Finite Group Databasebeta

← #29583037

Subgroups — #29583037

Order 15616.

500 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 125 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#21446746C212yesmaximal
2#21446624C211yesmaximal, characteristic
3#21443702C211yesmaximal, characteristic, complemented
4#21443660C212yesmaximal, complemented
5C61×#2571C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7C2×C4⋊#92418C2212yes—
8C2×#14519697—22no—
9C2×#14519697—22no—
10C2×#14519697—22no—
11C2×C4×#92418C2211yescharacteristic
12C61×#400—22no—
13C61×#389—22no—
14C61×#384C2211yescharacteristic
15C61×#382—22no—
16C4×C4⋊C4×C61C2212yes—
17C22×D244—21no—
18C22×D244—22no—
19C22×#92418—21no—
20C22×#92418—22no—
21#14519774—22no—
22#14519774—21no—
23#14519774Q812yes—
24C4⋊#92418—22no—
25C4⋊#92418—22no—
26#14519697—44no—
27#14519697—44no—
28#14519697—44no—
29C4×#92418—22no—
30C2×C4⋊C4×C61—22no—
31C2×C4⋊C4×C61—22no—
32C2×C4⋊C4×C61D412yes—
33C2×C4⋊C4×C61D412yes—
34C2×C4⋊C4×C61D412yes—
35C2×C42×C61—22no—
36C2×C42×C61C2311yescharacteristic
37#95×C61—22no—
38#95×C61—24no—
39#95×C61—22no—
40C23×D61—41no—
41C22×C4×C61—22no—
42C22×C4×C61—22no—
43C22×C4×C61—22no—
44C22×C4×C61C2×D411yescharacteristic
45C22×C4×C61C2×D412yes—
46C2×D244—22no—
47C2×D244—24no—
48C2×D244—42no—
49C2×D244—42no—
50C2×D244—42no—
51C2×D244—42no—
52C2×D244—42no—
53C2×D244—42no—
54C2×D244—41no—
55C2×D244—42no—
56C4⋊C4×C61—22no—
57C4⋊C4×C61—22no—
58C4⋊C4×C61—22no—
59C4⋊C4×C61—22no—
60C4⋊C4×C61—42no—
61C4⋊C4×C61—42no—
62C4⋊C4×C61—42no—
63C4⋊C4×C61—82no—
64C42×C61—42no—
65C42×C61—42no—
66C42×C61C2×Q811yescharacteristic
67C42×C61#4112yes—
68C42×C61#4111yescharacteristic
69C2×#92418—22no—
70C2×#92418—41no—
71C2×#92418—41no—
72C2×#92418—42no—
73C2×#92418—42no—
74C2×#92418—42no—
75C23×C61#12011yescharacteristic
76C22×D61—42no—
77C22×D61—82no—
78C22×D61—82no—
79C22×D61—81no—
80C22×D61—81no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—44no—
88C2×C4×C61—42no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61C22⋊Q811yescharacteristic
92C2×C4×C61C22⋊Q812yes—
93C2×C4×C61#12411yescharacteristic
94C2×C4×C61#12412yes—
95C2×C4×C61#13512yes—
96C2×C4×C61#13512yes—
97C2×C4×C61AGL(1,ℤ/8)12yes—
98C2×C4×C61AGL(1,ℤ/8)12yes—
99D244—44no—
100D244—44no—
101D244—44no—
102D244—82no—
103D244—82no—
104D244—82no—
105#92418—42no—
106#92418—82no—
107#10351—611nomaximal, Sylow
108C22×C61#39311yescharacteristic
109C22×C61#44912yes—
110C22×C61#44912yes—
111C22×C61#45311yescharacteristic
112C22×C61#45411yescharacteristic
113C2×D61—84no—
114C2×D61—82no—
115C2×D61—161no—
116C2×D61—161no—
117C2×D61—162no—
118C4×C61—21no—
119C4×C61—21no—
120C4×C61—82no—
121C4×C61—82no—
122C4×C61—82no—
123C4×C61#47812yes—
124C4×C61#47812yes—
125C4×C61#48812yes—
126C4×C61#49012yes—
127#2571D6111yescharacteristic, complemented
128#1982—612no—
129#1957—611no—
130#1296—611no—
131#1279—612no—
132C2×C61#162011yescharacteristic
133C2×C61#162411yescharacteristic
