Finite Group Databasebeta

← #29575518

Subgroups — #29575518

Order 15616.

630 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 139 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#21448541C211yesmaximal, characteristic
2#21444211C212yesmaximal, complemented
3#21444140C211yesmaximal, characteristic, complemented
4#21443647C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic, complemented
6C2×#14519773C411yescharacteristic
7C2×C4×D244C2211yescharacteristic
8#16839815C2212yes—
9#16839665C411yescharacteristic
10#16839663C411yescharacteristic
11#16839641—22no—
12#16839638—22no—
13#16839627C411yescharacteristic
14#16839608C2211yescharacteristic
15#16839491—24no—
16#16839486—22no—
17C2×C4×D4×C61C2211yescharacteristic
18C8×D4×C61—24no—
19C61×#431C2212yes—
20C2×C22⋊C8×C61—22no—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—22no—
23C22×C4×D61—21no—
24#14519773—22no—
25#14519773C814yescomplemented
26C4×D244—22no—
27C4×D244—22no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×C22⋊Dic61—22no—
30C2×#11758198—21no—
31C2×#11758198—22no—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61C2×C411yescharacteristic
34C4⋊#92418—22no—
35C22⋊#92418—22no—
36C4×#92418—22no—
37#14519684—22no—
38#14519684—21no—
39#14519684—21no—
40C22×D4×C61C2×C411yescharacteristic
41C23×C4×C61D412yes—
42C22×C8×C61—24no—
43C4×D4×C61—22no—
44C4×D4×C61—22no—
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61D412yes—
47C2×C42×C61C2311yescharacteristic
48C4⋊C8×C61—22no—
49C22⋊C8×C61—24no—
50C22⋊C8×C61—22no—
51C4×C8×C61D414yescomplemented
52C24×C61C22⋊C412yes—
53C23×D61—21no—
54C2×D4×C61—22no—
55C2×D4×C61—22no—
56C2×D4×C61C2×C812yes—
57C2×D4×C61M1612yes—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—82no—
61C22×C4×C61C22⋊C412yes—
62C22×C4×C61C22×C411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C2×C22⋊D61—28no—
66C2×C22⋊D61—82no—
67C2×C22⋊D61—82no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C2×D244—22no—
72C2×D244—22no—
73C2×D244—21no—
74C2×D244C2×C812yescomplemented
75C2×C4×D61—42no—
76C2×C4×D61—82no—
77C2×C8×C61—24no—
78C2×C8×C61—84no—
79C4⋊C4×C61—22no—
80C22⋊C4×C61—24no—
81C42×C61C2×D412yes—
82C42×C61#4112yes—
83C22⋊Dic61—82no—
84#11758198—41no—
85#11758198—41no—
86#11758198—82no—
87C4⋊Dic61—22no—
88C4×Dic61—21no—
89C4×Dic61—22no—
90C4×Dic61C2×C812yescomplemented
91C2×#92418—42no—
92C23×C61—22no—
93C23×C61—22no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61C22⋊C814yes—
97C23×C61C2×C22⋊C411yescharacteristic
98C22×D61—22no—
99C22×D61—42no—
100C22×D61—42no—
101C22×D61—41no—
102C22×D61—41no—
103D4×C61—24no—
104D4×C61—24no—
105C2×C4×C61—22no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—44no—
109C2×C4×C61—42no—
110C2×C4×C61—82no—
111C2×C4×C61C4×D412yes—
112C2×C4×C61#12011yescharacteristic
113C2×C4×C61#12112yes—
114C2×C4×C61#12311yescharacteristic
115C2×C4×C61C22×C811yescharacteristic
116C2×C4×C61C2×M1611yescharacteristic
117C2×C4×C61#13112yes—
118C22⋊D61—88no—
119C2×Dic61—22no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—42no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127D244—24no—
128D244—42no—
129C4×D61—84no—
130C8×C61—44no—
131#92418—82no—
132#9101—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—24no—
136C22×C61—44no—
137C22×C61—42no—
138C22×C61—42no—
139C22×C61—82no—
140C22×C61#38611yescharacteristic
141C22×C61C2×C22⋊C811yescharacteristic
142C22×C61#40811yescharacteristic
143C2×D61—44no—
144C2×D61—42no—
145C2×D61—82no—
146C2×D61—81no—
147C2×D61—81no—
148C4×C61—22no—
149C4×C61—21no—
150C4×C61—21no—
151C4×C61—42no—
152C4×C61C8×D412yes—
153C4×C61#43512yes—
154Dic61—42no—
155Dic61—81no—
156Dic61—81no—
157#2972D6111yescharacteristic, complemented
158#2328—611no—
159#1478—612no—
