Finite Group Databasebeta

← #29596834

Subgroups — #29596834

Order 15616.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#21454890C211yesmaximal, characteristic
2#21446203C211yesmaximal, characteristic
3#21446107C212yesmaximal, complemented
4#21443939C212yesmaximal, complemented
5C61×#2126C211yesmaximal, characteristic
6#16840529C2211yescharacteristic
7#16839834—22no—
8#16839834—22no—
9#16839819—21no—
10#16839789—22no—
11#16839779—21no—
12#16839736C2211yescharacteristic
13#16839729—21no—
14#16839729C2212yes—
15#16839532—22no—
16#16839522—21no—
17C61×#532C2211yescharacteristic
18C61×#410C2212yes—
19C61×#354C412yescomplemented
20C61×#353C412yescomplemented
21#14519784—42no—
22C2×#11758194—22no—
23C2×#11758194—21no—
24C2×#11758194—21no—
25C2×#11758194D412yes—
26C22×#92418—21no—
27C22×#92418D412yes—
28#14519736—21no—
29C4⋊#92418—21no—
30C4⋊#92418—21no—
31C4⋊#92418—44no—
32#14519704—42no—
33#14519692—22no—
34C22⋊#92418—22no—
35C22⋊#92418—22no—
36C22⋊#92418—22no—
37#14519688—22no—
38C2×#41×C61D411yescharacteristic
39C2×#41×C61C2311yescharacteristic
40C4⋊Q8×C61Q811yescharacteristic
41#127×C61C2×C411yescharacteristic
42#124×C61C2×C411yescharacteristic
43#124×C61Q811yescharacteristic
44#121×C61—22no—
45C4×D4×C61—22no—
46#117×C61—22no—
47C2×C42×C61D411yescharacteristic
48#99×C61—22no—
49#99×C61C2×C414yes—
50#41×C61—41no—
51#41×C61—41no—
52#41×C61—41no—
53#41×C61—41no—
54C2×Q8×C61C4⋊C411yescharacteristic
55C2×Q8×C61C2×Q811yescharacteristic
56C2×D4×C61—22no—
57C2×D4×C61—21no—
58C2×D4×C61C4⋊C411yescharacteristic
59C2×D4×C61C22×C411yescharacteristic
60C2×D4×C61#4112yes—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61C2×D412yes—
65C4⋊C4×C61—22no—
66C4⋊C4×C61—42no—
67C22⋊C4×C61—22no—
68C22⋊C4×C61—22no—
69C22⋊C4×C61—24no—
70C42×C61—21no—
71C42×C61—42no—
72C42×C61C4⋊C411yescharacteristic
73C42×C61C4⋊C411yescharacteristic
74C22⋊Dic61—41no—
75C4⋊Dic61—41no—
76C4×Dic61—41no—
77#11758194—22no—
78#11758194—42no—
79#11758194—41no—
80#11758194—42no—
81#11758194—81no—
82C2×#92418—24no—
83C2×#92418—22no—
84C2×#92418—22no—
85C2×#92418—41no—
86C2×#92418—42no—
87C2×#92418—81no—
88C23×C61—21no—
89C23×C61C4×D412yes—
90C23×C61#12311yescharacteristic
91Q8×C61—41no—
92Q8×C61—41no—
93D4×C61—41no—
94D4×C61—42no—
95D4×C61—41no—
96D4×C61—81no—
97D4×C61—82no—
98C2×C4×C61—22no—
99C2×C4×C61—21no—
100C2×C4×C61—22no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—44no—
108C2×C4×C61—81no—
109C2×C4×C61C2×C4⋊C411yescharacteristic
110C2×C4×C61#12011yescharacteristic
111C2×C4×C61C22⋊Q812yes—
112C2×Dic61—81no—
113#92418—42no—
114#92418—42no—
115#92418—42no—
116#12969—611nomaximal, Sylow
117C22×C61—21no—
118C22×C61—22no—
119C22×C61—41no—
120C22×C61—42no—
121C22×C61—42no—
122C22×C61—41no—
123C22×C61#38511yescharacteristic
124C4×C61—21no—
125C4×C61—22no—
126C4×C61—41no—
127C4×C61—41no—
128C4×C61—41no—
129C4×C61—84no—
130C4×C61B(4,𝔽2)11yescharacteristic
131C4×C61#45811yescharacteristic
132Dic61—161no—
133#2980—611no—
134#2126D6111yescharacteristic
135#1884—611no—
136#1868—612no—
137#1391—612no—
138C2×C61—21no—
139C2×C61—41no—
140C2×C61—42no—
141C2×C61—81no—
142C2×C61#186711yescharacteristic
