Finite Group Databasebeta

← #29577867

Subgroups — #29577867

Order 15616.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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#21450724C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic, complemented
3#21444245C212yesmaximal
4#21443678C212yesmaximal, complemented
5C61×#2982C211yesmaximal, characteristic, complemented
6C2×#14519753—21no—
7C2×C4×D244C2211yescharacteristic
8#16839786C2212yes—
9C2×#14519696—21no—
10#16839692C412yes—
11#16839679C412yes—
12#16839653—22no—
13#16839628C2211yescharacteristic
14#16839619—21no—
15#16839531—21no—
16#16839491—24no—
17C2×C4×Q8×C61C2211yescharacteristic
18C8×Q8×C61—24no—
19C61×#431—22no—
20C61×#431C2212yes—
21C22×D244C2×C411yescharacteristic
22C22×C4×D61—21no—
23C2×#11758194—22no—
24#14519753—42no—
25#14519753—42no—
26C4×D244—22no—
27C4×D244—22no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×#11758196—22no—
31C2×#11758198—22no—
32C2×#11758198—22no—
33C2×#11758198—21no—
34C4⋊#92418—22no—
35C4⋊#92418—22no—
36#14519696—42no—
37#14519684—22no—
38#14519684—21no—
39#14519684C2×C412yes—
40C22×Q8×C61D411yescharacteristic
41C4×Q8×C61—22no—
42C4×Q8×C61—22no—
43C2×C4⋊C4×C61C2×C412yes—
44C2×C4⋊C4×C61D411yescharacteristic
45C2×C42×C61D412yes—
46C2×C42×C61C2311yescharacteristic
47C4⋊C8×C61—24no—
48C4⋊C8×C61—22no—
49C22⋊C8×C61—22no—
50C4×C8×C61—24no—
51C4×C8×C61D414yes—
52C23×D61—21no—
53C2×Q8×C61—22no—
54C2×Q8×C61—22no—
55C2×Q8×C61D812yes—
56C2×Q8×C61SD1612yes—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×Dic61—21no—
63C22×Dic61—22no—
64C2×D244—22no—
65C2×D244—42no—
66C2×C4×D61—42no—
67C2×C4×D61—41no—
68C2×C4×D61—41no—
69C2×C4×D61—42no—
70C2×C8×C61—24no—
71C4⋊C4×C61—24no—
72C4⋊C4×C61—22no—
73C42×C61—22no—
74C42×C61—22no—
75C42×C61C2×D412yes—
76C42×C61#4112yes—
77#11758198—44no—
78#11758198—44no—
79#11758198—41no—
80#11758198—41no—
81C4⋊Dic61—22no—
82C4⋊Dic61—41no—
83C4⋊Dic61—41no—
84#11758196—44no—
85#11758194—42no—
86#11758194—42no—
87C2×#92418—42no—
88C23×C61C2×C22⋊C411yescharacteristic
89C22×D61—22no—
90C22×D61—41no—
91C22×D61—41no—
92C22×D61—41no—
93C22×D61—41no—
94C22×D61—41no—
95C22×D61—41no—
96Q8×C61—24no—
97Q8×C61—24no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61#10214yes—
105C2×C4×C61C4×D412yes—
106C2×C4×C61#12011yescharacteristic
107C2×C4×C61#12112yes—
108C2×C4×C61#12311yescharacteristic
109C2×C4×C61C2×D811yescharacteristic
110C2×C4×C61C2×SD1611yescharacteristic
111C2×C4×C61#13512yes—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—41no—
118D244—24no—
119D244—42no—
120C4×D61—82no—
121C4×D61—82no—
122C8×C61—44no—
123#92418—82no—
124#9486—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—21no—
127C22×C61#38611yescharacteristic
128C22×C61C2×#10211yescharacteristic
129C22×C61#41611yescharacteristic
130C2×D61—42no—
131C2×D61—42no—
132C2×D61—42no—
133C2×D61—81no—
134C2×D61—81no—
135C2×D61—81no—
136C2×D61—81no—
137C4×C61—22no—
138C4×C61—24no—
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—42no—
142C4×C61#46512yes—
143C4×C61#46612yes—
144Dic61—82no—
145Dic61—81no—
146#2982D6111yescharacteristic, complemented
147#2536—611no—
148#1801—611no—
149#1486—612no—
150#1286—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61#189111yescharacteristic
154C2×C61#189311yescharacteristic
