Finite Group Databasebeta

← #30606560

Subgroups — #30606560

Order 15616.

464 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#16840673C212yesmaximal
2#21461572C212yesmaximal
3#21460917C212yesmaximal
4#21448536C211yesmaximal, characteristic
5C2×#14519838C2214yes—
6C4×#11758217—22no—
7#16840673C418yescomplemented
8#16840642—22no—
9C2×#14519777C2211yescharacteristic
10C2×#14519742C411yescharacteristic
11C2×#14519742C2211yescharacteristic
12#16839667C411yescharacteristic
13#16839667C2211yescharacteristic
14#16839645—22no—
15#16839645—22no—
16#16839608C411yescharacteristic
17C61×#382C411yescharacteristic
18C2×#11758217—24no—
19#14519838—24no—
20#14519838C2×C418yes—
21#14519832—42no—
22C4×#92420—42no—
23C22×Dic122C2×C411yescharacteristic
24C22×Dic122C2311yescharacteristic
25#14519777—22no—
26#14519777C2×C414yes—
27#14519742—42no—
28#14519742—42no—
29C2×C4⋊Dic61—22no—
30C2×#11758196—21no—
31C2×#11758196D412yes—
32C2×C4×Dic61—21no—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35C2×C4×Dic61D412yes—
36#14519684—22no—
37#14519684—21no—
38#14519684—21no—
39#14519684—22no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43#95×C61—22no—
44C22×C4×C61—21no—
45C22×C4×C61—21no—
46C22×C4×C61C22⋊C412yes—
47C22×C4×C61C22×C411yescharacteristic
48C22×Dic61—21no—
49C22×Dic61—21no—
50C22×Dic61C22⋊C411yescharacteristic
51C22×Dic61C22⋊C411yescharacteristic
52C22×Dic61C2×D412yes—
53C2×Dic122—21no—
54C2×Dic122—41no—
55C2×Dic122—41no—
56C2×Dic122—41no—
57C2×Dic122—41no—
58C2×Dic122C4212yes—
59C2×Dic122C22×C412yes—
60C2×Dic122#4112yes—
61#11758217—28no—
62#11758217—44no—
63C2×#92420—44no—
64C4⋊C4×C61—41no—
65C4⋊C4×C61—41no—
66C42×C61—21no—
67C42×C61C4212yes—
68C4⋊Dic61—28no—
69#11758196—42no—
70#11758196—42no—
71C4×Dic61—22no—
72C4×Dic61—22no—
73C4×Dic61—22no—
74C4×Dic61—42no—
75C4×Dic61—41no—
76C23×C61C2×C22⋊C411yescharacteristic
77C2×C4×C61—21no—
78C2×C4×C61—21no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—41no—
82C2×C4×C61—41no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61—41no—
86C2×C4×C61C2×C4211yescharacteristic
87C2×C4×C61#11711yescharacteristic
88C2×Dic61—22no—
89C2×Dic61—22no—
90C2×Dic61—22no—
91C2×Dic61—21no—
92C2×Dic61—41no—
93C2×Dic61—41no—
94C2×Dic61—41no—
95C2×Dic61—41no—
96C2×Dic61—41no—
97C2×Dic61—41no—
98C2×Dic61—41no—
99C2×Dic61#9912yes—
100C2×Dic61C4×D414yes—
101Dic122—42no—
102Dic122—42no—
103Dic122—42no—
104Dic122—42no—
105#92420—48no—
106#12631—611nomaximal, Sylow
107C22×C61—21no—
108C22×C61—21no—
109C22×C61C4×C22⋊C411yescharacteristic
110C22×C61C2×#9911yescharacteristic
111C22×C61#41011yescharacteristic
112C4×C61—21no—
113C4×C61—21no—
114C4×C61—42no—
115C4×C61—42no—
116C4×C61—81no—
117Dic61—22no—
118Dic61—24no—
119Dic61—41no—
120Dic61—42no—
121Dic61—81no—
122#2327—611no—
123C2×#356—612no—
124C4×#101—612no—
125#1394—612no—
