Finite Group Databasebeta

← #29568254

Subgroups — #29568254

Order 15616.

730 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 105 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#21445511C212yesmaximal, complemented
3#21443644C212yesmaximal, complemented
4#21443640C211yesmaximal, characteristic, complemented
5C61×#1270C211yesmaximal, characteristic, complemented
6C22×#11758224C411yescharacteristic
7C22×#11758189—22no—
8C2×#14519735C411yescharacteristic
9C2×C42×D61C2211yescharacteristic
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—
28C2×#11758189—28no—
29C2×#11758189—44no—
30C2×#11758189—44no—
31C22×#92418—22no—
32#14519735—42no—
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×C22⋊Dic61—22no—
35C2×#11758198—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61D412yes—
39C42×D61—42no—
40#14519684—22no—
41#14519684—22no—
42C22×D4×C61C2×C411yescharacteristic
43C22×C8×C61—22no—
44C2×C4⋊C4×C61C2×C411yescharacteristic
45C2×C22⋊C4×C61—22no—
46C2×C42×C61C2311yescharacteristic
47C24×C61—21no—
48C23×D61C22⋊C411yescharacteristic
49C2×D4×C61—21no—
50C2×D4×C61—41no—
51C2×D4×C61—41no—
52C2×D4×C61M1612yescomplemented
53C22×C4×C61—21no—
54C22×C4×C61C22×C411yescharacteristic
55C22×C4×C61C2×D412yes—
56C2×C22⋊D61—24no—
57C2×C22⋊D61—42no—
58C2×C22⋊D61—42no—
59C22×Dic61—24no—
60C22×Dic61—21no—
61C22×Dic61—21no—
62C22×Dic61—42no—
63C22×Dic61—42no—
64C22×Dic61C22⋊C411yescharacteristic
65C22×Dic61C22⋊C412yes—
66#11758224—48no—
67#11758224—48no—
68C2×C4×D61—22no—
69C2×C4×D61—24no—
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—22no—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—24no—
76C2×Dic122—21no—
77C2×Dic122—41no—
78C2×Dic122—41no—
79C2×Dic122M1612yescomplemented
80C2×C8×C61—24no—
81C2×C8×C61—42no—
82C2×C8×C61—42no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—42no—
85C22⋊C4×C61—42no—
86C42×C61—41no—
87C22⋊Dic61—44no—
88#11758198—42no—
89C4⋊Dic61—41no—
90C4×Dic61—41no—
91C4×Dic61—42no—
92C4×Dic61—42no—
93C2×#92418—24no—
94C2×#92418—42no—
95C2×#92418—42no—
96#11758189—416no—
97C23×C61—22no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61C2×C22⋊C411yescharacteristic
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—22no—
106C22×D61—22no—
107C22×D61#9912yes—
108D4×C61—42no—
109D4×C61—42no—
110C2×C4×C61—22no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—22no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61C2×M1611yescharacteristic
117C2×C4×C61C2×M1611yescharacteristic
118C2×C4×C61AGL(1,ℤ/8)12yes—
119C2×C4×C61#13712yes—
120C22⋊D61—48no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—22no—
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—48no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—42no—
131C2×Dic61#10012yes—
132C2×Dic61#10414yes—
133C4×D61—28no—
134C4×D61—28no—
135C4×D61—24no—
136C4×D61—24no—
137Dic122—42no—
138Dic122—42no—
139C8×C61—44no—
140#92418—44no—
141#7631—611nomaximal, Sylow
142C22×C61—42no—
143C22×C61—44no—
144C22×C61—81no—
145C22×C61—82no—
146C22×C61—81no—
147C22×C61#40711yescharacteristic
148C22×C61C2×#9911yescharacteristic
149C22×C61C2×#10011yescharacteristic
150C22×C61#41712yes—
151C22×C61C2×#10412yes—
152C2×D61—22no—
153C2×D61—24no—
154C2×D61—41no—
155C2×D61—42no—
156C2×D61—41no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—22no—
160C4×C61—22no—
161C4×C61—81no—
162Dic61—22no—
163Dic61—24no—
164Dic61—42no—
165Dic61—81no—
166#2297—611no—
167#1728—612no—
168#1272—612no—
169#1270—611no—
170#1270D6111yescharacteristic, complemented
171C2×C61—82no—
172C2×C61#146211yescharacteristic
173C2×C61#148912yes—
174C2×C61#150412yes—
