Finite Group Databasebeta

← #30595446

Subgroups — #30595446

Order 15616.

846 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 110 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#21461711C212yesmaximal, complemented
2#21460636C212yesmaximal, complemented
3#21460635C212yesmaximal, complemented
4#390×D61C211yesmaximal, characteristic
5C2×#14519834—22no—
6#16840628—22no—
7#16840628C2214yescomplemented
8C22×D4×D61C411yescharacteristic
9C2×C4⋊C4×D61C411yescharacteristic
10C2×C22⋊C4×D61—22no—
11C2×C42×D61C2211yescharacteristic
12#16839665C2211yescharacteristic
13#16839664C411yescharacteristic
14#16839663C2211yescharacteristic
15#16839662—22no—
16C61×#390C411yescharacteristic
17C22×#92420—24no—
18#14519834—44no—
19C24×D61—21no—
20C22×D244C2×C411yescharacteristic
21C22×C22⋊D61—21no—
22C2×D4×D61—22no—
23C2×D4×D61—42no—
24C2×D4×D61—42no—
25C2×D4×D61C814yescomplemented
26C22×C4×D61—21no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D412yes—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×#11758196—21no—
31C4⋊C4×D61—42no—
32C2×C22⋊Dic61—22no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35C22⋊C4×D61—48no—
36C2×C4×Dic61D412yes—
37C2×C4×Dic61C2311yescharacteristic
38C42×D61—42no—
39C22×D4×C61C2×C411yescharacteristic
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C22⋊C4×C61—22no—
42C2×C42×C61C2×C411yescharacteristic
43C24×C61—21no—
44C23×D61—21no—
45C23×D61—24no—
46C23×D61—44no—
47C23×D61—44no—
48C23×D61—42no—
49C23×D61—42no—
50C23×D61C22⋊C411yescharacteristic
51C2×D4×C61—21no—
52C2×D4×C61—41no—
53C2×D4×C61—41no—
54C2×D4×C61C2×C812yes—
55C22×C4×C61—21no—
56C22×C4×C61C22⋊C412yes—
57C22×C4×C61C22×C411yescharacteristic
58C2×C22⋊D61—24no—
59C2×C22⋊D61—42no—
60C2×C22⋊D61—42no—
61C22×Dic61—21no—
62C22×Dic61C22⋊C411yescharacteristic
63C22×Dic61C2×D412yes—
64D4×D61—48no—
65D4×D61—48no—
66C2×D244—21no—
67C2×D244—41no—
68C2×D244—41no—
69C2×D244C2×C812yes—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—24no—
73C2×C4×D61—24no—
74C2×C4×D61—42no—
75C2×C4×D61—44no—
76C2×C4×D61—42no—
77C2×C4×D61C2×C812yes—
78C2×C4×D61M1612yes—
79C2×#92420—44no—
80C2×#92420—44no—
81C2×#92420—44no—
82C4⋊C4×C61—41no—
83C22⋊C4×C61—44no—
84C42×C61—41no—
85C22⋊Dic61—44no—
86#11758198—44no—
87#11758198—44no—
88C4⋊Dic61—41no—
89#11758196—42no—
90C4×Dic61—42no—
91C4×Dic61—41no—
92C23×C61—22no—
93C23×C61—41no—
94C23×C61—41no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61C2×C22⋊C411yescharacteristic
98C22×D61—22no—
99C22×D61—22no—
100C22×D61—22no—
101C22×D61—416no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—42no—
105C22×D61—42no—
106C22×D61—48no—
107C22×D61—84no—
108C22×D61—88no—
109C22×D61—84no—
110C22×D61C22⋊C812yes—
111C22×D61#9912yes—
112C22×D61#10012yes—
113D4×C61—42no—
114D4×C61—42no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—41no—
121C2×C4×C61—42no—
122C2×C4×C61C22×C811yescharacteristic
123C2×C4×C61C2×M1611yescharacteristic
124C22⋊D61—48no—
125C2×Dic61—22no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—22no—
129C2×Dic61—41no—
130C2×Dic61—42no—
131C2×Dic61C22⋊C812yes—
132C2×Dic61AGL(1,ℤ/8)14yes—
133D244—42no—
134D244—42no—
135C4×D61—24no—
136C4×D61—24no—
137C4×D61—48no—
138C4×D61—84no—
139#92420—84no—
140#7592—611nomaximal, Sylow
141C22×C61—42no—
142C22×C61—44no—
143C22×C61—81no—
144C22×C61—82no—
145C22×C61—81no—
146C22×C61C2×C22⋊C811yescharacteristic
147C22×C61C2×#9911yescharacteristic
148C22×C61C2×#10011yescharacteristic
149C22×C61#41812yes—
150C22×C61#42112yes—
151C2×D61—22no—
152C2×D61—24no—
153C2×D61—24no—
154C2×D61—22no—
155C2×D61—42no—
156C2×D61—44no—
157C2×D61—816no—
158C2×D61—81no—
159C2×D61—82no—
160C2×D61—81no—
161C2×D61#32314yes—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—42no—
165C4×C61—81no—
166Dic61—22no—
167Dic61—24no—
168Dic61—81no—
