Finite Group Databasebeta

← #29589238

Subgroups — #29589238

Order 15616.

438 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16839825C211yesmaximal, characteristic
2#21447428C212yesmaximal, complemented
3#21443758C212yesmaximal, complemented
4#21443743C211yesmaximal, characteristic, complemented
5C61×#1314C211yesmaximal, characteristic, complemented
6C23×#92418C411yescharacteristic
7C22×C8×D61C2211yescharacteristic
8C2×C4×C22⋊D61C411yescharacteristic
9C2×#14519716—22no—
10#16839825—22no—
11#16839825—22no—
12C2×#14519706C2211yescharacteristic
13C2×#14519689C411yescharacteristic
14C2×C8×Dic61C411yescharacteristic
15C22⋊#11758185C2212yes—
16#16839541—22no—
17C2×C22⋊C8×C61C2211yescharacteristic
18C61×C22⋊C16C2212yes—
19C22×C22⋊D61C811yescharacteristic
20C22×C4×D61C2×C411yescharacteristic
21C22×#92418—21no—
22C22×#92418—21no—
23C22×#92418—42no—
24C22×#92418C2×C411yescharacteristic
25C22×#92418C2×C411yescharacteristic
26C2×C8×D61—42no—
27C2×C8×D61D412yes—
28C2×C8×D61D412yes—
29C4×C22⋊D61—22no—
30C4×C22⋊D61—22no—
31C2×#11758196C811yescharacteristic
32C2×C22⋊Dic61C811yescharacteristic
33C2×#11758198C811yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35#14519716—44no—
36C2×#11758185—22no—
37#14519706—22no—
38#14519689—22no—
39C8×Dic61—21no—
40C8×Dic61—21no—
41C23×C4×C61C2×C411yescharacteristic
42C22×C8×C61C2×C411yescharacteristic
43C22×C8×C61C2311yescharacteristic
44C2×C16×C61—22no—
45C22⋊C8×C61—22no—
46C24×C61C2×C811yescharacteristic
47C23×D61C2×C811yescharacteristic
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61—41no—
51C22×C4×C61—41no—
52C22×C4×C61C2×C811yescharacteristic
53C22×C4×C61C22×C411yescharacteristic
54C2×C22⋊D61—22no—
55C2×C22⋊D61—22no—
56C2×C22⋊D61—22no—
57C22×Dic61C2×C811yescharacteristic
58C22×Dic61C2×C811yescharacteristic
59C22×Dic61C2×C811yescharacteristic
60C2×C4×D61—41no—
61C2×C4×D61—41no—
62C2×C4×D61C22⋊C412yes—
63C2×C4×D61M1612yes—
64C2×C8×C61—21no—
65C2×C8×C61—41no—
66C2×C8×C61—41no—
67C2×C8×C61C2×D412yes—
68C22⋊Dic61—22no—
69#11758198—22no—
70#11758196—22no—
71C4×Dic61—21no—
72C4×Dic61—21no—
73C2×#92418—22no—
74C2×#92418—21no—
75C2×#92418—42no—
76C2×#92418—42no—
77C2×#92418—41no—
78C2×#92418C22⋊C411yescharacteristic
79C2×#92418C22⋊C411yescharacteristic
80C8×D61—24no—
81C8×D61—42no—
82C16×C61—42no—
83#11758185—42no—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—41no—
87C23×C61—41no—
88C23×C61M3212yescomplemented
89C23×C61C22×C811yescharacteristic
90C22×D61—21no—
91C22×D61—21no—
92C22×D61—41no—
93C22×D61—41no—
94C22×D61C22⋊C812yes—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—42no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61C2×C22⋊C411yescharacteristic
103C2×C4×C61C2×M1611yescharacteristic
104C22⋊D61—28no—
105C2×Dic61—21no—
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61—41no—
109C2×Dic61—41no—
110C2×Dic61C22⋊C812yes—
111C2×Dic61M3212yescomplemented
112C4×D61—22no—
113C4×D61—22no—
114C4×D61—42no—
115C8×C61—22no—
116C8×C61—41no—
117#92418—21no—
118#92418—41no—
119#92418#9912yes—
120#11466—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—22no—
124C22×C61—42no—
125C22×C61—42no—
126C22×C61—41no—
127C22×C61—41no—
128C22×C61C2×C22⋊C811yescharacteristic
129C22×C61C2×M3211yescharacteristic
130C22×C61C2×M3211yescharacteristic
131C2×D61—22no—
132C2×D61—22no—
133C2×D61—22no—
134C2×D61—42no—
135C2×D61—42no—
136C4×C61—21no—
137C4×C61—41no—
138C4×C61C2×#9911yescharacteristic
139C4×C61#41311yescharacteristic
140Dic61—22no—
141Dic61—22no—
142Dic61—41no—