160#1460—611no—
161#1274—612no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61—44no—
165C2×C61#186911yescharacteristic
166C2×C61#191011yescharacteristic
167C2×C61#191111yescharacteristic
168D61—82no—
169C2×#127—611no—
170C2×C4×D4—611no—
171C2×C4×D4C2×D6111yescharacteristic
172C8×D4—24no—
173#431—612no—
174#431C2×D6112yes—
175C2×C22⋊C8—22no—
176#390—611no—
177#390—611no—
178#386—1222no—
179#386—1222no—
180#382—611no—
181#376—611no—
182D4⋊C8—1224no—
183#323—1222no—
184C61#910111yescharacteristic, complemented, Sylow
185C22×D4—611no—
186C22×D4—1222no—
187C22×D4C4×D6111yescharacteristic
188C23×C4—1221no—
189C23×C4C22⋊D6112yes—
190C22×C8—24no—
191#127—614no—
192#127—1222no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—1222no—
196C4×D4—1222no—
197C2×C4⋊C4—611no—
198C2×C4⋊C4C4×D6111yescharacteristic
199C2×C22⋊C4—1222no—
200C2×C22⋊C4—1222no—
201C2×C22⋊C4—1221no—
202C2×C22⋊C4D24412yes—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42C22×D6111yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—1222no—
208C22⋊C8—24no—
209C22⋊C8—22no—
210C22⋊C8—1222no—
211C4×C8—1222no—
212C4×C8C22⋊D6114yescomplemented
213#95—1222no—
214#95—1221no—
215#95—1221no—
216C24—1221no—
217C24#1175819812yes—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—612no—
221C2×D4—1228no—
222C2×D4—1222no—
223C2×D4—1221no—
224C2×D4—1222no—
225C2×D4—4882no—
226C2×D4—4882no—
227C2×D4C8×D6112yes—
228C2×D4#1175818912yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4—4882no—
236C22×C4—2442no—
237C22×C4#1175819812yes—
238C22×C4C2×C4×D6111yescharacteristic
239C22×C4C2×D24411yescharacteristic
240C22×C4C2×C22⋊D6111yescharacteristic
241C2×C8—24no—
242C2×C8—84no—
243C2×C8—2442no—
244C4⋊C4—22no—
245C4⋊C4—1222no—
246C22⋊C4—24no—
247C22⋊C4—4882no—
248C22⋊C4—4882no—
249C22⋊C4—2441no—
250C22⋊C4—2441no—
251C42—612no—
252C42—1221no—
253C42—1222no—
254C42#1175822212yes—
255C42C2×C22⋊D6112yes—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—1222no—
261C23—2442no—
262C23—2442no—
263C23—2441no—
264C23—2441no—
265C23#1451968914yes—
266C23C2×#1175819811yescharacteristic
267D4—24no—
268D4—24no—
269D4—1224no—
270D4—4888no—
271D4—2442no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—1222no—
279C2×C4—4884no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2442no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4C4×C22⋊D6112yes—
288C2×C4#1451974611yescharacteristic
289C2×C4#1451975112yes—
290C2×C4#1451976011yescharacteristic
291C2×C4C2×C8×D6111yescharacteristic
292C2×C4C2×#1175818911yescharacteristic
293C2×C4#1451978312yes—
294C8—44no—
295C8—4882no—
296C22—21no—
297C22—21no—
298C22—24no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—4882no—
304C22—4881no—
305C22—4881no—
306C22—2444no—
307C22—2442no—
308C22#1683964111yescharacteristic
309C22C2×#1451968911yescharacteristic
310C22#1683972811yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—4881no—
316C4—4881no—
317C4—2442no—
318C4C8×C22⋊D6112yes—
319C4#1683983312yes—
320C2—21no—
321C2—21no—
322C2—44no—
323C2—4882no—
324C2#2144611111yescharacteristic
325C2#2144634611yescharacteristic
326C2#2144635111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC61×#2972C2
socleC22×C61#386
prime radicalC2×#14519773C4
cosocleC2×C42×C61C23
hypercenter#2972D61
hypocenterC61#9101
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21444211C2semidirect
3. #21444140C2semidirect
4. #21443647C2semidirect
5. C61×#2972C2semidirect
25. #14519773C8semidirect
51. C4×C8×C61D4semidirect
74. C2×D244C2×C8semidirect
90. C4×Dic61C2×C8semidirect
157. #2972D61semidirect
184. C61#9101semidirect
212. C4×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214485412
#214442112
#214441402
#214436472
C61×#29722
#910161