Finite Group Databasebeta

← #29589328

Subgroups — #29589328

Order 15616.

430 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 109 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#21447615C212yesmaximal, complemented
3#21443743C211yesmaximal, characteristic, complemented
4#21443742C212yesmaximal, complemented
5C61×#1314C211yesmaximal, characteristic, complemented
6C23×#92418C411yescharacteristic
7C22×C8×D61C411yescharacteristic
8C2×C4×C22⋊D61C411yescharacteristic
9#16839825—22no—
10#16839825—22no—
11C2×#14519706C411yescharacteristic
12C2×#14519689C2211yescharacteristic
13C2×C8×Dic61C2211yescharacteristic
14#16839580—22no—
15C22⋊#11758185C2212yes—
16C16×Dic61—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—21no—
27C2×C8×D61—21no—
28C2×C8×D61—42no—
29C4×C22⋊D61—22no—
30C4×C22⋊D61—22no—
31C2×#11758196C811yescharacteristic
32C2×C22⋊Dic61C811yescharacteristic
33C2×#11758198C811yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35C2×#11758185—22no—
36#14519706—22no—
37#14519689—22no—
38C8×Dic61D412yes—
39C8×Dic61D412yes—
40C23×C4×C61C2×C411yescharacteristic
41C22×C8×C61C2×C411yescharacteristic
42C22×C8×C61C2311yescharacteristic
43C2×C16×C61—22no—
44C22⋊C8×C61—22no—
45C24×C61C2×C811yescharacteristic
46C23×D61C2×C811yescharacteristic
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—41no—
50C22×C4×C61—41no—
51C22×C4×C61C2×C811yescharacteristic
52C22×C4×C61C22×C411yescharacteristic
53C2×C22⋊D61—22no—
54C2×C22⋊D61—22no—
55C2×C22⋊D61C1614yescomplemented
56C22×Dic61C2×C811yescharacteristic
57C22×Dic61C2×C811yescharacteristic
58C22×Dic61C2×C811yescharacteristic
59C2×C4×D61—21no—
60C2×C4×D61—21no—
61C2×C4×D61—41no—
62C2×C4×D61—41no—
63C2×C8×C61—21no—
64C2×C8×C61—41no—
65C2×C8×C61—41no—
66C2×C8×C61C2×D412yes—
67C22⋊Dic61—22no—
68#11758198—22no—
69#11758196—22no—
70C4×Dic61C22⋊C412yes—
71C4×Dic61M1612yes—
72C2×#92418—22no—
73C2×#92418—21no—
74C2×#92418—42no—
75C2×#92418—42no—
76C2×#92418—41no—
77C2×#92418C22⋊C411yescharacteristic
78C2×#92418C22⋊C411yescharacteristic
79C8×D61—42no—
80C8×D61—42no—
81C16×C61—42no—
82#11758185—42no—
83C23×C61—21no—
84C23×C61—21no—
85C23×C61—41no—
86C23×C61—41no—
87C23×C61C2×C1612yes—
88C23×C61C22×C811yescharacteristic
89C22×D61—21no—
90C22×D61—21no—
91C22×D61—41no—
92C22×D61—41no—
93C22×D61C2×C1612yes—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—42no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61C2×C22⋊C411yescharacteristic
102C2×C4×C61C2×M1611yescharacteristic
103C22⋊D61—28no—
104C2×Dic61—21no—
105C2×Dic61—41no—
106C2×Dic61—41no—
107C2×Dic61C22⋊C812yes—
108C2×Dic61C22⋊C812yes—
109C2×Dic61C2×C1612yes—
110C2×Dic61M3212yes—
111C4×D61—42no—
112C4×D61—41no—
113C4×D61—41no—
114C8×C61—22no—
115C8×C61—41no—
116#92418—21no—
117#92418—41no—
118#92418#10012yes—
119#11487—611nomaximal, Sylow
120C22×C61—21no—
121C22×C61—21no—
122C22×C61—22no—
123C22×C61—42no—
124C22×C61—42no—
125C22×C61—41no—
126C22×C61—41no—
127C22×C61C2×C22⋊C811yescharacteristic
128C22×C61C22×C1611yescharacteristic
129C22×C61C2×M3211yescharacteristic
130C2×D61—22no—
131C2×D61—42no—
132C2×D61—41no—
133C2×D61—41no—
134C2×D61—42no—
135C4×C61—21no—
136C4×C61—41no—
137C4×C61#41011yescharacteristic
138C4×C61C2×#10011yescharacteristic
139Dic61—22no—
140Dic61—41no—
141Dic61C22⋊C1614yes—
142C2×C8×D4—611no—
143#2162—612no—
144#1314—611no—
145#1314—612no—
