Finite Group Databasebeta

← #30605368

Subgroups — #30605368

Order 15616.

396 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 72 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#21462056C211yesmaximal, characteristic, complemented
2#21461701C211yesmaximal, characteristic, complemented
3#21460790C214yesmaximal, complemented
4#21455468C211yesmaximal, characteristic
5D61⋊M32—21no—
6#16840772—21no—
7#16840741—21no—
8#16840659C2212yescomplemented
9#16840658—21no—
10#16840630C2212yescomplemented
11C22×D488C411yescharacteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840331—21no—
14#16840016C412yescomplemented
15#16840016C2212yes—
16#16839999—21no—
17#16839999—21no—
18#16839766—21no—
19C61×#427C411yescharacteristic, complemented
20#14519856—22no—
21#14519852—41no—
22#14519851—22no—
23D61⋊C16—41no—
24#14519846—41no—
25#14519845—41no—
26#14519836—41no—
27#14519835—41no—
28C2×#11758222D411yescharacteristic
29C22×D244—21no—
30C2×D488—21no—
31C2×D488—41no—
32C2×D488—41no—
33C2×D488C2×C412yes—
34#14519783—41no—
35#14519783—41no—
36C2×#11758194D411yescharacteristic
37C2×#11758189C2×C411yescharacteristic
38C2×#11758189C2311yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61D411yescharacteristic
41#14519774—41no—
42#14519765—41no—
43#14519752—22no—
44#14519752—22no—
45#14519747—41no—
46#14519747—41no—
47#14519697—22no—
48#14519697—22no—
49C22×C8×C61C2×C411yescharacteristic
50#117×C61—21no—
51#107×C61—21no—
52#106×C61C2×C412yes—
53C23×D61—41no—
54C22×C4×C61C22⋊C411yescharacteristic
55C2×C22⋊D61D811yescharacteristic
56C2×C22⋊D61SD1611yescharacteristic
57#11758222—41no—
58#11758222—41no—
59C2×D244—22no—
60C2×D244—42no—
61C2×D244—42no—
62C2×D244—41no—
63C2×D244—42no—
64C2×D244—42no—
65C2×D244D811yescharacteristic
66C2×C4×D61C22⋊C411yescharacteristic
67C2×C4×D61C2×D411yescharacteristic
68C2×Dic122SD1611yescharacteristic
69C22⋊#6923—42no—
70C4⋊#6923—42no—
71C4×#6923—42no—
72C2×C8×C61—21no—
73C2×C8×C61—21no—
74C2×C8×C61C22⋊C411yescharacteristic
75C2×C8×C61C22×C411yescharacteristic
76C4⋊C4×C61—22no—
77C22⋊C4×C61—41no—
78C42×C61—41no—
79#11758198—42no—
80#11758198—42no—
81#11758194—41no—
82C2×#92418C22⋊C411yescharacteristic
83C2×#92418C2×D411yescharacteristic
84D488—42no—
85D488—41no—
86D488—41no—
87#11758189—41no—
88#11758189—41no—
89C8×D61—21no—
90C8×D61—21no—
91C8×D61—21no—
92C8×D61—21no—
93#11758187—42no—
94C23×C61#10211yescharacteristic
95C22×D61—42no—
96C22×D61—82no—
97C22×D61—162no—
98C22×D61#10211yescharacteristic
99C22×D61C2×D811yescharacteristic
100C2×#6923—82no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—42no—
104C2×C4×C61#10211yescharacteristic
105C2×C4×C61C2×C22⋊C411yescharacteristic
106C22⋊D61—41no—
107C22⋊D61—41no—
108C2×Dic61#10211yescharacteristic
109C2×Dic61C2×SD1611yescharacteristic
110D244—41no—
111D244—42no—
112D244—42no—
113D244—82no—
114D244—162no—
115D244—162no—
116C4×D61—21no—
117C4×D61—21no—
118C4×D61—21no—
119C4×D61—21no—
120Dic122—41no—
121C8×C61—21no—
122C8×C61#9911yescharacteristic
123C8×C61#9911yescharacteristic
124#92418—21no—
125#92418—21no—
126#11868—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—41no—
129C22×C61C2×#10211yescharacteristic
130C2×D61—21no—
131C2×D61—21no—
132C2×D61—41no—
133C2×D61—41no—
134C2×D61—82no—
135C2×D61—82no—
136C2×D61—82no—
137C2×D61—164no—
138#6923—162no—
139C4×C61—21no—
140C4×C61—82no—
141C4×C61C2×#9911yescharacteristic
142C4×C61#42111yescharacteristic
143Dic61—21no—
144Dic61—21no—
