Finite Group Databasebeta

← #29572927

Subgroups — #29572927

Order 15616.

460 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16839830C211yesmaximal, characteristic
2#21445855C212yesmaximal, complemented
3#21444155C212yesmaximal
4#21443696C211yesmaximal, characteristic, complemented
5C61×#1293C211yesmaximal, characteristic, complemented
6C22×#11758189C411yescharacteristic
7C23×#92418C2211yescharacteristic
8C2×C4×C22⋊D61C2211yescharacteristic
9#16839830—22no—
10#16839830—22no—
11C2×#14519712—22no—
12C2×#14519706C411yescharacteristic
13#16839726C2212yes—
14C2×#14519689C411yescharacteristic
15C2×#14519687C411yescharacteristic
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—
24C2×#11758189—21no—
25C2×#11758189—21no—
26C2×#11758189—42no—
27C22×#92418—42no—
28C22×#92418C2×C411yescharacteristic
29C22×#92418C2×C411yescharacteristic
30C22×#92418D412yes—
31C22×#92418D412yes—
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—
43#14519687—21no—
44#14519687—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—42no—
62C22×Dic61C4⋊C411yescharacteristic
63C22×Dic61C4⋊C411yescharacteristic
64C22×Dic61C4⋊C411yescharacteristic
65C2×C4×D61—21no—
66C2×C4×D61—21no—
67C2×C4×D61—41no—
68C2×C4×D61—41no—
69M16×C61—42no—
70M16×C61—42no—
71C2×C8×C61—21no—
72C2×C8×C61—21no—
73C2×C8×C61—41no—
74C2×C8×C61—41no—
75C2×C8×C61—42no—
76C22⋊Dic61—22no—
77#11758198—22no—
78#11758196—22no—
79C4×Dic61—21no—
80C4×Dic61—21no—
81#11758194—42no—
82#11758194—42no—
83C2×#92418—24no—
84C2×#92418—22no—
85C2×#92418—41no—
86C2×#92418—42no—
87C2×#92418—42no—
88C2×#92418C22⋊C411yescharacteristic
89C2×#92418C22⋊C412yes—
90C2×#92418C22⋊C411yescharacteristic
91#11758189—42no—
92#11758189—42no—
93C23×C61—22no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61C2×C4⋊C411yescharacteristic
97C23×C61C22⋊Q812yes—
98C22×D61—21no—
99C22×D61—22no—
100C22×D61—41no—
101C22×D61—41no—
102C2×C4×C61—22no—
103C2×C4×C61—21no—
104C2×C4×C61—22no—
105C2×C4×C61—21no—
106C2×C4×C61—41no—
107C2×C4×C61—42no—
108C2×C4×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61#11711yescharacteristic
110C2×C4×C61#12112yes—
111C22⋊D61—44no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—22no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C4×D61—42no—
119C4×D61—41no—
120C4×D61—41no—
121C8×C61—41no—
122C8×C61—41no—
123C8×C61—82no—
124#92418—21no—
125#92418—22no—
126#92418—82no—
127#92418#9912yes—
128#8613—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—22no—
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—44no—
135C22×C61#38011yescharacteristic
136C22×C61#43012yes—
137C2×D61—41no—
138C2×D61—41no—
139C2×D61—41no—
140C2×D61—82no—
141C4×C61—22no—
142C4×C61—21no—
143C4×C61C2×#9911yescharacteristic
144C4×C61#41011yescharacteristic
145Dic61—41no—
146Dic61—41no—
147Dic61—41no—
148C2×#435—611no—
149#1810—612no—
150#1465—612no—
151#1293—611no—
152#1293D6111yescharacteristic, complemented
153C2×C61—22no—
154C2×C61—42no—
155C2×C61#179911yescharacteristic
156C2×C61#181111yescharacteristic
157C2×C61#181311yescharacteristic
158D61—81no—
159C22×M16—611no—
160C23×C8—611no—
161C2×C4×D4—611no—
162#435—1222no—
163#435—1222no—
164C2×#108—1222no—
165C2×C4⋊C8—611no—
166#407—612no—
167#407C2×D6112yes—
168C2×C22⋊C8—611no—
169C2×C22⋊C8C2×D6111yescharacteristic
170C2×#97—611no—
171#341—1222no—
172#324—1222no—
173#323—1222no—
174C61#861311yescharacteristic, complemented, Sylow
175C22×D4—611no—
176C23×C4—611no—
177C23×C4C22×D6111yescharacteristic
178C2×M16—22no—
179C2×M16—1221no—
180C2×M16—1221no—
181C2×M16—1222no—
182C2×M16—2442no—
183C22×C8—611no—
184C22×C8—612no—
185C22×C8—611no—
186C22×C8—612no—
187C22×C8—2442no—
188C22×C8C4×D6111yescharacteristic
189C22×C8C4×D6111yescharacteristic
190C4×D4—1222no—
191C4×D4—1222no—
192C2×C4⋊C4—611no—
193C2×C22⋊C4—611no—
194C2×C22⋊C4—611no—
195C2×C42—611no—
196#108—2444no—
197C4⋊C8—1222no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—1222no—
202C22⋊C8—1222no—
203C22⋊C8—1222no—
204#97—1221no—
205#97—1221no—
206C24—611no—
207C24Q8×D6111yescharacteristic
208C2×D4—1222no—
209C2×D4—2442no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—611no—
213C22×C4—611no—
214C22×C4—611no—
215C22×C4—1221no—
216C22×C4—1221no—
217C22×C4—2441no—
218C22×C4—2441no—
219C22×C4C2×C4×D6111yescharacteristic
220C22×C4#1175822212yes—
221C22×C4D4×D6112yes—
222C22×C4D4×D6111yescharacteristic
223M16—42no—
224M16—42no—
225M16—2442no—
226M16—2442no—
227M16—2442no—
228M16—2442no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—611no—
235C2×C8—611no—
236C2×C8—612no—
237C2×C8—1224no—
238C2×C8—1222no—
239C2×C8—2441no—
240C2×C8—2442no—
241C2×C8—2442no—
242C4⋊C4—1222no—
243C22⋊C4—1222no—
244C22⋊C4—1222no—
245C42—1221no—
246C42—1221no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—1222no—
251C23—1221no—
252C23—2441no—
253C23—2441no—
254C23C4⋊C4×D6111yescharacteristic
255C23#1451975512yes—
256D4—2444no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—1222no—
264C2×C4—1221no—
265C2×C4—1221no—
266C2×C4—1221no—
267C2×C4—2442no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4C22⋊C4×D6111yescharacteristic
273C2×C4#1451973411yescharacteristic
274C2×C4#1451975012yes—
275C8—41no—
276C8—41no—
277C8—82no—
278C8—612no—
279C8—1222no—
280C8—1221no—
281C8—4882no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—44no—
288C22—4882no—
289C22—2441no—
290C22—2441no—
291C22—2441no—
292C22#1683962111yescharacteristic
293C22#1683980912yes—
294C4—22no—
295C4—21no—
296C4—2441no—
297C4—2441no—
298C4—2441no—
299C4#99×D6111yescharacteristic
300C4#1683973411yescharacteristic
301C2—22no—
302C2—42no—
303C2—4881no—
304C2#2144580611yescharacteristic
305C2#2144586011yescharacteristic
306C2#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#8613
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
152. #1293D61semidirect
174. C61#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#168398302
#214458552
#214441552
#214436962
C61×#12932
#861361