Finite Group Databasebeta

← #30595926

Subgroups — #30595926

Order 15616.

494 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 100 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#21461711C211yesmaximal, characteristic
2#21461676C211yesmaximal, characteristic
3#21460637C214yesmaximal, complemented
4#21454172C211yesmaximal, characteristic
5C2×#14519844—21no—
6C2×#14519834—21no—
7C2×#14519834—21no—
8#16840628—21no—
9#16840628C2212yes—
10#16840628C2212yes—
11C2×#14519777C411yescharacteristic
12C2×#14519757—21no—
13C2×C42×D61C2211yescharacteristic
14#16839667—21no—
15#16839661—21no—
16#16839661C2212yes—
17#16839619—21no—
18#16839619C412yes—
19C61×#384C411yescharacteristic
20C22×#92420—22no—
21C22×#92420—22no—
22#14519844—42no—
23#14519834—42no—
24#14519834—42no—
25C22×Dic122—21no—
26C22×C4×D61C2×C411yescharacteristic
27C22×C4×D61D411yescharacteristic
28C22×C4×D61D411yescharacteristic
29#14519777—42no—
30#14519757—42no—
31#14519757—42no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—22no—
34C2×C4⋊Dic61C2×C412yes—
35C2×#11758198—22no—
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61C2311yescharacteristic
39C42×D61—42no—
40#14519684—22no—
41#14519684—22no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61C2×C412yes—
45C2×C42×C61C2×C411yescharacteristic
46C23×D61C22⋊C411yescharacteristic
47C22×C4×C61—22no—
48C22×C4×C61C22⋊C411yescharacteristic
49C22×C4×C61C22⋊C411yescharacteristic
50C22×C4×C61C22×C411yescharacteristic
51C22×Dic61—22no—
52C22×Dic61C22⋊C411yescharacteristic
53C22×Dic61C2×D411yescharacteristic
54C22×Dic61C2×D411yescharacteristic
55C2×C4×D61—22no—
56C2×C4×D61—22no—
57C2×C4×D61—22no—
58C2×C4×D61—22no—
59C2×C4×D61—41no—
60C2×C4×D61—41no—
61C2×C4×D61—41no—
62C2×C4×D61—41no—
63C2×C4×D61D812yes—
64C2×C4×D61SD1612yes—
65C2×Dic122—42no—
66C2×Dic122—42no—
67C2×Dic122—42no—
68C2×#92420—42no—
69C2×#92420—42no—
70C2×#92420—42no—
71C2×#92420—42no—
72C2×#92420—42no—
73C2×#92420—42no—
74C4⋊C4×C61—24no—
75C4⋊C4×C61—42no—
76C4⋊C4×C61—42no—
77C42×C61—22no—
78#11758198—44no—
79C4⋊Dic61—42no—
80C4⋊Dic61—42no—
81C4⋊Dic61—42no—
82C4⋊Dic61—42no—
83C4×Dic61—22no—
84C4×Dic61—41no—
85C4×Dic61—41no—
86C23×C61C2×C22⋊C411yescharacteristic
87C22×D61—21no—
88C22×D61—21no—
89C22×D61—21no—
90C22×D61—21no—
91C22×D61—21no—
92C22×D61—21no—
93C22×D61#10212yes—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61#9912yes—
104C2×C4×C61#10212yes—
105C2×C4×C61#13112yes—
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61—21no—
109C2×Dic61—21no—
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61#10112yes—
116C2×Dic61C2×D811yescharacteristic
117C2×Dic61C2×SD1611yescharacteristic
118C2×Dic61#13712yes—
119C4×D61—42no—
120C4×D61—42no—
121C4×D61—44no—
122C4×D61—44no—
123Dic122—84no—
124#92420—82no—
125#92420—82no—
126#7728—611nomaximal, Sylow
127C22×C61#40811yescharacteristic
128C22×C61C2×#9911yescharacteristic
129C22×C61C2×#10111yescharacteristic
130C22×C61C2×#10211yescharacteristic
131C22×C61#41911yescharacteristic
132C22×C61#42111yescharacteristic
133C22×C61#42111yescharacteristic
134C2×D61—22no—
135C2×D61—22no—
136C2×D61—22no—
137C2×D61—41no—
138C2×D61—41no—
139C2×D61—41no—
140C2×D61—41no—
141C4×C61—22no—
142C4×C61—22no—
143C4×C61—22no—
144C4×C61—82no—
145Dic61—22no—
146Dic61—22no—
147Dic61—21no—
148Dic61—21no—
149Dic61—82no—
150#2857—611no—
151#1817—611no—
152#1797—611no—
153#1271—614no—
154C2×C61#147211yescharacteristic
