Finite Group Databasebeta

← #29577368

Subgroups — #29577368

Order 15616.

538 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 135 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#21450992C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic
3#21444499C212yesmaximal, complemented
4#21443721C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7#16839786C2212yes—
8C2×#14519694—21no—
9#16839674C414yes—
10#16839665—21no—
11#16839627—21no—
12#16839622—22no—
13C4×C4⋊Dic61C2211yescharacteristic
14#16839531—21no—
15#16839510—24no—
16C2×C4×D4×C61C2211yescharacteristic
17C8×D4×C61—24no—
18C61×#431C2212yes—
19C2×C22⋊C8×C61—22no—
20C22×Dic122C2×C412yes—
21C2×#11758194—22no—
22#14519777—22no—
23#14519777—22no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C22⋊Dic61—24no—
28C2×C4×Dic61—21no—
29C4⋊#92418—22no—
30C4⋊#92418—22no—
31#14519694—42no—
32#14519684—24no—
33#14519684C2×C412yes—
34C22×D4×C61D411yescharacteristic
35C23×C4×C61D412yes—
36C22×C8×C61—24no—
37C4×D4×C61—22no—
38C4×D4×C61—22no—
39C2×C4⋊C4×C61D411yescharacteristic
40C2×C22⋊C4×C61C2×C412yes—
41C2×C42×C61C2311yescharacteristic
42C4⋊C8×C61D414yes—
43C22⋊C8×C61—24no—
44C22⋊C8×C61—22no—
45C4×C8×C61—22no—
46C24×C61C22⋊C412yes—
47C2×D4×C61—22no—
48C2×D4×C61—22no—
49C2×D4×C61SD1612yes—
50C2×D4×C61Q1612yes—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—44no—
54C22×C4×C61C22⋊C412yes—
55C22×C4×C61C22×C411yescharacteristic
56C22×C4×C61C2×D411yescharacteristic
57C22×C4×C61C2×D411yescharacteristic
58C22×Dic61—22no—
59C22×Dic61—22no—
60C2×Dic122—24no—
61C2×Dic122—82no—
62C2×C8×C61—24no—
63C2×C8×C61—84no—
64C4⋊C4×C61—22no—
65C22⋊C4×C61—24no—
66C42×C61C2×D412yes—
67C42×C61#4112yes—
68C22⋊Dic61—84no—
69C4⋊Dic61—42no—
70C4⋊Dic61—41no—
71C4⋊Dic61—42no—
72C4⋊Dic61—41no—
73C4×Dic61—42no—
74#11758194—42no—
75#11758194—42no—
76C2×#92418—42no—
77C23×C61—22no—
78C23×C61—22no—
79C23×C61—42no—
80C23×C61—42no—
81C23×C61#10314yes—
82C23×C61C2×C22⋊C411yescharacteristic
83D4×C61—24no—
84D4×C61—24no—
85C2×C4×C61—22no—
86C2×C4×C61—22no—
87C2×C4×C61—41no—
88C2×C4×C61—44no—
89C2×C4×C61—42no—
90C2×C4×C61—82no—
91C2×C4×C61C4×D412yes—
92C2×C4×C61#12011yescharacteristic
93C2×C4×C61#12112yes—
94C2×C4×C61#12311yescharacteristic
95C2×C4×C61C2×SD1611yescharacteristic
96C2×C4×C61C2×Q1611yescharacteristic
97C2×C4×C61#13512yes—
98C2×Dic61—42no—
99C2×Dic61—42no—
100C2×Dic61—44no—
101C2×Dic61—42no—
102C2×Dic61—42no—
103Dic122—84no—
104C8×C61—44no—
105#92418—82no—
106#9391—611nomaximal, Sylow
107C22×C61—21no—
108C22×C61—21no—
109C22×C61—24no—
110C22×C61—44no—
111C22×C61—42no—
112C22×C61—42no—
113C22×C61—82no—
114C22×C61#38611yescharacteristic
115C22×C61C2×#10311yescharacteristic
116C22×C61#41611yescharacteristic
117C4×C61—21no—
118C4×C61—22no—
119C4×C61—42no—
120C4×C61#45512yes—
121C4×C61#46512yes—
122C4×C61#46712yes—
123Dic61—82no—
124Dic61—82no—
125#2972D6111yescharacteristic
126#2563—611no—
127#1800—611no—
128#1548—612no—
129#1303—612no—
130C2×C61—21no—
131C2×C61—21no—
