Finite Group Databasebeta

← #29568246

Subgroups — #29568246

Order 15616.

658 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 109 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#391×D61C211yesmaximal, characteristic
2#21445580C212yesmaximal
3#21443645C211yesmaximal, characteristic, complemented
4#21443644C212yesmaximal, complemented
5C61×#1272C211yesmaximal, characteristic, complemented
6C22×Q8×D61C411yescharacteristic
7C2×C4⋊C4×D61C411yescharacteristic
8C2×C42×D61C2211yescharacteristic
9C2×#14519689—22no—
10C2×#14519687—22no—
11#16839669C411yescharacteristic
12#16839668C2211yescharacteristic
13#16839667C411yescharacteristic
14#16839666—22no—
15#95×D61—22no—
16#16839517C2212yes—
17C61×#391C2211yescharacteristic
18C61×#336C2212yes—
19C22×Dic122—21no—
20C22×Dic122C2×C411yescharacteristic
21C2×Q8×D61—22no—
22C2×Q8×D61—22no—
23C2×Q8×D61—42no—
24C2×Q8×D61—42no—
25C22×C4×D61—21no—
26C22×C4×D61—21no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D412yes—
29C22×#92418—22no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C4⋊C4×D61—42no—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61D412yes—
35C42×D61—42no—
36#14519689—28no—
37#14519687—44no—
38#14519684—22no—
39#14519684—22no—
40#14519684—22no—
41C22×Q8×C61C2×C411yescharacteristic
42C22×C8×C61—22no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C42×C61C2311yescharacteristic
45#95×C61—22no—
46C23×D61C22⋊C411yescharacteristic
47C2×Q8×C61—21no—
48C2×Q8×C61—41no—
49C2×Q8×C61—41no—
50C2×Q8×C61M1612yes—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61C22×C411yescharacteristic
54C22×C4×C61C2×D412yes—
55C22×Dic61—21no—
56C22×Dic61—21no—
57C22×Dic61C22⋊C412yes—
58C22×Dic61C22⋊C411yescharacteristic
59Q8×D61—48no—
60Q8×D61—48no—
61C2×C4×D61—22no—
62C2×C4×D61—21no—
63C2×C4×D61—21no—
64C2×C4×D61—22no—
65C2×C4×D61—24no—
66C2×C4×D61—24no—
67C2×C4×D61—22no—
68C2×C4×D61—22no—
69C2×C4×D61—42no—
70C2×C4×D61—42no—
71C2×C4×D61—42no—
72C2×C4×D61—42no—
73C2×C4×D61—44no—
74C2×Dic122—24no—
75C2×Dic122—21no—
76C2×Dic122—42no—
77C2×Dic122—42no—
78C2×Dic122—41no—
79C2×Dic122—41no—
80C2×Dic122M1612yes—
81C2×C8×C61—24no—
82C2×C8×C61—42no—
83C2×C8×C61—42no—
84C4⋊C4×C61—41no—
85C42×C61—41no—
86C4⋊Dic61—41no—
87#11758196—42no—
88C4×Dic61—24no—
89C4×Dic61—41no—
90C2×#92418—24no—
91C2×#92418—42no—
92C2×#92418—42no—
93C23×C61C2×C22⋊C411yescharacteristic
94C22×D61—21no—
95C22×D61—21no—
96C22×D61—22no—
97C22×D61#10012yes—
98C22×D61#10414yes—
99Q8×C61—42no—
100Q8×C61—42no—
101C2×C4×C61—22no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61C2×M1611yescharacteristic
111C2×C4×C61C2×M1611yescharacteristic
112C2×C4×C61#13712yes—
113C2×C4×C61#13712yes—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—22no—
117C2×Dic61—22no—
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—42no—
124C2×Dic61#9912yes—
125C4×D61—24no—
126C4×D61—24no—
127C4×D61—44no—
128C4×D61—44no—
129C4×D61—48no—
130C4×D61—84no—
131Dic122—48no—
132Dic122—42no—
133Dic122—42no—
134C8×C61—44no—
135#92418—44no—
136#7629—611nomaximal, Sylow
137C22×C61#40711yescharacteristic
138C22×C61C2×#9911yescharacteristic
139C22×C61C2×#10011yescharacteristic
140C22×C61#41912yes—
141C22×C61C2×#10412yes—
142C2×D61—22no—
143C2×D61—22no—
144C2×D61—24no—
145C2×D61—24no—
146C2×D61#32914yes—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—22no—
150C4×C61—22no—
151C4×C61—42no—
152C4×C61—81no—
153Dic61—22no—
154Dic61—42no—
155Dic61—42no—
156Dic61—81no—
157C2×#391—611no—
158#1749—612no—
159#1272—611no—
160#1272—612no—
