Finite Group Databasebeta

← #29575302

Subgroups — #29575302

Order 15616.

654 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16840136C211yesmaximal, characteristic
2#21446746C212yesmaximal
3C2×#16839511C211yesmaximal, characteristic, complemented
4#21443660C212yesmaximal, complemented
5C61×#2369C211yesmaximal, characteristic, complemented
6C22×#11758201—22no—
7C2×#14519774C2211yescharacteristic
8#16840136—22no—
9#16840136C414yescomplemented
10C2×C4⋊#92418C2212yes—
11C2×#14519697—22no—
12C2×#14519697—22no—
13C2×C4×#92418C2211yescharacteristic
14#16839511—22no—
15C2×C4⋊Q8×C61C2211yescharacteristic
16C61×#391—22no—
17C61×#389—22no—
18C4×C4⋊C4×C61—22no—
19C4×C4⋊C4×C61C2212yes—
20C22×D244—22no—
21C22×D244—21no—
22C2×#11758201—28no—
23C2×#11758201—44no—
24C2×#11758201—44no—
25C22×#92418—21no—
26C22×#92418—22no—
27#14519774—21no—
28#14519774—22no—
29#14519774C2×C412yes—
30C4⋊#92418—22no—
31C4⋊#92418—22no—
32#14519697—44no—
33#14519697—44no—
34C4×#92418—21no—
35C4×#92418C2×C412yes—
36C22×Q8×C61D412yes—
37C4⋊Q8×C61—22no—
38C4⋊Q8×C61C2×C412yes—
39C4⋊Q8×C61D412yes—
40C2×C4⋊C4×C61—22no—
41C2×C4⋊C4×C61—22no—
42C2×C4⋊C4×C61D412yes—
43C2×C4⋊C4×C61D412yes—
44C2×C42×C61—22no—
45C2×C42×C61—22no—
46C2×C42×C61C2311yescharacteristic
47#95×C61—24no—
48#95×C61—22no—
49C23×D61—41no—
50C2×Q8×C61—22no—
51C2×Q8×C61—42no—
52C2×Q8×C61—42no—
53C2×Q8×C61C22⋊C414yes—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D412yes—
59C2×D244—22no—
60C2×D244—24no—
61C2×D244—42no—
62C2×D244—42no—
63C2×D244—42no—
64C2×D244—42no—
65C2×D244—42no—
66C2×D244—42no—
67C2×D244—42no—
68C2×D244—41no—
69C4⋊C4×C61—24no—
70C4⋊C4×C61—24no—
71C4⋊C4×C61—22no—
72C4⋊C4×C61—22no—
73C4⋊C4×C61—22no—
74C4⋊C4×C61—82no—
75C4⋊C4×C61SD1614yescomplemented
76C42×C61—42no—
77C42×C61—42no—
78C42×C61—82no—
79C42×C61C22×C411yescharacteristic
80C42×C61C2×D411yescharacteristic
81C42×C61#4112yes—
82#11758201—416no—
83C2×#92418—22no—
84C2×#92418—22no—
85C2×#92418—22no—
86C2×#92418—42no—
87C2×#92418—42no—
88C2×#92418—42no—
89C23×C61#12011yescharacteristic
90C22×D61—42no—
91C22×D61—82no—
92C22×D61—82no—
93C22×D61—81no—
94C22×D61—81no—
95Q8×C61—44no—
96Q8×C61—82no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—24no—
100C2×C4×C61—24no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61C4×D412yes—
110C2×C4×C61#12112yes—
111C2×C4×C61#12311yescharacteristic
112C2×C4×C61C2×SD1612yes—
113C2×C4×C61#13512yes—
114C2×C4×C61AGL(1,ℤ/8)12yes—
115C2×C4×C61#13712yes—
116D244—44no—
117D244—44no—
118D244—44no—
119D244—82no—
120D244—82no—
121D244—82no—
122#92418—24no—
123#92418—82no—
124#9063—611nomaximal, Sylow
125C22×C61#38611yescharacteristic
126C22×C61#44812yes—
127C22×C61#44912yes—
128C22×C61#45311yescharacteristic
129C22×C61#45311yescharacteristic
130C2×D61—84no—
131C2×D61—82no—
132C2×D61—162no—
133C2×D61—161no—
134C2×D61—161no—
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—44no—
138C4×C61—44no—
139C4×C61—82no—
140C4×C61C4×SD1612yes—
141C4×C61#44012yes—
142C4×C61#46012yes—
143C4×C61#46412yes—
144C2×#492—611no—
145#2369D6111yescharacteristic, complemented
146#1982—612no—
147C2×#333—611no—
148#1279—612no—
149C2×C61#162212yes—
150C2×C61#162311yescharacteristic
151C2×C61#162411yescharacteristic
152C2×C61#188012yes—
153C2×C61#188711yescharacteristic
154D61—162no—
