Finite Group Databasebeta

← #30599900

Subgroups — #30599900

Order 15616.

566 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 112 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#21462714C211yesmaximal, characteristic, complemented
2#21461120C212yesmaximal
3#21461063C211yesmaximal, characteristic, complemented
4#21460647C212yesmaximal, complemented
5#21448358C211yesmaximal, characteristic
6#16840811—24no—
7#16840779—22no—
8#16840778C2212yes—
9#16840778C2212yes—
10#16840610—24no—
11#16840608—22no—
12C2×#14519765C411yescharacteristic
13C2×#14519742C2211yescharacteristic
14C2×#14519739C2211yescharacteristic
15#16839667C411yescharacteristic
16#16839663C411yescharacteristic
17#16839651—22no—
18#16839609—22no—
19C42×Dic61C2211yescharacteristic
20C61×#382C411yescharacteristic
21#14519844—24no—
22C4⋊#92420—22no—
23C4⋊#92420D414yes—
24C22⋊#92420—22no—
25C22⋊#92420—22no—
26C4×#92420—24no—
27C4×#92420—22no—
28C4×#92420—22no—
29C22×Dic122C2×C411yescharacteristic
30C22×D244C2×C411yescharacteristic
31C22×C4×D61—21no—
32#14519765—22no—
33#14519765C814yescomplemented
34#14519742—22no—
35#14519742—22no—
36#14519739—22no—
37#14519739—22no—
38C2×#11758196D412yes—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61D412yes—
46C2×C4×Dic61C2311yescharacteristic
47#14519684—22no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C42×C61C2×C411yescharacteristic
51#95×C61—22no—
52C23×D61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61C22⋊C412yes—
56C22×C4×C61C22×C411yescharacteristic
57C22×Dic61—21no—
58C22×Dic61C22⋊C412yes—
59C22×Dic61C2×D411yescharacteristic
60C22×Dic61C2×D411yescharacteristic
61C2×D244—22no—
62C2×D244—22no—
63C2×D244—21no—
64C2×D244C2×C812yes—
65C2×C4×D61—44no—
66C2×C4×D61—41no—
67C2×C4×D61—41no—
68C2×Dic122—22no—
69C2×Dic122—22no—
70C2×Dic122C2×C812yes—
71C2×Dic122M1612yes—
72C2×#92420—24no—
73C2×#92420—42no—
74C4⋊C4×C61—22no—
75C4⋊C4×C61—22no—
76C42×C61—21no—
77C42×C61C2×C812yes—
78#11758198—28no—
79#11758198—44no—
80#11758198—42no—
81#11758196—24no—
82C4×Dic61—22no—
83C4×Dic61—22no—
84C4×Dic61—22no—
85C4×Dic61—42no—
86C4×Dic61—42no—
87C4×Dic61C2×D412yes—
88C4×Dic61#4112yes—
89C23×C61C2×C22⋊C411yescharacteristic
90C22×D61—22no—
91C22×D61—42no—
92C22×D61—42no—
93C22×D61—41no—
94C22×D61—41no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61—41no—
102C2×C4×C61C4×D412yes—
103C2×C4×C61C22×C811yescharacteristic
104C2×C4×C61C2×M1611yescharacteristic
105C2×C4×C61#13112yes—
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—41no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61C22⋊C814yes—
114C2×Dic61#12011yescharacteristic
115C2×Dic61#12112yes—
116C2×Dic61#12311yescharacteristic
117D244—24no—
118D244—42no—
119C4×D61—82no—
120C4×D61—82no—
121Dic122—24no—
122Dic122—24no—
123#92420—44no—
124#92420—82no—
125#8960—611nomaximal, Sylow
126C22×C61—21no—
127C22×C61—21no—
128C22×C61#38611yescharacteristic
129C22×C61C2×C22⋊C811yescharacteristic
130C22×C61#40811yescharacteristic
131C2×D61—44no—
132C2×D61—42no—
133C2×D61—82no—
134C2×D61—81no—
135C2×D61—81no—
136C4×C61—42no—
137C4×C61—42no—
138C4×C61—81no—
139C4×C61C8×D412yes—
140C4×C61#43512yes—
141Dic61—24no—
142Dic61—22no—
143Dic61—21no—
144Dic61—42no—
145Dic61#45412yes—
146#2982—611no—
147C4×#124—611no—
148#1478—612no—
149#1461—611no—
