Finite Group Databasebeta

← #29656321

Subgroups — #29656321

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#21456199C211yesmaximal, characteristic, complemented
3#21455919C212yesmaximal, complemented
4#21445048C218yesmaximal, complemented
5#21444261C212yesmaximal, complemented
6C61×#1558C211yesmaximal, characteristic, complemented
7#16840478—24no—
8#16840478—22no—
9#16840478C2214yes—
10#16840319C2211yescharacteristic
11C2×#14519766—22no—
12C2×#14519752—22no—
13C2×C4×D244C2212yes—
14C2×#14519735—22no—
15#16840130—22no—
16#16840112C2214yescomplemented
17#16840098—22no—
18#16840043C2214yescomplemented
19#16839968C2214yescomplemented
20#16839915—22no—
21#16839864—24no—
22#16839771—22no—
23C4×#11758194C2211yescharacteristic
24#16839489C2218yescomplemented
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#432C2212yes—
27C61×#333C2214yescomplemented
28C22×D244—22no—
29C22×D244D412yes—
30C22×C4×D61—22no—
31C2×#11758190—24no—
32C2×#11758194—21no—
33#14519777C2312yes—
34#14519769—21no—
35#14519766—28no—
36#14519766—28no—
37#14519756—22no—
38#14519752—28no—
39#14519752—28no—
40C4×Dic122—22no—
41C4×D244—22no—
42C4×D244—22no—
43C4×D244D414yes—
44C4×D244C2314yes—
45#14519736—21no—
46#14519735—28no—
47#14519735—28no—
48C2×C4⋊Dic61D412yes—
49C2×#11758198—24no—
50C2×#11758198—22no—
51C2×C4×Dic61—22no—
52#14519711—24no—
53#14519702—24no—
54#14519701—28no—
55#14519699—24no—
56#14519688—21no—
57C4×#92418C2312yes—
58C22×Q8×C61D412yes—
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61D412yes—
61C4⋊Q8×C61C2312yes—
62C2×C4⋊C4×C61—22no—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61C2314yes—
65C22⋊C8×C61—22no—
66#97×C61—22no—
67C23×D61—22no—
68C2×Q8×C61—22no—
69C2×Q8×C61—44no—
70C2×Q8×C61C2×D414yes—
71C22×C4×C61—22no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D412yes—
74C22×Dic61—22no—
75C2×D244—22no—
76C2×D244—28no—
77C2×D244—44no—
78C2×D244—84no—
79C2×D244C2×D414yes—
80C2×C4×D61—24no—
81C2×C4×D61—24no—
82C2×C4×D61—48no—
83C2×Dic122—22no—
84C2×C8×C61—24no—
85C4⋊C4×C61—24no—
86C4⋊C4×C61—24no—
87C42×C61C2×D411yescharacteristic
88C42×C61C2×D412yes—
89C42×C61C2411yescharacteristic
90C22⋊Dic61—41no—
91#11758198—216no—
92#11758198—24no—
93#11758198—24no—
94C4⋊Dic61—22no—
95C4⋊Dic61—22no—
96C4⋊Dic61—22no—
97C4⋊Dic61C2×D414yes—
98#11758196—42no—
99C4×Dic61—28no—
100C4×Dic61—41no—
101#11758194—42no—
102C2×#92418—22no—
103#11758190—48no—
104C23×C61#12011yescharacteristic
105C22×D61—24no—
106C22×D61—44no—
107C22×D61—44no—
108C22×D61—44no—
109Q8×C61—44no—
110Q8×C61—82no—
111C2×C4×C61—22no—
112C2×C4×C61—24no—
113C2×C4×C61—24no—
114C2×C4×C61—44no—
115C2×C4×C61#12011yescharacteristic
116C2×C4×C61#12012yes—
117C2×C4×C61C22×D411yescharacteristic
118C2×C4×C61C22×D412yes—
119C2×Dic61—24no—
120C2×Dic61—44no—
121C2×Dic61—42no—
122D244—24no—
123D244—24no—
124D244—44no—
125D244—88no—
126C4×D61—48no—
127C4×D61—48no—
128Dic122—44no—
129C8×C61—44no—
130#92418—42no—
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#57514yes—
144Dic61—44no—
145Dic61—82no—
146C2×#555—611no—
147#3139—611no—
148#3110—612no—
149#1646—618no—
150#1558D6111yescharacteristic, complemented
