Finite Group Databasebeta

← #29589478

Subgroups — #29589478

Order 15616.

596 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 77 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#21448383C211yesmaximal, characteristic
2#21444156C212yesmaximal, complemented
3#21443764C212yesmaximal, complemented
4#21443751C211yesmaximal, characteristic, complemented
5C61×#1318C211yesmaximal, characteristic, complemented
6C2×#14519773C411yescharacteristic
7C2×D4×Dic61C411yescharacteristic
8C2×#14519739C2211yescharacteristic
9C2×#14519737—22no—
10#16839724—22no—
11#16839664C411yescharacteristic
12#16839662—22no—
13#16839631C2211yescharacteristic
14C42×Dic61C411yescharacteristic
15#16839488C2212yes—
16#16839485—22no—
17C61×#390C2211yescharacteristic
18C61×#324C2212yes—
19C22×D244C2×C411yescharacteristic
20C22×C22⋊D61—22no—
21C23×Dic61—21no—
22C22×C4×D61D412yes—
23C2×#11758189—42no—
24#14519773—22no—
25#14519773—22no—
26D4×Dic61—41no—
27D4×Dic61—41no—
28#14519739—41no—
29#14519739—41no—
30#14519737—48no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758196—22no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—22no—
35C2×#11758198D412yes—
36C2×C4×Dic61—21no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61C2×C411yescharacteristic
40C22⋊#92418—22no—
41#14519689—42no—
42C22×D4×C61C2×C411yescharacteristic
43C2×C4⋊C4×C61C2311yescharacteristic
44C2×C22⋊C4×C61—22no—
45C2×C42×C61C2×C411yescharacteristic
46C22⋊C8×C61—22no—
47C24×C61—21no—
48C23×D61C22⋊C412yes—
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—28no—
57C2×C22⋊D61—44no—
58C2×C22⋊D61—44no—
59C22×Dic61—21no—
60C22×Dic61—21no—
61C22×Dic61—44no—
62C22×Dic61—41no—
63C22×Dic61—41no—
64C22×Dic61C22⋊C411yescharacteristic
65C22×Dic61C22⋊C411yescharacteristic
66C2×D244—21no—
67C2×D244—21no—
68C2×D244—41no—
69C2×D244—41no—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—44no—
73C2×C8×C61—42no—
74C4⋊C4×C61—41no—
75C22⋊C4×C61—44no—
76C42×C61—41no—
77C22⋊Dic61—42no—
78#11758198—42no—
79#11758198—44no—
80C4⋊Dic61—41no—
81#11758196—44no—
82C4×Dic61—24no—
83C4×Dic61—21no—
84C4×Dic61—21no—
85C4×Dic61—21no—
86C4×Dic61—42no—
87C4×Dic61M1612yescomplemented
88C2×#92418—42no—
89#11758189—84no—
90C23×C61—22no—
91C23×C61—42no—
92C23×C61—42no—
93C23×C61—41no—
94C23×C61—41no—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—22no—
97C22×D61—22no—
98C22×D61—22no—
99C22×D61—42no—
100C22×D61—42no—
101D4×C61—42no—
102D4×C61—42no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61C2×M1611yescharacteristic
111C2×C4×C61C2×M1611yescharacteristic
112C22⋊D61—416no—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—22no—
117C2×Dic61—21no—
118C2×Dic61—41no—
119C2×Dic61—42no—
120C2×Dic61—41no—
121C2×Dic61—82no—
122C2×Dic61—82no—
123C2×Dic61#9911yescharacteristic
124C2×Dic61#9911yescharacteristic
125D244—42no—
126D244—42no—
127C4×D61—44no—
128C4×D61—82no—
129C8×C61—82no—
130#92418—82no—
131#11529—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—44no—
135C22×C61—42no—
136C22×C61—82no—
137C22×C61—81no—
138C22×C61—81no—
139C22×C61#40711yescharacteristic
140C22×C61C2×#9911yescharacteristic
141C22×C61#41311yescharacteristic
142C2×D61—44no—
143C2×D61—42no—
144C2×D61—42no—
145C2×D61—42no—
146C2×D61—82no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—42no—
150C4×C61—42no—
151Dic61—24no—
152Dic61—21no—
153Dic61—41no—
154Dic61—81no—
155Dic61#35312yes—
156C4×#127—611no—
157#1465—612no—
158#1325—612no—
159#1318—611no—
160#1318D6111yescharacteristic, complemented
161C2×C61—21no—
162C2×C61—21no—
163C2×C61—82no—
