Finite Group Databasebeta

← #30600015

Subgroups — #30600015

Order 15616.

398 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 74 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#21462732C211yesmaximal, characteristic, complemented
2#21461325C211yesmaximal, characteristic, complemented
3#21461179C212yesmaximal
4#21460713C212yesmaximal, complemented
5#21456163C211yesmaximal, characteristic
6#16840814—22no—
7#16840812—22no—
8#16840787—22no—
9#16840784C2212yes—
10#16840780C2212yes—
11#16840621—22no—
12D4⋊#92420—22no—
13D4⋊#92420—22no—
14#16840318C2211yescharacteristic
15#16840310C2211yescharacteristic
16#16840132C411yescharacteristic
17#16840131C411yescharacteristic
18#16840126C411yescharacteristic
19#16840120—22no—
20#16839755—22no—
21#97×D61C2211yescharacteristic
22C61×#490C411yescharacteristic
23#14519844—22no—
24#14519844—22no—
25C4⋊#92420—22no—
26C4⋊#92420D412yes—
27C4⋊#92420D412yes—
28#14519834—22no—
29#14519834—22no—
30#14519832—22no—
31#14519832—22no—
32C4×#92420—22no—
33C4×#92420—22no—
34C2×#11758224—21no—
35C2×C8×D61—21no—
36#14519777C2×C411yescharacteristic
37#14519776D412yes—
38#14519769C2×C411yescharacteristic
39#14519764—21no—
40#14519755—22no—
41#14519748—22no—
42#14519742—22no—
43D4×Dic61—22no—
44#14519735D412yes—
45C42×D61C2311yescharacteristic
46#14519702—22no—
47#14519700—22no—
48#14519698—22no—
49#14519688C2×C411yescharacteristic
50#14519687—21no—
51#127×C61C2×C411yescharacteristic
52#125×C61C2×C411yescharacteristic
53#102×C61—22no—
54#97×C61C2×C411yescharacteristic
55C2×D4×C61—21no—
56C2×D4×C61—22no—
57C2×C22⋊D61—41no—
58C22×Dic61—41no—
59#11758224—42no—
60#11758224—42no—
61C2×C4×D61C2×D411yescharacteristic
62C2×C4×D61#4112yes—
63C2×Dic122—21no—
64C2×Dic122—22no—
65C2×#92420—22no—
66C2×#92420—22no—
67C2×#92420—42no—
68C2×C8×C61—21no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61C22⋊C412yes—
71C42×C61C22×C411yescharacteristic
72C22⋊Dic61—42no—
73#11758198—22no—
74C4⋊Dic61—22no—
75C4⋊Dic61C22⋊C412yes—
76#11758196—22no—
77#11758196—22no—
78C4×Dic61—22no—
79C4×Dic61C2×D412yes—
80C4×Dic61C2×D411yescharacteristic
81C2×#92418—21no—
82C8×D61—41no—
83C8×D61—41no—
84C23×C61—41no—
85C22×D61#12311yescharacteristic
86D4×C61—42no—
87D4×C61—44no—
88C2×C4×C61—22no—
89C2×C4×C61C2×C22⋊C411yescharacteristic
90C2×C4×C61C4×D412yes—
91C22⋊D61—82no—
92C2×Dic61—22no—
93C2×Dic61—41no—
94C2×Dic61—81no—
95C2×Dic61—81no—
96C2×Dic61#12011yescharacteristic
97C2×Dic61#12112yes—
98C4×D61—21no—
99C4×D61—21no—
100C4×D61—42no—
101Dic122—42no—
102Dic122—82no—
103#92420—42no—
104#92420—42no—
105#92420—82no—
106C8×C61—41no—
107#92418—41no—
108#9008—611nomaximal, Sylow
109C22×C61—82no—
110C22×C61—81no—
111C22×C61#38611yescharacteristic
112C2×D61—22no—
113C2×D61—21no—
114C4×C61—22no—
115C4×C61—21no—
116C4×C61—42no—
117Dic61—22no—
