Finite Group Databasebeta

← #29572256

Subgroups — #29572256

Order 15616.

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

normal subgroups only

This group has 682 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#21459706C211yesmaximal, characteristic
2#21446808C212yesmaximal, complemented
3#21444557C211yesmaximal, characteristic, complemented
4#21444259C212yesmaximal
5C61×#1526C211yesmaximal, characteristic, complemented
6#16840491—22no—
7#16840485—22no—
8#16840454—24no—
9#16840375—22no—
10#16840338C412yes—
11#16840317—22no—
12C2×#14519777C2211yescharacteristic
13#16840239C412yes—
14C4×#11758222C2211yescharacteristic
15#16839786C2212yes—
16#16839530—24no—
17#16839508—22no—
18#16839494—22no—
19#16839491—22no—
20C61×#533C2211yescharacteristic
21C4×M16×C61C2212yes—
22C61×#330—22no—
23C2×#11758222C2×C411yescharacteristic
24C22×Dic122C2×C412yes—
25C2×#11758194—22no—
26#14519777—22no—
27#14519777—22no—
28#14519777—21no—
29#14519777—21no—
30#14519772—28no—
31#14519768—24no—
32#14519768—28no—
33#14519765—22no—
34#14519761—24no—
35#14519757—24no—
36#14519753—24no—
37Q8×Dic61—24no—
38C4×Dic122—24no—
39C4×Dic122D412yes—
40#14519742—24no—
41C4×D244D412yes—
42C4×C22⋊D61—22no—
43#14519736C2×C411yescharacteristic
44#14519735—24no—
45#14519734—21no—
46#14519733—24no—
47C2×C4⋊Dic61D414yes—
48C42×D61—21no—
49C4⋊#92418—22no—
50C4⋊#92418—22no—
51C2×M16×C61—22no—
52C4⋊Q8×C61D412yes—
53#125×C61D412yes—
54C22⋊Q8×C61—22no—
55C4×Q8×C61—22no—
56#117×C61C2×C412yes—
57C2×C42×C61C2311yescharacteristic
58#97×C61—22no—
59C4×C8×C61—22no—
60C2×Q8×C61—22no—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D412yes—
63C2×C22⋊D61—21no—
64C22×Dic61C22⋊C414yes—
65#11758222—42no—
66C2×D244C22⋊C411yescharacteristic
67C2×C4×D61—21no—
68C2×C4×D61—22no—
69C2×Dic122—22no—
70C2×Dic122—22no—
71C2×Dic122—44no—
72C2×Dic122C22⋊C411yescharacteristic
73M16×C61—24no—
74M16×C61—42no—
75C2×C8×C61—42no—
76C4⋊C4×C61—24no—
77C4⋊C4×C61—22no—
78C4⋊C4×C61C22⋊C414yes—
79C22⋊C4×C61—22no—
80C42×C61—22no—
81C42×C61C2×D411yescharacteristic
82C42×C61C2×D412yes—
83C42×C61C2×D411yescharacteristic
84C22⋊Dic61—21no—
85#11758198—28no—
86#11758198—22no—
87C4⋊Dic61—28no—
88C4⋊Dic61—24no—
89C4⋊Dic61—24no—
90C4⋊Dic61C22⋊C412yes—
91#11758196—28no—
92#11758196—22no—
93C4×Dic61—28no—
94C4×Dic61—21no—
95C4×Dic61—22no—
96#11758194—44no—
97C2×#92418—42no—
98C23×C61C2×C22⋊C411yescharacteristic
99C22×D61—21no—
100Q8×C61—42no—
101Q8×C61—42no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—24no—
106C2×C4×C61C2×C22⋊C411yescharacteristic
107C2×C4×C61C2×C22⋊C411yescharacteristic
108C2×C4×C61#12012yes—
109C2×C4×C61#12012yes—
110C22⋊D61—42no—
111C2×Dic61—22no—
112C2×Dic61—21no—
113C2×Dic61—24no—
114C2×Dic61—44no—
115D244—22no—
116C4×D61—42no—
117C4×D61—42no—
118C4×D61—42no—
119Dic122—22no—
120Dic122—44no—
121Dic122—44no—
122C8×C61—44no—
123#92418—82no—
124#8474—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—21no—
127C22×C61#37911yescharacteristic
128C2×D61—42no—
129C2×D61—41no—
130C4×C61—22no—
131C4×C61—21no—
132C4×C61—44no—
133C4×C61#41312yes—
134C4×C61#45412yes—
135C4×C61#45612yes—
136Dic61—42no—
137Dic61—41no—
138Dic61—44no—
139#3509—611no—
