Finite Group Databasebeta

← #29656318

Subgroups — #29656318

Order 15616.

982 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16840478C211yesmaximal, characteristic
2#21455914C212yesmaximal, complemented
3#21445044C218yesmaximal, complemented
4#21444545C211yesmaximal, characteristic, complemented
5#21444263C212yesmaximal, complemented
6C61×#3139C211yesmaximal, characteristic, complemented
7#16840478—24no—
8#16840478—22no—
9#16840478C2214yes—
10C2×#14519766—22no—
11C2×#14519752—22no—
12C2×C4×D244C2212yes—
13C2×#14519735—22no—
14#16840095—22no—
15#16840039C2214yescomplemented
16#16839964C2214yescomplemented
17#16839912—22no—
18#16839860—24no—
19#16839819C2212yes—
20#16839511C2214yescomplemented
21#16839491C2218yescomplemented
22C61×#533C2211yescharacteristic
23C2×C4⋊Q8×C61C2211yescharacteristic
24C61×#491—22no—
25C61×#487C2214yescomplemented
26C61×#419—22no—
27C4×M16×C61C2211yescharacteristic
28C22×D244—22no—
29C22×D244D412yes—
30C22×C4×D61—22no—
31C2×#11758201—24no—
32#14519766—28no—
33#14519766—28no—
34#14519752—28no—
35#14519752—28no—
36C4×D244—22no—
37C4×D244—22no—
38C4×D244D414yes—
39C4×D244C2314yes—
40#14519735—28no—
41#14519735—28no—
42C2×C4⋊Dic61D412yes—
43C2×#11758198—24no—
44C2×#11758198—22no—
45C2×C4×Dic61—22no—
46#14519710—24no—
47C4⋊#92418C2314yes—
48#14519701—24no—
49#14519697—24no—
50C22⋊#92418—22no—
51#14519688—22no—
52C22×Q8×C61D412yes—
53C2×M16×C61—21no—
54C4⋊Q8×C61—22no—
55C4⋊Q8×C61D412yes—
56C4⋊Q8×C61C2312yes—
57C4⋊Q8×C61C2312yes—
58#125×C61—21no—
59C22⋊Q8×C61—22no—
60C4×Q8×C61—22no—
61#117×C61—21no—
62C2×C4⋊C4×C61—22no—
63C2×C42×C61C2311yescharacteristic
64#103×C61—28no—
65#97×C61—21no—
66C4×C8×C61C2312yes—
67C23×D61—22no—
68C2×Q8×C61—22no—
69C2×Q8×C61—22no—
70C2×Q8×C61—44no—
71C2×Q8×C61C2×D414yes—
72C22×C4×C61—22no—
73C22×C4×C61C2×D412yes—
74C22×C4×C61C2×D411yescharacteristic
75C22×Dic61—22no—
76C2×D244—22no—
77C2×D244—28no—
78C2×D244—44no—
79C2×D244—84no—
80C2×D244C2×D414yes—
81C2×C4×D61—24no—
82C2×C4×D61—24no—
83C2×C4×D61—48no—
84M16×C61—42no—
85C2×C8×C61—22no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—42no—
91C22⋊C4×C61—41no—
92C42×C61—41no—
93C42×C61C2×D412yes—
94C42×C61C2×D411yescharacteristic
95C42×C61C2411yescharacteristic
96#11758201—48no—
97#11758198—216no—
98#11758198—24no—
99#11758198—24no—
100C4⋊Dic61—22no—
101C4⋊Dic61C2×D414yes—
102C4×Dic61—28no—
103C2×#92418—24no—
104C23×C61#12011yescharacteristic
105C22×D61—24no—
106C22×D61—44no—
107C22×D61—44no—
108C22×D61—44no—
109Q8×C61—44no—
110Q8×C61—44no—
111Q8×C61—82no—
112C2×C4×C61—22no—
113C2×C4×C61—24no—
114C2×C4×C61—24no—
115C2×C4×C61—44no—
116C2×C4×C61—42no—
117C2×C4×C61#12012yes—
118C2×C4×C61#12011yescharacteristic
119C2×C4×C61C22×D411yescharacteristic
120C2×C4×C61C22×D412yes—
121C2×Dic61—24no—
122C2×Dic61—44no—
123D244—24no—
124D244—24no—
125D244—44no—
126D244—88no—
127C4×D61—48no—
128C4×D61—48no—
129C8×C61—42no—
130#92418—44no—
131#18552—611nomaximal, Sylow
132C22×C61—22no—
133C22×C61#45512yes—
134C22×C61C2×#12011yescharacteristic
135C2×D61—44no—
136C2×D61—44no—
137C2×D61—44no—
138C2×D61—88no—
139C4×C61—22no—
140C4×C61—21no—
141C4×C61—21no—
142C4×C61—44no—
143C4×C61—82no—
144C4×C61#57514yes—
145Dic61—44no—
146C2×#555—611no—
147#3139D6111yescharacteristic, complemented
148#3110—612no—
149#1646—618no—
150#1558—611no—
151#1490—612no—
152C2×C61—22no—
153C2×C61#300512yes—
