Finite Group Databasebeta

← #29572575

Subgroups — #29572575

Order 15616.

800 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 115 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#21446647C212yesmaximal, complemented
3#21444557C211yesmaximal, characteristic, complemented
4#21444552C212yesmaximal, complemented
5C61×#1994C211yesmaximal, characteristic, complemented
6#16840510C414yescomplemented
7#16840472—22no—
8#16840434—22no—
9#16840415—22no—
10#16840367—22no—
11#16840342—24no—
12#16840312C2212yes—
13#16839786C2212yes—
14C2×#14519705—24no—
15#16839510—22no—
16#16839508—22no—
17#16839505—22no—
18C61×#533C2211yescharacteristic
19C61×#423C2211yescharacteristic
20C43×C61C2211yescharacteristic
21C61×#339—22no—
22C2×#11758226—24no—
23C2×#11758222D414yes—
24C2×#11758194—22no—
25#14519777D412yes—
26#14519774—22no—
27#14519774D412yes—
28#14519769—22no—
29#14519765C2×C414yes—
30#14519762—28no—
31#14519755—28no—
32#14519753—24no—
33#14519752—24no—
34#14519752—28no—
35#14519751—22no—
36#14519751—22no—
37#14519750—28no—
38C4×D244—24no—
39C4×D244—22no—
40C4×D244—22no—
41C4⋊C4×D61—24no—
42C4⋊#92418—22no—
43C4⋊#92418—22no—
44#14519705—48no—
45C2×M16×C61—21no—
46C4⋊Q8×C61D412yes—
47#125×C61C2×C412yes—
48C22⋊Q8×C61—22no—
49C4×Q8×C61—22no—
50#117×C61D412yes—
51C2×C42×C61—21no—
52C2×C42×C61—22no—
53C2×C42×C61C2311yescharacteristic
54C4⋊C8×C61—22no—
55C2×Q8×C61—22no—
56C22×C4×C61—41no—
57C22×C4×C61C2×D412yes—
58C22×C4×C61C2×D411yescharacteristic
59C2×C22⋊D61—24no—
60#11758226—84no—
61#11758226—84no—
62#11758222—44no—
63#11758222—44no—
64C2×D244—22no—
65C2×D244—22no—
66C2×D244—28no—
67C2×D244C22⋊C414yes—
68C2×C4×D61—24no—
69C2×C4×D61—28no—
70C2×Dic122C22⋊C414yes—
71M16×C61—42no—
72C2×C8×C61—41no—
73C4⋊C4×C61—24no—
74C4⋊C4×C61—22no—
75C4⋊C4×C61C22⋊C414yes—
76C22⋊C4×C61—22no—
77C42×C61—22no—
78C42×C61—22no—
79C42×C61—24no—
80C42×C61—41no—
81C42×C61—42no—
82C42×C61C22×C411yescharacteristic
83C42×C61C2×D411yescharacteristic
84C42×C61C2×D412yes—
85#11758198—28no—
86#11758198—24no—
87#11758198—24no—
88C4⋊Dic61—22no—
89C4⋊Dic61—22no—
90#11758196—28no—
91#11758194—44no—
92C2×#92418—42no—
93C23×C61#12011yescharacteristic
94C22×D61—24no—
95Q8×C61—42no—
96Q8×C61—42no—
97C2×C4×C61—22no—
98C2×C4×C61—21no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—24no—
102C2×C4×C61—44no—
103C2×C4×C61—42no—
104C2×C4×C61C2×C22⋊C412yes—
105C2×C4×C61C2×C22⋊C411yescharacteristic
106C2×C4×C61#12012yes—
107C2×C4×C61#12011yescharacteristic
108C22⋊D61—84no—
109C2×Dic61—24no—
110D244—44no—
111D244—42no—
112D244—42no—
113D244—42no—
114D244—42no—
115D244—88no—
116C4×D61—44no—
117C4×D61—44no—
118C4×D61—88no—
119Dic122—44no—
120C8×C61—81no—
121#92418—82no—
122#8532—611nomaximal, Sylow
123C22×C61—22no—
124C22×C61—21no—
125C22×C61#37911yescharacteristic
126C2×D61—44no—
127C2×D61—44no—
128C2×D61—44no—
129C4×C61—22no—
130C4×C61—21no—
131C4×C61—44no—
132C4×C61—44no—
133C4×C61#45312yes—
134C4×C61#45411yescharacteristic
135C4×C61#45512yes—
136C4×C61#45611yescharacteristic
137Dic61—44no—
138#3548—611no—
139#1994D6111yescharacteristic, complemented
140#1962—612no—
141#1561—611no—