143#568—611no—
144#532C2×D6111yescharacteristic
145#435—1222no—
146#435—1222no—
147#432—1221no—
148#424—1222no—
149#421—1221no—
150#410—611no—
151#410C2×D6112yes—
152#408—612no—
153#408—1221no—
154#354Dic6112yescomplemented
155#353Dic6112yescomplemented
156#344—1222no—
157#339—1221no—
158C61#1296911yescharacteristic, complemented, Sylow
159C2×#41D24411yescharacteristic
160C2×#41C22×D6111yescharacteristic
161#131—2442no—
162C2×M16—612no—
163C2×M16—1221no—
164C2×M16—1222no—
165C2×M16—1221no—
166C22×C8—612no—
167C22×C8—1221no—
168C4⋊Q8Dic12211yescharacteristic
169#127C2×Dic6111yescharacteristic
170#124Dic12211yescharacteristic
171#124C2×Dic6111yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—22no—
175#117—1221no—
176C2×C42D24411yescharacteristic
177C4⋊C8—1221no—
178C4⋊C8—1221no—
179C4⋊C8—2444no—
180#104—2442no—
181#99—22no—
182#99—1222no—
183#99C2×Dic6114yes—
184C22⋊C8—1222no—
185C22⋊C8—1222no—
186C22⋊C8—1222no—
187#97—1222no—
188#41—41no—
189#41—41no—
190#41—41no—
191#41—41no—
192C2×Q8C4⋊Dic6111yescharacteristic
193C2×Q8C2×Dic12211yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4C4⋊Dic6111yescharacteristic
197C2×D4#1175822412yes—
198C2×D4C22×Dic6111yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4C2×D24411yescharacteristic
202C22×C4D4×D6112yes—
203M16—1222no—
204M16—4881no—
205M16—2442no—
206M16—2441no—
207M16—2442no—
208C2×C8—1224no—
209C2×C8—1222no—
210C2×C8—1222no—
211C2×C8—4881no—
212C2×C8—2441no—
213C2×C8—2442no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—2441no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—24no—
220C22⋊C4—2441no—
221C42—21no—
222C42—42no—
223C42—2441no—
224C42C4⋊Dic6111yescharacteristic
225C42C4⋊Dic6111yescharacteristic
226C23—21no—
227C23D4×Dic6112yes—
228C23#1451976111yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—42no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—82no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—44no—
246C2×C4—81no—
247C2×C4—4881no—
248C2×C4C2×C4⋊Dic6111yescharacteristic
249C2×C4#1451974711yescharacteristic
250C2×C4#1451975412yes—
251C8—2442no—
252C8—2442no—
253C8—2442no—
254C22—22no—
255C22—21no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22#1683963711yescharacteristic
261C4—21no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—84no—
267C4—9761no—
268C4#1683995011yescharacteristic
269C4#1683995311yescharacteristic
270C2—21no—
271C2—42no—
272C2—41no—
273C2—81no—
274C2#2144610211yescharacteristic
subgroupG/H
centerC4#16839950
commutatorC2×D4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#2126C2
socleC2×C61#1867
prime radicalC61×#532C22
cosocleC2×#41×C61C23
hypercenter#2126D61
hypocenterC61#12969
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2126

Complements

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

NH ≅ G/Ntype
3. #21446107C2semidirect
4. #21443939C2semidirect
19. C61×#354C4semidirect
20. C61×#353C4semidirect
154. #354Dic61semidirect
155. #353Dic61semidirect
158. C61#12969semidirect

Maximal subgroups

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

subgroupindex
#214548902
#214462032
#214461072
#214439392
C61×#21262
#1296961