Finite Group Databasebeta

← #29568308

Subgroups — #29568308

Order 15616.

630 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 101 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#21448306C211yesmaximal, characteristic
2#21445886C212yesmaximal, complemented
3#21443644C212yesmaximal, complemented
4#21443640C211yesmaximal, characteristic, complemented
5C61×#1270C211yesmaximal, characteristic, complemented
6C22×#11758224C411yescharacteristic
7C2×#14519735C411yescharacteristic
8C2×C42×D61C2211yescharacteristic
9C2×#14519689—22no—
10#16839668C2211yescharacteristic
11#16839667C411yescharacteristic
12#16839665C411yescharacteristic
13#16839638—22no—
14C22⋊C4×Dic61—22no—
15#16839517C2212yes—
16#16839516—22no—
17C61×#390C2211yescharacteristic
18C61×#336C2212yes—
19C2×#11758224—22no—
20C2×#11758224—22no—
21C2×#11758224—42no—
22C2×#11758224—42no—
23C22×Dic122C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C23×Dic61—21no—
26C22×C4×D61C2×C411yescharacteristic
27C22×C4×D61D412yes—
28C22×#92418—22no—
29#14519735—42no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×C22⋊Dic61—22no—
32C2×#11758198—21no—
33C2×C4×Dic61—21no—
34C2×C4×Dic61C2×C411yescharacteristic
35C2×C4×Dic61D412yes—
36C42×D61—42no—
37#14519689—44no—
38#14519684—22no—
39#14519684—22no—
40C22×D4×C61C2×C411yescharacteristic
41C22×C8×C61—22no—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C22⋊C4×C61—22no—
44C2×C42×C61C2311yescharacteristic
45C24×C61—21no—
46C23×D61C22⋊C411yescharacteristic
47C2×D4×C61—21no—
48C2×D4×C61—41no—
49C2×D4×C61—41no—
50C2×D4×C61M1612yes—
51C22×C4×C61—21no—
52C22×C4×C61C22×C411yescharacteristic
53C22×C4×C61C2×D412yes—
54C2×C22⋊D61—24no—
55C2×C22⋊D61—42no—
56C2×C22⋊D61—42no—
57C22×Dic61—24no—
58C22×Dic61—21no—
59C22×Dic61—21no—
60C22×Dic61—42no—
61C22×Dic61—42no—
62C22×Dic61C22⋊C412yes—
63C22×Dic61C22⋊C411yescharacteristic
64#11758224—48no—
65#11758224—48no—
66C2×C4×D61—22no—
67C2×C4×D61—21no—
68C2×C4×D61—21no—
69C2×C4×D61—22no—
70C2×C4×D61—24no—
71C2×C4×D61—22no—
72C2×C4×D61—22no—
73C2×C4×D61—42no—
74C2×Dic122—21no—
75C2×Dic122—41no—
76C2×Dic122—41no—
77C2×Dic122M1612yes—
78C2×C8×C61—42no—
79C2×C8×C61—42no—
80C2×C8×C61—42no—
81C4⋊C4×C61—41no—
82C22⋊C4×C61—42no—
83C22⋊C4×C61—42no—
84C42×C61—41no—
85C22⋊Dic61—44no—
86#11758198—42no—
87C4⋊Dic61—41no—
88C4×Dic61—41no—
89C4×Dic61—42no—
90C4×Dic61—42no—
91C2×#92418—42no—
92C2×#92418—42no—
93C2×#92418—42no—
94C23×C61—22no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—22no—
103C22×D61—22no—
104C22×D61#9912yes—
105D4×C61—42no—
106D4×C61—42no—
107C2×C4×C61—22no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—22no—
111C2×C4×C61—41no—
112C2×C4×C61—42no—
113C2×C4×C61C2×M1611yescharacteristic
114C2×C4×C61C2×M1611yescharacteristic
115C2×C4×C61#13512yes—
116C2×C4×C61#13512yes—
117C22⋊D61—48no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—22no—
123C2×Dic61—22no—
124C2×Dic61—48no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61#10012yes—
130C4×D61—24no—
131C4×D61—24no—
132C4×D61—44no—
133C4×D61—44no—
134Dic122—42no—
135Dic122—42no—
136C8×C61—82no—
137#92418—82no—
138#7645—611nomaximal, Sylow
139C22×C61—42no—
140C22×C61—44no—
141C22×C61—81no—
142C22×C61—81no—
143C22×C61—82no—
144C22×C61#40711yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C22×C61C2×#10011yescharacteristic
147C22×C61#41612yes—
148C22×C61#42112yes—
149C2×D61—22no—
150C2×D61—24no—
151C2×D61—41no—
152C2×D61—41no—
153C2×D61—42no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—81no—
159Dic61—22no—
160Dic61—42no—
161Dic61—42no—
162Dic61—81no—
163#2297—611no—
164#1817—612no—
165#1272—612no—
166#1270—611no—
