Finite Group Databasebeta

← #29599698

Subgroups — #29599698

Order 15616.

416 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 79 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#21453812C211yesmaximal, characteristic
2#21446034C212yesmaximal
3#21443946C211yesmaximal, characteristic
4#21443945C212yesmaximal
5C61×#1857C211yesmaximal, characteristic
6#16839805—22no—
7C2×#14519696C2212yes—
8#16839683—22no—
9#16839682—21no—
10#16839669C2212yes—
11#16839668—21no—
12#16839666—22no—
13#16839521—22no—
14C61×#519C2211yescharacteristic
15C61×#428—21no—
16C2×#101×C61C2211yescharacteristic
17C61×#391C2211yescharacteristic
18C61×#339—21no—
19C22×Dic122—21no—
20C22×Dic122—21no—
21C2×#11758194—22no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61D412yes—
24C2×#11758196—22no—
25C2×C4×Dic61D412yes—
26#14519710—42no—
27#14519708—42no—
28#14519696—22no—
29#14519696Q814yescomplemented
30#14519684—21no—
31#14519684—22no—
32#14519684—22no—
33#14519684—22no—
34C22×Q8×C61C2311yescharacteristic
35C2×M16×C61—21no—
36C4×Q8×C61—41no—
37C4×Q8×C61—41no—
38#117×C61—21no—
39#117×C61—22no—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C42×C61D411yescharacteristic
42#107×C61—41no—
43#106×C61—41no—
44#101×C61—21no—
45#101×C61Q812yescomplemented
46#95×C61—22no—
47C2×Q8×C61—41no—
48C2×Q8×C61—41no—
49C2×Q8×C61C2×Q811yescharacteristic
50C2×Q8×C61C2×Q812yes—
51C2×Q8×C61#4111yescharacteristic
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61C2×D412yes—
55C22×C4×C61C2×D411yescharacteristic
56C22×Dic61—22no—
57C22×Dic61—22no—
58C2×Dic122—24no—
59C2×Dic122—81no—
60C2×Dic122—81no—
61C2×Dic122—82no—
62C2×Dic122—81no—
63M16×C61—22no—
64M16×C61—41no—
65C2×C8×C61—41no—
66C4⋊C4×C61—21no—
67C4⋊C4×C61—21no—
68C4⋊C4×C61—22no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61—41no—
71C22⋊C4×C61—41no—
72C42×C61—42no—
73C42×C61—41no—
74C42×C61—41no—
75C4⋊Dic61—42no—
76C4⋊Dic61—81no—
77#11758196—82no—
78C4×Dic61—42no—
79#11758194—24no—
80#11758194—42no—
81C2×#92418—42no—
82C23×C61#12011yescharacteristic
83Q8×C61—42no—
84Q8×C61—41no—
85Q8×C61—41no—
86C2×C4×C61—22no—
87C2×C4×C61—21no—
88C2×C4×C61—22no—
89C2×C4×C61—21no—
90C2×C4×C61—22no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61—42no—
94C2×C4×C61—41no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61C22⋊Q812yes—
98C2×C4×C61C22⋊Q811yescharacteristic
99C2×C4×C61C4⋊Q811yescharacteristic
100C2×C4×C61C4⋊Q812yes—
101C2×Dic61—24no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107Dic122—84no—
108Dic122—84no—
109C8×C61—42no—
110#92418—44no—
111#13464—611nomaximal, Sylow
112C22×C61—21no—
113C22×C61—21no—
114C22×C61#39511yescharacteristic
115C22×C61B(4,𝔽2)11yescharacteristic
116C22×C61#45811yescharacteristic
117C4×C61—22no—
118C4×C61—21no—
119C4×C61—21no—
120C4×C61—22no—
121C4×C61—42no—
122C4×C61—81no—
123C4×C61—81no—
124C4×C61—81no—
125Dic61—44no—
126Dic61—82no—
127#2811—611no—
128#1857—612no—
129#1857D6111yescharacteristic
130#1394—612no—
131#1394—611no—
132C2×C61—21no—
133C2×C61—21no—
134C2×C61#206211yescharacteristic
135C2×C61#220211yescharacteristic
136C2×C61#220511yescharacteristic
137#519C2×D6111yescharacteristic
138#428—21no—
139#428—1222no—
140C2×#101—612no—
141C2×#101C2×D6111yescharacteristic
142#395—1222no—
143#395—1221no—
144#391—612no—
145#391—1222no—
146#391—1221no—
147#391C2×D6111yescharacteristic
148#339—21no—
149#339—1222no—
150C61#1346411yescharacteristic, complemented, Sylow
151C22×Q8—1221no—
152C22×Q8—1221no—
153C22×Q8C22×D6111yescharacteristic
154C2×M16—21no—
155C2×M16—1222no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—22no—
159#117—21no—
160C2×C4⋊C4—612no—
161C2×C4⋊C4—1222no—
162C2×C4⋊C4—1221no—
163C2×C4⋊C4C22⋊D6111yescharacteristic
164C2×C42—612no—
165C2×C42C22⋊D6111yescharacteristic
166#107—41no—
167#107—2442no—
168#106—41no—
169#106—2442no—
170#101—21no—
171#101—614no—
172#101—1222no—
173#101Dic12212yescomplemented
174#95—22no—
175#95—1222no—
176#95—1221no—
177#95—1222no—
178#95—1222no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—1224no—
182C2×Q8—4881no—
183C2×Q8—4881no—
184C2×Q8—4882no—
185C2×Q8—4881no—
186C2×Q8C2×Dic12211yescharacteristic
187C2×Q8#1175822211yescharacteristic
188C2×Q8Q8×D6112yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—1222no—
192C22×C4—1222no—
193C22×C4D4×D6112yes—
194C22×C4C2×C22⋊D6111yescharacteristic
195M16—22no—
196M16—41no—
197M16—1224no—
198M16—2442no—
199C2×C8—41no—
200C2×C8—2442no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—4882no—
207C4⋊C4—4881no—
208C4⋊C4—2442no—
209C22⋊C4—41no—
210C42—41no—
211C42—41no—
212C42—42no—
213C42—2442no—
214C23#1451974511yescharacteristic
215Q8—41no—
216Q8—41no—
217Q8—42no—
218Q8—4884no—
219Q8—4884no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—1224no—
232C2×C4—2442no—
233C2×C4—2442no—
234C2×C4—2442no—
235C2×C4—2442no—
236C2×C4—2442no—
237C2×C4#1451975512yes—
238C2×C4#1451975611yescharacteristic
239C2×C4#1451977511yescharacteristic
240C2×C4#1451977612yes—
241C8—42no—
242C8—2444no—
243C22—21no—
244C22—21no—
245C22#1683968211yescharacteristic
246C22#1683994811yescharacteristic
247C22#1683995111yescharacteristic
248C4—21no—
249C4—22no—
250C4—21no—
251C4—22no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4—4882no—
257C4—2444no—
258C2—21no—
259C2—21no—
260C2#2144717211yescharacteristic
261C2#2144783511yescharacteristic
262C2#2144784811yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC22×Q8×C61C23
FrattiniC22×Q8C22×D61
FittingC61×#1857C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
hypercenter#1857D61
hypocenterC61#13464
LCSG ▷ C22×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
29. #14519696Q8semidirect
45. #101×C61Q8semidirect
150. C61#13464semidirect
173. #101Dic122semidirect

Maximal subgroups

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

subgroupindex
#214538122
#214460342
#214439462
#214439452
C61×#18572
#1346461