143C2×C8×D4—611no—
144#2112—612no—
145#1322—612no—
146#1314—611no—
147#1314D6111yescharacteristic, complemented
148C2×C61—21no—
149C2×C61—21no—
150C2×C61—42no—
151C2×C61#145911yescharacteristic
152C2×C61#211211yescharacteristic
153C2×C61#211511yescharacteristic
154D61—42no—
155C23×C8—611no—
156C23×C8—611no—
157C2×C4×D4—611no—
158C2×M32—1222no—
159C8×D4—1222no—
160C8×D4—1222no—
161C2×C4⋊C8—611no—
162C2×C22⋊C8—611no—
163C2×C22⋊C8C2×D6111yescharacteristic
164C2×C4×C8—611no—
165C22⋊C16—612no—
166C22⋊C16—1222no—
167C22⋊C16C2×D6112yes—
168C61#1146611yescharacteristic, complemented, Sylow
169C22×D4—611no—
170C23×C4—611no—
171C23×C4C4×D6111yescharacteristic
172C22×C8—611no—
173C22×C8—612no—
174C22×C8—612no—
175C22×C8—611no—
176C22×C8—1221no—
177C22×C8—1221no—
178C22×C8—2442no—
179C22×C8—2442no—
180C22×C8C4×D6111yescharacteristic
181C22×C8C22×D6111yescharacteristic
182C4×D4—1222no—
183C4×D4—1222no—
184C2×C4⋊C4—611no—
185C2×C22⋊C4—611no—
186C2×C22⋊C4—611no—
187C2×C42—611no—
188M32—2444no—
189C2×C16—22no—
190C2×C16—1222no—
191C4⋊C8—1222no—
192C22⋊C8—22no—
193C22⋊C8—1222no—
194C4×C8—1221no—
195C4×C8—1221no—
196C24—611no—
197C24C8×D6111yescharacteristic
198C2×D4—1222no—
199C2×D4—1222no—
200C2×D4—1222no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—611no—
206C22×C4—611no—
207C22×C4—612no—
208C22×C4—612no—
209C22×C4—611no—
210C22×C4—2441no—
211C22×C4—2441no—
212C22×C4C8×D6111yescharacteristic
213C22×C4C2×C4×D6111yescharacteristic
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—611no—
218C2×C8—611no—
219C2×C8—1224no—
220C2×C8—1222no—
221C2×C8—1221no—
222C2×C8—2442no—
223C2×C8—2442no—
224C2×C8—2442no—
225C2×C8—2441no—
226C2×C8D4×D6112yes—
227C4⋊C4—1222no—
228C22⋊C4—1222no—
229C22⋊C4—1222no—
230C42—1221no—
231C42—1221no—
232C16—42no—
233C16—2442no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—612no—
239C23—1221no—
240C23—1221no—
241C23—2441no—
242C23—2441no—
243C23#1451971612yescomplemented
244C23C2×C8×D6111yescharacteristic
245D4—1228no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—612no—
254C2×C4—612no—
255C2×C4—1221no—
256C2×C4—1221no—
257C2×C4—1222no—
258C2×C4—1222no—
259C2×C4—1221no—
260C2×C4—2442no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4C22⋊C4×D6111yescharacteristic
264C2×C4M16×D6111yescharacteristic
265C8—22no—
266C8—41no—
267C8—612no—
268C8—1221no—
269C8—2441no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—1222no—
278C22—1222no—
279C22—1222no—
280C22—2442no—
281C22—2442no—
282C22C22⋊C8×D6111yescharacteristic
283C22M32×D6111yescharacteristic
284C22C2×#1451971611yescharacteristic
285C4—21no—
286C4—41no—
287C4—1222no—
288C4—1222no—
289C4—2441no—
290C4#99×D6111yescharacteristic
291C4#1683974311yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—2442no—
296C2#2144413411yescharacteristic
297C2#2144742811yescharacteristic
298C2#2144743811yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC23×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#1314C2
socleC22×C61C2×C22⋊C8
prime radicalC22×C22⋊D61C8
cosocleC22×C8×C61C23
hypercenter#1314D61
hypocenterC61#11466
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #21447428C2semidirect
3. #21443758C2semidirect
4. #21443743C2semidirect
5. C61×#1314C2semidirect
88. C23×C61M32semidirect
111. C2×Dic61M32semidirect
147. #1314D61semidirect
168. C61#11466semidirect
243. C23#14519716semidirect

Maximal subgroups

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

subgroupindex
C2×#168398252
#214474282
#214437582
#214437432
C61×#13142
#1146661