145#3058—611no—
146#2115—611no—
147#1812—611no—
148#1339—614no—
149C2×C61—21no—
150C2×C61—41no—
151C2×C61#151911yescharacteristic
152D61—41no—
153D61—41no—
154D61—162no—
155C22×D8—611no—
156C2×#131—611no—
157#535—1221no—
158C2×M32—1221no—
159#469—612no—
160#469—612no—
161#466—1221no—
162#466—1221no—
163#427—1221no—
164#427#692311yescharacteristic, complemented
165#418—1221no—
166#410—1221no—
167#349—612no—
168#349—1221no—
169H(3,ℤ/4)—612no—
170C61#1186811yescharacteristic, complemented, Sylow
171C2×#41—611no—
172C22×D4—1221no—
173C2×D8—612no—
174C2×D8—1221no—
175C2×D8—2441no—
176C2×D8—2441no—
177#131—2441no—
178#131—2441no—
179C2×M16—611no—
180C2×M16—611no—
181C2×M16—611no—
182C22×C8—611no—
183C22×C8—611no—
184C22×C8C2×#692311yescharacteristic
185#127—2441no—
186#124—2441no—
187#121—1222no—
188#121—1222no—
189#120—2441no—
190#120—2441no—
191#117—21no—
192#117—1222no—
193M32—1222no—
194M32—2441no—
195C2×C16—2441no—
196#107—21no—
197#107—2441no—
198#106—2441no—
199#106C2×#692312yes—
200#102—1222no—
201#102—1222no—
202#99—2441no—
203#99—2441no—
204C24—2441no—
205#41—2441no—
206#41—2441no—
207C2×Q8—611no—
208C2×D4—611no—
209C2×D4—611no—
210C2×D4—611no—
211C2×D4—1222no—
212C2×D4—2442no—
213C2×D4—2442no—
214C2×D4—2442no—
215C2×D4—2442no—
216C2×D4—2441no—
217C22×C4—611no—
218C22×C4—611no—
219C22×C4C22⋊#692311yescharacteristic
220D8—2442no—
221D8—2441no—
222D8—2441no—
223M16—2441no—
224M16—2441no—
225M16—2441no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—611no—
229C2×C8—611no—
230C2×C8—1221no—
231C2×C8—1221no—
232C2×C8—1221no—
233C2×C8—1221no—
234C2×C8C22⋊#692311yescharacteristic
235C2×C8C22×#692311yescharacteristic
236C4⋊C4—22no—
237C4⋊C4—2442no—
238C22⋊C4—41no—
239C22⋊C4—2442no—
240C22⋊C4—2442no—
241C22⋊C4—2442no—
242C42—41no—
243C42—2442no—
244C16—2442no—
245C23—611no—
246C23—611no—
247C23—9762no—
248C23—4882no—
249C23—2442no—
250C23#1451983911yescharacteristic
251Q8—2441no—
252D4—9762no—
253D4—9762no—
254D4—4882no—
255D4—2441no—
256D4—2441no—
257D4—2441no—
258D4—2442no—
259D4—2442no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—611no—
264C2×C4—611no—
265C2×C4—1221no—
266C2×C4—1221no—
267C2×C4—1221no—
268C2×C4—1221no—
269C2×C4—4882no—
270C2×C4#1451983911yescharacteristic
271C2×C4C2×C22⋊#692311yescharacteristic
272C8—21no—
273C8—1221no—
274C8—1221no—
275C8#1451983611yescharacteristic
276C8#1451983611yescharacteristic
277C22—21no—
278C22—41no—
279C22—1221no—
280C22—1221no—
281C22—9764no—
282C22—4882no—
283C22—4882no—
284C22—4882no—
285C22—2441no—
286C22—2441no—
287C22C2×#1451983911yescharacteristic
288C4—21no—
289C4—82no—
290C4—1221no—
291C4—1221no—
292C4—9762no—
293C4C2×#1451983611yescharacteristic
294C4#1684076111yescharacteristic
295C2—21no—
296C2—41no—
297C2—9762no—
298C2—2441no—
299C2—2441no—
300C2#2146125611yescharacteristic
subgroupG/H
centerC2#21461256
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#427C4
socleC2×C61#1519
prime radical#21455468C2
cosocleC2×#11758189C23
hypercenter#427#6923
hypocenterC61#11868
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #21462056C2semidirect
2. #21461701C2semidirect
3. #21460790C2semidirect
8. #16840659C22semidirect
10. #16840630C22semidirect
11. C22×D488C4semidirect
14. #16840016C4semidirect
19. C61×#427C4semidirect
164. #427#6923semidirect
170. C61#11868semidirect

Maximal subgroups

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

subgroupindex
#214620562
#214617012
#214607902
#214554682
#1186861