Finite Group Databasebeta

← #29578858

Subgroups — #29578858

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839660C211yesmaximal, characteristic
2#21445581C211yesmaximal, characteristic
3C2×#16839487C214yesmaximal
4C61×#1797C211yesmaximal, characteristic
5C22×C4⋊Dic61C411yescharacteristic
6C22×C4⋊Dic61C411yescharacteristic
7C22×#11758196C2212yes—
8C2×C22⋊#92418C2212yes—
9C2×#14519688—21no—
10#16839660—21no—
11#16839660—21no—
12#16839660—21no—
13#16839660—21no—
14C2×#14519684C412yes—
15#16839517—21no—
16#16839487—28no—
17C22×C42×C61C2211yescharacteristic
18C2×C4⋊C8×C61—21no—
19C2×C22⋊C8×C61C2212yes—
20C61×#336—21no—
21C23×Dic61C2×C412yes—
22C23×Dic61C2×C412yes—
23C22×#92418—22no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—24no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—42no—
30C2×#11758196—24no—
31C2×#11758196—22no—
32C2×#11758196—22no—
33C2×#11758196—24no—
34C22⋊#92418—22no—
35C22⋊#92418D414yes—
36#14519688—42no—
37#14519684—22no—
38#14519684—22no—
39C23×C4×C61C2×C411yescharacteristic
40C23×C4×C61C2×C411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C22×C8×C61—22no—
43C2×C42×C61—42no—
44C2×C42×C61D411yescharacteristic
45C2×C42×C61D411yescharacteristic
46C2×C42×C61D411yescharacteristic
47C2×C42×C61D411yescharacteristic
48C4⋊C8×C61—42no—
49C22⋊C8×C61—24no—
50C24×C61C22×C411yescharacteristic
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—41no—
57C22×C4×C61—41no—
58C22×C4×C61—41no—
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C22×C4×C61C2×D412yes—
66C22×C4×C61#4112yes—
67C22×Dic61—22no—
68C22×Dic61—24no—
69C22×Dic61—22no—
70C22×Dic61—24no—
71C22×Dic61—22no—
72C22×Dic61—24no—
73C22×Dic61—42no—
74C22×Dic61—42no—
75C22×Dic61M1614yes—
76C2×C8×C61—42no—
77C2×C8×C61—42no—
78C2×C8×C61—42no—
79C42×C61—22no—
80C42×C61—22no—
81C42×C61—41no—
82C42×C61—41no—
83C4⋊Dic61—24no—
84C4⋊Dic61—24no—
85C4⋊Dic61—42no—
86C4⋊Dic61—42no—
87#11758196—44no—
88#11758196—42no—
89#11758196—42no—
90C2×#92418—24no—
91C2×#92418—42no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61C2×C22⋊C411yescharacteristic
98C23×C61C4×D412yes—
99C23×C61C2×M1612yes—
100C23×C61#13112yes—
101C2×C4×C61—28no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—44no—
118C2×C4×C61#9912yes—
119C2×C4×C61#10112yes—
120C2×C4×C61#10112yes—
121C2×C4×C61#12112yes—
122C2×C4×C61#12411yescharacteristic
123C2×C4×C61#12711yescharacteristic
124C2×Dic61—24no—
125C2×Dic61—28no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—44no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—84no—
133C8×C61—82no—
134#92418—44no—
135#9654—611nomaximal, Sylow
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—22no—
142C22×C61#39011yescharacteristic
143C22×C61#40711yescharacteristic
144C22×C61#40811yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C22×C61#41011yescharacteristic
147C22×C61C2×#10111yescharacteristic
148C22×C61C2×#10111yescharacteristic
149C22×C61#43514yes—
150C4×C61—22no—
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—22no—
154C4×C61—22no—
155C4×C61—22no—
156C4×C61—41no—
157Dic61—44no—
158Dic61—82no—
159C2×#389—611no—
160#1797D6111yescharacteristic
161#1749—611no—
162C2×#324—614no—
163C2×C61—24no—
164C2×C61#146311yescharacteristic
165C2×C61#146711yescharacteristic
