Finite Group Databasebeta

← #30598257

Subgroups — #30598257

Order 15616.

802 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 132 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#21461675C212yesmaximal
2C2×#16840621C214yesmaximal, complemented
3C2×#16840315C211yesmaximal, characteristic
4C2×#14519844C2214yes—
5C2×C22⋊#92420—22no—
6#16840628—22no—
7#16840621—28no—
8#16840315—28no—
9C2×#14519776C2212yes—
10C2×#14519768C412yes—
11C2×#14519754—22no—
12C2×C4×Dic122—22no—
13C2×#14519736C411yescharacteristic
14C22×C4×Dic61C2211yescharacteristic
15C2×#117×C61C411yescharacteristic
16C22×#92420—24no—
17#14519844—28no—
18C22⋊#92420—44no—
19C22×Dic122—22no—
20C23×Dic61D412yes—
21#14519776—24no—
22#14519776D418yes—
23#14519768—24no—
24#14519768—24no—
25#14519754—216no—
26C4×Dic122—216no—
27#14519736—24no—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61C2×C412yes—
30C2×#11758196—22no—
31C2×#11758196—22no—
32C2×C22⋊Dic61—21no—
33C2×C4×Dic61—21no—
34C2×C4×Dic61—24no—
35C2×C4×Dic61C2×C411yescharacteristic
36C2×C4×Dic61D412yes—
37C2×C4×Dic61C2311yescharacteristic
38C23×C4×C61C2×C411yescharacteristic
39#117×C61—24no—
40C2×C4⋊C4×C61C2×C412yes—
41C2×C22⋊C4×C61—21no—
42C2×C42×C61—21no—
43C24×C61C22⋊C411yescharacteristic
44C22×C4×C61—22no—
45C22×C4×C61—22no—
46C22×C4×C61—22no—
47C22×C4×C61—22no—
48C22×C4×C61C22⋊C411yescharacteristic
49C22×C4×C61C22×C411yescharacteristic
50C22×Dic61—22no—
51C22×Dic61—24no—
52C22×Dic61—24no—
53C22×Dic61—22no—
54C22×Dic61—22no—
55C22×Dic61C22⋊C412yes—
56C22×Dic61C2×D412yes—
57C2×Dic122—28no—
58C2×Dic122—44no—
59C2×Dic122—42no—
60C2×Dic122—42no—
61C2×#92420—44no—
62C2×#92420—44no—
63C2×#92420—44no—
64C4⋊C4×C61—22no—
65C4⋊C4×C61C22⋊C414yes—
66C22⋊C4×C61—24no—
67C42×C61—24no—
68C22⋊Dic61—24no—
69C4⋊Dic61—28no—
70C4⋊Dic61—22no—
71C4⋊Dic61C22⋊C414yes—
72#11758196—28no—
73#11758196—28no—
74C4×Dic61—24no—
75C4×Dic61—22no—
76C4×Dic61—24no—
77C4×Dic61C2×D414yes—
78C23×C61—21no—
79C23×C61—21no—
80C23×C61—22no—
81C23×C61—22no—
82C23×C61#10112yes—
83C23×C61C2×C22⋊C411yescharacteristic
84C2×C4×C61—28no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—24no—
90C2×C4×C61—42no—
91C2×C4×C61—42no—
92C2×C4×C61#10112yes—
93C2×C4×C61C2×C22⋊C412yes—
94C2×Dic61—22no—
95C2×Dic61—24no—
96C2×Dic61—22no—
97C2×Dic61—22no—
98C2×Dic61—42no—
99C2×Dic61—42no—
100C2×Dic61—44no—
101C2×Dic61—44no—
102C2×Dic61#12014yes—
103C2×Dic61#13514yes—
104C2×Dic61AGL(1,ℤ/8)12yes—
105C2×Dic61#13712yes—
106Dic122—48no—
107Dic122—48no—
108#92420—84no—
109#8345—611nomaximal, Sylow
110C22×C61—24no—
111C22×C61—22no—
112C22×C61—41no—
113C22×C61—41no—
114C22×C61—42no—
115C22×C61#37911yescharacteristic
116C22×C61C2×#10111yescharacteristic
117C22×C61C2×#10111yescharacteristic
118C22×C61#41612yes—
119C22×C61#41712yes—
120C4×C61—22no—
121C4×C61—22no—
122C4×C61—44no—
123Dic61—44no—
124Dic61—42no—
125Dic61#44914yes—
126Dic61#45214yes—
127C2×#533—611no—
