Finite Group Databasebeta

← #29573238

Subgroups — #29573238

Order 15616.

418 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 115 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#21454532C211yesmaximal, characteristic
2#21445950C211yesmaximal, characteristic, complemented
3#21444380C212yesmaximal
4#21443678C212yesmaximal, complemented
5C61×#1564C211yesmaximal, characteristic, complemented
6C2×#14519783—21no—
7#16840312C2211yescharacteristic
8#16839841—21no—
9#16839841C412yescomplemented
10C8×D244—21no—
11C8×D244C412yescomplemented
12#16839786C2211yescharacteristic
13#16839728—21no—
14#16839628C2212yes—
15#16839525—21no—
16#16839525—21no—
17#16839497—24no—
18C61×#431C2212yes—
19C2×C4×C8×C61C2211yescharacteristic
20C61×#335—21no—
21C61×#334—21no—
22C2×#11758222—21no—
23#14519783—44no—
24C2×#11758194—21no—
25C2×#11758189—22no—
26C2×C8×D61—22no—
27#14519777Q811yescharacteristic
28#14519774Q811yescharacteristic
29#14519765—21no—
30#14519751—21no—
31#14519751D412yes—
32C4×D244—21no—
33C4×D244C2×C412yes—
34C2×C4⋊Dic61—22no—
35C4⋊#92418—21no—
36C4⋊#92418C2×C412yes—
37#14519689—22no—
38#14519689—22no—
39#14519684—22no—
40#14519684—22no—
41C22×C8×C61D411yescharacteristic
42C22×C8×C61D411yescharacteristic
43C2×C42×C61C2311yescharacteristic
44C4⋊C8×C61—22no—
45C22⋊C8×C61—22no—
46C4×C8×C61—22no—
47C4×C8×C61C2×C411yescharacteristic
48C4×C8×C61C2×C411yescharacteristic
49C4×C8×C61D411yescharacteristic
50C4×C8×C61D411yescharacteristic
51C22×C4×C61C2×D411yescharacteristic
52C22×C4×C61C2×D411yescharacteristic
53C22×C4×C61#4111yescharacteristic
54C2×C22⋊D61—22no—
55C22×Dic61—42no—
56#11758222—42no—
57#11758222—42no—
58C2×D244—21no—
59C2×D244—21no—
60C2×D244C4⋊C412yes—
61C2×C4×D61—22no—
62C2×Dic122—21no—
63C2×C8×C61—22no—
64C2×C8×C61—22no—
65C2×C8×C61—42no—
66C2×C8×C61C22⋊C411yescharacteristic
67C2×C8×C61C22⋊C411yescharacteristic
68C2×C8×C61C22⋊C411yescharacteristic
69C2×C8×C61C22⋊C411yescharacteristic
70C42×C61C22×C411yescharacteristic
71C42×C61C2×D411yescharacteristic
72C42×C61C2×Q811yescharacteristic
73C42×C61#4111yescharacteristic
74#11758198—22no—
75#11758198—22no—
76C4⋊Dic61—21no—
77C4⋊Dic61—42no—
78C4⋊Dic61—42no—
79C4⋊Dic61C4⋊C412yes—
80#11758194—42no—
81C2×#92418—22no—
82#11758189—44no—
83C8×D61—44no—
84C23×C61#12111yescharacteristic
85C22×D61—22no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61C2×C22⋊C411yescharacteristic
89C2×C4×C61C2×C4⋊C411yescharacteristic
90C2×C4×C61#11711yescharacteristic
91C2×C4×C61#12111yescharacteristic
92C2×C4×C61C22⋊Q811yescharacteristic
93C2×C4×C61C22⋊Q811yescharacteristic
94C2×C4×C61#13512yes—
95C2×C4×C61AGL(1,ℤ/8)12yes—
96C22⋊D61—44no—
97C2×Dic61—22no—
98C2×Dic61—82no—
99C2×Dic61—82no—
100C2×Dic61—82no—
101D244—42no—
102D244—42no—
103D244—42no—
104C4×D61—42no—
105C4×D61—42no—
106Dic122—42no—
107C8×C61—21no—
108C8×C61—21no—
109C8×C61—21no—
110C8×C61—21no—
111C8×C61—82no—
112#92418—42no—
113#8678—611nomaximal, Sylow
114C22×C61—21no—
115C22×C61—21no—
116C22×C61#38011yescharacteristic
117C22×C61#46312yes—
118C2×D61—42no—
119C2×D61—42no—
120C2×D61—42no—
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—21no—
124C4×C61—21no—
125C4×C61#41611yescharacteristic
126C4×C61#41811yescharacteristic
127C4×C61#47812yes—
128Dic61—42no—
129Dic61—162no—
130#2931—611no—
131#1835—611no—
132#1564D6111yescharacteristic, complemented
133#1517—612no—
134#1286—612no—
135C2×C61—22no—
136C2×C61#180411yescharacteristic
