Finite Group Databasebeta

← #29577612

Subgroups — #29577612

Order 15616.

504 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 107 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#21456279C211yesmaximal, characteristic
2#21444554C211yesmaximal, characteristic
3#21444503C212yesmaximal, complemented
4#21443710C212yesmaximal, complemented
5C61×#2979C211yesmaximal, characteristic
6C2×#14519807—22no—
7C2×#14519777C2211yescharacteristic
8#16840157C414yes—
9#16840141—21no—
10#16840134—21no—
11#16839786C2212yes—
12C4×#11758194C2211yescharacteristic
13#16839511—21no—
14#16839509—22no—
15#16839509—22no—
16#16839506—21no—
17C61×#532C2211yescharacteristic
18C61×#436—22no—
19C8×D4×C61—22no—
20C61×#431C2212yes—
21C61×#408—22no—
22C22×Dic122—21no—
23#14519807—44no—
24#14519807—44no—
25C2×#11758202—24no—
26C2×#11758200—24no—
27C2×#11758194—21no—
28C2×#11758194—22no—
29#14519777—22no—
30#14519777—21no—
31#14519777C2×C412yes—
32C2×C4⋊Dic61—22no—
33C4⋊#92418—22no—
34C4⋊#92418—22no—
35#14519688C2×C412yes—
36C4×#92418—21no—
37C2×#41×C61D412yes—
38C2×M16×C61—22no—
39C22×C8×C61—22no—
40C4⋊Q8×C61D411yescharacteristic
41#127×C61D411yescharacteristic
42#124×C61C2×C412yes—
43#121×C61—22no—
44C4×D4×C61—22no—
45C2×C42×C61C2311yescharacteristic
46C4⋊C8×C61D412yes—
47C4⋊C8×C61D412yes—
48C22⋊C8×C61—24no—
49C22⋊C8×C61—22no—
50C4×C8×C61—22no—
51#41×C61—42no—
52#41×C61—42no—
53C2×Q8×C61C22⋊C412yes—
54C2×D4×C61—22no—
55C2×D4×C61—22no—
56C2×D4×C61C22⋊C412yes—
57C22×C4×C61—22no—
58C22×C4×C61C2×D411yescharacteristic
59C22×C4×C61#4112yes—
60C22×Dic61—41no—
61C2×Dic122—22no—
62C2×Dic122—42no—
63C2×Dic122—42no—
64C2×Dic122—41no—
65M16×C61—82no—
66C2×C8×C61—22no—
67C2×C8×C61—22no—
68C2×C8×C61—82no—
69C4⋊C4×C61—22no—
70C22⋊C4×C61—24no—
71C42×C61C22×C411yescharacteristic
72C42×C61C2×D411yescharacteristic
73C42×C61C2×D412yes—
74#11758202—48no—
75#11758200—48no—
76C4⋊Dic61—24no—
77C4⋊Dic61—42no—
78#11758194—22no—
79#11758194—22no—
80#11758194—82no—
81C2×#92418—22no—
82C2×#92418—42no—
83C23×C61—22no—
84C23×C61#12311yescharacteristic
85Q8×C61—42no—
86D4×C61—42no—
87D4×C61—42no—
88D4×C61—42no—
89D4×C61—82no—
90C2×C4×C61—21no—
91C2×C4×C61—21no—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—22no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61C2×C22⋊C411yescharacteristic
98C2×C4×C61C4×D412yes—
99C2×C4×C61#12011yescharacteristic
100C2×C4×C61#12112yes—
101C2×Dic61—42no—
102C2×Dic61—82no—
103Dic122—44no—
104Dic122—81no—
105Dic122—81no—
106C8×C61—42no—
107C8×C61—42no—
108#92418—24no—
109#92418—82no—
110#9435—611nomaximal, Sylow
111C22×C61—21no—
112C22×C61—22no—
113C22×C61—42no—
114C22×C61—44no—
115C22×C61#38611yescharacteristic
116C4×C61—21no—
117C4×C61—22no—
118C4×C61—42no—
119C4×C61#45311yescharacteristic
120C4×C61#45311yescharacteristic
121C4×C61#47112yes—
122C4×C61#47312yes—
123Dic61—82no—
124#3152—611no—