161#1272D6111yescharacteristic, complemented
162C2×C61#146211yescharacteristic
163C2×C61#149112yes—
164C2×C61#150812yes—
165C2×C61C2×#32911yescharacteristic
166C2×C61#155811yescharacteristic
167D61—24no—
168C23×Q8—611no—
169C22×C4⋊C4—611no—
170C22×C42—611no—
171C2×C22⋊C8—1222no—
172C2×#97—1222no—
173#391—611no—
174#391—611no—
175#391—611no—
176#391—1222no—
177#391C2×D6111yescharacteristic
178C2×#95—1222no—
179#336—612no—
180#336C2×D6112yes—
181C61#762911yescharacteristic, complemented, Sylow
182C22×Q8—611no—
183C22×Q8—1221no—
184C22×Q8—1222no—
185C22×Q8—1222no—
186C22×Q8—2442no—
187C22×Q8—2442no—
188C22×Q8C4×D6111yescharacteristic
189C23×C4—611no—
190C23×C4—612no—
191C23×C4—1221no—
192C23×C4—1221no—
193C22×C8—22no—
194C22×C8—1222no—
195C2×C4⋊C4—611no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—2442no—
198C2×C4⋊C4C4×D6111yescharacteristic
199C2×C42—612no—
200C2×C42—611no—
201C2×C42—2442no—
202C2×C42C22×D6111yescharacteristic
203C22⋊C8—1228no—
204#97—2444no—
205#95—22no—
206#95—1222no—
207#95—1222no—
208#95—1222no—
209C24—611no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—612no—
214C2×Q8—1224no—
215C2×Q8—1221no—
216C2×Q8—2441no—
217C2×Q8—2441no—
218C2×Q8—2442no—
219C2×Q8—2442no—
220C2×Q8—2448no—
221C2×Q8—2448no—
222C2×Q8#1175818912yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—611no—
226C22×C4—612no—
227C22×C4—1222no—
228C22×C4—1222no—
229C22×C4—1224no—
230C22×C4—1224no—
231C22×C4—1221no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1222no—
235C22×C4—1222no—
236C22×C4—1221no—
237C22×C4—2442no—
238C22×C4—2442no—
239C22×C4—2444no—
240C22×C4—2442no—
241C22×C4—2442no—
242C22×C4C2×C4×D6111yescharacteristic
243C22×C4D4×D6112yes—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—1224no—
248C2×C8—2442no—
249C2×C8—2442no—
250C4⋊C4—41no—
251C4⋊C4—2441no—
252C4⋊C4—2442no—
253C42—41no—
254C42—1224no—
255C42—2441no—
256C23—612no—
257C23—614no—
258C23—1221no—
259C23—1221no—
260C23—1222no—
261C23C22⋊C4×D6111yescharacteristic
262Q8—42no—
263Q8—42no—
264Q8—2442no—
265Q8—2442no—
266Q8—2448no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—612no—
277C2×C4—1224no—
278C2×C4—1224no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1222no—
282C2×C4—1222no—
283C2×C4—1222no—
284C2×C4—1222no—
285C2×C4—4884no—
286C2×C4—2444no—
287C2×C4—2444no—
288C2×C4—2448no—
289C2×C4—2441no—
290C2×C4—2442no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4M16×D6111yescharacteristic
294C2×C4C2×#1175818911yescharacteristic
295C2×C4#1451980612yes—
296C2×C4#1451980912yes—
297C8—44no—
298C8—2444no—
299C22—614no—
300C22—1222no—
301C22—1224no—
302C22—1222no—
303C22—1224no—
304C22#1683972411yescharacteristic
305C22#99×D6111yescharacteristic
306C22#100×D6111yescharacteristic
307C22#1683976912yes—
308C22#104×D6112yes—
309C4—21no—
310C4—21no—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—81no—
315C4—1222no—
316C4—4881no—
317C4—2442no—
318C4—2442no—
319C2—1224no—
320C2#2144414411yescharacteristic
321C2#2144426412yes—
322C2#2144433712yes—
323C2#329×D6111yescharacteristic
324C2#2144454311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1272C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1272D61
hypocenterC61#7629
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
3. #21443645C2semidirect
4. #21443644C2semidirect
5. C61×#1272C2semidirect
161. #1272D61semidirect
181. C61#7629semidirect

Maximal subgroups

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

subgroupindex
#391×D612
#214455802
#214436452
#214436442
C61×#12722
#762961