167#1270D6111yescharacteristic, complemented
168C2×C61—82no—
169C2×C61#146211yescharacteristic
170C2×C61#148812yes—
171C2×C61#151612yes—
172C2×C61#153611yescharacteristic
173C2×C61#154311yescharacteristic
174D61—42no—
175C22×#41—611no—
176C2×#117—611no—
177C22×C42—611no—
178C2×C22⋊C8—1222no—
179#391—611no—
180#391—611no—
181#390—611no—
182#390C2×D6111yescharacteristic
183#386—1222no—
184C4×C22⋊C4—1222no—
185#336—612no—
186#336—1222no—
187#336C2×D6112yes—
188C61#764511yescharacteristic, complemented, Sylow
189C2×#41—1222no—
190C2×#41—1222no—
191C2×#41—2442no—
192C2×#41—2442no—
193C22×Q8—611no—
194C22×D4—1221no—
195C22×D4C4×D6111yescharacteristic
196C23×C4—611no—
197C23×C4—612no—
198C23×C4—1221no—
199C22×C8—22no—
200C22×C8—1222no—
201#117—2442no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4C4×D6111yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—1221no—
206C2×C22⋊C4—1222no—
207C2×C42—611no—
208C2×C42—612no—
209C2×C42—1221no—
210C2×C42—2442no—
211C2×C42C22×D6111yescharacteristic
212C22⋊C8—2444no—
213#95—1222no—
214#95—1222no—
215C24—21no—
216C24—611no—
217#41—2448no—
218#41—2448no—
219C2×Q8—612no—
220C2×Q8—1221no—
221C2×Q8—2441no—
222C2×Q8—2441no—
223C2×D4—21no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—1224no—
227C2×D4—2442no—
228C2×D4—2442no—
229C2×D4#1175818912yes—
230C22×C4—21no—
231C22×C4—611no—
232C22×C4—612no—
233C22×C4—1222no—
234C22×C4—1222no—
235C22×C4—1224no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4—1224no—
239C22×C4—1221no—
240C22×C4—1222no—
241C22×C4—1222no—
242C22×C4—1221no—
243C22×C4—2442no—
244C22×C4—2442no—
245C22×C4—2442no—
246C22×C4C2×C4×D6111yescharacteristic
247C22×C4D4×D6112yes—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—2442no—
252C2×C8—2442no—
253C2×C8—2442no—
254C4⋊C4—41no—
255C4⋊C4—2441no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—2442no—
259C22⋊C4—2444no—
260C42—41no—
261C42—2442no—
262C42—2441no—
263C42—2442no—
264C23—22no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—612no—
270C23—1221no—
271C23—1221no—
272C23—1222no—
273C23—1222no—
274C23C22⋊C4×D6111yescharacteristic
275Q8—2442no—
276Q8—2442no—
277D4—42no—
278D4—42no—
279D4—2448no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—612no—
287C2×C4—1224no—
288C2×C4—1224no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—2444no—
296C2×C4—2444no—
297C2×C4—2448no—
298C2×C4—2441no—
299C2×C4—2442no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4M16×D6111yescharacteristic
303C2×C4C2×#1175818911yescharacteristic
304C2×C4#1451979612yes—
305C2×C4#1451979812yes—
306C8—82no—
307C8—4882no—
308C22—42no—
309C22—44no—
310C22—82no—
311C22—81no—
312C22—81no—
313C22—1224no—
314C22—1222no—
315C22—2441no—
316C22—2441no—
317C22—2442no—
318C22#1683972411yescharacteristic
319C22#99×D6111yescharacteristic
320C22#100×D6111yescharacteristic
321C22#1683975512yes—
322C22#1683977712yes—
323C4—22no—
324C4—22no—
325C4—21no—
326C4—21no—
327C4—81no—
328C4—1222no—
329C4—4881no—
330C4—2442no—
331C4—2442no—
332C2—82no—
333C2—2442no—
334C2#2144414411yescharacteristic
335C2#2144425012yes—
336C2#2144437212yes—
337C2#2144445211yescharacteristic
338C2#2144447811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61C2×C22⋊C4
prime radicalC22×#11758224C4
cosocleC2×C42×C61C23
hypercenter#1270D61
hypocenterC61#7645
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21445886C2semidirect
3. #21443644C2semidirect
4. #21443640C2semidirect
5. C61×#1270C2semidirect
167. #1270D61semidirect
188. C61#7645semidirect

Maximal subgroups

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

subgroupindex
#214483062
#214458862
#214436442
#214436402
C61×#12702
#764561