Finite Group Databasebeta

← #29572923

Subgroups — #29572923

Order 15616.

486 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#16839825C211yesmaximal, characteristic
2#21445855C212yesmaximal, complemented
3#21444155C212yesmaximal
4#21443696C211yesmaximal, characteristic, complemented
5C61×#1293C211yesmaximal, characteristic, complemented
6C23×#92418C2211yescharacteristic
7C22×C8×D61C411yescharacteristic
8C2×C4×C22⋊D61C2211yescharacteristic
9#16839825—22no—
10#16839825—22no—
11C2×#14519712—22no—
12C2×#14519706C411yescharacteristic
13#16839726C2212yes—
14C2×#14519689C411yescharacteristic
15C2×C8×Dic61C411yescharacteristic
16#16839526—22no—
17#16839487—22no—
18#16839485—22no—
19C61×#407C2212yes—
20C2×C22⋊C8×C61C2211yescharacteristic
21C22×C22⋊D61Q811yescharacteristic
22C22×C4×D61C2×C411yescharacteristic
23C2×#11758194—22no—
24C22×#92418—42no—
25C22×#92418C2×C411yescharacteristic
26C22×#92418C2×C411yescharacteristic
27C22×#92418D412yes—
28C22×#92418D412yes—
29C2×C8×D61—21no—
30C2×C8×D61—21no—
31C2×C8×D61—42no—
32C4×C22⋊D61—22no—
33C4×C22⋊D61—22no—
34C2×#11758196D411yescharacteristic
35C2×C22⋊Dic61Q811yescharacteristic
36C2×#11758198D411yescharacteristic
37C2×C4×Dic61C2×C411yescharacteristic
38#14519712—44no—
39#14519706—22no—
40C22⋊#92418—22no—
41C22⋊#92418—22no—
42#14519689—22no—
43C8×Dic61—21no—
44C8×Dic61—21no—
45C23×C4×C61C2311yescharacteristic
46C2×M16×C61—22no—
47C22×C8×C61C2×C411yescharacteristic
48C22×C8×C61C2×C411yescharacteristic
49C22⋊C8×C61—22no—
50C22⋊C8×C61—22no—
51C22⋊C8×C61—22no—
52C24×C61C2×Q811yescharacteristic
53C23×D61C4⋊C411yescharacteristic
54C22×C4×C61—41no—
55C22×C4×C61—41no—
56C22×C4×C61C22×C411yescharacteristic
57C22×C4×C61C2×D412yes—
58C22×C4×C61C2×D411yescharacteristic
59C22×C4×C61#4112yes—
60C2×C22⋊D61—22no—
61C2×C22⋊D61—22no—
62C2×C22⋊D61—22no—
63C22×Dic61C4⋊C411yescharacteristic
64C22×Dic61C4⋊C411yescharacteristic
65C22×Dic61C4⋊C411yescharacteristic
66C2×C4×D61—21no—
67C2×C4×D61—21no—
68C2×C4×D61—41no—
69C2×C4×D61—41no—
70M16×C61—42no—
71M16×C61—42no—
72C2×C8×C61—21no—
73C2×C8×C61—21no—
74C2×C8×C61—41no—
75C2×C8×C61—41no—
76C2×C8×C61—42no—
77C22⋊Dic61—22no—
78#11758198—22no—
79#11758196—22no—
80C4×Dic61—21no—
81C4×Dic61—21no—
82#11758194—42no—
83#11758194—42no—
84C2×#92418—24no—
85C2×#92418—22no—
86C2×#92418—41no—
87C2×#92418—42no—
88C2×#92418—42no—
89C2×#92418C22⋊C411yescharacteristic
90C2×#92418C22⋊C412yes—
91C2×#92418C22⋊C411yescharacteristic
92C8×D61—42no—
93C8×D61—42no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61C2×C4⋊C411yescharacteristic
99C23×C61C22⋊Q812yes—
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61#10812yescomplemented
105C2×C4×C61—22no—
106C2×C4×C61—21no—
107C2×C4×C61—22no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61C2×C22⋊C411yescharacteristic
112C2×C4×C61#11711yescharacteristic
113C2×C4×C61#12112yes—
114C22⋊D61—28no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—41no—
120C2×Dic61—41no—
121C2×Dic61#10812yescomplemented
122C4×D61—42no—
123C4×D61—41no—
124C4×D61—41no—
125C8×C61—41no—
126C8×C61—41no—
127C8×C61—82no—
128#92418—21no—
129#92418—22no—
130#92418—82no—
131#92418#9912yes—
132#8612—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61#38011yescharacteristic
141C22×C61C2×#10811yescharacteristic
142C22×C61C2×#10811yescharacteristic
143C2×D61—22no—
144C2×D61—42no—
145C2×D61—41no—
146C2×D61—41no—
147C2×D61—42no—
148C4×C61—22no—
149C4×C61—21no—
150C4×C61C2×#9911yescharacteristic
151C4×C61#41011yescharacteristic
152Dic61—22no—
153Dic61—41no—
154Dic61—41no—
155C2×C8×D4—611no—
