Finite Group Databasebeta

← #29658302

Subgroups — #29658302

Order 15616.

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

normal subgroups only

This group has 808 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#21460314C211yesmaximal, characteristic
2#21456970C212yesmaximal
3#21444974C218yesmaximal, complemented
4#21444484C211yesmaximal, characteristic
5#21444241C212yesmaximal
6C61×#3139C211yesmaximal, characteristic
7C61⋊#564—22no—
8#16840491—22no—
9#16840490—22no—
10#16840487C2214yes—
11#16840374—22no—
12#16840372—24no—
13#16840319C2212yes—
14#16840165—22no—
15#16840114—22no—
16#16840097—24no—
17#16840054C2218yes—
18#16839815C2212yes—
19#16839508C2214yes—
20#16839494C2218yes—
21C61×#533C2211yescharacteristic
22C2×C4⋊Q8×C61C2211yescharacteristic
23C61×#491—22no—
24C61×#487C2214yes—
25C61×#419—22no—
26C4×M16×C61C2211yescharacteristic
27#14519777—22no—
28#14519777D412yes—
29#14519777D412yes—
30#14519775—24no—
31#14519770—28no—
32#14519769—22no—
33#14519769—22no—
34#14519768—28no—
35#14519768—24no—
36#14519767—28no—
37#14519767—28no—
38#14519756—22no—
39#14519756D414yes—
40Q8×Dic61—24no—
41C4×Dic122—24no—
42C4×Dic122—22no—
43C4×Dic122C2314yes—
44#14519742—24no—
45#14519736—22no—
46#14519710—28no—
47C4⋊#92418C2314yes—
48#14519700—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×Dic122—22no—
74C2×Dic122C2×D414yes—
75M16×C61—42no—
76C2×C8×C61—22no—
77C4⋊C4×C61—24no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—24no—
80C4⋊C4×C61—42no—
81C4⋊C4×C61C2×D414yes—
82C22⋊C4×C61—22no—
83C42×C61—22no—
84C42×C61C2×D412yes—
85C42×C61C2×D411yescharacteristic
86C42×C61C2411yescharacteristic
87C22⋊Dic61—24no—
88C4⋊Dic61—28no—
89C4⋊Dic61—22no—
90C4⋊Dic61—22no—
91C4⋊Dic61—22no—
92C4⋊Dic61C2×D414yes—
93#11758196—28no—
94#11758196—28no—
95#11758196—24no—
96#11758196—24no—
97C4×Dic61—28no—
98C4×Dic61—24no—
99C2×#92418—24no—
100C23×C61#12011yescharacteristic
101Q8×C61—44no—
102Q8×C61—44no—
103Q8×C61—82no—
104C2×C4×C61—24no—
105C2×C4×C61—22no—
106C2×C4×C61—24no—
107C2×C4×C61—42no—
108C2×C4×C61—82no—
109C2×C4×C61#12012yes—
110C2×C4×C61#12011yescharacteristic
111C2×C4×C61C22×D411yescharacteristic
112C2×C4×C61C22×D412yes—
113C2×Dic61—24no—
114C2×Dic61—24no—
115Dic122—28no—
116Dic122—44no—
117C8×C61—42no—
118#92418—44no—
119#18732—611nomaximal, Sylow
120C22×C61—22no—
121C22×C61—21no—
122C22×C61C2×#12011yescharacteristic
123C4×C61—22no—
124C4×C61—21no—
125C4×C61—44no—
126C4×C61—82no—
127C4×C61#45412yes—
128C4×C61#57814yes—
129Dic61—44no—
130Dic61—44no—
131#3556—611no—
132#3230—612no—
133#3139D6111yescharacteristic
134#1636—618no—
135#1544—611no—
136#1485—612no—
137C2×C61—41no—
138C2×C61#301112yes—
139C2×C61C2×#45411yescharacteristic
140#564—1222no—
141#557—614no—