155C2×C61#191211yescharacteristic
156D61—82no—
157C2×C22⋊Q8—1221no—
158C2×C4×Q8C2×D6111yescharacteristic
159C2×C4×D4—611no—
160C8×Q8—24no—
161#431—22no—
162#431C2×D6112yes—
163#423—612no—
164C2×#101—1221no—
165#398—612no—
166#394—612no—
167#388—1222no—
168#382—611no—
169#380—1221no—
170#343—1221no—
171D4⋊C8—1224no—
172C61#948611yescharacteristic, complemented, Sylow
173C22×Q8D24411yescharacteristic
174C22×D4—611no—
175C23×C4—1221no—
176C2×M16—1222no—
177C22⋊Q8—2442no—
178C22⋊Q8—2442no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—1222no—
182C4×D4—1222no—
183C2×C4⋊C4—611no—
184C2×C4⋊C4—1222no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4C4×D6112yes—
187C2×C4⋊C4D24411yescharacteristic
188C2×C22⋊C4—1222no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1221no—
191C2×C42C22⋊D6112yes—
192C2×C42C22×D6111yescharacteristic
193C4⋊C8—24no—
194C4⋊C8—22no—
195C4⋊C8—1222no—
196C4⋊C8—1222no—
197#101—2442no—
198C22⋊C8—22no—
199C4×C8—24no—
200C4×C8C22⋊D6114yes—
201#95—612no—
202#95—1222no—
203#95—1221no—
204C24—1221no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1175819012yes—
208C2×Q8D48812yes—
209C2×D4—1222no—
210C2×D4—2442no—
211C22×C4—1222no—
212C22×C4—1221no—
213C22×C4—2442no—
214C22×C4—2441no—
215C22×C4—2441no—
216C22×C4—2442no—
217C22×C4#1175819812yes—
218C22×C4#1175819812yes—
219C22×C4C2×C4×D6111yescharacteristic
220C22×C4C2×D24411yescharacteristic
221C22×C4C2×C22⋊D6111yescharacteristic
222M16—2442no—
223M16—2442no—
224C2×C8—24no—
225C2×C8—2442no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—1222no—
229C4⋊C4—2444no—
230C4⋊C4—2441no—
231C4⋊C4—2441no—
232C22⋊C4—2444no—
233C22⋊C4—2444no—
234C22⋊C4—2441no—
235C22⋊C4—2441no—
236C42—22no—
237C42—22no—
238C42#1175822212yes—
239C42C2×C22⋊D6112yes—
240C23—1222no—
241C23—2441no—
242C23—2441no—
243C23—2441no—
244C23—2441no—
245C23—2441no—
246C23—2441no—
247C23C2×#1175819811yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—1224no—
251D4—2442no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—4882no—
259C2×C4—4882no—
260C2×C4—2442no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4#1451969914yes—
267C2×C4C4×C22⋊D6112yes—
268C2×C4#1451974611yescharacteristic
269C2×C4#1451975112yes—
270C2×C4#1451976011yescharacteristic
271C2×C4C2×D48811yescharacteristic
272C2×C4C2×#1175819011yescharacteristic
273C2×C4#1451979912yes—
274C8—44no—
275C8—4882no—
276C22—21no—
277C22—21no—
278C22—4881no—
279C22—4881no—
280C22—4881no—
281C22—4881no—
282C22—2442no—
283C22—2442no—
284C22—2442no—
285C22#1683964111yescharacteristic
286C22C2×#1451969911yescharacteristic
287C22#1683975911yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—4882no—
294C4—4881no—
295C4#1683999112yes—
296C4#1683999812yes—
297C2—21no—
298C2—21no—
299C2—4882no—
300C2#2144623711yescharacteristic
301C2#2144624711yescharacteristic
302C2#2144635611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2982C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#2982D61
hypocenterC61#9486
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #21445815C2semidirect
4. #21443678C2semidirect
5. C61×#2982C2semidirect
146. #2982D61semidirect
172. C61#9486semidirect

Maximal subgroups

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

subgroupindex
#214507242
#214458152
#214442452
#214436782
C61×#29822
#948661