Finite Group Databasebeta

← #29581004

Subgroups — #29581004

Order 15616.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#21456343C211yesmaximal, characteristic, complemented
2#21455828C212yesmaximal
3#21446617C211yesmaximal, characteristic
4#21444223C212yesmaximal, complemented
5C61×#1275C211yesmaximal, characteristic, complemented
6#16840319C2212yes—
7C2×C4×D244C2211yescharacteristic
8#16840171—22no—
9#16840171—22no—
10#16840093—22no—
11#16840053—22no—
12#16839968—22no—
13#16839915—22no—
14C4×#11758190—24no—
15#16839800—22no—
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519689—22no—
18#16839494—24no—
19C61×#431C2212yes—
20C2×#97×C61C2211yescharacteristic
21C22×D244D411yescharacteristic
22C22×C4×D61—21no—
23C2×#11758190—24no—
24C22×#92418—21no—
25#14519777D412yes—
26#14519777D412yes—
27#14519769—22no—
28#14519756—22no—
29#14519756—22no—
30C4×Dic122—22no—
31C4×Dic122—22no—
32C4×D244—22no—
33C4×D244—22no—
34#14519736—22no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×#11758198—21no—
37#14519711—22no—
38#14519711—22no—
39#14519709—24no—
40#14519709—24no—
41C4⋊#92418—22no—
42#14519702—24no—
43#14519699—24no—
44#14519689—44no—
45C22×C8×C61D412yes—
46C2×C42×C61C2311yescharacteristic
47C4⋊C8×C61—22no—
48C22⋊C8×C61—22no—
49#97×C61—22no—
50C4×C8×C61D414yes—
51C23×D61—21no—
52C22×C4×C61C2×D411yescharacteristic
53C22×C4×C61C2×D411yescharacteristic
54C22×C4×C61#4111yescharacteristic
55C22×Dic61—21no—
56C2×D244—22no—
57C2×D244—42no—
58C2×C4×D61—41no—
59C2×C4×D61—41no—
60C2×C4×D61—42no—
61C2×C4×D61—42no—
62C2×Dic122—22no—
63C2×Dic122—22no—
64C2×C8×C61—22no—
65C2×C8×C61—22no—
66C2×C8×C61—24no—
67C2×C8×C61SD1614yescomplemented
68C42×C61C2×D412yes—
69C42×C61C2×D412yes—
70C22⋊Dic61—42no—
71#11758198—42no—
72C4⋊Dic61—22no—
73C4⋊Dic61—22no—
74C4⋊Dic61—22no—
75C4⋊Dic61—22no—
76C4⋊Dic61—22no—
77#11758196—42no—
78#11758196—42no—
79C4×Dic61—42no—
80C2×#92418—41no—
81C2×#92418—42no—
82#11758190—48no—
83C23×C61#12111yescharacteristic
84C22×D61—22no—
85C22×D61—41no—
86C22×D61—42no—
87C22×D61—41no—
88C22×D61—42no—
89C2×C4×C61—41no—
90C2×C4×C61—41no—
91C2×C4×C61#12012yes—
92C2×C4×C61#12011yescharacteristic
93C2×C4×C61#12112yes—
94C2×C4×C61#12711yescharacteristic
95C2×C4×C61C2×SD1612yes—
96C2×C4×C61#13712yes—
97C2×Dic61—42no—
98C2×Dic61—42no—
99C2×Dic61—41no—
100C2×Dic61—42no—
101D244—24no—
102D244—42no—
103C4×D61—82no—
104C4×D61—81no—
105C4×D61—81no—
106Dic122—44no—
107Dic122—44no—
108C8×C61—24no—
109C8×C61—44no—
110#92418—81no—
111#10036—611nomaximal, Sylow
112C22×C61—21no—
113C22×C61—21no—
114C22×C61#39211yescharacteristic
115C22×C61#46112yes—
116C2×D61—44no—
117C2×D61—42no—
118C2×D61—82no—
119C2×D61—81no—
120C2×D61—81no—
121C4×C61—22no—
122C4×C61—22no—
123C4×C61#44812yes—
124C4×C61#49211yescharacteristic
125C4×C61#49711yescharacteristic
126Dic61—82no—
127Dic61—81no—
128Dic61—82no—
129#3162—611no—
130#3101—612no—
131#1955—611no—
132#1481—612no—
133#1275D6111yescharacteristic, complemented
134C2×C61—22no—
135C2×C61#201011yescharacteristic
136C2×C61#201311yescharacteristic
137C2×C61#201711yescharacteristic
