Finite Group Databasebeta

← #29568110

Subgroups — #29568110

Order 15616.

654 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 117 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#21448308C211yesmaximal, characteristic
2#21445798C212yesmaximal, complemented
3#21443645C211yesmaximal, characteristic, complemented
4#21443639C212yesmaximal, complemented
5C61×#1272C211yesmaximal, characteristic, complemented
6C22×#11758226C411yescharacteristic
7C2×#14519735C411yescharacteristic
8C2×C42×D61C2211yescharacteristic
9C2×#14519706—22no—
10C2×#14519689—22no—
11#16839669C411yescharacteristic
12#16839664C2211yescharacteristic
13#16839663C411yescharacteristic
14#16839640—22no—
15#16839609—22no—
16#16839517C2212yes—
17C61×#391C2211yescharacteristic
18C61×#336C2212yes—
19C2×#11758226—22no—
20C2×#11758226—42no—
21C2×#11758226—42no—
22C2×#11758226C814yescomplemented
23C22×D244—21no—
24C22×D244C2×C411yescharacteristic
25C22×C4×D61—21no—
26C22×C4×D61C2×C411yescharacteristic
27C22×C4×D61D412yes—
28C22×#92418—22no—
29#14519735—42no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758198—21no—
32C2×#11758198—22no—
33C2×#11758198—22no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61C2×C411yescharacteristic
36C2×C4×Dic61D412yes—
37C42×D61—42no—
38#14519706—44no—
39#14519689—44no—
40#14519684—22no—
41C22×Q8×C61C2×C411yescharacteristic
42C22×C8×C61—22no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C42×C61C2311yescharacteristic
45#95×C61—22no—
46C23×D61—21no—
47C23×D61C22⋊C411yescharacteristic
48C2×Q8×C61—21no—
49C2×Q8×C61—41no—
50C2×Q8×C61—41no—
51C2×Q8×C61C2×C812yes—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61C22×C411yescharacteristic
55C22×C4×C61C2×D412yes—
56C22×Dic61—21no—
57C22×Dic61C22⋊C412yes—
58C22×Dic61C22⋊C411yescharacteristic
59#11758226—48no—
60#11758226—48no—
61C2×D244—24no—
62C2×D244—21no—
63C2×D244—41no—
64C2×D244—41no—
65C2×D244—42no—
66C2×D244—42no—
67C2×D244C2×C812yes—
68C2×C4×D61—22no—
69C2×C4×D61—22no—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—24no—
73C2×C4×D61—42no—
74C2×C4×D61—42no—
75C2×C4×D61—42no—
76C2×C4×D61—42no—
77C2×C4×D61C2×C812yes—
78C2×C4×D61M1612yes—
79C2×C8×C61—42no—
80C2×C8×C61—42no—
81C2×C8×C61—42no—
82C4⋊C4×C61—41no—
83C42×C61—41no—
84#11758198—44no—
85#11758198—42no—
86#11758198—42no—
87#11758198—42no—
88C4⋊Dic61—41no—
89C4×Dic61—24no—
90C4×Dic61—41no—
91C4×Dic61—42no—
92C2×#92418—42no—
93C2×#92418—42no—
94C2×#92418—42no—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—22no—
99C22×D61—22no—
100C22×D61—22no—
101C22×D61—41no—
102C22×D61—41no—
103C22×D61—42no—
104C22×D61—42no—
105C22×D61C22⋊C812yes—
106Q8×C61—42no—
107Q8×C61—42no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61C22×C811yescharacteristic
118C2×C4×C61C2×M1611yescharacteristic
119C2×C4×C61AGL(1,ℤ/8)12yes—
120C2×C4×C61#13712yes—
121C2×Dic61—22no—
122C2×Dic61—22no—
123C2×Dic61—22no—
124C2×Dic61—41no—
125C2×Dic61—42no—
126C2×Dic61C22⋊C812yes—
127C2×Dic61#9912yes—
128C2×Dic61#10012yes—
129D244—42no—
130D244—42no—
131D244—48no—
132C4×D61—24no—
133C4×D61—24no—
134C4×D61—44no—
135C4×D61—44no—
136C4×D61—48no—
137C8×C61—82no—
138#92418—82no—
139#7597—611nomaximal, Sylow
140C22×C61C2×C22⋊C811yescharacteristic
141C22×C61C2×#9911yescharacteristic
142C22×C61C2×#10011yescharacteristic
143C22×C61#41712yes—
144C22×C61#42112yes—
145C2×D61—22no—
146C2×D61—24no—
147C2×D61—41no—
148C2×D61—41no—
149C2×D61—42no—
150C2×D61—42no—
151C2×D61—44no—
152C2×D61—81no—
153C2×D61—81no—
154C2×D61—82no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—42no—
160C4×C61—81no—
161Dic61—42no—
162Dic61—81no—
163Dic61#32314yes—
164#2297—611no—
165#1797—612no—
166#1272—611no—
167#1272D6111yescharacteristic, complemented