164C2×C61#146511yescharacteristic
165C2×C61C2×#35311yescharacteristic
166C2×C61#213211yescharacteristic
167D61—82no—
168C2×#127—611no—
169C2×C4×D4—611no—
170C2×C4×D4—611no—
171C2×C4×D4—1222no—
172#407—1222no—
173#390—611no—
174#390—1222no—
175#390C2×D6111yescharacteristic
176#384—611no—
177C43—611no—
178#324—612no—
179#324C2×D6112yes—
180#323—1222no—
181C61#1152911yescharacteristic, complemented, Sylow
182C22×D4—611no—
183C22×D4—1222no—
184C22×D4C4×D6111yescharacteristic
185C23×C4—612no—
186C23×C4—1221no—
187C2×M16—2442no—
188#127—1222no—
189#127—1222no—
190C4×D4—2448no—
191C4×D4—2441no—
192C4×D4—2441no—
193C4×D4—2441no—
194C4×D4—2441no—
195C2×C4⋊C4—611no—
196C2×C4⋊C4—1222no—
197C2×C4⋊C4C22×D6111yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—612no—
200C2×C22⋊C4—1222no—
201C2×C22⋊C4—1221no—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—1221no—
205C2×C42—1221no—
206C2×C42C4×D6111yescharacteristic
207C22⋊C8—22no—
208C22⋊C8—1222no—
209C22⋊C8—2442no—
210C24—21no—
211C24—612no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—1228no—
216C2×D4—1221no—
217C2×D4—1221no—
218C2×D4—2441no—
219C2×D4—2441no—
220C2×D4—2444no—
221C2×D4—2444no—
222C2×D4#1175818912yescomplemented
223C22×C4—21no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—1222no—
227C22×C4—1222no—
228C22×C4—1221no—
229C22×C4—1221no—
230C22×C4—2444no—
231C22×C4—2441no—
232C22×C4—2441no—
233C22×C4—2444no—
234C22×C4C2×C4×D6111yescharacteristic
235C22×C4D4×D6112yes—
236M16—4884no—
237C2×C8—42no—
238C2×C8—2442no—
239C4⋊C4—41no—
240C4⋊C4—2441no—
241C4⋊C4—2444no—
242C22⋊C4—44no—
243C22⋊C4—2442no—
244C22⋊C4—2444no—
245C22⋊C4—2442no—
246C42—41no—
247C42—612no—
248C42—1221no—
249C42—1221no—
250C42—1224no—
251C42—1221no—
252C42—2442no—
253C23—22no—
254C23—42no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23—1222no—
259C23—1222no—
260C23—1222no—
261C23—2442no—
262C23—2442no—
263C23C22⋊C4×D6111yescharacteristic
264D4—42no—
265D4—42no—
266D4—2442no—
267D4—2442no—
268D4—24416no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—611no—
277C2×C4—611no—
278C2×C4—1222no—
279C2×C4—1222no—
280C2×C4—1222no—
281C2×C4—1222no—
282C2×C4—1221no—
283C2×C4—4882no—
284C2×C4—4882no—
285C2×C4—4882no—
286C2×C4—2444no—
287C2×C4—2442no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4M16×D6111yescharacteristic
291C2×C4C2×#1175818911yescharacteristic
292C8—82no—
293C8—4882no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—42no—
298C22—82no—
299C22—81no—
300C22—81no—
301C22—4882no—
302C22—2444no—
303C22—2442no—
304C22—2442no—
305C22—2442no—
306C22#1683972411yescharacteristic
307C22#99×D6111yescharacteristic
308C22#1683974311yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—612no—
314C4—1224no—
315C4—1221no—
316C4—4881no—
317C4—2441no—
318C2—21no—
319C2—21no—
320C2—82no—
321C2—4882no—
322C2#2144415411yescharacteristic
323C2#353×D6111yescharacteristic
324C2#2144750111yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1318C2
socleC22×C61C2×#99
prime radicalC2×#14519773C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1318D61
hypocenterC61#11529
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #21444156C2semidirect
3. #21443764C2semidirect
4. #21443751C2semidirect
5. C61×#1318C2semidirect
52. C2×D4×C61M16semidirect
87. C4×Dic61M16semidirect
160. #1318D61semidirect
181. C61#11529semidirect
222. C2×D4#11758189semidirect

Maximal subgroups

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

subgroupindex
#214483832
#214441562
#214437642
#214437512
C61×#13182
#1152961