138D61—82no—
139#533—612no—
140C2×C4×D4—611no—
141#501—1222no—
142#501—1222no—
143#483—1222no—
144#476—1222no—
145#461—1222no—
146#450—1222no—
147C4×SD16—1224no—
148#431C2×D6112yes—
149#427—1222no—
150C2×C4⋊C8—611no—
151C2×C22⋊C8—1222no—
152C2×#97C2×D6111yescharacteristic
153Q8⋊C8—1224no—
154C61#1003611yescharacteristic, complemented, Sylow
155C22×D4—611no—
156C23×C4—1221no—
157C2×SD16—1224no—
158C22×C8—1221no—
159C22×C8D24412yes—
160C4⋊Q8—612no—
161C4⋊Q8—612no—
162#125—1222no—
163C22⋊Q8—1222no—
164C22⋊Q8—1222no—
165C4×Q8—1222no—
166C4×Q8—1222no—
167C4×D4—1222no—
168C4×D4—1222no—
169#117—1222no—
170C2×C4⋊C4—611no—
171C2×C22⋊C4—1221no—
172C2×C42C22×D6111yescharacteristic
173#107—1222no—
174#107—1222no—
175#106—1224no—
176#106—1224no—
177C4⋊C8—22no—
178C4⋊C8—1222no—
179#103—1224no—
180#102—1224no—
181C22⋊C8—22no—
182C22⋊C8—2444no—
183#97—22no—
184C4×C8D24414yes—
185C24—1221no—
186C2×Q8—1222no—
187C2×Q8—1222no—
188C2×D4—1222no—
189C2×D4—2442no—
190C22×C4—1221no—
191C22×C4—2442no—
192C22×C4—2441no—
193C22×C4—2441no—
194C22×C4—2442no—
195C22×C4C2×D24411yescharacteristic
196C22×C4D4×D6111yescharacteristic
197C22×C4#1175822611yescharacteristic
198SD16—2448no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—2441no—
203C2×C8—2442no—
204C2×C8#1175819014yescomplemented
205C4⋊C4—1222no—
206C4⋊C4—1222no—
207C4⋊C4—1222no—
208C4⋊C4—1222no—
209C4⋊C4—1222no—
210C4⋊C4—2442no—
211C4⋊C4—2442no—
212C22⋊C4—2442no—
213C22⋊C4—2442no—
214C42—2442no—
215C42C2×D24412yes—
216C42D4×D6112yes—
217C23—1222no—
218C23—2441no—
219C23—2442no—
220C23—2442no—
221C23—2441no—
222C23#1451975211yescharacteristic
223Q8—2444no—
224Q8—2444no—
225D4—1224no—
226D4—2442no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—4881no—
230C2×C4—4881no—
231C2×C4—4882no—
232C2×C4—2442no—
233C2×C4—2442no—
234C2×C4—2442no—
235C2×C4—2441no—
236C2×C4#1451974712yes—
237C2×C4#1451974711yescharacteristic
238C2×C4#1451975212yes—
239C2×C4#1451977411yescharacteristic
240C2×C4C2×#1175819012yes—
241C2×C4#1451981012yes—
242C8—24no—
243C8—44no—
244C8—4881no—
245C22—21no—
246C22—21no—
247C22—4882no—
248C22—4881no—
249C22—4881no—
250C22—2444no—
251C22—2442no—
252C22#1683967211yescharacteristic
253C22#1683996812yes—
254C4—22no—
255C4—22no—
256C4—4882no—
257C4—4882no—
258C4—4881no—
259C4#1683990312yes—
260C4#1684013511yescharacteristic
261C4#1684015611yescharacteristic
262C2—22no—
263C2—4882no—
264C2#2144688711yescharacteristic
265C2#2144690411yescharacteristic
266C2#2144692411yescharacteristic
subgroupG/H
centerC2×C4#14519774
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1275C2
socleC22×C61#392
prime radicalC22×D244D4
hypercenter#1275D61
hypocenterC61#10036
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
1. #21456343C2semidirect
4. #21444223C2semidirect
5. C61×#1275C2semidirect
67. C2×C8×C61SD16semidirect
133. #1275D61semidirect
154. C61#10036semidirect
204. C2×C8#11758190semidirect

Maximal subgroups

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

subgroupindex
#214563432
#214558282
#214466172
#214442232
C61×#12752
#1003661