Finite Group Databasebeta

← #29567962

Subgroups — #29567962

Order 15616.

690 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 121 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#21445576C212yesmaximal, complemented
3#21443644C212yesmaximal, complemented
4#21443640C211yesmaximal, characteristic, complemented
5C61×#1270C211yesmaximal, characteristic, complemented
6C22×#11758224C411yescharacteristic
7C2×#14519735C411yescharacteristic
8C2×C42×D61C2211yescharacteristic
9C2×#14519689—22no—
10C2×C8×Dic61—22no—
11#16839668C2211yescharacteristic
12#16839667C411yescharacteristic
13#16839665C411yescharacteristic
14#16839638—22no—
15C22⋊C4×Dic61—22no—
16#16839517C2212yes—
17C61×#390C2211yescharacteristic
18C61×#336C2212yes—
19C2×#11758224—22no—
20C2×#11758224—42no—
21C2×#11758224—42no—
22C2×#11758224C814yescomplemented
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—28no—
38C8×Dic61—44no—
39#14519684—22no—
40#14519684—22no—
41C22×D4×C61C2×C411yescharacteristic
42C22×C8×C61—22no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C22⋊C4×C61—22no—
45C2×C42×C61C2311yescharacteristic
46C24×C61—21no—
47C23×D61C22⋊C411yescharacteristic
48C2×D4×C61—21no—
49C2×D4×C61—41no—
50C2×D4×C61—41no—
51C2×D4×C61C2×C812yes—
52C22×C4×C61—21no—
53C22×C4×C61C22×C411yescharacteristic
54C22×C4×C61C2×D412yes—
55C2×C22⋊D61—24no—
56C2×C22⋊D61—42no—
57C2×C22⋊D61—42no—
58C22×Dic61—21no—
59C22×Dic61—24no—
60C22×Dic61—21no—
61C22×Dic61—42no—
62C22×Dic61—42no—
63C22×Dic61C22⋊C412yes—
64C22×Dic61C22⋊C411yescharacteristic
65#11758224—48no—
66#11758224—48no—
67C2×C4×D61—22no—
68C2×C4×D61—22no—
69C2×C4×D61—22no—
70C2×C4×D61—22no—
71C2×C4×D61—24no—
72C2×C4×D61—42no—
73C2×C4×D61C2×C812yes—
74C2×C4×D61M1612yes—
75C2×Dic122—21no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×Dic122C2×C812yes—
79C2×C8×C61—24no—
80C2×C8×C61—42no—
81C2×C8×C61—42no—
82C4⋊C4×C61—41no—
83C22⋊C4×C61—42no—
84C22⋊C4×C61—42no—
85C42×C61—41no—
86C22⋊Dic61—44no—
87#11758198—42no—
88C4⋊Dic61—41no—
89C4×Dic61—24no—
90C4×Dic61—41no—
91C4×Dic61—42no—
92C2×#92418—24no—
93C2×#92418—42no—
94C2×#92418—42no—
95C23×C61—22no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—22no—
104C22×D61—22no—
105C22×D61C22⋊C812yes—
106D4×C61—42no—
107D4×C61—42no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—22no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61C22×C811yescharacteristic
115C2×C4×C61C2×M1611yescharacteristic
116C2×C4×C61#13512yes—
117C2×C4×C61#13512yes—
118C22⋊D61—48no—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—48no—
125C2×Dic61—41no—
126C2×Dic61—42no—
127C2×Dic61C22⋊C812yes—
128C2×Dic61#9912yes—
129C2×Dic61#10012yes—
130C2×Dic61#10414yes—
131C4×D61—24no—
132C4×D61—24no—
133C4×D61—44no—
134C4×D61—44no—
135Dic122—42no—
136Dic122—42no—
137C8×C61—44no—
138#92418—44no—
139#7563—611nomaximal, Sylow
140C22×C61—42no—
141C22×C61—44no—
142C22×C61—81no—
143C22×C61—82no—
144C22×C61—81no—
145C22×C61C2×C22⋊C811yescharacteristic
146C22×C61C2×#9911yescharacteristic
147C22×C61C2×#10011yescharacteristic
148C22×C61#41612yes—
149C22×C61C2×#10412yes—
150C2×D61—22no—
151C2×D61—24no—
152C2×D61—41no—
153C2×D61—42no—
154C2×D61—41no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—81no—
160Dic61—24no—
161Dic61—42no—
162Dic61—81no—
163Dic61#32314yes—
164#2297—611no—
165C4×C22⋊C8—612no—
166#1272—612no—
167#1270—611no—
168#1270D6111yescharacteristic, complemented
169C2×C61—82no—
170C2×C61C2×#32311yescharacteristic
171C2×C61#147612yes—
