Finite Group Databasebeta

← #29658272

Subgroups — #29658272

Order 15616.

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

normal subgroups only

This group has 920 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
1#21460219C211yesmaximal, characteristic
2#21456947C212yesmaximal, complemented
3#21444967C218yesmaximal, complemented
4#21444484C211yesmaximal, characteristic, complemented
5#21444237C212yesmaximal, complemented
6C61×#3139C211yesmaximal, characteristic, complemented
7#16840510—22no—
8#16840472—22no—
9#16840434—22no—
10#16840415C2214yes—
11#16840367—22no—
12#16840342—24no—
13#16840312C2212yes—
14#16840158—22no—
15#16840110—22no—
16#16840080—24no—
17#16840049C2218yes—
18#16839815C2212yes—
19#16839508C2214yes—
20#16839491C2218yes—
21C61×#533C2211yescharacteristic
22C2×C4⋊Q8×C61C2211yescharacteristic
23C61×#491—22no—
24C61×#487C2214yes—
25C61×#419—22no—
26C4×M16×C61C2211yescharacteristic
27C2×#11758226—24no—
28C2×#11758222—22no—
29#14519777D412yes—
30#14519774—22no—
31#14519774D412yes—
32#14519769—22no—
33#14519765—22no—
34#14519762—28no—
35#14519755—28no—
36#14519753—24no—
37#14519752—28no—
38#14519752—24no—
39#14519751—22no—
40#14519751D414yes—
41#14519750—28no—
42C4×D244—24no—
43C4×D244—22no—
44C4×D244C2314yes—
45C4⋊C4×D61—24no—
46#14519708—28no—
47C4⋊#92418C2314yes—
48#14519697—28no—
49C22⋊#92418—22no—
50C4×#92418—22no—
51C22×Q8×C61—21no—
52C2×M16×C61—21no—
53C4⋊Q8×C61—22no—
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61C2312yes—
56C4⋊Q8×C61C2312yes—
57#125×C61D412yes—
58C22⋊Q8×C61—22no—
59C4×Q8×C61—22no—
60#117×C61D412yes—
61C2×C4⋊C4×C61—22no—
62C2×C42×C61C2311yescharacteristic
63#103×C61—28no—
64#97×C61—21no—
65C4×C8×C61C2312yes—
66C2×Q8×C61—22no—
67C2×Q8×C61—22no—
68C2×Q8×C61—44no—
69C2×Q8×C61—41no—
70C22×C4×C61—41no—
71C22×C4×C61C2×D412yes—
72C22×C4×C61C2×D411yescharacteristic
73C2×C22⋊D61—24no—
74#11758226—88no—
75#11758222—84no—
76C2×D244—22no—
77C2×D244—28no—
78C2×D244—22no—
79C2×D244C2×D414yes—
80C2×C4×D61—24no—
81C2×C4×D61—28no—
82C2×Dic122—22no—
83M16×C61—42no—
84C2×C8×C61—22no—
85C4⋊C4×C61—24no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—42no—
89C4⋊C4×C61C2×D414yes—
90C22⋊C4×C61—22no—
91C42×C61—22no—
92C42×C61C2×D412yes—
93C42×C61C2×D411yescharacteristic
94C42×C61C2411yescharacteristic
95#11758198—28no—
96#11758198—24no—
97#11758198—24no—
98C4⋊Dic61—22no—
99C4⋊Dic61C2×D414yes—
100#11758196—28no—
101C2×#92418—24no—
102C23×C61#12011yescharacteristic
103C22×D61—24no—
104Q8×C61—44no—
105Q8×C61—44no—
106Q8×C61—82no—
107C2×C4×C61—24no—
108C2×C4×C61—22no—
109C2×C4×C61—24no—
110C2×C4×C61—42no—
111C2×C4×C61—82no—
112C2×C4×C61#12012yes—
113C2×C4×C61#12011yescharacteristic
114C2×C4×C61C22×D411yescharacteristic
115C2×C4×C61C22×D412yes—
116C22⋊D61—84no—
117C2×Dic61—24no—
118D244—28no—
119D244—44no—
120D244—82no—
121D244—88no—
122C4×D61—48no—
123C4×D61—88no—
124Dic122—82no—
125C8×C61—42no—
126#92418—44no—
127#18728—611nomaximal, Sylow
128C22×C61—22no—
129C22×C61—21no—
130C22×C61C2×#12011yescharacteristic
131C2×D61—44no—
132C2×D61—44no—
133C2×D61—44no—
134C4×C61—22no—
135C4×C61—21no—
136C4×C61—44no—
137C4×C61—82no—
138C4×C61#45412yes—
139C4×C61#57714yes—
140Dic61—44no—
141#3548—611no—
142#3227—612no—