151#1490—612no—
152C2×C61—22no—
153C2×C61#300512yes—
154C2×C61C2×#45511yescharacteristic
155D61—84no—
156#555—614no—
157#555—1224no—
158#555—1222no—
159#533—611no—
160C2×C4⋊Q8C2×D6111yescharacteristic
161C2×#124—1222no—
162C2×#121—1222no—
163C2×C4×D4—612no—
164C2×#117—1222no—
165#491—1222no—
166#487—614no—
167#484—1222no—
168#474—614no—
169#461—614no—
170#450—1222no—
171#440—1224no—
172#432C2×D6112yes—
173#419—1222no—
174C4×M16—611no—
175#333C2×D6114yescomplemented
176D4⋊C8—618no—
177C61#1855211yescharacteristic, complemented, Sylow
178C22×Q8D24412yes—
179C22×D4—612no—
180C22×D4—1222no—
181C23×C4—1222no—
182C2×SD16—1224no—
183C2×M16—1221no—
184C4⋊Q8—22no—
185C4⋊Q8—612no—
186C4⋊Q8D24412yes—
187C4⋊Q8C22×D6112yes—
188#125—1221no—
189#124—1228no—
190#124—1228no—
191C22⋊Q8—1222no—
192#121—1228no—
193#121—1228no—
194C4×Q8—1222no—
195C4×D4—614no—
196C4×D4—614no—
197C4×D4—1222no—
198C4×D4—1222no—
199#117—1228no—
200#117—1228no—
201#117—1221no—
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⋊C8C22×D6114yes—
210#103—1224no—
211#103—1228no—
212#102—1224no—
213C22⋊C8—22no—
214#97—22no—
215#97—1221no—
216C4×C8—612no—
217C24—1222no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—1222no—
221C2×Q8C2×D24414yes—
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×C4C2×D24411yescharacteristic
233C22×C4D4×D6112yes—
234SD16—2448no—
235M16—2442no—
236C2×C8—24no—
237C2×C8—1222no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—614no—
241C4⋊C4—1222no—
242C4⋊C4—1222no—
243C4⋊C4—1222no—
244C4⋊C4—2442no—
245C22⋊C4—12216no—
246C22⋊C4—1224no—
247C22⋊C4—1224no—
248C22⋊C4—2441no—
249C42—1228no—
250C42—2441no—
251C42C2×D24411yescharacteristic
252C42D4×D6112yes—
253C42C23×D6111yescharacteristic
254C23—1224no—
255C23—2444no—
256C23—2444no—
257C23—2444no—
258C23#1451974711yescharacteristic
259Q8—44no—
260Q8—82no—
261Q8—2444no—
262D4—1224no—
263D4—1224no—
264D4—4888no—
265D4—2444no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—44no—
270C2×C4—1224no—
271C2×C4—2448no—
272C2×C4—2442no—
273C2×C4—2444no—
274C2×C4—2448no—
275C2×C4#1451974711yescharacteristic
276C2×C4#1451974712yes—
277C2×C4C2×D4×D6112yes—
278C2×C4C22×D24411yescharacteristic
279C8—44no—
280C8—2442no—
281C22—22no—
282C22—4888no—
283C22—2444no—
284C22—2444no—
285C22—2444no—
286C22#1683994212yes—
287C22C2×#1451974711yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—44no—
292C4—4882no—
293C4—2444no—
294C4#1684056114yes—
295C2—22no—
296C2—4884no—
297C2#2145506512yes—
298C2C2×#1683994211yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1558C2
socleC22×C61C2×#120
prime radicalC2×#16840478C2
hypercenter#1558D61
hypocenterC61#18552
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #21456199C2semidirect
3. #21455919C2semidirect
4. #21445048C2semidirect
5. #21444261C2semidirect
6. C61×#1558C2semidirect
16. #16840112C22semidirect
18. #16840043C22semidirect
19. #16839968C22semidirect
24. #16839489C22semidirect
27. C61×#333C22semidirect
150. #1558D61semidirect
175. #333C2×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
#214561992
#214559192
#214450482
#214442612
C61×#15582
#1855261