Finite Group Databasebeta

← #29577853

Subgroups — #29577853

Order 15616.

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

normal subgroups only

This group has 542 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#21454843C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic
3#21444248C212yesmaximal, complemented
4#21443677C212yesmaximal, complemented
5C61×#2580C211yesmaximal, characteristic
6D4×#92418—24no—
7#16839815C2212yes—
8#16839786C2212yes—
9C2×#14519694—21no—
10C2×C22⋊#92418—22no—
11#16839531—21no—
12Q8⋊#92418—24no—
13C2×C4×Q8×C61C2211yescharacteristic
14C2×C4×D4×C61C2211yescharacteristic
15C61×#399C412yes—
16C61×#393C412yes—
17C61×#390—21no—
18C61×#389—21no—
19C61×#385—22no—
20C61×#382C2211yescharacteristic
21C2×#11758194—22no—
22C22×#92418—24no—
23C4⋊#92418—22no—
24C4⋊#92418—22no—
25C4⋊#92418—22no—
26#14519694—42no—
27C22⋊#92418—24no—
28C22⋊#92418—22no—
29C4×#92418D414yes—
30C22×Q8×C61C2×C411yescharacteristic
31C22×D4×C61D411yescharacteristic
32C23×C4×C61D412yes—
33C4×Q8×C61—22no—
34C4×Q8×C61—22no—
35C4×D4×C61—22no—
36C4×D4×C61—22no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61C2×C412yes—
45C2×C42×C61—21no—
46C2×C42×C61C2311yescharacteristic
47#95×C61—22no—
48#95×C61—21no—
49#95×C61C2×C412yes—
50C24×C61C22⋊C412yes—
51C2×Q8×C61—22no—
52C2×Q8×C61—42no—
53C2×D4×C61—22no—
54C2×D4×C61—22no—
55C2×D4×C61SD1612yes—
56C2×D4×C61Q1612yes—
57C22×C4×C61—21no—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—22no—
61C22×C4×C61—21no—
62C22×C4×C61—82no—
63C22×C4×C61C22⋊C412yes—
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C4⋊C4×C61—22no—
68C4⋊C4×C61—22no—
69C4⋊C4×C61—41no—
70C4⋊C4×C61—41no—
71C4⋊C4×C61—42no—
72C4⋊C4×C61—82no—
73C22⋊C4×C61—24no—
74C22⋊C4×C61—44no—
75C22⋊C4×C61—82no—
76C42×C61—42no—
77C42×C61C2×D412yes—
78C42×C61#4112yes—
79#11758194—42no—
80#11758194—42no—
81C2×#92418—24no—
82C2×#92418—42no—
83C2×#92418—84no—
84C23×C61—22no—
85C23×C61—22no—
86C23×C61—42no—
87C23×C61—42no—
88C23×C61#10314yes—
89C23×C61C2×C22⋊C411yescharacteristic
90Q8×C61—24no—
91Q8×C61—42no—
92D4×C61—24no—
93D4×C61—24no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—41no—
97C2×C4×C61—41no—
98C2×C4×C61—44no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—41no—
104C2×C4×C61—42no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—42no—
108C2×C4×C61—82no—
109C2×C4×C61C4×D412yes—
110C2×C4×C61#12011yescharacteristic
111C2×C4×C61#12112yes—
112C2×C4×C61#12311yescharacteristic
113C2×C4×C61C2×SD1611yescharacteristic
114C2×C4×C61C2×Q1611yescharacteristic
115C2×C4×C61#13512yes—
116#92418—44no—
117#92418—82no—
118#9483—611nomaximal, Sylow
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—24no—
122C22×C61—44no—
123C22×C61—42no—
124C22×C61—42no—
125C22×C61—82no—
126C22×C61#38611yescharacteristic
127C22×C61C2×#10311yescharacteristic
128C22×C61#41611yescharacteristic
129C4×C61—22no—
130C4×C61—21no—
131C4×C61—21no—
132C4×C61—42no—
133C4×C61—42no—
134C4×C61—82no—
135C4×C61—81no—
136C4×C61#46512yes—
137C4×C61#46712yes—
138#2972—611no—
139#2580D6111yescharacteristic
140#1800—611no—
141#1487—612no—
142#1286—612no—
143C2×C61—21no—
144C2×C61—21no—
145C2×C61—44no—
146C2×C61#189111yescharacteristic
147C2×C61#189211yescharacteristic
148C2×C61#191111yescharacteristic
149C2×C4×Q8C2×D6111yescharacteristic
150C2×C4×D4C2×D6111yescharacteristic
151C8×D4—1224no—
152#431—612no—
153#423—612no—
154C2×#100—1221no—
155C2×C22⋊C8—1222no—
156#399Dic6112yes—
157#393Dic6112yes—
158#390—21no—
159#389—21no—
160#385—22no—
161#382C2×D6111yescharacteristic
162#343—1221no—
163Q8⋊C8—1224no—
164C61#948311yescharacteristic, complemented, Sylow
165C22×Q8C2×Dic6111yescharacteristic
166C22×D4C22⋊D6111yescharacteristic
167C23×C4C22⋊D6112yes—
168C2×M16—1222no—
169C22×C8—1224no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×Dic6111yescharacteristic
178C2×C4⋊C4C22⋊D6111yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×Dic6112yes—
182C2×C42—21no—
183C2×C42C22×D6111yescharacteristic
184C4⋊C8—1222no—
185C4⋊C8—1222no—
186C4⋊C8—1222no—
187#100—2442no—
188C22⋊C8—1224no—
189C22⋊C8—1222no—
190C4×C8—614no—
191#95—22no—
192#95—21no—
193#95C2×Dic6112yes—
194C24C22⋊Dic6112yes—
195C2×Q8—22no—
196C2×Q8—42no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4#1175820112yes—
200C2×D4#1175820212yes—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—82no—
207C22×C4C22⋊Dic6112yes—
208C22×C4C22×Dic6111yescharacteristic
209C22×C4C2×C22⋊D6111yescharacteristic
210C22×C4C2×C22⋊D6111yescharacteristic
211M16—2442no—
212M16—2442no—
213C2×C8—1224no—
214C2×C8—4884no—
215C2×C8—2442no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C22⋊C4—24no—
223C22⋊C4—44no—
224C22⋊C4—82no—
225C42—42no—
226C42D4×D6112yes—
227C42#1175822412yes—
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23#1451970114yes—
233C23C2×C22⋊Dic6111yescharacteristic
234Q8—24no—
235Q8—42no—
236D4—24no—
237D4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4D4×Dic6112yes—
254C2×C4#1451974511yescharacteristic
255C2×C4#1451974912yes—
256C2×C4#1451975811yescharacteristic
257C2×C4C2×#1175820111yescharacteristic
258C2×C4C2×#1175820211yescharacteristic
259C2×C4#1451979712yes—
260C8—4882no—
261C8—2444no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#1683964211yescharacteristic
270C22C2×#1451970111yescharacteristic
271C22#1683975811yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—42no—
277C4—82no—
278C4—81no—
279C4#1683999312yes—
280C4#1684000112yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#2144623811yescharacteristic
285C2#2144624311yescharacteristic
286C2#2144635211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2580C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#2580D61
hypocenterC61#9483
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #21444248C2semidirect
4. #21443677C2semidirect
164. C61#9483semidirect

Maximal subgroups

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

subgroupindex
#214548432
#214458112
#214442482
#214436772
C61×#25802
#948361