118Dic61—42no—
119Dic61—81no—
120Dic61#45411yescharacteristic
121Dic61#45411yescharacteristic
122#3135—611no—
123#2987—611no—
124#1544—611no—
125#1496—612no—
126#1302—612no—
127C2×C61—161no—
128C2×C61#188112yes—
129C2×C61#188811yescharacteristic
130D61—41no—
131#533—611no—
132#532—611no—
133#491—611no—
134#491—611no—
135#490—611no—
136#490#692311yescharacteristic
137#489—1222no—
138#446—1222no—
139#446—1222no—
140#433—1222no—
141#432—612no—
142#431—612no—
143#416—1222no—
144C2×#97—611no—
145#332—1222no—
146D4⋊C8—1222no—
147D4⋊C8—1222no—
148C61#900811yescharacteristic, complemented, Sylow
149C2×#41—1221no—
150C22×C8—1221no—
151C4⋊Q8—612no—
152C4⋊Q8—611no—
153#127C2×#692311yescharacteristic
154#125—611no—
155#125C2×#692311yescharacteristic
156#124—1221no—
157C22⋊Q8—1222no—
158#121—1222no—
159C4×Q8—1222no—
160C4×D4—1222no—
161#117—612no—
162C2×C42—611no—
163C4⋊C8—612no—
164C4⋊C8—612no—
165C4⋊C8—1222no—
166C4⋊C8—1222no—
167C4⋊C8—1222no—
168#103—1222no—
169#103—1222no—
170#102—22no—
171#102—1222no—
172C22⋊C8—1222no—
173C22⋊C8—1222no—
174#97—611no—
175#97—1221no—
176#97—1222no—
177#97—1222no—
178#97C2×#692311yescharacteristic
179C4×C8—1222no—
180C4×C8—1222no—
181#41—2442no—
182#41—2442no—
183C2×Q8—1222no—
184C2×Q8—1221no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4—2441no—
188C22×C4—611no—
189C22×C4—612no—
190C22×C4—2441no—
191C2×C8—21no—
192C2×C8—1222no—
193C2×C8—1222no—
194C2×C8—1221no—
195C2×C8—2441no—
196C2×C8—2441no—
197C2×C8—2442no—
198C4⋊C4—22no—
199C4⋊C4—612no—
200C4⋊C4—1222no—
201C4⋊C4—1222no—
202C4⋊C4—1222no—
203C4⋊C4C22⋊#692312yes—
204C22⋊C4—1222no—
205C22⋊C4—2442no—
206C42—611no—
207C42—612no—
208C42—1222no—
209C42C22×#692311yescharacteristic
210C23—41no—
211C23—611no—
212Q8—4882no—
213Q8—2442no—
214D4—44no—
215D4—42no—
216D4—4882no—
217C2×C4—22no—
218C2×C4—611no—
219C2×C4—612no—
220C2×C4—1221no—
221C2×C4—1221no—
222C2×C4—1222no—
223C2×C4—4881no—
224C2×C4—4881no—
225C2×C4—2442no—
226C2×C4—2441no—
227C2×C4C2×C22⋊#692311yescharacteristic
228C2×C4D4×#692312yes—
229C8—41no—
230C8—4882no—
231C8—2441no—
232C8—2442no—
233C8—2442no—
234C22—82no—
235C22—81no—
236C22—1222no—
237C22—1221no—
238C22#1684071011yescharacteristic
239C4—22no—
240C4—21no—
241C4—42no—
242C4—611no—
243C4—611no—
244C4—1222no—
245C4—4881no—
246C4—2442no—
247C2—161no—
248C2—2441no—
249C2#2146181712yes—
250C2#2146182711yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#490C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#490#6923
hypocenterC61#9008
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #21462732C2semidirect
2. #21461325C2semidirect
4. #21460713C2semidirect
148. C61#9008semidirect

Maximal subgroups

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

subgroupindex
#214627322
#214613252
#214611792
#214607132
#214561632
#900861