132C2×C61—44no—
133C2×C61#188511yescharacteristic
134C2×C61#189211yescharacteristic
135C2×C61#191411yescharacteristic
136C2×C4⋊Q8—611no—
137C2×C4×D4C2×D6111yescharacteristic
138C8×D4—24no—
139#431C2×D6112yes—
140#423—612no—
141C2×#100—1221no—
142C2×C22⋊C8—22no—
143#393—614no—
144#390—1221no—
145#382—1221no—
146#380—1222no—
147C4×C4⋊C4—611no—
148#343—1221no—
149#333—1224no—
150C61#939111yescharacteristic, complemented, Sylow
151C22×Q8—612no—
152C22×D4D24411yescharacteristic
153C23×C4C22⋊D6112yes—
154C2×M16—1222no—
155C22×C8—24no—
156C4⋊Q8—1222no—
157C4⋊Q8—1222no—
158C4×D4—22no—
159C4×D4—22no—
160C2×C4⋊C4—1221no—
161C2×C4⋊C4—1221no—
162C2×C4⋊C4—1221no—
163C2×C4⋊C4D24411yescharacteristic
164C2×C22⋊C4—1224no—
165C2×C22⋊C4C4×D6112yes—
166C2×C42—1221no—
167C2×C42C22×D6111yescharacteristic
168C4⋊C8—1222no—
169C4⋊C8—1222no—
170C4⋊C8C22⋊D6114yes—
171#100—2442no—
172C22⋊C8—24no—
173C22⋊C8—22no—
174C4×C8—22no—
175#95—612no—
176#95—1224no—
177C24#1175819812yes—
178C2×Q8—1224no—
179C2×Q8—4882no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4#1175819012yes—
183C2×D4Dic24412yes—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—44no—
187C22×C4—1222no—
188C22×C4—1222no—
189C22×C4#1175819812yes—
190C22×C4C2×C4×D6111yescharacteristic
191C22×C4C2×D24411yescharacteristic
192C22×C4C2×C22⋊D6111yescharacteristic
193M16—2442no—
194M16—2442no—
195C2×C8—24no—
196C2×C8—84no—
197C2×C8—2442no—
198C4⋊C4—22no—
199C4⋊C4—2442no—
200C4⋊C4—2441no—
201C4⋊C4—2442no—
202C4⋊C4—2441no—
203C22⋊C4—24no—
204C22⋊C4—4884no—
205C42—2442no—
206C42#1175822212yes—
207C42C2×C22⋊D6112yes—
208C23—22no—
209C23—22no—
210C23—42no—
211C23—42no—
212C23#1451970214yes—
213C23C2×#1175819811yescharacteristic
214Q8—4884no—
215D4—24no—
216D4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—2442no—
224C2×C4—2444no—
225C2×C4—2442no—
226C2×C4—2442no—
227C2×C4—2442no—
228C2×C4C4×C22⋊D6112yes—
229C2×C4#1451974611yescharacteristic
230C2×C4#1451975112yes—
231C2×C4#1451976011yescharacteristic
232C2×C4C2×#1175819011yescharacteristic
233C2×C4C2×Dic24411yescharacteristic
234C2×C4#1451979912yes—
235C8—44no—
236C8—4882no—
237C22—21no—
238C22—21no—
239C22—24no—
240C22—44no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22#1683964111yescharacteristic
245C22C2×#1451970211yescharacteristic
246C22#1683975911yescharacteristic
247C4—21no—
248C4—22no—
249C4—42no—
250C4—4882no—
251C4—4882no—
252C4#1683994012yes—
253C4#1683999112yes—
254C4#1684000312yes—
255C2—21no—
256C2—21no—
257C2—44no—
258C2#2144620711yescharacteristic
259C2#2144624211yescharacteristic
260C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2972C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#2972D61
hypocenterC61#9391
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
3. #21444499C2semidirect
4. #21443721C2semidirect
150. C61#9391semidirect

Maximal subgroups

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

subgroupindex
#214509922
#214458112
#214444992
#214437212
C61×#29722
#939161