
204C22×C4D4×D6112yes—
205M16—24no—
206M16—42no—
207M16—2441no—
208M16—2441no—
209C2×C8—42no—
210C2×C8—2441no—
211C4⋊C4—1222no—
212C4⋊C4—1222no—
213C4⋊C4—1222no—
214C4⋊C4—2442no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—4881no—
218C22⋊C4—2444no—
219C22⋊C4—2441no—
220C42—21no—
221C42—2442no—
222C42—2442no—
223C42—2441no—
224C42—2444no—
225C42C4⋊Dic6112yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1451974711yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—41no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—1224no—
244C2×C4—4884no—
245C2×C4—2442no—
246C2×C4—2442no—
247C2×C4—2442no—
248C2×C4—2442no—
249C2×C4—2442no—
250C2×C4—2442no—
251C2×C4C2×C4⋊Dic6111yescharacteristic
252C2×C4D4×Dic6112yes—
253C2×C4#1451975412yes—
254C2×C4#1451976111yescharacteristic
255C8—44no—
256C8—4881no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1683963711yescharacteristic
264C22#1683995011yescharacteristic
265C22#1683995311yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—4882no—
272C4—4882no—
273C4—2444no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2144610211yescharacteristic
278C2#2144784612yes—
subgroupG/H
centerC22#16839950
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC2×C61×#355C2
socleC22×C61B(4,𝔽2)
prime radicalC22×D4×C61D4
cosocleC22×Q8×C61C23
hypercenterC2×#355D61
hypocenterC61#12972
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #21446036C2semidirect
4. #21443945C2semidirect
19. C61×#355C4semidirect
47. #101×C61C2×C4semidirect
156. #355Dic61semidirect
159. C61#12972semidirect
183. #101C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214489312
#214462232
#214460362
#214439452
C2×C61×#3552
#1297261