155C22×SD16—1222no—
156C2×C4⋊Q8C2×D6111yescharacteristic
157C2×#127—611no—
158#492—614no—
159#492—1222no—
160C2×C4⋊C8—612no—
161C2×#102—1222no—
162C2×#102—1222no—
163C2×C4×C8—611no—
164#391—22no—
165#389—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4C2×D6112yes—
168#333—1222no—
169C61#906311yescharacteristic, complemented, Sylow
170C22×Q8C22⋊D6112yes—
171C22×D4—1221no—
172C22×D4—1222no—
173C2×SD16—1228no—
174C2×SD16—2444no—
175C2×SD16—2444no—
176C22×C8—1221no—
177C22×C8—1222no—
178C4⋊Q8—22no—
179C4⋊Q8C4×D6112yes—
180C4⋊Q8D24412yes—
181#127—612no—
182#127—1221no—
183#127—1222no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C22⋊D6112yes—
187C2×C4⋊C4C22⋊D6112yes—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42C22×D6111yescharacteristic
191C4⋊C8—1222no—
192C4⋊C8—1222no—
193#102—2444no—
194#102—2444no—
195C4×C8—612no—
196C4×C8—1221no—
197#95—24no—
198#95—22no—
199C24—2441no—
200C2×Q8—22no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8#1175819814yes—
204C2×D4—1222no—
205C2×D4—1224no—
206C2×D4—2442no—
207C2×D4—2442no—
208C2×D4—2442no—
209C2×D4—2442no—
210C2×D4—2441no—
211C2×D4—2442no—
212C2×D4—2442no—
213C2×D4—2442no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4C2×C22⋊D6111yescharacteristic
218C22×C4C2×C22⋊D6112yes—
219SD16—24416no—
220C2×C8—1222no—
221C2×C8—1222no—
222C2×C8—1222no—
223C2×C8—2442no—
224C2×C8—2442no—
225C2×C8—2442no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—82no—
232C4⋊C4#1175820114yescomplemented
233C42—42no—
234C42—42no—
235C42—82no—
236C42C2×C4×D6111yescharacteristic
237C42C2×D24411yescharacteristic
238C42#1175822212yes—
239C23—4882no—
240C23—4882no—
241C23—4881no—
242C23—4881no—
243C23—2442no—
244C23#1451974611yescharacteristic
245Q8—44no—
246Q8—82no—
247D4—4882no—
248D4—4882no—
249D4—4882no—
250D4—2444no—
251D4—2444no—
252D4—2444no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4C2×#1175819811yescharacteristic
265C2×C4C4×C22⋊D6112yes—
266C2×C4#1451975112yes—
267C2×C4#1451976011yescharacteristic
268C2×C4C2×#1175820112yes—
269C2×C4#1451979712yes—
270C2×C4#1451980212yes—
271C2×C4#1451980812yes—
272C8—1224no—
273C8—4882no—
274C22—9762no—
275C22—9761no—
276C22—9761no—
277C22—4884no—
278C22—4882no—
279C22#1683964111yescharacteristic
280C22#1683990112yes—
281C22#1683990712yes—
282C22#1683993011yescharacteristic
283C22#1683993011yescharacteristic
284C4—21no—
285C4—21no—
286C4—44no—
287C4—44no—
288C4—82no—
289C4C4×#1175820112yes—
290C4#1683986012yes—
291C4#1683996212yes—
292C4#1683998412yes—
293C2—9762no—
294C2#2144489112yes—
295C2#2144489711yescharacteristic
296C2#2144490211yescharacteristic
297C2#2144617412yes—
298C2#2144621711yescharacteristic
subgroupG/H
centerC23#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2369C2
socleC23×C61#120
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#2369D61
hypocenterC61#9063
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2369

Complements

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

NH ≅ G/Ntype
3. C2×#16839511C2semidirect
4. #21443660C2semidirect
5. C61×#2369C2semidirect
9. #16840136C4semidirect
75. C4⋊C4×C61SD16semidirect
145. #2369D61semidirect
169. C61#9063semidirect
232. C4⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168401362
#214467462
C2×#168395112
#214436602
C61×#23692
#906361