137C2×C61#181211yescharacteristic
138C2×C61#181411yescharacteristic
139D61—82no—
140C2×#131—1221no—
141#532—611no—
142#436—612no—
143#436—1221no—
144C8×D4—612no—
145C8×D4—1221no—
146#431C2×D6112yes—
147#423—611no—
148#408—1221no—
149C2×C4×C8C2×D6111yescharacteristic
150#382—612no—
151#340—1221no—
152#340—1221no—
153#335—21no—
154#334—21no—
155#327—1224no—
156C61#867811yescharacteristic, complemented, Sylow
157C2×#41—1221no—
158#131—2444no—
159C2×M16—1221no—
160C2×M16—1222no—
161C22×C8—1222no—
162C22×C8D24411yescharacteristic
163C22×C8D24411yescharacteristic
164C4⋊Q8—611no—
165#127—611no—
166#124—1221no—
167#121—612no—
168#121—1221no—
169C4×D4—612no—
170C4×D4—1221no—
171C2×C4⋊C4—1222no—
172C2×C42C22×D6111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—612no—
175C4⋊C8—1221no—
176C22⋊C8—22no—
177C22⋊C8—1222no—
178C22⋊C8—1222no—
179C4×C8—22no—
180C4×C8C4×D6111yescharacteristic
181C4×C8C4×D6111yescharacteristic
182C4×C8C22⋊D6111yescharacteristic
183C4×C8C22⋊D6111yescharacteristic
184#95—1222no—
185#95—1222no—
186#41—2442no—
187#41—2442no—
188C2×Q8—1221no—
189C2×D4—612no—
190C2×D4—1222no—
191C2×D4—1221no—
192C2×D4—1221no—
193C22×C4—1222no—
194C22×C4—2442no—
195C22×C4C2×D24411yescharacteristic
196C22×C4D4×D6111yescharacteristic
197C22×C4#1175822611yescharacteristic
198M16—2444no—
199M16—2442no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—42no—
203C2×C8—1222no—
204C2×C8—2444no—
205C2×C8#1175819811yescharacteristic
206C2×C8#1175819811yescharacteristic
207C2×C8#1175819811yescharacteristic
208C2×C8#1175819811yescharacteristic
209C4⋊C4—612no—
210C4⋊C4—1221no—
211C4⋊C4—2442no—
212C4⋊C4—2442no—
213C22⋊C4—1222no—
214C22⋊C4—1222no—
215C42C2×C4×D6111yescharacteristic
216C42#1175822411yescharacteristic
217C42Q8×D6111yescharacteristic
218C42C2×C22⋊D6111yescharacteristic
219C23—1222no—
220C23#1451975211yescharacteristic
221Q8—2442no—
222D4—2442no—
223D4—2442no—
224D4—2444no—
225D4—2442no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—1222no—
229C2×C4—4882no—
230C2×C4—4882no—
231C2×C4—4882no—
232C2×C4—2442no—
233C2×C4—2442no—
234C2×C4C2×#1175819811yescharacteristic
235C2×C4C4⋊C4×D6111yescharacteristic
236C2×C4#1451973511yescharacteristic
237C2×C4#1451974811yescharacteristic
238C2×C4#1451975311yescharacteristic
239C2×C4#1451975711yescharacteristic
240C2×C4#1451979912yes—
241C2×C4#1451980512yes—
242C8—21no—
243C8—21no—
244C8—21no—
245C8—21no—
246C8—82no—
247C8—2442no—
248C22—21no—
249C22—21no—
250C22—2442no—
251C22—2442no—
252C22—2442no—
253C22#1683961911yescharacteristic
254C22#1683997812yes—
255C4—21no—
256C4—21no—
257C4—21no—
258C4—21no—
259C4—9762no—
260C4—2442no—
261C4#1683975911yescharacteristic
262C4#1683976811yescharacteristic
263C4#1684006312yes—
264C2—22no—
265C2—4882no—
266C2#2144583011yescharacteristic
267C2#2144586211yescharacteristic
268C2#2144587411yescharacteristic
subgroupG/H
centerC22#16839619
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1564C2
socleC22×C61#380
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1564D61
hypocenterC61#8678
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
2. #21445950C2semidirect
4. #21443678C2semidirect
5. C61×#1564C2semidirect
9. #16839841C4semidirect
11. C8×D244C4semidirect
132. #1564D61semidirect
156. C61#8678semidirect

Maximal subgroups

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

subgroupindex
#214545322
#214459502
#214443802
#214436782
C61×#15642
#867861