125#2979D6111yescharacteristic
126#1560—611no—
127#1549—612no—
128#1299—612no—
129C2×C61—41no—
130C2×C61—82no—
131C2×C61#188711yescharacteristic
132C2×C61#190812yes—
133C2×#137—1222no—
134#532C2×D6111yescharacteristic
135C2×C4⋊Q8—611no—
136#497—614no—
137#494—1221no—
138#492—1221no—
139#436—22no—
140C8×D4—22no—
141#431C2×D6112yes—
142#423—612no—
143#408—22no—
144C4×M16—611no—
145#333—1221no—
146#332—1222no—
147#332—1222no—
148#331—1221no—
149C61#943511yescharacteristic, complemented, Sylow
150C2×#41C22⋊D6112yes—
151C22×Q8—1221no—
152#137—2444no—
153#137—2444no—
154C2×Q16—1224no—
155C2×SD16—1224no—
156C2×M16—22no—
157C2×M16—1221no—
158C2×M16—1222no—
159C22×C8—22no—
160C4⋊Q8—612no—
161C4⋊Q8—1221no—
162C4⋊Q8—1222no—
163C4⋊Q8D24411yescharacteristic
164#127D24411yescharacteristic
165#124C4×D6112yes—
166#121—22no—
167C4×D4—22no—
168C2×C4⋊C4—1222no—
169C2×C42C22×D6111yescharacteristic
170C4⋊C8—1222no—
171C4⋊C8—1222no—
172C4⋊C8C22⋊D6112yes—
173C4⋊C8C22⋊D6112yes—
174C22⋊C8—24no—
175C22⋊C8—22no—
176#97—612no—
177C4×C8—22no—
178C4×C8—1221no—
179#41—42no—
180#41—42no—
181C2×Q8—1222no—
182C2×Q8—2442no—
183C2×Q8—2442no—
184C2×Q8—2441no—
185C2×Q8#1175819812yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4#1175819812yes—
189C22×C4—22no—
190C22×C4—2441no—
191C22×C4#1175822212yes—
192C22×C4C2×C22⋊D6111yescharacteristic
193Q16—2448no—
194SD16—2448no—
195M16—82no—
196M16—1222no—
197M16—1222no—
198M16—4882no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—82no—
202C2×C8—1222no—
203C2×C8—2442no—
204C4⋊C4—22no—
205C4⋊C4—1224no—
206C4⋊C4—2442no—
207C22⋊C4—24no—
208C42C2×C4×D6111yescharacteristic
209C42C2×D24411yescharacteristic
210C42C2×C22⋊D6112yes—
211C23—22no—
212C23#1451976011yescharacteristic
213Q8—42no—
214Q8—4881no—
215Q8—4881no—
216Q8—2444no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—82no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—4882no—
229C2×C4—2442no—
230C2×C4C2×#1175819811yescharacteristic
231C2×C4C4×C22⋊D6112yes—
232C2×C4#1451974611yescharacteristic
233C2×C4#1451975112yes—
234C8—42no—
235C8—42no—
236C8—1224no—
237C8—4882no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—44no—
242C22#1683964111yescharacteristic
243C4—21no—
244C4—22no—
245C4—42no—
246C4—4882no—
247C4#1683993011yescharacteristic
248C4#1683993011yescharacteristic
249C4#1684002512yes—
250C4#1684003712yes—
251C2—41no—
252C2—82no—
253C2#2144621711yescharacteristic
254C2#2144633412yes—
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2979C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#2979D61
hypocenterC61#9435
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
3. #21444503C2semidirect
4. #21443710C2semidirect
149. C61#9435semidirect

Maximal subgroups

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

subgroupindex
#214562792
#214445542
#214445032
#214437102
C61×#29792
#943561