Finite Group Databasebeta

← #29577378

Subgroups — #29577378

Order 15616.

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

normal subgroups only

This group has 558 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#21450811C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic
3#21444490C212yesmaximal, complemented
4#21443721C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7C2×#14519764—21no—
8#16839786C2212yes—
9C2×#14519694—21no—
10#16839687C414yes—
11#16839667—21no—
12#16839622—22no—
13C4×C4⋊Dic61C2211yescharacteristic
14#16839531—21no—
15#16839507—24no—
16C2×C4×D4×C61C2211yescharacteristic
17C8×D4×C61—24no—
18C61×#431C2212yes—
19C2×C22⋊C8×C61—22no—
20C22×Dic122—21no—
21C2×#11758194—22no—
22#14519777—22no—
23#14519777—22no—
24#14519764—42no—
25#14519764—42no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61C2×C412yes—
29C2×C22⋊Dic61—24no—
30C2×C4×Dic61—21no—
31C4⋊#92418—22no—
32C4⋊#92418—22no—
33#14519694—42no—
34#14519684—24no—
35#14519684C2×C412yes—
36C22×D4×C61D411yescharacteristic
37C23×C4×C61D412yes—
38C22×C8×C61—24no—
39C4×D4×C61—22no—
40C4×D4×C61—22no—
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61C2×C412yes—
43C2×C42×C61C2311yescharacteristic
44C4⋊C8×C61D414yes—
45C22⋊C8×C61—24no—
46C22⋊C8×C61—22no—
47C4×C8×C61—22no—
48C24×C61C22⋊C412yes—
49C2×D4×C61—22no—
50C2×D4×C61—22no—
51C2×D4×C61D812yes—
52C2×D4×C61SD1612yes—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—44no—
56C22×C4×C61C22⋊C412yes—
57C22×C4×C61C22×C411yescharacteristic
58C22×C4×C61C2×D411yescharacteristic
59C22×C4×C61C2×D411yescharacteristic
60C22×Dic61—22no—
61C22×Dic61—22no—
62C2×Dic122—42no—
63C2×Dic122—41no—
64C2×Dic122—41no—
65C2×Dic122—42no—
66C2×C8×C61—24no—
67C2×C8×C61—84no—
68C4⋊C4×C61—22no—
69C22⋊C4×C61—24no—
70C42×C61C2×D412yes—
71C42×C61#4112yes—
72C22⋊Dic61—48no—
73C4⋊Dic61—24no—
74C4⋊Dic61—42no—
75C4⋊Dic61—41no—
76C4⋊Dic61—41no—
77C4×Dic61—41no—
78C4×Dic61—41no—
79#11758194—42no—
80#11758194—42no—
81C2×#92418—42no—
82C23×C61—22no—
83C23×C61—22no—
84C23×C61—42no—
85C23×C61—42no—
86C23×C61#10214yes—
87C23×C61C2×C22⋊C411yescharacteristic
88D4×C61—24no—
89D4×C61—24no—
90C2×C4×C61—22no—
91C2×C4×C61—22no—
92C2×C4×C61—41no—
93C2×C4×C61—44no—
94C2×C4×C61—42no—
95C2×C4×C61—82no—
96C2×C4×C61C4×D412yes—
97C2×C4×C61#12011yescharacteristic
98C2×C4×C61#12112yes—
99C2×C4×C61#12311yescharacteristic
100C2×C4×C61C2×D811yescharacteristic
101C2×C4×C61C2×SD1611yescharacteristic
102C2×C4×C61#13512yes—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107C2×Dic61—42no—
108C2×Dic61—42no—
109Dic122—82no—
110Dic122—82no—
111C8×C61—44no—
112#92418—82no—
113#9393—611nomaximal, Sylow
114C22×C61—21no—
115C22×C61—21no—
116C22×C61—24no—
117C22×C61—44no—
118C22×C61—42no—
119C22×C61—42no—
120C22×C61—82no—
121C22×C61#38611yescharacteristic
122C22×C61C2×#10211yescharacteristic
123C22×C61#41611yescharacteristic
124C4×C61—21no—
125C4×C61—22no—
126C4×C61—42no—
127C4×C61#45512yes—
128C4×C61#46512yes—
129C4×C61#46612yes—
130Dic61—82no—
131Dic61—82no—
132#2972D6111yescharacteristic
133#2547—611no—
134#1800—611no—
135#1546—612no—