169C2×#390—611no—
170#1817—612no—
171#1270—612no—
172#1270—612no—
173C2×C61—82no—
174C2×C61C2×#32311yescharacteristic
175C2×C61#147812yes—
176C2×C61#151012yes—
177C2×C61#153311yescharacteristic
178C2×C61#155511yescharacteristic
179D61—24no—
180D61—82no—
181C23×D4—611no—
182C22×C4⋊C4—611no—
183C22×C22⋊C4—1222no—
184C22×C42—611no—
185C2×C22⋊C8—1222no—
186#390—611no—
187#390—611no—
188#390—611no—
189#390—1222no—
190#390#692311yescharacteristic
191#336—614no—
192#336—1222no—
193C61#759211yescharacteristic, complemented, Sylow
194C25—1221no—
195C22×D4—614no—
196C22×D4—611no—
197C22×D4—1222no—
198C22×D4—1221no—
199C22×D4—2442no—
200C22×D4—2442no—
201C22×D4C2×#692311yescharacteristic
202C23×C4—611no—
203C23×C4—612no—
204C23×C4—1221no—
205C22×C8—1224no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—2442no—
209C2×C4⋊C4C2×#692311yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—1222no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—2448no—
215C2×C42—612no—
216C2×C42—611no—
217C2×C42—2442no—
218C2×C42C2×#692311yescharacteristic
219C22⋊C8—2444no—
220C24—21no—
221C24—611no—
222C24—1224no—
223C24—1221no—
224C24—2442no—
225C24—2442no—
226C24—2444no—
227C24—2444no—
228C2×D4—21no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—612no—
232C2×D4—1224no—
233C2×D4—1221no—
234C2×D4—2441no—
235C2×D4—2441no—
236C2×D4—2448no—
237C2×D4—2448no—
238C2×D4—2442no—
239C2×D4—2442no—
240C2×D4D61⋊C812yes—
241C22×C4—21no—
242C22×C4—612no—
243C22×C4—611no—
244C22×C4—612no—
245C22×C4—612no—
246C22×C4—1224no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4—1224no—
250C22×C4—1221no—
251C22×C4—2442no—
252C22×C4—2442no—
253C22×C4—2444no—
254C22×C4C22⋊#692312yes—
255C22×C4C22×#692311yescharacteristic
256C2×C8—2444no—
257C2×C8—2444no—
258C2×C8—2444no—
259C4⋊C4—41no—
260C4⋊C4—2441no—
261C4⋊C4—2442no—
262C22⋊C4—44no—
263C22⋊C4—2444no—
264C22⋊C4—2444no—
265C22⋊C4—2444no—
266C42—41no—
267C42—2441no—
268C42—2442no—
269C23—22no—
270C23—41no—
271C23—41no—
272C23—42no—
273C23—42no—
274C23—612no—
275C23—612no—
276C23—612no—
277C23—1222no—
278C23—1222no—
279C23—1222no—
280C23—4884no—
281C23—4884no—
282C23—4888no—
283C23—2448no—
284C23—24416no—
285C23—2441no—
286C23—2441no—
287C23—2442no—
288C23—2442no—
289C23C2×C22⋊#692311yescharacteristic
290D4—42no—
291D4—42no—
292D4—2442no—
293D4—2442no—
294D4—2448no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—612no—
303C2×C4—614no—
304C2×C4—1224no—
305C2×C4—1224no—
306C2×C4—1222no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—1222no—
310C2×C4—4884no—
311C2×C4—2448no—
312C2×C4—2441no—
313C2×C4—2442no—
314C2×C4C2×D61⋊C811yescharacteristic
315C2×C4D61⋊M1611yescharacteristic
316C8—4884no—
317C22—42no—
318C22—44no—
319C22—81no—
320C22—81no—
321C22—82no—
322C22—614no—
323C22—1222no—
324C22—1222no—
325C22—1224no—
326C22—1224no—
327C22—48816no—
328C22—4881no—
329C22—4881no—
330C22—4882no—
331C22—2442no—
332C22—2444no—
333C22D61⋊(C22⋊C8)11yescharacteristic
334C22D61⋊#9911yescharacteristic
335C22D61⋊#10011yescharacteristic
336C22#1684075412yes—
337C22#1684075912yes—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—81no—
342C4—1222no—
343C4—1224no—
344C4—4881no—
345C2—82no—
346C2—1224no—
347C2—4882no—
348C2D61⋊#32311yescharacteristic
349C2#2146111912yes—
350C2#2146122512yes—
351C2#2146129911yescharacteristic
352C2#2146134711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×D61C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7592
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461711C2semidirect
2. #21460636C2semidirect
3. #21460635C2semidirect
7. #16840628C22semidirect
25. C2×D4×D61C8semidirect
193. C61#7592semidirect

Maximal subgroups

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

subgroupindex
#214617112
#214606362
#214606352
#390×D612
#759261