Finite Group Databasebeta

← #30597833

Subgroups — #30597833

Order 15616.

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

normal subgroups only

This group has 510 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#21461555C212yesmaximal, complemented
2#21461194C212yesmaximal
3#21460701C211yesmaximal, characteristic, complemented
4#21460699C211yesmaximal, characteristic, complemented
5#21454511C211yesmaximal, characteristic
6D61⋊(C22⋊C8)C2212yes—
7C2×#14519831—22no—
8C4×C22⋊#6923C2212yes—
9(C2×#6923)⋊C8—22no—
10#16840616C414yescomplemented
11#16840613C414yescomplemented
12C22×C8×D61C2211yescharacteristic
13#16840269C2211yescharacteristic
14#16839831—22no—
15C8×C22⋊D61—22no—
16#16839790C411yescharacteristic, complemented
17#16839729C411yescharacteristic, complemented
18#16839725C2211yescharacteristic
19#16839717C411yescharacteristic, complemented
20C2×#131×C61C411yescharacteristic, complemented
21C2×D61⋊C8—22no—
22C2×C22⋊#6923—22no—
23C2×C4×#6923—22no—
24#14519834C2×C414yes—
25C22⋊#92420—22no—
26#14519831—44no—
27(C2×#6923)⋊C4C2×C414yes—
28C22×C4×D61C2311yescharacteristic
29C2×#11758194C2×C411yescharacteristic
30C22×#92418C2×C411yescharacteristic
31C2×C8×D61—42no—
32C2×C8×D61D412yes—
33C2×C8×D61D412yes—
34#14519758—21no—
35#14519753C2×C412yes—
36#14519748C2×C412yes—
37#14519745—21no—
38C4×C22⋊D61—22no—
39#14519736C2×C411yescharacteristic
40#14519706—22no—
41C22⋊#92418—22no—
42#14519689—22no—
43#14519687—21no—
44C8×Dic61—21no—
45C2×#41×C61C2×C411yescharacteristic
46#131×C61—42no—
47C2×M16×C61—21no—
48C2×M16×C61C2×C411yescharacteristic
49C22×C8×C61—21no—
50C22×C8×C61C2×C411yescharacteristic
51C23×D61#4111yescharacteristic
52C22×#6923—24no—
53#41×C61—41no—
54#41×C61—41no—
55C2×Q8×C61C4211yescharacteristic
56C2×D4×C61—21no—
57C2×D4×C61C4211yescharacteristic
58C22×C4×C61—21no—
59C22×C4×C61C22×C411yescharacteristic
60C2×C22⋊D61—22no—
61C22×Dic61#4111yescharacteristic
62C2×C4×D61—41no—
63C2×C4×D61—41no—
64C2×C4×D61C22×C412yes—
65C2×C4×D61C2×D412yes—
66C22⋊#6923—44no—
67C2×#92420—24no—
68C4×#6923—42no—
69C4×#6923—42no—
70D61⋊C8—42no—
71D61⋊C8—42no—
72M16×C61—42no—
73M16×C61—41no—
74C2×C8×C61—22no—
75C2×C8×C61—21no—
76C2×C8×C61—42no—
77C2×C8×C61C22⋊C411yescharacteristic
78C2×C8×C61C22⋊C411yescharacteristic
79C22⋊Dic61—21no—
80C22⋊Dic61—22no—
81#11758198—22no—
82C4⋊Dic61C4212yes—
83#11758196—22no—
84C4×Dic61—21no—
85#11758194—41no—
86C2×#92418—21no—
87C2×#92418—41no—
88C2×#92418C22⋊C412yes—
89C8×D61—24no—
90C8×D61—42no—
91C23×C61—21no—
92C23×C61#11711yescharacteristic
93C22×D61—24no—
94C22×D61—41no—
95C22×D61C4×D412yes—
96C2×#6923—44no—
97C2×#6923—44no—
98C2×#6923—44no—
99Q8×C61—41no—
100D4×C61—41no—
101D4×C61—42no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61C2×C4211yescharacteristic
108C2×C4×C61C2×C22⋊C411yescharacteristic
109C22⋊D61—44no—
110C2×Dic61—22no—
111C2×Dic61—41no—
112C2×Dic61C4×D412yes—
113C4×D61—22no—
114C4×D61—22no—
115C4×D61—42no—
116#92420—44no—
117C8×C61—21no—
118C8×C61—41no—
119C8×C61#9912yes—
120#92418—22no—
121#92418—41no—
122#8239—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—22no—
125C22×C61—41no—
126C22×C61—42no—
127C22×C61C4×C22⋊C411yescharacteristic
128C2×D61—22no—
129C2×D61—22no—
130C2×D61—44no—
131C2×D61#44414yes—
132#6923—84no—
133C4×C61—21no—
134C4×C61—41no—
135C4×C61C2×#9911yescharacteristic
136C4×C61#41011yescharacteristic
137Dic61—22no—
138Dic61—42no—