154C2×C61C2×#45511yescharacteristic
155D61—84no—
156#555—614no—
157#555—1224no—
158#555—1222no—
159#533C2×D6111yescharacteristic
160C2×C4⋊Q8C2×D6111yescharacteristic
161C2×#124—1222no—
162C2×#121—1222no—
163C2×C4×D4—612no—
164C2×#117—1222no—
165#491—22no—
166#487C2×D6114yescomplemented
167#484—1222no—
168#474—614no—
169#461—614no—
170#450—1222no—
171#440—1224no—
172#432—612no—
173#419—22no—
174C4×M16C2×D6111yescharacteristic
175#333—614no—
176D4⋊C8—618no—
177C61#1855211yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D6112yes—
179C22×D4—612no—
180C22×D4—1222no—
181C23×C4—1222no—
182C2×SD16—1224no—
183C2×M16—21no—
184C4⋊Q8—22no—
185C4⋊Q8C22⋊D6112yes—
186C4⋊Q8C22×D6112yes—
187C4⋊Q8C22×D6112yes—
188#125—21no—
189#124—1228no—
190#124—1228no—
191C22⋊Q8—22no—
192#121—1228no—
193#121—1228no—
194C4×Q8—22no—
195C4×D4—614no—
196C4×D4—614no—
197C4×D4—1222no—
198C4×D4—1222no—
199#117—21no—
200#117—1228no—
201#117—1228no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—612no—
204C2×C22⋊C4—1224no—
205C2×C22⋊C4—1222no—
206C2×C42—1222no—
207C2×C42C22×D6111yescharacteristic
208#107—1224no—
209C4⋊C8—614no—
210#103—28no—
211#103—1224no—
212#102—1224no—
213C22⋊C8—1222no—
214#97—21no—
215#97—1222no—
216C4×C8C22×D6112yes—
217C24—1222no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—44no—
221C2×Q8C2×C22⋊D6114yes—
222C2×D4—614no—
223C2×D4—1228no—
224C2×D4—1222no—
225C2×D4—4884no—
226C2×D4—2444no—
227C22×C4—22no—
228C22×C4—1222no—
229C22×C4—1224no—
230C22×C4—1224no—
231C22×C4—2448no—
232C22×C4D4×D6112yes—
233C22×C4C2×C22⋊D6111yescharacteristic
234SD16—2448no—
235M16—42no—
236C2×C8—22no—
237C2×C8—1224no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—614no—
244C4⋊C4—1222no—
245C22⋊C4—41no—
246C22⋊C4—12216no—
247C22⋊C4—1224no—
248C22⋊C4—1224no—
249C42—41no—
250C42—1228no—
251C42D4×D6112yes—
252C42C2×C22⋊D6111yescharacteristic
253C42C23×D6111yescharacteristic
254C23—1224no—
255C23—2444no—
256C23—2444no—
257C23—2444no—
258C23#1451974511yescharacteristic
259Q8—44no—
260Q8—44no—
261Q8—82no—
262D4—1224no—
263D4—1224no—
264D4—4888no—
265D4—2444no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—1224no—
272C2×C4—2448no—
273C2×C4—2444no—
274C2×C4—2448no—
275C2×C4#1451974511yescharacteristic
276C2×C4#1451974512yes—
277C2×C4C2×D4×D6112yes—
278C2×C4C22×C22⋊D6111yescharacteristic
279C8—42no—
280C8—2444no—
281C22—22no—
282C22—4888no—
283C22—2444no—
284C22—2444no—
285C22—2444no—
286C22#1683993912yes—
287C22C2×#1451974511yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—44no—
292C4—82no—
293C4—2444no—
294C4#1684056114yes—
295C2—22no—
296C2—4884no—
297C2#2145506012yes—
298C2C2×#1683993911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3139C2
socleC22×C61C2×#120
prime radicalC2×#16840478C2
hypercenter#3139D61
hypocenterC61#18552
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #21455914C2semidirect
3. #21445044C2semidirect
4. #21444545C2semidirect
5. #21444263C2semidirect
6. C61×#3139C2semidirect
15. #16840039C22semidirect
16. #16839964C22semidirect
20. #16839511C22semidirect
21. #16839491C22semidirect
25. C61×#487C22semidirect
147. #3139D61semidirect
166. #487C2×D61semidirect
177. C61#18552semidirect

Maximal subgroups

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

subgroupindex
C2×#168404782
#214559142
#214450442
#214445452
#214442632
C61×#31392
#1855261