Finite Group Databasebeta

← #29583085

Subgroups — #29583085

Order 15616.

400 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 93 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#21446791C212yesmaximal
2#21446638C211yesmaximal, characteristic
3#21443700C211yesmaximal, characteristic
4#21443700C212yesmaximal
5C61×#2844C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7C2×#14519700—21no—
8C2×#14519700—21no—
9C2×#14519688C2212yes—
10C2×#14519688C2211yescharacteristic
11#16839524—22no—
12#16839524—22no—
13C61×#400—21no—
14C61×#400—22no—
15C61×#395—21no—
16C61×#395—22no—
17C61×#382C2212yes—
18C61×#382C2211yescharacteristic
19C22×Dic122—21no—
20C22×#92418—21no—
21C22×#92418—22no—
22#14519777—22no—
23#14519777—21no—
24#14519777Q812yescomplemented
25C2×C4⋊Dic61—22no—
26#14519700—42no—
27#14519700—42no—
28#14519688—22no—
29#14519688—22no—
30#14519688—22no—
31C22×Q8×C61—21no—
32C2×C4⋊C4×C61—21no—
33C2×C4⋊C4×C61—22no—
34C2×C4⋊C4×C61D411yescharacteristic
35C2×C4⋊C4×C61D411yescharacteristic
36C2×C4⋊C4×C61D412yes—
37C2×C4⋊C4×C61D412yes—
38C2×C42×C61C2311yescharacteristic
39#95×C61—22no—
40#95×C61—21no—
41#95×C61—21no—
42#95×C61—22no—
43#95×C61—22no—
44#95×C61—22no—
45C2×Q8×C61—81no—
46C2×Q8×C61—82no—
47C22×C4×C61—21no—
48C22×C4×C61—22no—
49C22×C4×C61—21no—
50C22×C4×C61—22no—
51C22×C4×C61C2×D411yescharacteristic
52C22×C4×C61C2×D412yes—
53C22×Dic61—41no—
54C2×Dic122—22no—
55C2×Dic122—41no—
56C2×Dic122—41no—
57C2×Dic122—41no—
58C2×Dic122—41no—
59C2×Dic122—41no—
60C4⋊C4×C61—42no—
61C4⋊C4×C61—42no—
62C4⋊C4×C61—41no—
63C4⋊C4×C61—41no—
64C4⋊C4×C61—81no—
65C4⋊C4×C61—82no—
66C42×C61C2×Q811yescharacteristic
67C42×C61#4112yes—
68C42×C61#4111yescharacteristic
69C4⋊Dic61—24no—
70C4⋊Dic61—42no—
71C2×#92418—22no—
72C2×#92418—24no—
73C2×#92418—41no—
74C2×#92418—42no—
75C2×#92418—42no—
76C2×#92418—41no—
77C23×C61#12011yescharacteristic
78Q8×C61—84no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—22no—
82C2×C4×C61—22no—
83C2×C4×C61—41no—
84C2×C4×C61—41no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61—42no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61—42no—
92C2×C4×C61—42no—
93C2×C4×C61C22⋊Q812yes—
94C2×C4×C61C22⋊Q811yescharacteristic
95C2×C4×C61#12412yes—
96C2×C4×C61#12411yescharacteristic
97C2×Dic61—42no—
98C2×Dic61—81no—
99C2×Dic61—81no—
100Dic122—42no—
101Dic122—42no—
102Dic122—81no—
103Dic122—81no—
104#92418—42no—
105#92418—44no—
106#10359—611nomaximal, Sylow
107C22×C61#39311yescharacteristic
108C22×C61#45311yescharacteristic
109C22×C61#45411yescharacteristic
110C22×C61#45512yes—
111C22×C61#45612yes—
112C4×C61—21no—
113C4×C61—21no—
114C4×C61—22no—
115C4×C61—22no—
116C4×C61—81no—
117C4×C61—82no—
118C4×C61—81no—
119C4×C61—82no—
120Dic61—82no—
121#2844D6111yescharacteristic
122#1993—612no—
123#1960—611no—
124#1295—611no—
125#1295—612no—
126C2×C61#165911yescharacteristic
127C2×C61#166011yescharacteristic
128C2×C61#166412yes—
129C2×C61#202711yescharacteristic
130C2×C61#202812yes—
131C2×C4⋊Q8—611no—
132C2×#103—1221no—
133C2×#103—1221no—
134C2×#97—611no—
135C2×#97—612no—
136#400—22no—
137#400—21no—
138#395—22no—
139#395—21no—
140#382C2×D6111yescharacteristic
141#382C2×D6112yes—
142#340—1222no—
143#340—1222no—
144C61#1035911yescharacteristic, complemented, Sylow
145C22×Q8—21no—
146C22×Q8—1221no—
147C22×C8—1222no—
148C22×C8—1221no—
149C4⋊Q8—612no—
150C4⋊Q8—1222no—
151C4⋊Q8—1221no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—1222no—
155C2×C4⋊C4C22⋊D6112yes—
156C2×C4⋊C4C22⋊D6112yes—
157C2×C4⋊C4C22⋊D6111yescharacteristic
158C2×C4⋊C4C22⋊D6111yescharacteristic
159C2×C42C22×D6111yescharacteristic
160#103—2442no—
161#103—2442no—
162#97—1222no—
163#97—1222no—
164#97—1222no—
165#95—21no—
166#95—22no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—82no—
172C2×Q8—81no—
173C2×Q8—1222no—
174C2×Q8—2441no—
175C2×Q8—2441no—
176C2×Q8—2441no—
177C2×Q8—2441no—
178C2×Q8—2441no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—2441no—
184C22×C4C2×C22⋊D6112yes—
185C22×C4C2×C22⋊D6111yescharacteristic
186C2×C8—1222no—
187C2×C8—1224no—
188C2×C8—2442no—
189C2×C8—2441no—
190C2×C8—2442no—
191C2×C8—2441no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—82no—
197C4⋊C4—81no—
198C4⋊C4—1224no—
199C4⋊C4—2442no—
200C42Q8×D6111yescharacteristic
201C42#1175822612yes—
202C42#1175822611yescharacteristic
203C23#1451974611yescharacteristic
204Q8—84no—
205Q8—4881no—
206Q8—4881no—
207Q8—2442no—
208Q8—2442no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—4881no—
224C2×C4—4881no—
225C2×C4—2442no—
226C2×C4#1451975312yes—
227C2×C4#1451975311yescharacteristic
228C2×C4#1451976612yes—
229C2×C4#1451976611yescharacteristic
230C8—2442no—
231C8—2444no—
232C22#1683967311yescharacteristic
233C22#1683993011yescharacteristic
234C22#1683993511yescharacteristic
235C22#1683994012yes—
236C22#1683994512yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—81no—
243C4—81no—
244C4—82no—
245C4—4882no—
246C2#2144512511yescharacteristic
247C2#2144513011yescharacteristic
248C2#2144515312yes—
249C2#2144697211yescharacteristic
250C2#2144697712yes—
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2844C2
socle, prime radicalC23×C61#120
hypercenter#2844D61
hypocenterC61#10359
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
24. #14519777Q8semidirect
144. C61#10359semidirect

Maximal subgroups

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

subgroupindex
#214467912
#214466382
#214437002
C61×#28442
#1035961