139#2929—611no—
140#1752—612no—
141#1501—612no—
142#1298—611no—
143#1297—611no—
144C2×C61—41no—
145C2×C61—81no—
146C2×C61C4×#9911yescharacteristic
147C2×C61#176512yes—
148D61—24no—
149C2×#131#692311yescharacteristic, complemented
150C23×C8—611no—
151#525—611no—
152#435—1222no—
153C8×D4—1222no—
154#424—611no—
155#408—611no—
156#407—611no—
157C2×C22⋊C8—612no—
158#405—611no—
159C2×#97—1222no—
160C4×C22⋊C4—612no—
161#336—1222no—
162#328—614no—
163#327—614no—
164C61#823911yescharacteristic, complemented, Sylow
165C2×#41C2×#692311yescharacteristic
166C23×C4—611no—
167#131—42no—
168C2×M16—21no—
169C2×M16—611no—
170C2×M16C2×#692311yescharacteristic
171C22×C8—21no—
172C22×C8—611no—
173C22×C8—612no—
174C22×C8—612no—
175C22×C8—1222no—
176C22×C8—2442no—
177C22×C8C2×#692311yescharacteristic
178#123—1221no—
179C22⋊Q8—612no—
180#121—612no—
181#120—1221no—
182C4×D4—1222no—
183#117—611no—
184C2×C22⋊C4—1222no—
185C2×C42—1222no—
186C4⋊C8—1222no—
187C22⋊C8—614no—
188C22⋊C8—1222no—
189C22⋊C8—1222no—
190C22⋊C8—1222no—
191#97—1221no—
192#97—2444no—
193C4×C8—1221no—
194#95—614no—
195C24—611no—
196#41—41no—
197#41—41no—
198C2×Q8C4×#692311yescharacteristic
199C2×D4—21no—
200C2×D4—1222no—
201C2×D4C4×#692311yescharacteristic
202C22×C4—21no—
203C22×C4—611no—
204C22×C4—612no—
205C22×C4—612no—
206C22×C4—1224no—
207C22×C4—2441no—
208C22×C4—2441no—
209C22×C4C22×#692311yescharacteristic
210M16—42no—
211M16—41no—
212M16—2441no—
213C2×C8—22no—
214C2×C8—21no—
215C2×C8—42no—
216C2×C8—612no—
217C2×C8—1224no—
218C2×C8—1224no—
219C2×C8—1221no—
220C2×C8—2442no—
221C2×C8—2441no—
222C2×C8—2442no—
223C2×C8—2442no—
224C2×C8C22⋊#692311yescharacteristic
225C2×C8C22⋊#692311yescharacteristic
226C4⋊C4—612no—
227C4⋊C4—1222no—
228C22⋊C4—1222no—
229C22⋊C4—1222no—
230C22⋊C4—1221no—
231C22⋊C4—2444no—
232C42—1221no—
233C42—2442no—
234C42—2442no—
235C23—21no—
236C23—612no—
237C23—1224no—
238C23—2441no—
239C23#1451985611yescharacteristic
240Q8—41no—
241D4—41no—
242D4—42no—
243D4—2444no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—612no—
250C2×C4—1222no—
251C2×C4—1222no—
252C2×C4—1222no—
253C2×C4—2442no—
254C2×C4—2441no—
255C2×C4—2444no—
256C2×C4—2444no—
257C2×C4—2444no—
258C2×C4C2×C4×#692311yescharacteristic
259C2×C4C2×C22⋊#692311yescharacteristic
260C8—21no—
261C8—41no—
262C8—1222no—
263C8—2444no—
264C8—2441no—
265C8#1451983512yes—
266C22—21no—
267C22—22no—
268C22—41no—
269C22—42no—
270C22—614no—
271C22—1222no—
272C22—1222no—
273C22—2444no—
274C22C4×C22⋊#692311yescharacteristic
275C4—21no—
276C4—41no—
277C4—1222no—
278C4—4884no—
279C4—2442no—
280C4C2×#1451983511yescharacteristic
281C4#1684074111yescharacteristic
282C2—41no—
283C2—81no—
284C2—1224no—
285C2C4×#1451983511yescharacteristic
286C2#2146159712yes—
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×#131×C61C4
socleC22×C61C4×C22⋊C4
prime radical#16840269C22
cosocleC22×C4×D61C23
hypercenterC2×#131#6923
hypocenterC61#8239
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. #21461555C2semidirect
3. #21460701C2semidirect
4. #21460699C2semidirect
10. #16840616C4semidirect
11. #16840613C4semidirect
16. #16839790C4semidirect
17. #16839729C4semidirect
19. #16839717C4semidirect
20. C2×#131×C61C4semidirect
149. C2×#131#6923semidirect
164. C61#8239semidirect

Maximal subgroups

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

subgroupindex
#214615552
#214611942
#214607012
#214606992
#214545112
#823961