Finite Group Databasebeta

← #29592473

Subgroups — #29592473

Order 15616.

398 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 67 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#21457129C211yesmaximal, characteristic
2#21447738C212yesmaximal, complemented
3#21447142C211yesmaximal, characteristic, complemented
4#21443829C212yesmaximal, complemented
5C61×#1322C211yesmaximal, characteristic, complemented
6C22×Dic244C2211yescharacteristic
7#16840312—21no—
8C2×#14519722—22no—
9#16840043—22no—
10#16839998—21no—
11#16839998C2212yes—
12#16839958—22no—
13C2×#14519702—21no—
14#16839668—21no—
15#16839567—22no—
16#16839525C2211yescharacteristic
17C2×C4⋊C8×C61C2211yescharacteristic
18C61×C22⋊C16C2212yes—
19C2×#11758222—41no—
20C22×Dic122—21no—
21C2×Dic244—22no—
22C2×Dic244—42no—
23C2×D488—21no—
24C2×D488D412yes—
25#14519777—21no—
26#14519774—21no—
27#14519765—41no—
28#14519751—22no—
29#14519751—41no—
30C4×D244—22no—
31C4×D244—41no—
32C2×C4⋊Dic61—21no—
33#14519722—44no—
34#14519711—21no—
35#14519711D412yes—
36#14519702—42no—
37#14519699—22no—
38#14519699—22no—
39#14519684—41no—
40#14519684—41no—
41C22×C8×C61D411yescharacteristic
42C22×C8×C61C2311yescharacteristic
43C2×C42×C61D411yescharacteristic
44C2×C16×C61—22no—
45C4⋊C8×C61—22no—
46C22×C4×C61D811yescharacteristic
47C22×C4×C61D811yescharacteristic
48C22×C4×C61C2×D411yescharacteristic
49C2×C22⋊D61—42no—
50C22×Dic61—41no—
51C22×Dic61—41no—
52#11758222—82no—
53#11758222—82no—
54C2×D244—22no—
55C2×D244—41no—
56C2×D244—41no—
57C2×C4×D61—42no—
58C2×Dic122—22no—
59C2×Dic122—41no—
60C2×Dic122—41no—
61C2×Dic122—41no—
62C2×Dic122—41no—
63C2×Dic122—41no—
64C2×Dic122—41no—
65C2×C8×C61—21no—
66C2×C8×C61—41no—
67C2×C8×C61—41no—
68C2×C8×C61C2×D412yes—
69C42×C61—21no—
70C42×C61—21no—
71#11758198—42no—
72#11758198—42no—
73C4⋊Dic61—22no—
74C4⋊Dic61—41no—
75C4⋊Dic61—41no—
76C4⋊Dic61—41no—
77Dic244—24no—
78Dic244—42no—
79D488—42no—
80D488—81no—
81C16×C61—42no—
82C23×C61C2×D811yescharacteristic
83C22×D61—42no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—41no—
90C2×C4×C61SD3212yescomplemented
91C2×C4×C61#12011yescharacteristic
92C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
93C22⋊D61—82no—
94C22⋊D61—82no—
95C2×Dic61—42no—
96C2×Dic61—42no—
97C2×Dic61—81no—
98C2×Dic61—81no—
99C2×Dic61—81no—
100C2×Dic61—81no—
101C2×Dic61—81no—
102D244—42no—
103D244—42no—
104D244—82no—
105D244—82no—
106C4×D61—84no—
107Dic122—42no—
108Dic122—42no—
109Dic122—81no—
110Dic122—82no—
111Dic122—81no—
112C8×C61—22no—
113C8×C61—41no—
114#12161—611nomaximal, Sylow
115C22×C61—21no—
116C22×C61—21no—
117C22×C61#44711yescharacteristic
118C22×C61C2×SD3211yescharacteristic
119C22×C61#50911yescharacteristic
120C2×D61—82no—
121C2×D61—82no—
122C2×D61—82no—
123C4×C61—21no—
124C4×C61—22no—
125C4×C61—41no—
126C4×C61#45411yescharacteristic
127C4×C61B(4,𝔽2)11yescharacteristic
128Dic61—82no—
129Dic61—82no—
130Dic61—161no—
131#3245—611no—
132#2185—612no—
133#2056—611no—
134#1347—612no—
135#1322D6111yescharacteristic, complemented
136C2×C61—21no—
137C2×C61—21no—
138C2×C61#159811yescharacteristic