140#1997—612no—
141#1561—611no—
142#1526D6111yescharacteristic, complemented
143#1489—612no—
144C2×C61—41no—
145C2×C61#178511yescharacteristic
146C2×C61#179312yes—
147D61—81no—
148#557—1222no—
149#556—1222no—
150#552—1224no—
151#541—1222no—
152#536—612no—
153#533—1222no—
154#533C2×D6111yescharacteristic
155C2×C4⋊Q8—611no—
156#519—612no—
157C4×#41—611no—
158#423—612no—
159C4×M16C2×D6112yes—
160#343—1224no—
161#332—1222no—
162#330—22no—
163Q8⋊C8—1222no—
164D4⋊C8—1222no—
165C61#847411yescharacteristic, complemented, Sylow
166C2×#41—611no—
167C22×Q8—612no—
168C2×M16—22no—
169C2×M16—1222no—
170C4⋊Q8—1222no—
171C4⋊Q8—1221no—
172C4⋊Q8—1222no—
173C4⋊Q8—1221no—
174C4⋊Q8D24412yes—
175#126—1228no—
176#125—1224no—
177#125—1228no—
178#125C22⋊D6112yes—
179#124—1222no—
180#123—1224no—
181C22⋊Q8—22no—
182C22⋊Q8—1224no—
183C22⋊Q8—1224no—
184C4×Q8—22no—
185C4×Q8—612no—
186C4×Q8—1224no—
187C4×Q8—1224no—
188C4×Q8—1224no—
189C4×D4—612no—
190C4×D4—1222no—
191#117—611no—
192#117—1224no—
193#117—1224no—
194#117—1221no—
195#117C4×D6112yes—
196C2×C4⋊C4—614no—
197C2×C42—1221no—
198C2×C42C22×D6111yescharacteristic
199C4⋊C8—1222no—
200C4⋊C8—1222no—
201#97—22no—
202C4×C8—22no—
203#41—2442no—
204C2×Q8—22no—
205C2×Q8—611no—
206C2×Q8—1222no—
207C2×Q8—1222no—
208C2×Q8—2444no—
209C2×D4—611no—
210C2×D4—1221no—
211C22×C4—614no—
212C22×C4—1221no—
213C22×C4—1222no—
214C22×C4C2×C4×D6111yescharacteristic
215C22×C4D4×D6112yes—
216M16—24no—
217M16—42no—
218M16—2444no—
219C2×C8—42no—
220C2×C8—2442no—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—612no—
224C4⋊C4—1228no—
225C4⋊C4—1228no—
226C4⋊C4—1224no—
227C4⋊C4—1222no—
228C4⋊C4—1224no—
229C4⋊C4#1175819814yes—
230C22⋊C4—22no—
231C22⋊C4—1228no—
232C22⋊C4—1221no—
233C22⋊C4—1222no—
234C42—22no—
235C42—1228no—
236C42—1221no—
237C42—1222no—
238C42C2×D24411yescharacteristic
239C42D4×D6112yes—
240C42C2×C22⋊D6111yescharacteristic
241C23—1221no—
242C23C22⋊C4×D6111yescharacteristic
243Q8—42no—
244Q8—42no—
245Q8—1222no—
246Q8—2444no—
247Q8—2444no—
248D4—1222no—
249D4—2442no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—1222no—
255C2×C4—1224no—
256C2×C4—1221no—
257C2×C4—2442no—
258C2×C4—2442no—
259C2×C4—2442no—
260C2×C4—2444no—
261C2×C4C22⋊C4×D6111yescharacteristic
262C2×C4C2×#1175819811yescharacteristic
263C2×C4#1451974512yes—
264C2×C4#1451974712yes—
265C8—44no—
266C8—4882no—
267C22—21no—
268C22—22no—
269C22—2441no—
270C22—2442no—
271C22#1683961711yescharacteristic
272C4—21no—
273C4—22no—
274C4—44no—
275C4—2441no—
276C4—2444no—
277C4—2442no—
278C4#1683974512yes—
279C4#1683993612yes—
280C4#1683994612yes—
281C2—41no—
282C2—4881no—
283C2#2144574611yescharacteristic
284C2#2144577912yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1526C2
socleC22×C61#379
prime radicalC2×#11758222C2×C4
cosocleC2×C42×C61C23
hypercenter#1526D61
hypocenterC61#8474
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #21446808C2semidirect
3. #21444557C2semidirect
5. C61×#1526C2semidirect
142. #1526D61semidirect
165. C61#8474semidirect

Maximal subgroups

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

subgroupindex
#214597062
#214468082
#214445572
#214442592
C61×#15262
#847461