126C2×C61—21no—
127C2×C61—21no—
128C2×C61C4×#9911yescharacteristic
129C2×C61#213311yescharacteristic
130C2×C61#213511yescharacteristic
131C2×C4⋊Q8—611no—
132C2×C4×Q8—611no—
133C2×C4×Q8—611no—
134C2×#101—614no—
135C4×M16—1222no—
136#391—611no—
137#391—611no—
138#387—1222no—
139#387—1222no—
140#382#692311yescharacteristic
141#376—611no—
142#356—618no—
143#343—1222no—
144C61#1263111yescharacteristic, complemented, Sylow
145C22×Q8—611no—
146C22×Q8—611no—
147C2×M16—1224no—
148C4⋊Q8—614no—
149C4⋊Q8—1222no—
150C4×Q8—2442no—
151C4×Q8—2442no—
152C2×C4⋊C4—612no—
153C2×C4⋊C4—1222no—
154C2×C4⋊C4—1221no—
155C2×C4⋊C4C2×#692311yescharacteristic
156C2×C4⋊C4C2×#692311yescharacteristic
157C2×C42—612no—
158C2×C42—611no—
159C2×C42—611no—
160C2×C42—1221no—
161C2×C42C2×#692311yescharacteristic
162#101—618no—
163#101—1224no—
164#97—2442no—
165C4×C8—2442no—
166#95—22no—
167#95—1221no—
168#95—1221no—
169#95—1222no—
170#95—1222no—
171C2×Q8—612no—
172C2×Q8—612no—
173C2×Q8—612no—
174C2×Q8—1221no—
175C2×Q8—2441no—
176C2×Q8—2441no—
177C2×Q8—2441no—
178C2×Q8—2441no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—612no—
182C22×C4—611no—
183C22×C4—611no—
184C22×C4—1221no—
185C22×C4—1221no—
186C22×C4C22⋊#692312yes—
187C22×C4C22×#692311yescharacteristic
188M16—1228no—
189M16—2444no—
190C2×C8—2444no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—1228no—
194C4⋊C4—2442no—
195C4⋊C4—2442no—
196C42—21no—
197C42—1222no—
198C42—1222no—
199C42—1222no—
200C42—2441no—
201C42—2442no—
202C42C4×#692312yes—
203C23C2×C22⋊#692311yescharacteristic
204Q8—2442no—
205Q8—2442no—
206Q8—2442no—
207Q8—2442no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—612no—
218C2×C4—614no—
219C2×C4—1222no—
220C2×C4—1222no—
221C2×C4—1221no—
222C2×C4—1222no—
223C2×C4—2441no—
224C2×C4—2441no—
225C2×C4—2441no—
226C2×C4—2441no—
227C2×C4—2441no—
228C2×C4—2441no—
229C2×C4—2441no—
230C2×C4C2×C4×#692311yescharacteristic
231C2×C4#1451985611yescharacteristic
232C8—2448no—
233C22—21no—
234C22—21no—
235C22C4×C22⋊#692311yescharacteristic
236C22D61⋊#9911yescharacteristic
237C22#1684074311yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—1224no—
244C4—1222no—
245C4—4881no—
246C4—2442no—
247C4—2441no—
248C2—21no—
249C2—21no—
250C2#2146156411yescharacteristic
251C2#2146211511yescharacteristic
252C2#2146212211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#382C4
socleC22×C61C2×#99
prime radicalC23×C61C2×C22⋊C4
cosocleC22×Dic122C23
hypercenter#382#6923
hypocenterC61#12631
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #382

Complements

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

NH ≅ G/Ntype
7. #16840673C4semidirect
144. C61#12631semidirect

Maximal subgroups

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

subgroupindex
C2×#168406732
#214615722
#214609172
#214485362
#1263161