172C2×C61#150112yes—
173C2×C61#152511yescharacteristic
174C2×C61#154311yescharacteristic
175D61—42no—
176C22×#41—611no—
177C2×#117—611no—
178C22×C42—611no—
179C2×C22⋊C8—1222no—
180C2×C4×C8—1222no—
181#391—611no—
182#391—611no—
183#390—611no—
184#390C2×D6111yescharacteristic
185#386—1222no—
186C4×C22⋊C4—1222no—
187#336—612no—
188#336C2×D6112yes—
189C61#756311yescharacteristic, complemented, Sylow
190C2×#41—614no—
191C2×#41—1222no—
192C2×#41—2442no—
193C2×#41—2442no—
194C22×Q8—611no—
195C22×D4—1221no—
196C22×D4C4×D6111yescharacteristic
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—22no—
206C2×C22⋊C4—1221no—
207C2×C22⋊C4—1222no—
208C2×C42—612no—
209C2×C42—611no—
210C2×C42—1221no—
211C2×C42—2442no—
212C2×C42C22×D6111yescharacteristic
213C22⋊C8—1228no—
214C4×C8—2444no—
215#95—1222no—
216#95—1222no—
217C24—21no—
218C24—611no—
219#41—2448no—
220#41—2448no—
221C2×Q8—612no—
222C2×Q8—1221no—
223C2×Q8—2441no—
224C2×Q8—2441no—
225C2×D4—21no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—1224no—
229C2×D4—2442no—
230C2×D4—2442no—
231C2×D4C8×D6112yes—
232C22×C4—21no—
233C22×C4—611no—
234C22×C4—612no—
235C22×C4—612no—
236C22×C4—612no—
237C22×C4—1222no—
238C22×C4—1222no—
239C22×C4—1224no—
240C22×C4—1224no—
241C22×C4—1221no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—1221no—
245C22×C4—2442no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4C2×C4×D6111yescharacteristic
249C22×C4D4×D6112yes—
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—1224no—
254C2×C8—2442no—
255C2×C8—2442no—
256C4⋊C4—41no—
257C4⋊C4—2441no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—2442no—
261C22⋊C4—2444no—
262C42—41no—
263C42—1224no—
264C42—2441no—
265C42—2442no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—612no—
272C23—1221no—
273C23—1221no—
274C23—1222no—
275C23—1222no—
276C23C22⋊C4×D6111yescharacteristic
277Q8—2442no—
278Q8—2442no—
279D4—42no—
280D4—42no—
281D4—2448no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—612no—
289C2×C4—612no—
290C2×C4—612no—
291C2×C4—614no—
292C2×C4—1224no—
293C2×C4—1224no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—2444no—
298C2×C4—2444no—
299C2×C4—2448no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4C2×C8×D6111yescharacteristic
305C2×C4M16×D6111yescharacteristic
306C2×C4#1451979612yes—
307C2×C4#1451979812yes—
308C8—44no—
309C8—2444no—
310C22—42no—
311C22—44no—
312C22—81no—
313C22—81no—
314C22—82no—
315C22—1222no—
316C22—1224no—
317C22—2441no—
318C22—2441no—
319C22—2442no—
320C22C22⋊C8×D6111yescharacteristic
321C22#99×D6111yescharacteristic
322C22#100×D6111yescharacteristic
323C22#1683975512yes—
324C22#104×D6112yes—
325C4—21no—
326C4—21no—
327C4—22no—
328C4—22no—
329C4—81no—
330C4—614no—
331C4—1224no—
332C4—4881no—
333C4—2442no—
334C2—82no—
335C2—2442no—
336C2#323×D6111yescharacteristic
337C2#2144419812yes—
338C2#2144430712yes—
339C2#2144440911yescharacteristic
340C2#2144447811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61C2×C22⋊C4
prime radicalC22×#11758224C4
cosocleC2×C42×C61C23
hypercenter#1270D61
hypocenterC61#7563
LCSG ▷ C2×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. #21445576C2semidirect
3. #21443644C2semidirect
4. #21443640C2semidirect
5. C61×#1270C2semidirect
22. C2×#11758224C8semidirect
168. #1270D61semidirect
189. C61#7563semidirect

Maximal subgroups

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

subgroupindex
#214483062
#214455762
#214436442
#214436402
C61×#12702
#756361