Finite Group Databasebeta

← #30604367

Subgroups — #30604367

Order 15616.

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

normal subgroups only

This group has 552 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#21461074C211yesmaximal, characteristic
2#21461058C211yesmaximal, characteristic
3#21460764C214yesmaximal, complemented
4#21449178C211yesmaximal, characteristic
5#16840779—21no—
6#16840733—21no—
7#16840608—21no—
8#16840606—21no—
9#16840605C2212yes—
10#16840605C2212yes—
11C2×#14519753—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839681—21no—
14#16839673—21no—
15#16839669—21no—
16#16839641C412yes—
17#16839635C2212yes—
18#16839622C411yescharacteristic
19C23×Q8×C61C411yescharacteristic
20C2×#11758213—41no—
21#14519844—41no—
22#14519834—41no—
23C22⋊#92420—22no—
24C22⋊#92420—22no—
25C4×#92420—41no—
26C22×C22⋊D61C2×C411yescharacteristic
27C22×C4×D61D411yescharacteristic
28#14519753—44no—
29C4×C22⋊D61—41no—
30C4×C22⋊D61—41no—
31C2×C4⋊Dic61—21no—
32C2×#11758196—22no—
33C2×#11758196D411yescharacteristic
34C2×C22⋊Dic61C2×C412yes—
35C2×C22⋊Dic61C2311yescharacteristic
36C2×#11758198—22no—
37C2×#11758198D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39#14519684—21no—
40#14519684—22no—
41#14519684—22no—
42C22×Q8×C61—21no—
43C22×Q8×C61—21no—
44C22×Q8×C61—41no—
45C22×Q8×C61—41no—
46C22×Q8×C61—44no—
47C23×C4×C61C2×C412yes—
48C23×C4×C61C2×C411yescharacteristic
49C24×C61C22×C411yescharacteristic
50C23×D61C22⋊C411yescharacteristic
51C2×Q8×C61—24no—
52C2×Q8×C61—42no—
53C2×Q8×C61—44no—
54C2×Q8×C61—42no—
55C2×Q8×C61—48no—
56C2×Q8×C61—44no—
57C2×Q8×C61—44no—
58C2×Q8×C61—41no—
59C2×Q8×C61—41no—
60C22×C4×C61—22no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—42no—
64C22×C4×C61—44no—
65C22×C4×C61—42no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C2×C22⋊D61—22no—
69C2×C22⋊D61—22no—
70C2×C22⋊D61—41no—
71C22×Dic61—22no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—42no—
78C2×#92420—42no—
79C2×#92420—42no—
80#11758213—82no—
81C22⋊Dic61—24no—
82C22⋊Dic61—41no—
83#11758198—41no—
84#11758198—44no—
85C4⋊Dic61—42no—
86#11758196—41no—
87#11758196—44no—
88C4×Dic61—21no—
89C4×Dic61—21no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61C2×C22⋊C411yescharacteristic
95C23×C61#13112yes—
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—21no—
99C22×D61—41no—
100C22×D61—41no—
101Q8×C61—44no—
102Q8×C61—48no—
103Q8×C61—44no—
104C2×C4×C61—22no—
105C2×C4×C61—24no—
106C2×C4×C61—21no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—44no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—41no—
113C2×C4×C61—44no—
114C2×C4×C61#9911yescharacteristic
115C2×C4×C61#9911yescharacteristic
116C22⋊D61—44no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61#10112yes—
127C4×D61—42no—
128C4×D61—81no—
129#92420—82no—
130#92420—82no—
131#11580—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—22no—
135C22×C61—42no—
136C22×C61—42no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61#40811yescharacteristic
140C22×C61C2×#9911yescharacteristic
141C22×C61C2×#10111yescharacteristic
142C2×D61—42no—
143C2×D61—41no—
144C2×D61—41no—
145C2×D61—41no—
146C2×D61—81no—
147C4×C61—21no—
148C4×C61—44no—
149C4×C61—41no—
150C4×C61AΣL(1,𝔽16)12yes—
151Dic61—22no—