175C2×C61#152511yescharacteristic
176C2×C61#156011yescharacteristic
177D61—42no—
178C22×#41—611no—
179C22×M16—1222no—
180C2×#117—611no—
181C22×C42—611no—
182#391—611no—
183#391—611no—
184#390—611no—
185#390C2×D6111yescharacteristic
186#386—1222no—
187C4×C22⋊C4—1222no—
188#336—612no—
189#336—1222no—
190#336C2×D6112yes—
191C61#763111yescharacteristic, complemented, Sylow
192C2×#41—1222no—
193C2×#41—1222no—
194C2×#41—2442no—
195C2×#41—2442no—
196C22×Q8—611no—
197C22×D4—1221no—
198C22×D4C4×D6111yescharacteristic
199C23×C4—611no—
200C23×C4—612no—
201C23×C4—1221no—
202C2×M16—1228no—
203C2×M16—2444no—
204C2×M16—2444no—
205C22×C8—22no—
206C22×C8—1222no—
207#117—2442no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4C4×D6111yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1222no—
213C2×C42—611no—
214C2×C42—612no—
215C2×C42—1221no—
216C2×C42—2442no—
217C2×C42C22×D6111yescharacteristic
218#95—1222no—
219#95—1222no—
220C24—21no—
221C24—611no—
222#41—2448no—
223#41—2448no—
224C2×Q8—612no—
225C2×Q8—1221no—
226C2×Q8—2441no—
227C2×Q8—2441no—
228C2×D4—21no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—1224no—
232C2×D4—2442no—
233C2×D4—2442no—
234C2×D4#1175818912yescomplemented
235C22×C4—21no—
236C22×C4—612no—
237C22×C4—611no—
238C22×C4—1222no—
239C22×C4—1224no—
240C22×C4—1224no—
241C22×C4—1222no—
242C22×C4—1221no—
243C22×C4—1221no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4—1224no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—2442no—
250C22×C4—2442no—
251C22×C4C2×C4×D6111yescharacteristic
252C22×C4D4×D6112yes—
253M16—24416no—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—1224no—
258C2×C8—2442no—
259C2×C8—2442no—
260C4⋊C4—41no—
261C4⋊C4—2441no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—2442no—
265C22⋊C4—2444no—
266C42—41no—
267C42—2442no—
268C42—2441no—
269C42—2442no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—42no—
275C23—612no—
276C23—1221no—
277C23—1221no—
278C23—1222no—
279C23—1222no—
280C23C22⋊C4×D6111yescharacteristic
281Q8—2442no—
282Q8—2442no—
283D4—42no—
284D4—42no—
285D4—2448no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—612no—
293C2×C4—614no—
294C2×C4—1228no—
295C2×C4—1228no—
296C2×C4—1224no—
297C2×C4—1224no—
298C2×C4—1221no—
299C2×C4—1222no—
300C2×C4—1221no—
301C2×C4—1222no—
302C2×C4—1222no—
303C2×C4—2448no—
304C2×C4—2441no—
305C2×C4—2442no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4M16×D6111yescharacteristic
309C2×C4C2×#1175818911yescharacteristic
310C2×C4#1451980112yes—
311C2×C4#1451980612yes—
312C8—44no—
313C8—2444no—
314C22—42no—
315C22—44no—
316C22—81no—
317C22—81no—
318C22—82no—
319C22—1222no—
320C22—1224no—
321C22—2441no—
322C22—2442no—
323C22—2441no—
324C22#1683972411yescharacteristic
325C22#99×D6111yescharacteristic
326C22#100×D6111yescharacteristic
327C22#1683976012yes—
328C22#104×D6112yes—
329C4—21no—
330C4—21no—
331C4—22no—
332C4—22no—
333C4—81no—
334C4—1222no—
335C4—1224no—
336C4—4881no—
337C4—2442no—
338C2—82no—
339C2—2442no—
340C2#2144414411yescharacteristic
341C2#2144425512yes—
342C2#2144432012yes—
343C2#2144440911yescharacteristic
344C2#2144455111yescharacteristic
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#7631
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. #21445511C2semidirect
3. #21443644C2semidirect
4. #21443640C2semidirect
5. C61×#1270C2semidirect
52. C2×D4×C61M16semidirect
79. C2×Dic122M16semidirect
170. #1270D61semidirect
191. C61#7631semidirect
234. C2×D4#11758189semidirect

Maximal subgroups

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

subgroupindex
#214483062
#214455112
#214436442
#214436402
C61×#12702
#763161