134C2×C61#162812yes—
135C2×C61#202512yes—
136C2×C61#202711yescharacteristic
137D61—162no—
138C2×#127—611no—
139C2×C4⋊C8—612no—
140C2×#102—1222no—
141C2×#102—1222no—
142C2×#102—1222no—
143C2×C4×C8—611no—
144#400—22no—
145#389—22no—
146#384C2×D6111yescharacteristic
147#382—22no—
148C4×C4⋊C4C2×D6112yes—
149C61#1035111yescharacteristic, complemented, Sylow
150C22×D4—1221no—
151C22×D4—1222no—
152C22×C8—1221no—
153C22×C8—1222no—
154#127—612no—
155#127—1222no—
156#127—1221no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C22⋊D6112yes—
160C2×C4⋊C4C22⋊D6112yes—
161C2×C4⋊C4C22⋊D6112yes—
162C2×C42—22no—
163C2×C42C22×D6111yescharacteristic
164C4⋊C8—1222no—
165C4⋊C8—1222no—
166#102—2444no—
167#102—2444no—
168#102—2444no—
169C4×C8—1222no—
170#95—22no—
171#95—24no—
172#95—22no—
173C24—2441no—
174C2×D4—1222no—
175C2×D4—1224no—
176C2×D4—2442no—
177C2×D4—2442no—
178C2×D4—2441no—
179C2×D4—2442no—
180C2×D4—2442no—
181C2×D4—2442no—
182C2×D4—2442no—
183C2×D4—2442no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C2×C22⋊D6111yescharacteristic
188C22×C4C2×C22⋊D6112yes—
189C2×C8—1222no—
190C2×C8—2442no—
191C2×C8—2442no—
192C2×C8—2442no—
193C2×C8—2441no—
194C2×C8—2441no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C42—42no—
204C42—42no—
205C42Q8×D6111yescharacteristic
206C42#1175822612yes—
207C42#1175822611yescharacteristic
208C23—4882no—
209C23—4882no—
210C23—4881no—
211C23—4881no—
212C23—2442no—
213C23#1451974611yescharacteristic
214D4—4882no—
215D4—4882no—
216D4—4882no—
217D4—2444no—
218D4—2444no—
219D4—2444no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4#1451975311yescharacteristic
231C2×C4#1451975312yes—
232C2×C4#1451976611yescharacteristic
233C2×C4#1451976612yes—
234C2×C4#1451979712yes—
235C2×C4#1451979712yes—
236C2×C4#1451980212yes—
237C2×C4#1451980212yes—
238C8—4882no—
239C8—2442no—
240C22—9761no—
241C22—9761no—
242C22—9762no—
243C22—4884no—
244C22—4882no—
245C22#1683967311yescharacteristic
246C22#1683990712yes—
247C22#1683990712yes—
248C22#1683993011yescharacteristic
249C22#1683993511yescharacteristic
250C4—21no—
251C4—21no—
252C4—82no—
253C4—82no—
254C4—82no—
255C4#1684005912yes—
256C4#1684005912yes—
257C4#1684011912yes—
258C4#1684012812yes—
259C2—9762no—
260C2#2144487811yescharacteristic
261C2#2144490211yescharacteristic
262C2#2144492912yes—
263C2#2144696212yes—
264C2#2144697211yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2571C2
socleC23×C61#120
prime radicalC2×#14519774C22
hypercenter#2571D61
hypocenterC61#10351
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

Complements

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

NH ≅ G/Ntype
3. #21443702C2semidirect
4. #21443660C2semidirect
5. C61×#2571C2semidirect
127. #2571D61semidirect
149. C61#10351semidirect

Maximal subgroups

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

subgroupindex
#214467462
#214466242
#214437022
#214436602
C61×#25712
#1035161