Finite Group Databasebeta

← #29577398

Subgroups — #29577398

Order 15616.

730 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 135 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#21450765C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic, complemented
3#21444486C212yesmaximal, complemented
4#21443721C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7C2×#14519773—21no—
8C2×#14519751—22no—
9#16839786C2212yes—
10C2×#14519694—21no—
11#16839671C414yes—
12#16839663—21no—
13C4×C4⋊Dic61C2211yescharacteristic
14#16839531—21no—
15#16839505—24no—
16C2×C4×D4×C61C2211yescharacteristic
17C8×D4×C61—24no—
18C61×#431C2212yes—
19C2×C22⋊C8×C61—22no—
20C22×D244—21no—
21C22×D244—21no—
22C22×D244C2×C412yes—
23C22×C22⋊D61—24no—
24C2×#11758194—22no—
25#14519774—22no—
26#14519774—22no—
27#14519773—42no—
28#14519773—42no—
29#14519751—48no—
30C2×C4⋊Dic61—21no—
31C2×#11758198—24no—
32C2×C4×Dic61—21no—
33C4⋊#92418—22no—
34C4⋊#92418—22no—
35#14519694—42no—
36#14519684C2×C412yes—
37C22×D4×C61D411yescharacteristic
38C23×C4×C61D412yes—
39C22×C8×C61—24no—
40C4×D4×C61—22no—
41C4×D4×C61—22no—
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61C2×C412yes—
44C2×C42×C61C2311yescharacteristic
45C4⋊C8×C61D414yescomplemented
46C22⋊C8×C61—24no—
47C22⋊C8×C61—22no—
48C4×C8×C61—22no—
49C24×C61C22⋊C412yes—
50C23×D61—22no—
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61D812yes—
54C2×D4×C61SD1612yes—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—44no—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C2×C22⋊D61—48no—
63C2×C22⋊D61—48no—
64C2×C22⋊D61—84no—
65C22×Dic61—22no—
66C2×D244—24no—
67C2×D244—44no—
68C2×D244—41no—
69C2×D244—42no—
70C2×D244—41no—
71C2×D244—41no—
72C2×D244—41no—
73C2×D244—42no—
74C2×D244—82no—
75C2×C8×C61—24no—
76C2×C8×C61—84no—
77C4⋊C4×C61—22no—
78C22⋊C4×C61—24no—
79C42×C61C2×D412yes—
80C42×C61#4112yes—
81#11758198—48no—
82C4⋊Dic61—24no—
83C4×Dic61—41no—
84C4×Dic61—41no—
85#11758194—42no—
86#11758194—42no—
87C2×#92418—42no—
88C23×C61—22no—
89C23×C61—22no—
90C23×C61—42no—
91C23×C61—42no—
92C23×C61#10214yes—
93C23×C61C2×C22⋊C411yescharacteristic
94C22×D61—44no—
95C22×D61—44no—
96C22×D61—42no—
97C22×D61—42no—
98C22×D61—42no—
99D4×C61—24no—
100D4×C61—24no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—41no—
104C2×C4×C61—44no—
105C2×C4×C61—42no—
106C2×C4×C61—82no—
107C2×C4×C61C4×D412yes—
108C2×C4×C61#12011yescharacteristic
109C2×C4×C61#12112yes—
110C2×C4×C61#12311yescharacteristic
111C2×C4×C61C2×D811yescharacteristic
112C2×C4×C61C2×SD1611yescharacteristic
113C2×C4×C61#13512yes—
114C22⋊D61—816no—
115C2×Dic61—24no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118D244—84no—
119D244—82no—
120D244—82no—
121D244—82no—
122D244—82no—
123C8×C61—44no—
124#92418—82no—
125#9397—611nomaximal, Sylow
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—24no—
129C22×C61—44no—
130C22×C61—42no—
131C22×C61—42no—
132C22×C61—82no—
133C22×C61#38611yescharacteristic
134C22×C61C2×#10211yescharacteristic
135C22×C61#41611yescharacteristic
136C2×D61—84no—
137C2×D61—84no—
138C2×D61—82no—
139C2×D61—82no—
140C2×D61—82no—
141C4×C61—21no—
142C4×C61—22no—
143C4×C61—42no—
144C4×C61#45312yes—
145C4×C61#46512yes—
146C4×C61#46612yes—
147Dic61—44no—
148#2972D6111yescharacteristic, complemented
149#2541—611no—
150#1800—611no—
151#1545—612no—
152#1303—612no—
153C2×C61—21no—
154C2×C61—21no—
155C2×C61—44no—
156C2×C61#188711yescharacteristic