128#1797—612no—
129C2×#332—614no—
130C2×C61—42no—
131C2×C61#154411yescharacteristic
132C2×C61#155212yes—
133C2×C61#156111yescharacteristic
134C2×C61#178411yescharacteristic
135C2×C61#178812yes—
136#533—1228no—
137C2×C4⋊Q8—612no—
138C2×#125—612no—
139C2×C22⋊Q8—1222no—
140C2×C4×Q8—1222no—
141C2×#117—611no—
142C2×#117#692311yescharacteristic
143C22×C42—611no—
144C2×C4⋊C8—614no—
145C2×C22⋊C8—1222no—
146#336—1222no—
147#332—1228no—
148C61#834511yescharacteristic, complemented, Sylow
149C22×Q8—1222no—
150C23×C4—612no—
151C23×C4C2×#692311yescharacteristic
152C22×C8—1224no—
153C4⋊Q8—618no—
154C4⋊Q8—1224no—
155#125—1224no—
156#125—1224no—
157C22⋊Q8—12216no—
158C4×Q8—12216no—
159#117—24no—
160#117—1224no—
161C2×C4⋊C4—612no—
162C2×C4⋊C4—1222no—
163C2×C4⋊C4—1222no—
164C2×C4⋊C4—1222no—
165C2×C4⋊C4C2×#692312yes—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—1221no—
168C2×C42—21no—
169C2×C42—611no—
170C2×C42—612no—
171C2×C42—611no—
172C2×C42—1224no—
173C2×C42—1221no—
174C4⋊C8—1228no—
175C22⋊C8—2444no—
176C24C22⋊#692311yescharacteristic
177C2×Q8—1228no—
178C2×Q8—2444no—
179C2×Q8—2442no—
180C2×Q8—2442no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—612no—
186C22×C4—612no—
187C22×C4—1222no—
188C22×C4—1224no—
189C22×C4—1224no—
190C22×C4—1222no—
191C22×C4—1222no—
192C22×C4C22⋊#692311yescharacteristic
193C22×C4C22×#692311yescharacteristic
194C2×C8—2444no—
195C2×C8—2444no—
196C2×C8—2444no—
197C4⋊C4—22no—
198C4⋊C4—614no—
199C4⋊C4—1228no—
200C4⋊C4—1228no—
201C4⋊C4—1228no—
202C4⋊C4—1222no—
203C4⋊C4C22⋊#692314yes—
204C22⋊C4—24no—
205C22⋊C4—1224no—
206C42—24no—
207C42—614no—
208C42—1224no—
209C42—1224no—
210C42—1222no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23—21no—
215C23#1451983812yes—
216C23C2×C22⋊#692311yescharacteristic
217Q8—2448no—
218Q8—2448no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—614no—
228C2×C4—612no—
229C2×C4—612no—
230C2×C4—614no—
231C2×C4—1222no—
232C2×C4—1222no—
233C2×C4—1222no—
234C2×C4—1224no—
235C2×C4—2444no—
236C2×C4—2444no—
237C2×C4—2442no—
238C2×C4—2442no—
239C2×C4#1451983812yes—
240C2×C4C2×C22⋊#692312yes—
241C8—4884no—
242C22—24no—
243C22—22no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22#1684070311yescharacteristic
248C22C2×#1451983811yescharacteristic
249C22C2×#1451983811yescharacteristic
250C22#1684075212yes—
251C22#1684075312yes—
252C4—22no—
253C4—22no—
254C4—44no—
255C4—614no—
256C4—614no—
257C4—2442no—
258C4—2444no—
259C2—42no—
260C2#2146132411yescharacteristic
261C2#2146134012yes—
262C2#2146135811yescharacteristic
263C2#2146165311yescharacteristic
264C2#2146165912yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×#117#6923
hypocenterC61#8345
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#16840621C2semidirect
148. C61#8345semidirect

Maximal subgroups

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

subgroupindex
#214616752
C2×#168406212
C2×#168403152
#834561