Finite Group Databasebeta

← #29593233

Subgroups — #29593233

Order 15616.

396 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 67 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#21456570C211yesmaximal, characteristic
2#21448005C212yesmaximal, complemented
3#21447247C211yesmaximal, characteristic, complemented
4#21443829C212yesmaximal, complemented
5C61×#1314C211yesmaximal, characteristic, complemented
6C2×#14519751—21no—
7#16840076—22no—
8#16839998—21no—
9#16839998C2212yes—
10#16839898—22no—
11C2×#14519711C2211yescharacteristic, complemented
12#16839622—21no—
13#16839590—22no—
14#16839564—22no—
15#16839525—21no—
16#16839525C2211yescharacteristic, complemented
17C2×C22⋊C8×C61C2211yescharacteristic
18C61×C22⋊C16C2212yescomplemented
19C22×D244—21no—
20C22×C22⋊D61—41no—
21C2×D488—21no—
22C2×D488D412yes—
23#14519751—22no—
24#14519751—22no—
25#14519751—41no—
26#14519751—41no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—21no—
30C2×C22⋊Dic61—41no—
31C2×#11758198—41no—
32#14519711—21no—
33#14519711—22no—
34#14519711D412yes—
35#14519699—22no—
36#14519699—22no—
37#14519684—41no—
38C23×C4×C61D411yescharacteristic
39C22×C8×C61D411yescharacteristic
40C22×C8×C61C2311yescharacteristic
41C2×C16×C61—22no—
42C22⋊C8×C61—22no—
43C24×C61D811yescharacteristic
44C23×D61—41no—
45C22×C4×C61—21no—
46C22×C4×C61—21no—
47C22×C4×C61—41no—
48C22×C4×C61—41no—
49C22×C4×C61D811yescharacteristic
50C22×C4×C61C2×D411yescharacteristic
51C2×C22⋊D61—42no—
52C2×C22⋊D61—42no—
53C2×C22⋊D61—82no—
54C2×C22⋊D61—82no—
55C22×Dic61—41no—
56C22×Dic61—41no—
57C22×Dic61—41no—
58C2×D244—22no—
59C2×D244—41no—
60C2×D244—81no—
61C2×D244—81no—
62C2×C8×C61—21no—
63C2×C8×C61—41no—
64C2×C8×C61—41no—
65C2×C8×C61C2×D412yes—
66C22⋊Dic61—82no—
67#11758198—42no—
68#11758198—42no—
69C4⋊Dic61—22no—
70C4⋊Dic61—41no—
71C4⋊Dic61—41no—
72C4⋊Dic61—41no—
73C4⋊Dic61—41no—
74C4⋊Dic61—41no—
75D488—42no—
76D488—81no—
77C16×C61—42no—
78C23×C61—21no—
79C23×C61—21no—
80C23×C61—41no—
81C23×C61—41no—
82C23×C61SD3212yes—
83C23×C61C2×D811yescharacteristic
84C22×D61—42no—
85C22×D61—82no—
86C22×D61—82no—
87C22×D61—82no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—21no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61—81no—
95C2×C4×C61#12011yescharacteristic
96C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
97C22⋊D61—84no—
98C22⋊D61—82no—
99C22⋊D61—82no—
100C2×Dic61—42no—
101C2×Dic61—82no—
102C2×Dic61—81no—
103C2×Dic61—81no—
104C2×Dic61—81no—
105C2×Dic61—81no—
106C2×Dic61—81no—
107C2×Dic61—81no—
108D244—42no—
109D244—42no—
110D244—82no—
111C8×C61—22no—
112C8×C61—41no—
113#12281—611nomaximal, Sylow
114C22×C61—21no—
115C22×C61—21no—
116C22×C61—22no—
117C22×C61—42no—
118C22×C61—41no—
119C22×C61—41no—
120C22×C61—81no—
121C22×C61#44711yescharacteristic
122C22×C61C2×SD3211yescharacteristic
123C22×C61#50911yescharacteristic
124C2×D61—82no—
125C2×D61—82no—
126C2×D61—82no—
127C2×D61—164no—
128C4×C61—21no—
129C4×C61—41no—
130C4×C61#45511yescharacteristic
131C4×C61#45811yescharacteristic
132Dic61—82no—
133Dic61—161no—
134Dic61—161no—
135#3186—611no—
136#2233—612no—
137#2076—611no—
138#1347—612no—
139#1314D6111yescharacteristic, complemented