157C2×C61#189311yescharacteristic
158C2×C61#191411yescharacteristic
159D61—162no—
160C2×#127—611no—
161C2×#127—1221no—
162C2×#121—1222no—
163C2×C4×D4C2×D6111yescharacteristic
164C8×D4—24no—
165#431C2×D6112yes—
166#423—612no—
167C2×#100—1221no—
168C2×C22⋊C8—22no—
169#392—614no—
170#390—1221no—
171C4×C4⋊C4—611no—
172#343—1221no—
173#331—1224no—
174C61#939711yescharacteristic, complemented, Sylow
175C22×D4—612no—
176C22×D4—1224no—
177C22×D4—1221no—
178C22×D4—1221no—
179C22×D4D24411yescharacteristic
180C23×C4C22⋊D6112yes—
181C2×M16—1222no—
182C22×C8—24no—
183#127—1222no—
184#127—1222no—
185#127—2442no—
186#127—2442no—
187#121—2448no—
188C4×D4—22no—
189C4×D4—22no—
190C2×C4⋊C4—1221no—
191C2×C4⋊C4D24411yescharacteristic
192C2×C22⋊C4—1224no—
193C2×C22⋊C4C4×D6112yes—
194C2×C42—1221no—
195C2×C42C22×D6111yescharacteristic
196C4⋊C8—1222no—
197C4⋊C8—1222no—
198C4⋊C8C22⋊D6114yescomplemented
199#100—2442no—
200C22⋊C8—24no—
201C22⋊C8—22no—
202C4×C8—22no—
203#95—612no—
204C24—1222no—
205C24#1175819812yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—1224no—
209C2×D4—4882no—
210C2×D4—4884no—
211C2×D4—2444no—
212C2×D4—2448no—
213C2×D4—2448no—
214C2×D4—2441no—
215C2×D4—2442no—
216C2×D4—2441no—
217C2×D4—2441no—
218C2×D4—2441no—
219C2×D4—2442no—
220C2×D4#1175819012yes—
221C2×D4D48812yes—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—44no—
225C22×C4—1222no—
226C22×C4#1175819812yes—
227C22×C4C2×C4×D6111yescharacteristic
228C22×C4C2×D24411yescharacteristic
229C22×C4C2×C22⋊D6111yescharacteristic
230M16—2442no—
231M16—2442no—
232C2×C8—24no—
233C2×C8—84no—
234C2×C8—2442no—
235C4⋊C4—22no—
236C4⋊C4—1224no—
237C22⋊C4—24no—
238C22⋊C4—2448no—
239C42—2441no—
240C42—2441no—
241C42#1175822212yes—
242C42C2×C22⋊D6112yes—
243C23—22no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—2444no—
248C23—2444no—
249C23—2442no—
250C23—2442no—
251C23—2442no—
252C23#1451969914yes—
253C23C2×#1175819811yescharacteristic
254D4—24no—
255D4—24no—
256D4—4884no—
257D4—4882no—
258D4—48816no—
259D4—4882no—
260D4—4882no—
261D4—4882no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—1224no—
269C2×C4—2442no—
270C2×C4—2442no—
271C2×C4C4×C22⋊D6112yes—
272C2×C4#1451974611yescharacteristic
273C2×C4#1451975112yes—
274C2×C4#1451976011yescharacteristic
275C2×C4C2×D48811yescharacteristic
276C2×C4C2×#1175819011yescharacteristic
277C2×C4#1451979912yes—
278C8—44no—
279C8—4882no—
280C22—21no—
281C22—21no—
282C22—24no—
283C22—44no—
284C22—42no—
285C22—42no—
286C22—82no—
287C22—4884no—
288C22—4884no—
289C22—4882no—
290C22—4882no—
291C22—4882no—
292C22#1683964111yescharacteristic
293C22C2×#1451969911yescharacteristic
294C22#1683975911yescharacteristic
295C4—21no—
296C4—22no—
297C4—42no—
298C4—2444no—
299C4#1683993012yes—
300C4#1683999112yes—
301C4#1683999812yes—
302C2—21no—
303C2—21no—
304C2—44no—
305C2—9762no—
306C2#2144621711yescharacteristic
307C2#2144624711yescharacteristic
308C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2972C2
socleC22×C61#386
prime radical#21450765C2
cosocleC2×C42×C61C23
hypercenter#2972D61
hypocenterC61#9397
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21445811C2semidirect
3. #21444486C2semidirect
4. #21443721C2semidirect
5. C61×#2972C2semidirect
45. C4⋊C8×C61D4semidirect
148. #2972D61semidirect
174. C61#9397semidirect
198. C4⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214507652
#214458112
#214444862
#214437212
C61×#29722
#939761