139C2×C61#218711yescharacteristic
140C2×C61#219111yescharacteristic
141D61—162no—
142C22×Q16—611no—
143#532—1221no—
144C2×SD32—1222no—
145#474—1222no—
146#466—612no—
147#466—1221no—
148#459—1222no—
149C2×C4⋊C8C2×D6111yescharacteristic
150C2×#103—1221no—
151#391—1221no—
152#358—1222no—
153C22⋊C16C2×D6112yes—
154#340—611no—
155C61#1216111yescharacteristic, complemented, Sylow
156C2×#41—2441no—
157C22×Q8—1221no—
158C2×Q16—1222no—
159C2×Q16—2442no—
160C2×D8—612no—
161C2×D8—1221no—
162C22×C8D24411yescharacteristic
163C22×C8C22×D6111yescharacteristic
164C4⋊Q8—1221no—
165#127—1221no—
166#124—2441no—
167#121—1222no—
168#121—2441no—
169C4×D4—1222no—
170C4×D4—2441no—
171C2×C4⋊C4—1221no—
172C2×C42D24411yescharacteristic
173SD32—2444no—
174C2×C16—22no—
175#107—612no—
176#107—1221no—
177C4⋊C8—22no—
178#103—2442no—
179#102—1222no—
180#102—1222no—
181#95—2441no—
182#95—2441no—
183#41—4882no—
184#41—4882no—
185C2×Q8—1222no—
186C2×Q8—2441no—
187C2×Q8—2441no—
188C2×Q8—2441no—
189C2×Q8—2441no—
190C2×Q8—2441no—
191C2×Q8—2441no—
192C2×D4—1222no—
193C2×D4—2441no—
194C2×D4—2441no—
195C2×D4—2442no—
196C22×C4—2442no—
197C22×C4—2441no—
198C22×C4—2441no—
199C22×C4D48811yescharacteristic
200C22×C4D48811yescharacteristic
201C22×C4C2×D24411yescharacteristic
202Q16—1224no—
203Q16—2442no—
204D8—4881no—
205D8—2442no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D6112yes—
210C4⋊C4—1222no—
211C4⋊C4—2441no—
212C4⋊C4—2441no—
213C4⋊C4—2441no—
214C22⋊C4—2442no—
215C22⋊C4—2442no—
216C42—21no—
217C42—21no—
218C16—42no—
219C23—2442no—
220C23C2×D48811yescharacteristic
221Q8—4881no—
222Q8—4882no—
223Q8—4881no—
224Q8—2442no—
225Q8—2442no—
226D4—4882no—
227D4—4882no—
228D4—4882no—
229D4—4882no—
230D4—2442no—
231D4—2442no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—4884no—
239C2×C4—4881no—
240C2×C4—4881no—
241C2×C4—4881no—
242C2×C4—4881no—
243C2×C4—4881no—
244C2×C4—2442no—
245C2×C4—2442no—
246C2×C4#1451972212yescomplemented
247C2×C4#1451974711yescharacteristic
248C2×C4#1451980511yescharacteristic
249C8—22no—
250C8—41no—
251C22—21no—
252C22—21no—
253C22—4882no—
254C22—4882no—
255C22—4882no—
256C22#1683989811yescharacteristic
257C22C2×#1451972211yescharacteristic
258C22#1684020011yescharacteristic
259C4—22no—
260C4—21no—
261C4—41no—
262C4—9761no—
263C4—4882no—
264C4—4882no—
265C4#1683993711yescharacteristic
266C4#1683995011yescharacteristic
267C2—21no—
268C2—21no—
269C2—9762no—
270C2#2144475611yescharacteristic
271C2#2144775011yescharacteristic
272C2#2144777211yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1322C2
socleC22×C61#447
prime radical#21457129C2
hypercenter#1322D61
hypocenterC61#12161
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #21447738C2semidirect
3. #21447142C2semidirect
4. #21443829C2semidirect
5. C61×#1322C2semidirect
90. C2×C4×C61SD32semidirect
135. #1322D61semidirect
155. C61#12161semidirect
246. C2×C4#14519722semidirect

Maximal subgroups

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

subgroupindex
#214571292
#214477382
#214471422
#214438292
C61×#13222
#1216161