136#1303—612no—
137C2×C61—21no—
138C2×C61—21no—
139C2×C61—44no—
140C2×C61#188511yescharacteristic
141C2×C61#189311yescharacteristic
142C2×C61#191511yescharacteristic
143C2×C4⋊Q8—611no—
144C2×#124—1221no—
145C2×C4×D4C2×D6111yescharacteristic
146C8×D4—24no—
147#431C2×D6112yes—
148#423—612no—
149C2×#100—1221no—
150C2×C22⋊C8—22no—
151#397—614no—
152#391—1221no—
153#380—1222no—
154C4×C4⋊C4—611no—
155#343—1221no—
156#332—1224no—
157C61#939311yescharacteristic, complemented, Sylow
158C22×Q8—1221no—
159C22×D4D24411yescharacteristic
160C23×C4C22⋊D6112yes—
161C2×M16—1222no—
162C22×C8—24no—
163C4⋊Q8—1222no—
164C4⋊Q8—1222no—
165#124—2442no—
166#124—2442no—
167C4×D4—22no—
168C4×D4—22no—
169C2×C4⋊C4—612no—
170C2×C4⋊C4—1221no—
171C2×C4⋊C4—1221no—
172C2×C4⋊C4D24411yescharacteristic
173C2×C22⋊C4—1224no—
174C2×C22⋊C4C4×D6112yes—
175C2×C42—1221no—
176C2×C42C22×D6111yescharacteristic
177C4⋊C8—1222no—
178C4⋊C8—1222no—
179C4⋊C8C22⋊D6114yes—
180#100—2442no—
181C22⋊C8—24no—
182C22⋊C8—22no—
183C4×C8—22no—
184#95—612no—
185#95—1224no—
186C24#1175819812yes—
187C2×Q8—2442no—
188C2×Q8—2441no—
189C2×Q8—2441no—
190C2×Q8—2442no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4#1175819012yes—
194C2×D4D48812yes—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—44no—
198C22×C4—1222no—
199C22×C4—1222no—
200C22×C4#1175819812yes—
201C22×C4C2×C4×D6111yescharacteristic
202C22×C4C2×D24411yescharacteristic
203C22×C4C2×C22⋊D6111yescharacteristic
204M16—2442no—
205M16—2442no—
206C2×C8—24no—
207C2×C8—84no—
208C2×C8—2442no—
209C4⋊C4—22no—
210C4⋊C4—1224no—
211C4⋊C4—2442no—
212C4⋊C4—2441no—
213C4⋊C4—2441no—
214C22⋊C4—24no—
215C22⋊C4—2448no—
216C42—2441no—
217C42—2441no—
218C42#1175822212yes—
219C42C2×C22⋊D6112yes—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23#1451969914yes—
225C23C2×#1175819811yescharacteristic
226Q8—4882no—
227Q8—4882no—
228D4—24no—
229D4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—2442no—
237C2×C4—2442no—
238C2×C4—2442no—
239C2×C4—2442no—
240C2×C4—2442no—
241C2×C4—2442no—
242C2×C4C4×C22⋊D6112yes—
243C2×C4#1451974611yescharacteristic
244C2×C4#1451975112yes—
245C2×C4#1451976011yescharacteristic
246C2×C4C2×D48811yescharacteristic
247C2×C4C2×#1175819011yescharacteristic
248C2×C4#1451979912yes—
249C8—44no—
250C8—4882no—
251C22—21no—
252C22—21no—
253C22—24no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22—82no—
258C22#1683964111yescharacteristic
259C22C2×#1451969911yescharacteristic
260C22#1683975911yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4—4882no—
265C4—4882no—
266C4#1683994012yes—
267C4#1683999112yes—
268C4#1683999812yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#2144620711yescharacteristic
273C2#2144624711yescharacteristic
274C2#2144637111yescharacteristic
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#9393
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. #21444490C2semidirect
4. #21443721C2semidirect
157. C61#9393semidirect

Maximal subgroups

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

subgroupindex
#214508112
#214458112
#214444902
#214437212
C61×#29722
#939361