156#1810—612no—
157#1465—612no—
158#1293—611no—
159#1293D6111yescharacteristic, complemented
160C2×C61—21no—
161C2×C61—21no—
162C2×C61—42no—
163C2×C61#179911yescharacteristic
164C2×C61#181011yescharacteristic
165C2×C61#181311yescharacteristic
166D61—42no—
167C23×C8—611no—
168C23×C8—611no—
169C2×C4×D4—611no—
170C8×D4—1222no—
171C8×D4—1222no—
172C2×#108—1222no—
173C2×C4⋊C8—611no—
174#407—612no—
175#407C2×D6112yes—
176C2×C22⋊C8—611no—
177C2×C22⋊C8C2×D6111yescharacteristic
178C2×C4×C8—611no—
179#341—1222no—
180#324—1222no—
181#323—1222no—
182C61#861211yescharacteristic, complemented, Sylow
183C22×D4—611no—
184C23×C4—611no—
185C23×C4C22×D6111yescharacteristic
186C2×M16—22no—
187C2×M16—1222no—
188C22×C8—611no—
189C22×C8—612no—
190C22×C8—611no—
191C22×C8—612no—
192C22×C8—1221no—
193C22×C8—1221no—
194C22×C8—2442no—
195C22×C8—2442no—
196C22×C8C4×D6111yescharacteristic
197C22×C8C4×D6111yescharacteristic
198C4×D4—1222no—
199C4×D4—1222no—
200C2×C4⋊C4—611no—
201C2×C22⋊C4—611no—
202C2×C22⋊C4—611no—
203C2×C42—611no—
204#108—2444no—
205C4⋊C8—1222no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—1222no—
210C22⋊C8—1222no—
211C22⋊C8—1222no—
212C4×C8—1221no—
213C4×C8—1221no—
214C24—611no—
215C24Q8×D6111yescharacteristic
216C2×D4—1222no—
217C2×D4—1222no—
218C2×D4—1222no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—611no—
222C22×C4—611no—
223C22×C4—611no—
224C22×C4—1221no—
225C22×C4—1221no—
226C22×C4—2441no—
227C22×C4—2441no—
228C22×C4C2×C4×D6111yescharacteristic
229C22×C4#1175822212yes—
230C22×C4D4×D6112yes—
231C22×C4D4×D6111yescharacteristic
232M16—42no—
233M16—42no—
234M16—2442no—
235M16—2442no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—611no—
242C2×C8—611no—
243C2×C8—612no—
244C2×C8—1224no—
245C2×C8—1222no—
246C2×C8—2442no—
247C2×C8—2442no—
248C2×C8—2441no—
249C2×C8—2442no—
250C2×C8—2442no—
251C4⋊C4—1222no—
252C22⋊C4—1222no—
253C22⋊C4—1222no—
254C42—1221no—
255C42—1221no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—612no—
261C23—1221no—
262C23—1221no—
263C23—2441no—
264C23—2441no—
265C23C4⋊C4×D6111yescharacteristic
266C23#1451975512yes—
267D4—1228no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—612no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—2442no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4C22⋊C4×D6111yescharacteristic
285C2×C4#1451973411yescharacteristic
286C2×C4#1451975012yes—
287C8—41no—
288C8—41no—
289C8—82no—
290C8—612no—
291C8—1222no—
292C8—1221no—
293C8—4882no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—1222no—
302C22—2442no—
303C22—2442no—
304C22—2441no—
305C22—2441no—
306C22#1683962111yescharacteristic
307C22#108×D6111yescharacteristic
308C22#108×D6111yescharacteristic
309C4—22no—
310C4—21no—
311C4—1222no—
312C4—2441no—
313C4—2441no—
314C4#99×D6111yescharacteristic
315C4#1683973411yescharacteristic
316C2—21no—
317C2—21no—
318C2—42no—
319C2—2442no—
320C2#2144580611yescharacteristic
321C2#2144585611yescharacteristic
322C2#2144586811yescharacteristic
subgroupG/H
centerC22#16839621
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1293C2
socleC22×C61#380
prime radicalC22×C22⋊D61Q8
cosocleC23×C4×C61C23
hypercenter#1293D61
hypocenterC61#8612
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #21445855C2semidirect
4. #21443696C2semidirect
5. C61×#1293C2semidirect
104. C22×D61#108semidirect
121. C2×Dic61#108semidirect
159. #1293D61semidirect
182. C61#8612semidirect

Maximal subgroups

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

subgroupindex
C2×#168398252
#214458552
#214441552
#214436962
C61×#12932
#861261