Finite Group Databasebeta

← #30605169

Subgroups — #30605169

Order 15616.

586 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 74 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#21461718C211yesmaximal, characteristic
2#21461069C211yesmaximal, characteristic
3#21460820C214yesmaximal, complemented
4C2×#16839681C211yesmaximal, characteristic
5#16840732—21no—
6#16840732—21no—
7#16840642—21no—
8#16840642C2212yes—
9#16840608—21no—
10#16840608C2212yes—
11C22×#11758196—21no—
12C22×#11758196C2211yescharacteristic
13#16839681—21no—
14#16839681—21no—
15#16839681C412yes—
16#16839681C2212yes—
17C2×#14519684—21no—
18C2×#14519684C411yescharacteristic
19C23×Q8×C61C411yescharacteristic
20C2×#11758217—22no—
21C2×#11758217—41no—
22C22⋊#92420—22no—
23C22⋊#92420—41no—
24C22⋊#92420—41no—
25C22⋊#92420—41no—
26C23×Dic61—21no—
27C23×Dic61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C2×#11758196—22no—
30C2×#11758196—22no—
31C2×#11758196—41no—
32C2×#11758196—41no—
33C2×#11758196—44no—
34C2×#11758196C2×C411yescharacteristic
35C2×#11758196D411yescharacteristic
36C2×#11758196D411yescharacteristic
37C2×#11758196C2311yescharacteristic
38#14519684—21no—
39#14519684—22no—
40#14519684—22no—
41#14519684C2×C412yes—
42C22×Q8×C61—21no—
43C22×Q8×C61—44no—
44C22×Q8×C61—41no—
45C22×Q8×C61—41no—
46C22×Q8×C61C2×C412yes—
47C23×C4×C61—21no—
48C23×C4×C61C2×C411yescharacteristic
49C24×C61C22⋊C411yescharacteristic
50C2×Q8×C61—44no—
51C2×Q8×C61—42no—
52C2×Q8×C61—44no—
53C2×Q8×C61—42no—
54C2×Q8×C61—48no—
55C2×Q8×C61—44no—
56C2×Q8×C61—44no—
57C22×C4×C61—24no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—22no—
61C22×C4×C61—42no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—42no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—24no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—22no—
72C22×Dic61—21no—
73C22×Dic61—22no—
74C22×Dic61—41no—
75C22×Dic61—42no—
76C22×Dic61—42no—
77C22×Dic61—42no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C2×D411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82#11758217—42no—
83#11758217—42no—
84#11758217—82no—
85C2×#92420—42no—
86C2×#92420—42no—
87#11758196—41no—
88#11758196—41no—
89#11758196—41no—
90#11758196—41no—
91#11758196—44no—
92#11758196—44no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61#9911yescharacteristic
98C23×C61#9911yescharacteristic
99C23×C61C2×C22⋊C411yescharacteristic
100Q8×C61—48no—
101Q8×C61—44no—
102Q8×C61—44no—
103C2×C4×C61—24no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—42no—
107C2×C4×C61—44no—
108C2×C4×C61—42no—
109C2×C4×C61—44no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—24no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—48no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—81no—
131C2×Dic61#13712yes—
132#92420—82no—
133#92420—82no—
134#11815—611nomaximal, Sylow
135C22×C61—24no—
136C22×C61—24no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61AΣL(1,𝔽16)12yes—
144C22×C61#35612yes—
145C22×C61C2×#9911yescharacteristic
146C22×C61#41911yescharacteristic
147C22×C61#42111yescharacteristic
148C4×C61—44no—
149C4×C61—41no—
150C4×C61—41no—
151Dic61—22no—
152Dic61—44no—
153Dic61—41no—
154Dic61—41no—
155Dic61—41no—
156C2×#395—611no—
157#1821—611no—
158#1463—611no—
159#1354—614no—
160C2×C61—21no—
161C2×C61—22no—
162C2×C61—22no—
163C2×C61—21no—
164C2×C61#151111yescharacteristic
165C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
166C2×C61C2×#35611yescharacteristic
167C23×Q8#692311yescharacteristic
168C22×C4⋊C4—611no—
169C22×C4⋊C4—1221no—
170#407—1221no—
171#407—1221no—
172#395—612no—
173#395—612no—
174#395—1221no—
175#395—1221no—
176C2×#95—611no—
177C2×#95—1221no—
178#343—612no—
179#343—1221no—
180#324—612no—
181#324—1221no—
182C61#1181511yescharacteristic, complemented, Sylow
183C22×Q8—21no—
184C22×Q8—44no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8C2×#692312yes—
188C23×C4—21no—
189C23×C4—611no—
190C23×C4—611no—
191C23×C4—1221no—
192C23×C4C2×#692311yescharacteristic
193C2×M16—1222no—
194C2×M16—2441no—
195C2×C4⋊C4—611no—
196C2×C4⋊C4—611no—
197C2×C4⋊C4—611no—
198C2×C4⋊C4—611no—
199C2×C4⋊C4—1222no—
200C2×C4⋊C4—1222no—
201C2×C4⋊C4—2441no—
202C2×C4⋊C4—2441no—
203C2×C4⋊C4—2444no—
204C22⋊C8—1222no—
205C22⋊C8—2441no—
206C22⋊C8—2441no—
207C22⋊C8—2441no—
208#95—612no—
209#95—1222no—
210#95—1222no—
211#95—1221no—
212C24C22⋊#692311yescharacteristic
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—48no—
218C2×Q8—44no—
219C2×Q8—44no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—611no—
229C22×C4—611no—
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—1222no—
233C22×C4—1221no—
234C22×C4—1224no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4—1222no—
239C22×C4—2441no—
240C22×C4—2442no—
241C22×C4—2442no—
242C22×C4—2442no—
243C22×C4C22⋊#692311yescharacteristic
244C22×C4C22×#692311yescharacteristic
245M16—4882no—
246M16—2442no—
247M16—2442no—
248C2×C8—2442no—
249C2×C8—2442no—
250C4⋊C4—2441no—
251C4⋊C4—2441no—
252C4⋊C4—2441no—
253C4⋊C4—2441no—
254C4⋊C4—2444no—
255C4⋊C4—2444no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1451983511yescharacteristic
261C23#1451983511yescharacteristic
262C23C2×C22⋊#692311yescharacteristic
263Q8—44no—
264Q8—44no—
265Q8—48no—
266C2×C4—24no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—612no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1224no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—4881no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2448no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2441no—
294C2×C4—2441no—
295C8—4882no—
296C8—4882no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1684066212yes—
306C22#1684067212yes—
307C22C2×#1451983511yescharacteristic
308C22#1684075511yescharacteristic
309C22#1684076111yescharacteristic
310C4—44no—
311C4—41no—
312C4—41no—
313C4—1222no—
314C4—2444no—
315C4—2441no—
316C4—2441no—
317C4—2441no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2146122911yescharacteristic
323C2C2×#1684066211yescharacteristic
324C2C2×#1684067211yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×#11758196C23
hypercenterC23×Q8#6923
hypocenterC61#11815
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #21460820C2semidirect
182. C61#11815semidirect

Maximal subgroups

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

subgroupindex
#214617182
#214610692
#214608202
C2×#168396812
#1181561