146#1314D6111yescharacteristic, complemented
147C2×C61—21no—
148C2×C61—21no—
149C2×C61—42no—
150C2×C61#146011yescharacteristic
151C2×C61C2×C22⋊C1611yescharacteristic
152C2×C61#211511yescharacteristic
153D61—42no—
154C23×C8—611no—
155C23×C8—611no—
156C2×C4×D4—611no—
157C8×D4—1222no—
158C8×D4—1222no—
159C2×C4⋊C8—611no—
160C2×C22⋊C8—611no—
161C2×C22⋊C8C2×D6111yescharacteristic
162C2×C4×C8—611no—
163C4⋊C16—1222no—
164C22⋊C16—612no—
165C22⋊C16C2×D6112yes—
166C4×C16—1222no—
167C61#1148711yescharacteristic, complemented, Sylow
168C22×D4—611no—
169C23×C4—611no—
170C23×C4C4×D6111yescharacteristic
171C22×C8—611no—
172C22×C8—611no—
173C22×C8—1221no—
174C22×C8—1221no—
175C22×C8—1221no—
176C22×C8—1221no—
177C22×C8—2442no—
178C22×C8—2442no—
179C22×C8C4×D6111yescharacteristic
180C22×C8C22×D6111yescharacteristic
181C4×D4—1222no—
182C4×D4—1222no—
183C2×C4⋊C4—611no—
184C2×C22⋊C4—611no—
185C2×C22⋊C4—611no—
186C2×C42—611no—
187C2×C16—22no—
188C2×C16—1222no—
189C4⋊C8—1222no—
190C22⋊C8—22no—
191C22⋊C8—1222no—
192C4×C8—612no—
193C4×C8—612no—
194C24—611no—
195C24C8×D6111yescharacteristic
196C2×D4—614no—
197C2×D4—1222no—
198C2×D4—1222no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—611no—
204C22×C4—611no—
205C22×C4—611no—
206C22×C4—1221no—
207C22×C4—1221no—
208C22×C4—2441no—
209C22×C4—2441no—
210C22×C4C8×D6111yescharacteristic
211C22×C4C2×C4×D6111yescharacteristic
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—611no—
216C2×C8—611no—
217C2×C8—1222no—
218C2×C8—1221no—
219C2×C8—2442no—
220C2×C8—2442no—
221C2×C8—2442no—
222C2×C8—2442no—
223C2×C8—2441no—
224C2×C8D4×D6112yes—
225C4⋊C4—1222no—
226C22⋊C4—1222no—
227C22⋊C4—1222no—
228C42—612no—
229C42—612no—
230C16—42no—
231C16—2442no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23—612no—
237C23—1221no—
238C23—1221no—
239C23—2441no—
240C23—2441no—
241C23C16×D6112yes—
242C23C2×C8×D6111yescharacteristic
243D4—1228no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—612no—
252C2×C4—612no—
253C2×C4—612no—
254C2×C4—612no—
255C2×C4—1221no—
256C2×C4—2442no—
257C2×C4—2441no—
258C2×C4—2441no—
259C2×C4—2441no—
260C2×C4—2441no—
261C2×C4C22⋊C4×D6111yescharacteristic
262C2×C4M16×D6111yescharacteristic
263C8—22no—
264C8—41no—
265C8—612no—
266C8—1221no—
267C8—2441no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—1222no—
276C22—2442no—
277C22—2442no—
278C22—2441no—
279C22—2441no—
280C22C22⋊C8×D6111yescharacteristic
281C22C2×C16×D6111yescharacteristic
282C22M32×D6111yescharacteristic
283C4—21no—
284C4—41no—
285C4—614no—
286C4—1222no—
287C4—2441no—
288C4#1683973411yescharacteristic
289C4#100×D6111yescharacteristic
290C2—21no—
291C2—21no—
292C2—42no—
293C2—2442no—
294C2#2144413811yescharacteristic
295C2C22⋊C16×D6111yescharacteristic
296C2#2144743811yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC22×C61C22×C16
FrattiniC22×C8C22×D61
FittingC61×#1314C2
socleC22×C61C2×C22⋊C8
prime radicalC22×C22⋊D61C8
cosocleC22×C8×C61C23
hypercenter#1314D61
hypocenterC61#11487
LCSG ▷ C22×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. #21447615C2semidirect
3. #21443743C2semidirect
4. #21443742C2semidirect
5. C61×#1314C2semidirect
55. C2×C22⋊D61C16semidirect
146. #1314D61semidirect
167. C61#11487semidirect

Maximal subgroups

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

subgroupindex
C2×#168398252
#214476152
#214437432
#214437422
C61×#13142
#1148761