150#1276—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61#186911yescharacteristic
154C2×C61#188411yescharacteristic
155C2×C61#188811yescharacteristic
156D61—82no—
157C2×#124—611no—
158C2×C4×Q8—611no—
159C2×C4×D4—611no—
160C8×Q8—1224no—
161#431—612no—
162#431—612no—
163#431—1222no—
164#391—611no—
165#390—611no—
166#388—1222no—
167#382#692311yescharacteristic
168C4×C22⋊C4—1222no—
169C43—611no—
170D4⋊C8—1224no—
171#324—1222no—
172C61#896011yescharacteristic, complemented, Sylow
173C22×Q8—611no—
174C22×D4—611no—
175C23×C4—1221no—
176#124—614no—
177#124—1222no—
178C4×Q8—1222no—
179C4×Q8—1222no—
180C4×D4—1222no—
181C4×D4—1222no—
182C2×C4⋊C4—612no—
183C2×C4⋊C4C2×#692311yescharacteristic
184C2×C4⋊C4C2×#692311yescharacteristic
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1221no—
187C2×C22⋊C4—1222no—
188C2×C42—611no—
189C2×C42—612no—
190C2×C42—611no—
191C2×C42—1221no—
192C2×C42—1221no—
193C2×C42C2×#692311yescharacteristic
194C4⋊C8—614no—
195C4⋊C8—1222no—
196C4⋊C8—1224no—
197C22⋊C8—1222no—
198C22⋊C8—1222no—
199C4×C8—1222no—
200C4×C8—1222no—
201C4×C8—1224no—
202#95—22no—
203#95—1222no—
204C24—1221no—
205C2×Q8—612no—
206C2×Q8—612no—
207C2×Q8—1222no—
208C2×Q8—1222no—
209C2×D4—612no—
210C2×D4—1222no—
211C2×D4—1221no—
212C2×D4—1222no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—612no—
216C22×C4—611no—
217C22×C4—611no—
218C22×C4—1221no—
219C22×C4—2441no—
220C22×C4—2441no—
221C22×C4—2444no—
222C22×C4C22⋊#692312yes—
223C22×C4C22×#692311yescharacteristic
224C2×C8—1224no—
225C2×C8—2442no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—1224no—
229C22⋊C4—1228no—
230C22⋊C4—2442no—
231C22⋊C4—2444no—
232C42—21no—
233C42—612no—
234C42—612no—
235C42—1222no—
236C42—1222no—
237C42—1222no—
238C42—2442no—
239C42—2442no—
240C42D61⋊C812yes—
241C23—1222no—
242C23—2442no—
243C23—2442no—
244C23—2441no—
245C23—2441no—
246C23C2×C22⋊#692311yescharacteristic
247Q8—1224no—
248Q8—1224no—
249D4—1224no—
250D4—2442no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—611no—
259C2×C4—612no—
260C2×C4—611no—
261C2×C4—614no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—1222no—
266C2×C4—4882no—
267C2×C4—4882no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4D4×#692312yes—
272C2×C4C2×D61⋊C811yescharacteristic
273C2×C4D61⋊M1611yescharacteristic
274C2×C4#1451986512yes—
275C8—4882no—
276C8—2444no—
277C22—21no—
278C22—21no—
279C22—4882no—
280C22—4881no—
281C22—4881no—
282C22—2444no—
283C22—2442no—
284C22#1684071011yescharacteristic
285C22D61⋊(C22⋊C8)11yescharacteristic
286C22#1684073711yescharacteristic
287C4—42no—
288C4—42no—
289C4—81no—
290C4—612no—
291C4—1224no—
292C4—1222no—
293C4—1221no—
294C4—2442no—
295C4#1684079012yes—
296C4#1684079212yes—
297C2—21no—
298C2—21no—
299C2—4882no—
300C2#2146179511yescharacteristic
301C2#2146182011yescharacteristic
302C2#2146182711yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#382C4
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C4×Dic61C23
hypercenter#382#6923
hypocenterC61#8960
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #21462714C2semidirect
3. #21461063C2semidirect
4. #21460647C2semidirect
33. #14519765C8semidirect
172. C61#8960semidirect

Maximal subgroups

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

subgroupindex
#214627142
#214611202
#214610632
#214606472
#214483582
#896061