143#3139D6111yescharacteristic, complemented
144#1635—618no—
145#1544—611no—
146#1484—612no—
147C2×C61—41no—
148C2×C61#300912yes—
149C2×C61C2×#45411yescharacteristic
150D61—84no—
151#562—1222no—
152#554—1222no—
153#550—1222no—
154#547—614no—
155#540—1222no—
156#537—1224no—
157#533C2×D6111yescharacteristic
158#532—612no—
159C2×C4⋊Q8C2×D6111yescharacteristic
160#498—1222no—
161#491—22no—
162#487C2×D6114yes—
163#486—1222no—
164#481—1224no—
165#476—618no—
166#431—612no—
167#419—22no—
168C4×M16C2×D6111yescharacteristic
169#332—614no—
170D4⋊C8—618no—
171C61#1872811yescharacteristic, complemented, Sylow
172C2×#41—1224no—
173C2×#41—1222no—
174C22×Q8—21no—
175C2×M16—21no—
176C4⋊Q8—22no—
177C4⋊Q8—21no—
178C4⋊Q8—612no—
179C4⋊Q8C22×D6112yes—
180C4⋊Q8C22×D6112yes—
181#127—612no—
182#127—1222no—
183#125—1222no—
184#125C22⋊D6112yes—
185#124—1222no—
186#123—1228no—
187C22⋊Q8—22no—
188C22⋊Q8—1228no—
189C22⋊Q8—1224no—
190#121—614no—
191#121—1228no—
192#121—1224no—
193#121—1228no—
194#121—1222no—
195C4×Q8—22no—
196C4×D4—614no—
197C4×D4—1224no—
198C4×D4—1222no—
199#117C22⋊D6112yes—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—1224no—
202C2×C42C22×D6111yescharacteristic
203#106—1228no—
204C4⋊C8—614no—
205#103—28no—
206#102—1228no—
207C22⋊C8—1222no—
208#97—21no—
209C4×C8—1222no—
210C4×C8C22×D6112yes—
211#41—4884no—
212#41—4888no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—44no—
216C2×Q8—41no—
217C2×Q8—1222no—
218C2×D4—614no—
219C2×D4—1228no—
220C2×D4—1224no—
221C2×D4—1222no—
222C2×D4—1222no—
223C22×C4—41no—
224C22×C4—1228no—
225C22×C4—1224no—
226C22×C4D4×D6112yes—
227C22×C4C2×C22⋊D6111yescharacteristic
228M16—42no—
229C2×C8—22no—
230C2×C8—1224no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—614no—
236C4⋊C4—1228no—
237C4⋊C4—1222no—
238C4⋊C4C2×C22⋊D6114yes—
239C22⋊C4—22no—
240C22⋊C4—1228no—
241C22⋊C4—1224no—
242C22⋊C4—1224no—
243C42—22no—
244C42D4×D6112yes—
245C42C2×C22⋊D6111yescharacteristic
246C42C23×D6111yescharacteristic
247C23—1224no—
248C23#1451974511yescharacteristic
249Q8—44no—
250Q8—44no—
251Q8—82no—
252Q8—4882no—
253D4—1228no—
254D4—4882no—
255D4—4888no—
256D4—4884no—
257D4—2444no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—1224no—
264C2×C4—4888no—
265C2×C4—2448no—
266C2×C4#1451974512yes—
267C2×C4#1451974511yescharacteristic
268C2×C4C2×D4×D6112yes—
269C2×C4C22×C22⋊D6111yescharacteristic
270C8—42no—
271C8—2444no—
272C22—21no—
273C22—22no—
274C22—2444no—
275C22—2444no—
276C22—2444no—
277C22C2×#1451974511yescharacteristic
278C4—21no—
279C4—22no—
280C4—44no—
281C4—82no—
282C4—2444no—
283C4#1683993412yes—
284C4#1684057114yes—
285C2—41no—
286C2—4884no—
287C2#2145509512yes—
288C2C2×#1683993411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3139C2
socleC22×C61C2×#120
prime radical#21460219C2
hypercenter#3139D61
hypocenterC61#18728
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. #21456947C2semidirect
3. #21444967C2semidirect
4. #21444484C2semidirect
5. #21444237C2semidirect
6. C61×#3139C2semidirect
143. #3139D61semidirect
171. C61#18728semidirect

Maximal subgroups

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

subgroupindex
#214602192
#214569472
#214449672
#214444842
#214442372
C61×#31392
#1872861