Finite Group Databasebeta

← #30598918

Subgroups — #30598918

Order 15616.

540 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 106 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#21461604C212yesmaximal, complemented
2C2×#16840618C212yesmaximal
3#21460695C212yesmaximal, complemented
4C4×#14519766C211yesmaximal, characteristic
5C2×#14519844—22no—
6C2×#14519832C2214yes—
7#16840628—22no—
8#16840618—24no—
9C2×#14519766C2211yescharacteristic
10C2×C4×D244C411yescharacteristic
11#16839669C411yescharacteristic
12#16839663C411yescharacteristic
13#16839652—22no—
14#16839627C2211yescharacteristic
15C4×#11758198—22no—
16C42×Dic61C2211yescharacteristic
17C2×C4×Q8×C61C411yescharacteristic
18C22×#92420—24no—
19#14519844—28no—
20#14519832—24no—
21#14519832—24no—
22C22×D244C2×C411yescharacteristic
23C22×C4×D61—21no—
24#14519766—22no—
25#14519766D412yes—
26#14519766Q812yes—
27C4×D244—42no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×#11758198—22no—
30C2×#11758198—21no—
31C2×#11758198—22no—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61D412yes—
34C2×C4×Dic61D412yes—
35C2×C4×Dic61C2311yescharacteristic
36#14519684—22no—
37#14519684—22no—
38C22×Q8×C61C2×C411yescharacteristic
39C4×Q8×C61—42no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C42×C61—21no—
43C2×C42×C61C2×C411yescharacteristic
44C23×D61—21no—
45C2×Q8×C61—21no—
46C2×Q8×C61—41no—
47C2×Q8×C61—41no—
48C2×Q8×C61C4⋊C412yes—
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61C22⋊C411yescharacteristic
52C22×C4×C61C22⋊C411yescharacteristic
53C22×C4×C61C22×C411yescharacteristic
54C22×Dic61—21no—
55C22×Dic61C22⋊C412yes—
56C22×Dic61C2×D412yes—
57C2×D244—21no—
58C2×D244—41no—
59C2×D244—41no—
60C2×D244C4⋊C412yes—
61C2×C4×D61—44no—
62C2×C4×D61—42no—
63C2×#92420—28no—
64C2×#92420—44no—
65C2×#92420—44no—
66C4⋊C4×C61—42no—
67C4⋊C4×C61—41no—
68C42×C61—22no—
69C42×C61—42no—
70#11758198—24no—
71#11758198—24no—
72#11758198—42no—
73#11758198—44no—
74C4⋊Dic61—41no—
75C4×Dic61—24no—
76C4×Dic61—41no—
77C4×Dic61—42no—
78C4×Dic61C2×D411yescharacteristic
79C4×Dic61C2×Q811yescharacteristic
80C4×Dic61#4112yes—
81C23×C61C2×C22⋊C411yescharacteristic
82C22×D61—22no—
83C22×D61—41no—
84C22×D61—41no—
85C22×D61—42no—
86C22×D61—42no—
87Q8×C61—42no—
88Q8×C61—42no—
89C2×C4×C61—21no—
90C2×C4×C61—21no—
91C2×C4×C61—22no—
92C2×C4×C61—22no—
93C2×C4×C61—21no—
94C2×C4×C61—22no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61—41no—
98C2×C4×C61—42no—
99C2×C4×C61#10012yes—
100C2×C4×C61C2×C4⋊C411yescharacteristic
101C2×C4×C61#11711yescharacteristic
102C2×Dic61—22no—
103C2×Dic61—22no—
104C2×Dic61—22no—
105C2×Dic61—22no—
106C2×Dic61—41no—
107C2×Dic61—42no—
108C2×Dic61#10414yes—
109C2×Dic61#12112yes—
110C2×Dic61C22⋊Q812yes—
111D244—42no—
112D244—42no—
113C4×D61—84no—
114#92420—48no—
115#8583—611nomaximal, Sylow
116C22×C61#38011yescharacteristic
117C22×C61C2×#10011yescharacteristic
118C22×C61#41311yescharacteristic
119C22×C61C2×#10412yes—
120C22×C61#42112yes—
121C2×D61—42no—
122C2×D61—44no—
123C2×D61—81no—
124C2×D61—82no—
125C2×D61—81no—
126C4×C61—21no—
127C4×C61—21no—
128C4×C61—22no—
129C4×C61—42no—
130C4×C61—41no—
131C4×C61—81no—