140C2×C61—21no—
141C2×C61—21no—
142C2×C61—42no—
143C2×C61#160711yescharacteristic
144C2×C61#218711yescharacteristic
145C2×C61#219011yescharacteristic
146D61—162no—
147C2×#121—1221no—
148#480—1222no—
149#466—612no—
150#466—1221no—
151#447—1222no—
152C2×#107—611no—
153C2×C22⋊C8C2×D6111yescharacteristic
154#380—1221no—
155#367—1222no—
156#357—1222no—
157C22⋊C16C2×D6112yescomplemented
158#340—611no—
159#340—1221no—
160C61#1228111yescharacteristic, complemented, Sylow
161C22×D4—1221no—
162C22×D4—2441no—
163C23×C4D24411yescharacteristic
164C2×D8—612no—
165C2×D8—1221no—
166C22×C8D24411yescharacteristic
167C22×C8C22×D6111yescharacteristic
168#121—1222no—
169#121—1222no—
170#121—2441no—
171#121—2441no—
172C2×C4⋊C4—1221no—
173C2×C4⋊C4—1221no—
174C2×C4⋊C4—1221no—
175C2×C22⋊C4—2441no—
176C2×C22⋊C4—2441no—
177C2×C16—22no—
178#107—612no—
179#107—1221no—
180#107—1222no—
181#102—1222no—
182#102—1222no—
183C22⋊C8—22no—
184#95—2441no—
185C24—2441no—
186C24D48811yescharacteristic
187C2×D4—1222no—
188C2×D4—4881no—
189C2×D4—4881no—
190C2×D4—4882no—
191C2×D4—4882no—
192C2×D4—2441no—
193C2×D4—2442no—
194C2×D4—2442no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—2441no—
200C22×C4—2441no—
201C22×C4—2441no—
202C22×C4D48811yescharacteristic
203C22×C4C2×D24411yescharacteristic
204D8—4881no—
205D8—2442no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D6112yes—
210C4⋊C4—1222no—
211C4⋊C4—2441no—
212C4⋊C4—2441no—
213C4⋊C4—2441no—
214C4⋊C4—2441no—
215C4⋊C4—2441no—
216C22⋊C4—4882no—
217C22⋊C4—2442no—
218C22⋊C4—2442no—
219C16—42no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—4882no—
225C23—4882no—
226C23—4882no—
227C23—2442no—
228C23#1451972212yes—
229C23C2×D48811yescharacteristic
230D4—4882no—
231D4—4884no—
232D4—4882no—
233D4—4882no—
234D4—2442no—
235D4—2442no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—4881no—
244C2×C4—4881no—
245C2×C4—4881no—
246C2×C4—4881no—
247C2×C4—4881no—
248C2×C4—4881no—
249C2×C4—4882no—
250C2×C4—2442no—
251C2×C4#1451974711yescharacteristic
252C2×C4#1451980511yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—9764no—
263C22—4882no—
264C22—4882no—
265C22—4882no—
266C22#1683989811yescharacteristic
267C22C2×#1451972211yescharacteristic
268C22#1684020011yescharacteristic
269C4—21no—
270C4—41no—
271C4—9761no—
272C4—9761no—
273C4—4882no—
274C4#1683994211yescharacteristic
275C4#1683995311yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—9762no—
280C2#2144480711yescharacteristic
281C2#2144775011yescharacteristic
282C2#2144776711yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1314C2
socleC22×C61#447
prime radical#21456570C2
hypercenter#1314D61
hypocenterC61#12281
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #21448005C2semidirect
3. #21447247C2semidirect
4. #21443829C2semidirect
5. C61×#1314C2semidirect
11. C2×#14519711C22semidirect
16. #16839525C22semidirect
18. C61×C22⋊C16C22semidirect
139. #1314D61semidirect
157. C22⋊C16C2×D61semidirect
160. C61#12281semidirect

Maximal subgroups

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

subgroupindex
#214565702
#214480052
#214472472
#214438292
C61×#13142
#1228161