142#557—1222no—
143#557—1222no—
144#541—1224no—
145#541—1222no—
146#533—612no—
147#533C2×D6111yescharacteristic
148C2×C4⋊Q8C2×D6111yescharacteristic
149#500—1222no—
150#491—22no—
151#487—1222no—
152#487C2×D6114yes—
153#484—1224no—
154#477—618no—
155#431—612no—
156#419—22no—
157C4×M16C2×D6111yescharacteristic
158#332—614no—
159Q8⋊C8—618no—
160C61#1873211yescharacteristic, complemented, Sylow
161C22×Q8—21no—
162C2×M16—21no—
163C4⋊Q8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—612no—
166C4⋊Q8—612no—
167C4⋊Q8—1222no—
168C4⋊Q8—1224no—
169C4⋊Q8C22×D6112yes—
170C4⋊Q8C22×D6112yes—
171#126—1228no—
172#125—1222no—
173#125—1224no—
174#125—1228no—
175#125—1228no—
176#125—1228no—
177#125—1222no—
178#125C22⋊D6112yes—
179C22⋊Q8—22no—
180C22⋊Q8—614no—
181C22⋊Q8—1222no—
182C4×Q8—22no—
183C4×Q8—614no—
184C4×Q8—1224no—
185C4×Q8—1224no—
186C4×Q8—1224no—
187C4×Q8—1222no—
188#117—1222no—
189#117C22⋊D6112yes—
190C2×C4⋊C4—22no—
191C2×C42C22×D6111yescharacteristic
192#107—1228no—
193C4⋊C8—614no—
194#103—28no—
195#103—1228no—
196C22⋊C8—1222no—
197#97—21no—
198C4×C8—1222no—
199C4×C8C22×D6112yes—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—44no—
203C2×Q8—41no—
204C2×Q8—614no—
205C2×Q8—1222no—
206C22×C4—41no—
207C22×C4D4×D6112yes—
208C22×C4C2×C22⋊D6111yescharacteristic
209M16—42no—
210C2×C8—22no—
211C2×C8—1224no—
212C4⋊C4—22no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—42no—
216C4⋊C4—614no—
217C4⋊C4—1228no—
218C4⋊C4—1228no—
219C4⋊C4—1228no—
220C4⋊C4—1224no—
221C4⋊C4—1224no—
222C4⋊C4—1222no—
223C4⋊C4—1222no—
224C4⋊C4—1222no—
225C4⋊C4C2×C22⋊D6114yes—
226C22⋊C4—22no—
227C22⋊C4—1224no—
228C42—22no—
229C42—1228no—
230C42—1224no—
231C42D4×D6112yes—
232C42C2×C22⋊D6111yescharacteristic
233C42C23×D6111yescharacteristic
234C23#1451974511yescharacteristic
235Q8—44no—
236Q8—44no—
237Q8—82no—
238Q8—1228no—
239Q8—2444no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—1224no—
246C2×C4—1224no—
247C2×C4#1451974512yes—
248C2×C4#1451974511yescharacteristic
249C2×C4C2×D4×D6112yes—
250C2×C4C22×C22⋊D6111yescharacteristic
251C8—42no—
252C8—2444no—
253C22—22no—
254C22—21no—
255C22C2×#1451974511yescharacteristic
256C4—22no—
257C4—21no—
258C4—44no—
259C4—82no—
260C4—2444no—
261C4—2444no—
262C4#1683993412yes—
263C4#1684057814yes—
264C2—41no—
265C2#2145511312yes—
266C2C2×#1683993411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3139C2
socleC22×C61C2×#120
prime radicalC23×C61#120
hypercenter#3139D61
hypocenterC61#18732
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
3. #21444974C2semidirect
160. C61#18732semidirect

Maximal subgroups

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

subgroupindex
#214603142
#214569702
#214449742
#214444842
#214442412
C61×#31392
#1873261