168#1270—612no—
169C2×C61C2×#32311yescharacteristic
170C2×C61#147712yes—
171C2×C61#151312yes—
172C2×C61#153611yescharacteristic
173C2×C61#156011yescharacteristic
174D61—42no—
175D61—82no—
176C22×#41—611no—
177C2×#117—611no—
178C22×C42—611no—
179C2×C4⋊C8—1222no—
180C2×C22⋊C8—1222no—
181#391—611no—
182#391C2×D6111yescharacteristic
183#390—611no—
184#390—611no—
185#386—1222no—
186C4×C22⋊C4—1222no—
187#336—612no—
188#336C2×D6112yes—
189C61#759711yescharacteristic, complemented, Sylow
190C2×#41—614no—
191C2×#41—1222no—
192C2×#41—2442no—
193C2×#41—2442no—
194C22×Q8C4×D6111yescharacteristic
195C22×D4—611no—
196C22×D4—1221no—
197C23×C4—611no—
198C23×C4—612no—
199C23×C4—1221no—
200C22×C8—22no—
201C22×C8—1222no—
202#117—2442no—
203C2×C4⋊C4—611no—
204C2×C4⋊C4C4×D6111yescharacteristic
205C2×C22⋊C4—1222no—
206C2×C22⋊C4—1221no—
207C2×C22⋊C4—1222no—
208C2×C42—612no—
209C2×C42—611no—
210C2×C42—1221no—
211C2×C42—2442no—
212C2×C42C22×D6111yescharacteristic
213C4⋊C8—2444no—
214C22⋊C8—2444no—
215#95—22no—
216#95—1222no—
217C24—611no—
218C24—1221no—
219#41—2448no—
220#41—2448no—
221C2×Q8—21no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8C8×D6112yes—
225C2×D4—612no—
226C2×D4—1224no—
227C2×D4—1221no—
228C2×D4—2441no—
229C2×D4—2441no—
230C2×D4—2442no—
231C2×D4—2442no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—611no—
235C22×C4—612no—
236C22×C4—612no—
237C22×C4—612no—
238C22×C4—1222no—
239C22×C4—1222no—
240C22×C4—1224no—
241C22×C4—1222no—
242C22×C4—1222no—
243C22×C4—1221no—
244C22×C4—2442no—
245C22×C4—2442no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4C2×C4×D6111yescharacteristic
249C22×C4D4×D6112yes—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—2442no—
254C2×C8—2442no—
255C2×C8—2442no—
256C4⋊C4—41no—
257C4⋊C4—2441no—
258C22⋊C4—2444no—
259C22⋊C4—2442no—
260C22⋊C4—2442no—
261C22⋊C4—2442no—
262C42—41no—
263C42—1224no—
264C42—2441no—
265C42—2442no—
266C23—612no—
267C23—1221no—
268C23—1221no—
269C23—1222no—
270C23—1222no—
271C23—1222no—
272C23—2441no—
273C23—2441no—
274C23—2442no—
275C23—2442no—
276C23C22⋊C4×D6111yescharacteristic
277Q8—42no—
278Q8—42no—
279D4—2442no—
280D4—2442no—
281D4—2448no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—612no—
292C2×C4—612no—
293C2×C4—612no—
294C2×C4—1224no—
295C2×C4—1224no—
296C2×C4—1222no—
297C2×C4—1222no—
298C2×C4—1222no—
299C2×C4—2444no—
300C2×C4—2444no—
301C2×C4—2448no—
302C2×C4—2441no—
303C2×C4—2442no—
304C2×C4C2×C8×D6111yescharacteristic
305C2×C4M16×D6111yescharacteristic
306C2×C4#1451980412yes—
307C2×C4#1451980912yes—
308C8—82no—
309C8—4882no—
310C22—1222no—
311C22—1224no—
312C22—4882no—
313C22—4881no—
314C22—4881no—
315C22—2442no—
316C22—2444no—
317C22—2442no—
318C22—2441no—
319C22—2441no—
320C22C22⋊C8×D6111yescharacteristic
321C22#99×D6111yescharacteristic
322C22#100×D6111yescharacteristic
323C22#1683976212yes—
324C22#1683977712yes—
325C4—22no—
326C4—22no—
327C4—21no—
328C4—21no—
329C4—42no—
330C4—81no—
331C4—614no—
332C4—4881no—
333C4—2442no—
334C2—4882no—
335C2—2442no—
336C2#323×D6111yescharacteristic
337C2#2144420512yes—
338C2#2144436112yes—
339C2#2144445211yescharacteristic
340C2#2144455311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC61×#1272C2
socleC23×C61C2×C22⋊C4
prime radicalC22×#11758226C4
cosocleC2×C42×C61C23
hypercenter#1272D61
hypocenterC61#7597
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #21445798C2semidirect
3. #21443645C2semidirect
4. #21443639C2semidirect
5. C61×#1272C2semidirect
22. C2×#11758226C8semidirect
167. #1272D61semidirect
189. C61#7597semidirect

Maximal subgroups

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

subgroupindex
#214483082
#214457982
#214436452
#214436392
C61×#12722
#759761