152Dic61—41no—
153Dic61—41no—
154Dic61—82no—
155#2409—611no—
156#1465—611no—
157#1460—611no—
158#1326—614no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61—42no—
162C2×C61#147211yescharacteristic
163C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
164C2×C61#212311yescharacteristic
165D61—81no—
166C23×Q8#692311yescharacteristic
167C2×C22⋊Q8—1221no—
168C2×C4×D4—611no—
169#431—1221no—
170#407—1221no—
171#395—1221no—
172#393—1221no—
173#391—1221no—
174#386—612no—
175#385—612no—
176#380—611no—
177#324—1221no—
178#323—612no—
179#323—612no—
180#323—1221no—
181C61#1158011yescharacteristic, complemented, Sylow
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—44no—
187C22×D4—611no—
188C23×C4—611no—
189C23×C4C2×#692312yes—
190C23×C4C2×#692311yescharacteristic
191C2×M16—2441no—
192C22⋊Q8—2444no—
193C4×D4—2441no—
194C4×D4—2441no—
195C2×C4⋊C4—611no—
196C2×C4⋊C4—1222no—
197C2×C4⋊C4—1221no—
198C2×C22⋊C4—612no—
199C2×C22⋊C4—611no—
200C2×C22⋊C4—611no—
201C2×C22⋊C4—1222no—
202C2×C42—611no—
203C4⋊C8—2441no—
204C22⋊C8—1222no—
205C22⋊C8—1222no—
206C22⋊C8—2441no—
207C4×C8—2441no—
208#95—1222no—
209#95—1222no—
210#95—1221no—
211C24—611no—
212C24C22×#692311yescharacteristic
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—1222no—
223C2×D4—1222no—
224C2×D4—2441no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—611no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—1221no—
235C22×C4—1221no—
236C22×C4—1222no—
237C22×C4—2442no—
238C22×C4C22⋊#692311yescharacteristic
239C22×C4C22⋊#692311yescharacteristic
240M16—4882no—
241C2×C8—2442no—
242C2×C8—2442no—
243C4⋊C4—2441no—
244C4⋊C4—2444no—
245C4⋊C4—2442no—
246C22⋊C4—1224no—
247C22⋊C4—2441no—
248C22⋊C4—2444no—
249C22⋊C4—2441no—
250C42—1221no—
251C42—1221no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—1221no—
257C23—1221no—
258C23—1221no—
259C23—2441no—
260C23—2441no—
261C23C2×C22⋊#692311yescharacteristic
262C23#1451986512yes—
263Q8—44no—
264Q8—48no—
265Q8—44no—
266D4—2444no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—612no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—4881no—
283C2×C4—2442no—
284C2×C4—2442no—
285C2×C4—2442no—
286C2×C4—2442no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4#1451983511yescharacteristic
290C2×C4#1451983511yescharacteristic
291C8—4882no—
292C8—4882no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—4881no—
301C22—2442no—
302C22—2441no—
303C22—2441no—
304C22—2441no—
305C22#1684073711yescharacteristic
306C22C2×#1451983511yescharacteristic
307C22D61⋊#10111yescharacteristic
308C4—21no—
309C4—44no—
310C4—41no—
311C4—1222no—
312C4—4882no—
313C4—2441no—
314C4—2441no—
315C4#1684066212yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—4881no—
320C2#2146109911yescharacteristic
321C2C2×#1684066211yescharacteristic
322C2#2146208111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC23×Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61C2×C4
cosocleC2×C22⋊Dic61C23
hypercenterC23×Q8#6923
hypocenterC61#11580
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #21460764C2semidirect
181. C61#11580semidirect

Maximal subgroups

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

subgroupindex
#214610742
#214610582
#214607642
#214491782
#1158061