132Dic61—22no—
133Dic61—22no—
134Dic61—24no—
135Dic61—81no—
136C4×#124—611no—
137#1767—612no—
138C2×#329—612no—
139#1295—612no—
140C2×C61#152511yescharacteristic
141C2×C61#153212yes—
142C2×C61#153311yescharacteristic
143C2×C61#180011yescharacteristic
144C2×C61#180612yes—
145D61—82no—
146C2×#124—611no—
147C2×C4×Q8#692311yescharacteristic
148C2×C4×D4—611no—
149C2×C4⋊C8—1222no—
150C2×#97—614no—
151#391—611no—
152#390—611no—
153#388—1222no—
154#382—611no—
155C4×C22⋊C4—1222no—
156C43—611no—
157#336—1222no—
158#329—1224no—
159C61#858311yescharacteristic, complemented, Sylow
160C22×Q8C2×#692311yescharacteristic
161C22×D4—611no—
162C23×C4—1221no—
163C22×C8—1224no—
164#124—612no—
165#124—612no—
166#124—1222no—
167C4×Q8—42no—
168C4×D4—2442no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—611no—
171C2×C4⋊C4C2×#692311yescharacteristic
172C2×C22⋊C4—1222no—
173C2×C22⋊C4—1222no—
174C2×C22⋊C4—1221no—
175C2×C42—21no—
176C2×C42—611no—
177C2×C42—612no—
178C2×C42—612no—
179C2×C42—611no—
180C2×C42C2×#692311yescharacteristic
181C4⋊C8—1228no—
182#97—1224no—
183#97—1224no—
184#95—1222no—
185#95—1222no—
186C24—1221no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8C4⋊#692312yes—
191C2×D4—612no—
192C2×D4—1221no—
193C2×D4—2441no—
194C2×D4—2441no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—612no—
198C22×C4—612no—
199C22×C4—1221no—
200C22×C4—2444no—
201C22×C4—2442no—
202C22×C4C22⋊#692311yescharacteristic
203C22×C4C22⋊#692311yescharacteristic
204C22×C4C22×#692311yescharacteristic
205C2×C8—1228no—
206C2×C8—2444no—
207C2×C8—2444no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—2441no—
211C22⋊C4—1224no—
212C22⋊C4—1224no—
213C22⋊C4—2442no—
214C22⋊C4—2444no—
215C42—22no—
216C42—42no—
217C42—611no—
218C42—611no—
219C42—612no—
220C42—1224no—
221C42—2441no—
222C42—2442no—
223C23—1222no—
224C23—2441no—
225C23—2441no—
226C23—2442no—
227C23—2442no—
228C23C2×C22⋊#692311yescharacteristic
229Q8—42no—
230Q8—42no—
231D4—2442no—
232D4—2442no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—612no—
244C2×C4—612no—
245C2×C4—614no—
246C2×C4—1222no—
247C2×C4—1222no—
248C2×C4—1222no—
249C2×C4—1222no—
250C2×C4—4884no—
251C2×C4—2441no—
252C2×C4—2442no—
253C2×C4#1451983712yes—
254C2×C4C2×C4⋊#692311yescharacteristic
255C2×C4#1451985611yescharacteristic
256C8—2448no—
257C22—4881no—
258C22—4882no—
259C22—4881no—
260C22—2442no—
261C22—2444no—
262C22#1684070411yescharacteristic
263C22C2×#1451983711yescharacteristic
264C22#1684074811yescharacteristic
265C22D61⋊#10412yes—
266C22#1684076112yes—
267C4—22no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—42no—
272C4—81no—
273C4—1224no—
274C4—1222no—
275C4—1222no—
276C4—4881no—
277C2—4882no—
278C2#2146127311yescharacteristic
279C2#2146129612yes—
280C2#2146130011yescharacteristic
281C2#2146168311yescharacteristic
282C2#2146169412yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8583
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21461604C2semidirect
3. #21460695C2semidirect
159. C61#8583semidirect

Maximal subgroups

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

subgroupindex
#214616042
C2×#168406182
#214606952
C4×#145197662
#858361