142#1560—612no—
143C2×C61—41no—
144C2×C61#179212yes—
145C2×C61#179311yescharacteristic
146D61—84no—
147#562—614no—
148#554—1222no—
149#550—1222no—
150#547—1222no—
151#540—1222no—
152#537—1224no—
153#533C2×D6111yescharacteristic
154#532—612no—
155#423—612no—
156#423C2×D6111yescharacteristic
157C2×#104—1224no—
158C43C2×D6111yescharacteristic
159#339—22no—
160#333—1222no—
161#332—1222no—
162#331—1222no—
163C61#853211yescharacteristic, complemented, Sylow
164C2×#41—614no—
165C2×#41—1224no—
166C2×M16—21no—
167C2×M16—1222no—
168C4⋊Q8—612no—
169C4⋊Q8D24412yes—
170#127—612no—
171#127—1222no—
172#125—1222no—
173#125C4×D6112yes—
174#124—614no—
175#123—1228no—
176C22⋊Q8—22no—
177C22⋊Q8—1228no—
178C22⋊Q8—1224no—
179#121—1228no—
180#121—1228no—
181#121—1224no—
182#121—1222no—
183#121—1222no—
184C4×Q8—22no—
185C4×D4—1224no—
186C4×D4—1222no—
187C4×D4—1222no—
188#117C22⋊D6112yes—
189C2×C4⋊C4—1224no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42C22×D6111yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—1222no—
195C4⋊C8—1222no—
196#104—2448no—
197#41—4884no—
198#41—4884no—
199#41—2444no—
200#41—2444no—
201C2×Q8—22no—
202C2×Q8—614no—
203C2×D4—614no—
204C2×D4—1228no—
205C2×D4—1224no—
206C2×D4—1222no—
207C2×D4—1222no—
208C22×C4—41no—
209C22×C4—1228no—
210C22×C4—1224no—
211C22×C4D4×D6112yes—
212C22×C4C2×C22⋊D6111yescharacteristic
213M16—42no—
214M16—2444no—
215C2×C8—41no—
216C2×C8—2442no—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—1228no—
220C4⋊C4—1222no—
221C4⋊C4—1222no—
222C4⋊C4#1175819814yes—
223C22⋊C4—22no—
224C22⋊C4—1228no—
225C22⋊C4—1224no—
226C22⋊C4—1224no—
227C42—22no—
228C42—24no—
229C42—22no—
230C42—42no—
231C42—41no—
232C42C2×C4×D6111yescharacteristic
233C42C2×D24411yescharacteristic
234C42D4×D6112yes—
235C23—1224no—
236C23#1451974511yescharacteristic
237Q8—42no—
238Q8—42no—
239Q8—2444no—
240D4—4888no—
241D4—4884no—
242D4—2444no—
243D4—2442no—
244D4—2442no—
245D4—2442no—
246D4—2442no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—1224no—
255C2×C4—4888no—
256C2×C4—2444no—
257C2×C4—2444no—
258C2×C4C22⋊C4×D6112yes—
259C2×C4C2×#1175819811yescharacteristic
260C2×C4#1451974511yescharacteristic
261C2×C4#1451974712yes—
262C8—81no—
263C8—4882no—
264C22—22no—
265C22—21no—
266C22—2444no—
267C22—2444no—
268C22—2444no—
269C22#1683961711yescharacteristic
270C4—22no—
271C4—21no—
272C4—44no—
273C4—44no—
274C4—2444no—
275C4#1683993112yes—
276C4#1683993411yescharacteristic
277C4#1683994112yes—
278C4#1683994411yescharacteristic
279C2—41no—
280C2—4884no—
281C2#2144577412yes—
282C2#2144577711yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1994C2
socleC22×C61#379
prime radical#21460219C2
cosocleC2×C42×C61C23
hypercenter#1994D61
hypocenterC61#8532
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
2. #21446647C2semidirect
3. #21444557C2semidirect
4. #21444552C2semidirect
5. C61×#1994C2semidirect
6. #16840510C4semidirect
139. #1994D61semidirect
163. C61#8532semidirect

Maximal subgroups

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

subgroupindex
#214602192
#214466472
#214445572
#214445522
C61×#19942
#853261