155C2×C61#148011yescharacteristic
156C2×C61#149711yescharacteristic
157C2×C61#151111yescharacteristic
158C2×C61#151911yescharacteristic
159C2×C61#153711yescharacteristic
160C2×C61#154611yescharacteristic
161D61—42no—
162C2×C4⋊Q8—611no—
163C2×C22⋊Q8—1221no—
164C22×C42—611no—
165C2×C4⋊C8—1221no—
166C2×C22⋊C8—1221no—
167C2×C22⋊C8—1221no—
168#391—1221no—
169#389—612no—
170#389—1221no—
171#384#692311yescharacteristic
172#380—612no—
173#380—1221no—
174#336—612no—
175#336—612no—
176#336—1221no—
177C61#772811yescharacteristic, complemented, Sylow
178C22×Q8—1221no—
179C23×C4—611no—
180C23×C4—611no—
181C23×C4—611no—
182C22×C8—1222no—
183C22×C8—1222no—
184C4⋊Q8—2442no—
185C22⋊Q8—2442no—
186C22⋊Q8—2442no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—612no—
190C2×C4⋊C4—1222no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4C2×#692312yes—
193C2×C22⋊C4—1222no—
194C2×C42—611no—
195C2×C42—611no—
196C2×C42—611no—
197C2×C42—2442no—
198C2×C42C2×#692311yescharacteristic
199C4⋊C8—2442no—
200C22⋊C8—2442no—
201C22⋊C8—2442no—
202#95—1222no—
203#95—1222no—
204C24—611no—
205C2×Q8—2442no—
206C2×Q8—2442no—
207C2×Q8—2442no—
208C22×C4—22no—
209C22×C4—611no—
210C22×C4—611no—
211C22×C4—611no—
212C22×C4—612no—
213C22×C4—612no—
214C22×C4—1222no—
215C22×C4—1222no—
216C22×C4—1222no—
217C22×C4—1222no—
218C22×C4—1222no—
219C22×C4—2441no—
220C22×C4—2441no—
221C22×C4—2441no—
222C22×C4—2441no—
223C22×C4C22⋊#692311yescharacteristic
224C22×C4C22⋊#692311yescharacteristic
225C22×C4C22×#692311yescharacteristic
226C2×C8—2442no—
227C2×C8—2442no—
228C2×C8—2442no—
229C2×C8—2442no—
230C2×C8—2442no—
231C2×C8—2442no—
232C4⋊C4—24no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—2442no—
236C4⋊C4—2442no—
237C4⋊C4—2442no—
238C4⋊C4—2442no—
239C22⋊C4—2444no—
240C42—22no—
241C42—1222no—
242C42—2441no—
243C42—2441no—
244C23—612no—
245C23—1221no—
246C23—1221no—
247C23—1221no—
248C23—1221no—
249C23—1221no—
250C23—1221no—
251C23C2×C22⋊#692311yescharacteristic
252Q8—4884no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—612no—
263C2×C4—611no—
264C2×C4—611no—
265C2×C4—612no—
266C2×C4—1221no—
267C2×C4—1221no—
268C2×C4—1221no—
269C2×C4—1221no—
270C2×C4—1221no—
271C2×C4—1221no—
272C2×C4—2442no—
273C2×C4—2442no—
274C2×C4—2444no—
275C2×C4—2444no—
276C2×C4—2442no—
277C2×C4—2442no—
278C2×C4—2442no—
279C2×C4#1451983612yes—
280C2×C4#1451983912yes—
281C2×C4#1451986412yes—
282C8—4882no—
283C8—4882no—
284C22—1222no—
285C22—1222no—
286C22—1222no—
287C22—2441no—
288C22—2441no—
289C22—2441no—
290C22—2441no—
291C22#1684073611yescharacteristic
292C22C2×#1451983611yescharacteristic
293C22D61⋊#10111yescharacteristic
294C22C2×#1451983911yescharacteristic
295C22#1684075511yescharacteristic
296C22#1684076011yescharacteristic
297C22#1684076111yescharacteristic
298C4—22no—
299C4—22no—
300C4—22no—
301C4—82no—
302C4—1221no—
303C4—1221no—
304C4—1222no—
305C4—1222no—
306C4—4882no—
307C2—2442no—
308C2#2146109711yescharacteristic
309C2#2146112511yescharacteristic
310C2#2146118211yescharacteristic
311C2#2146122711yescharacteristic
312C2#2146125611yescharacteristic
313C2#2146131111yescharacteristic
314C2#2146132911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#384#6923
hypocenterC61#7728
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #21460637C2semidirect
177. C61#7728semidirect

Maximal subgroups

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

subgroupindex
#214617112
#214616762
#214606372
#214541722
#772861