Finite Group Databasebeta

← #29581001

Subgroups — #29581001

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#21455823C212yesmaximal
2#21446617C211yesmaximal, characteristic
3#21444222C212yesmaximal, complemented
4#21443650C211yesmaximal, characteristic, complemented
5C61×#3162C211yesmaximal, characteristic, complemented
6C2×C4×D244C2211yescharacteristic
7#16840090—22no—
8#16840049—22no—
9#16839964—22no—
10#16839912—22no—
11C4×#11758201—24no—
12#16839815C2212yes—
13C2×C4⋊#92418C2211yescharacteristic
14C2×#14519689—22no—
15Q8⋊#92418—24no—
16C61×#533C2212yes—
17C61×#501—22no—
18C61×#501—22no—
19C61×#427—22no—
20C2×#97×C61C2211yescharacteristic
21C22×D244D411yescharacteristic
22C22×C4×D61—21no—
23C2×#11758201—24no—
24C22×#92418—21no—
25C4×D244—22no—
26C4×D244—22no—
27C2×C4⋊Dic61D411yescharacteristic
28C2×#11758198—21no—
29#14519708—24no—
30C4⋊#92418—22no—
31C4⋊#92418—22no—
32#14519701—24no—
33#14519697—24no—
34C22⋊#92418—22no—
35#14519689—44no—
36C4×#92418D414yes—
37C22×C8×C61D412yes—
38C4⋊Q8×C61D412yes—
39C4⋊Q8×C61D412yes—
40#125×C61—22no—
41C22⋊Q8×C61—22no—
42C22⋊Q8×C61—22no—
43C4×Q8×C61—22no—
44C4×Q8×C61—22no—
45#117×C61—22no—
46C2×C42×C61C2311yescharacteristic
47#107×C61—22no—
48#107×C61—22no—
49#106×C61—24no—
50#97×C61—22no—
51C23×D61—21no—
52C2×Q8×C61—22no—
53C2×Q8×C61—22no—
54C22×C4×C61C2×D411yescharacteristic
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61#4111yescharacteristic
57C22×Dic61—21no—
58C2×D244—22no—
59C2×D244—42no—
60C2×C4×D61—41no—
61C2×C4×D61—41no—
62C2×C4×D61—42no—
63C2×C4×D61—42no—
64C2×C8×C61—22no—
65C2×C8×C61—22no—
66C2×C8×C61SD1614yescomplemented
67C4⋊C4×C61—22no—
68C4⋊C4×C61—22no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61—22no—
71C4⋊C4×C61—42no—
72C4⋊C4×C61—42no—
73C22⋊C4×C61—42no—
74C42×C61—42no—
75C42×C61C2×D412yes—
76C42×C61C2×D412yes—
77#11758201—48no—
78#11758198—42no—
79C4⋊Dic61—22no—
80C2×#92418—24no—
81C2×#92418—41no—
82C2×#92418—42no—
83C23×C61#12111yescharacteristic
84C22×D61—22no—
85C22×D61—41no—
86C22×D61—42no—
87C22×D61—41no—
88C22×D61—42no—
89Q8×C61—44no—
90Q8×C61—44no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61—42no—
94C2×C4×C61—42no—
95C2×C4×C61#12012yes—
96C2×C4×C61#12011yescharacteristic
97C2×C4×C61#12112yes—
98C2×C4×C61#12711yescharacteristic
99C2×C4×C61C2×SD1612yes—
100C2×C4×C61#13712yes—
101C2×Dic61—41no—
102C2×Dic61—42no—
103D244—24no—
104D244—42no—
105C4×D61—82no—
106C4×D61—81no—
107C4×D61—81no—
108C8×C61—24no—
109#92418—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—82no—
124C4×C61—82no—
125C4×C61#44812yes—
126C4×C61#49211yescharacteristic
127C4×C61#49711yescharacteristic
128Dic61—81no—
129#3162D6111yescharacteristic, complemented
130#3101—612no—
131#1955—611no—
132#1481—612no—
133#1275—611no—
134C2×C61—22no—
135C2×C61#201011yescharacteristic
136C2×C61#201311yescharacteristic
137C2×C61#201711yescharacteristic
138D61—82no—
139#533C2×D6112yes—
140C2×C4×D4—611no—
141#501—22no—
142#501—22no—
143#483—1222no—
144#476—1222no—
145#461—1222no—
146#450—1222no—
147C4×SD16—1224no—
148#431—612no—
149#427—22no—
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×C8C22⋊D6112yes—
160C4⋊Q8C22⋊D6112yes—
161C4⋊Q8C22⋊D6112yes—
162#125—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×D4—1222no—
168C4×D4—1222no—
169#117—22no—
170C2×C4⋊C4—611no—
171C2×C22⋊C4—1221no—
172C2×C42C22×D6111yescharacteristic
173#107—22no—
174#107—22no—
175#106—24no—
176#106—1224no—
177C4⋊C8—1222no—
178C4⋊C8—1222no—
179#103—1224no—
180#102—1224no—
181C22⋊C8—1222no—
182C22⋊C8—2444no—
183#97—22no—
184C4×C8—614no—
185C24—1221no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×D4—1222no—
189C2×D4—2442no—
190C22×C4—1221no—
191C22×C4—2442no—
192C22×C4—2441no—
193C22×C4—2441no—
194C22×C4—2442no—
195C22×C4D4×D6111yescharacteristic
196C22×C4#1175822411yescharacteristic
197C22×C4C2×C22⋊D6111yescharacteristic
198SD16—2448no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—1224no—
202C2×C8—2441no—
203C2×C8—2442no—
204C2×C8#1175820114yescomplemented
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—1222no—
212C22⋊C4—42no—
213C22⋊C4—2442no—
214C42—42no—
215C42D4×D6112yes—
216C42C2×C22⋊D6112yes—
217C23—1222no—
218C23—2441no—
219C23—2442no—
220C23—2442no—
221C23—2441no—
222C23#1451974811yescharacteristic
223Q8—44no—
224Q8—44no—
225D4—1224no—
226D4—2442no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—4881no—
232C2×C4—4881no—
233C2×C4—4882no—
234C2×C4—2442no—
235C2×C4—2441no—
236C2×C4#1451974512yes—
237C2×C4#1451974611yescharacteristic
238C2×C4#1451974912yes—
239C2×C4#1451977311yescharacteristic
240C2×C4C2×#1175820112yes—
241C2×C4#1451980812yes—
242C8—24no—
243C8—4881no—
244C8—2444no—
245C22—21no—
246C22—21no—
247C22—4882no—
248C22—4881no—
249C22—4881no—
250C22—2444no—
251C22—2442no—
252C22#1683967011yescharacteristic
253C22#1683996412yes—
254C4—22no—
255C4—22no—
256C4—82no—
257C4—82no—
258C4—4881no—
259C4#1683990112yes—
260C4#1684013611yescharacteristic
261C4#1684015411yescharacteristic
262C2—22no—
263C2—4882no—
264C2#2144688311yescharacteristic
265C2#2144690211yescharacteristic
266C2#2144692211yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3162C2
socleC22×C61#392
prime radicalC22×D244D4
hypercenter#3162D61
hypocenterC61#10036
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #21444222C2semidirect
4. #21443650C2semidirect
5. C61×#3162C2semidirect
66. C2×C8×C61SD16semidirect
129. #3162D61semidirect
154. C61#10036semidirect
204. C2×C8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214558232
#214466172
#214442222
#214436502
C61×#31622
#1003661