166C2×C61#147011yescharacteristic
167C2×C61#147211yescharacteristic
168C2×C61#195511yescharacteristic
169C2×C61#196411yescharacteristic
170C2×C61#196611yescharacteristic
171C22×C4⋊C4—611no—
172C22×C4⋊C4—612no—
173C22×C4⋊C4—611no—
174C22×C42C2×D6111yescharacteristic
175C2×C4⋊C8—21no—
176C2×C22⋊C8—612no—
177C2×C22⋊C8C2×D6112yes—
178C2×#97—1221no—
179#389—1221no—
180#389—1221no—
181#389—1221no—
182#389—1221no—
183C2×#95—612no—
184#336—21no—
185#336—1221no—
186#324—1228no—
187C61#965411yescharacteristic, complemented, Sylow
188C23×C4—612no—
189C23×C4—612no—
190C23×C4C4×D6111yescharacteristic
191C23×C4C4×D6111yescharacteristic
192C23×C4C22×D6111yescharacteristic
193C22×C8—22no—
194C22×C8—1222no—
195C2×C4⋊C4—1224no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—1222no—
201C2×C4⋊C4—1224no—
202C2×C4⋊C4—1224no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—2442no—
205C2×C42—42no—
206C2×C42D24411yescharacteristic
207C2×C42D24411yescharacteristic
208C2×C42C22⋊D6111yescharacteristic
209C2×C42C22⋊D6111yescharacteristic
210C4⋊C8—42no—
211C22⋊C8—24no—
212C22⋊C8—614no—
213C22⋊C8—1222no—
214#97—2442no—
215#95—1222no—
216#95—1222no—
217C24C2×C4×D6111yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—614no—
227C22×C4—1222no—
228C22×C4—1224no—
229C22×C4—1222no—
230C22×C4—1224no—
231C22×C4—1224no—
232C22×C4—1222no—
233C22×C4—2442no—
234C22×C4—2442no—
235C22×C4#1175819811yescharacteristic
236C22×C4#1175819811yescharacteristic
237C22×C4#1175819811yescharacteristic
238C22×C4#1175819811yescharacteristic
239C22×C4C2×D24411yescharacteristic
240C22×C4D4×D6112yes—
241C22×C4#1175822612yes—
242C22×C4C2×C22⋊D6111yescharacteristic
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—1224no—
247C2×C8—2442no—
248C2×C8—2442no—
249C4⋊C4—1224no—
250C4⋊C4—1224no—
251C4⋊C4—2442no—
252C4⋊C4—2442no—
253C4⋊C4—2444no—
254C4⋊C4—2442no—
255C4⋊C4—2442no—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23C2×#1175819811yescharacteristic
265C23#1451973912yes—
266C23M16×D6112yes—
267C23#1451978312yes—
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—1224no—
286C2×C4—1228no—
287C2×C4—4884no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2444no—
293C2×C4—2442no—
294C2×C4#1451969112yes—
295C2×C4#1451969512yes—
296C2×C4#1451969512yes—
297C2×C4#1451975212yes—
298C2×C4#1451976611yescharacteristic
299C2×C4#1451977311yescharacteristic
300C8—82no—
301C8—2444no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22#1683966311yescharacteristic
309C22#1683972511yescharacteristic
310C22#1683972811yescharacteristic
311C22C2×#1451969111yescharacteristic
312C22#1683973511yescharacteristic
313C22C2×#1451969511yescharacteristic
314C22C2×#1451969511yescharacteristic
315C22#1683983214yes—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—21no—
320C4—22no—
321C4—21no—
322C4—41no—
323C4—4882no—
324C4—2444no—
325C2—24no—
326C2#2144414811yescharacteristic
327C2#2144416211yescharacteristic
328C2#2144417411yescharacteristic
329C2#2144418211yescharacteristic
330C2#2144661611yescharacteristic
331C2#2144665411yescharacteristic
332C2#2144666211yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutator, prime radicalC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1797C2
socleC23×C61C2×C22⋊C4
cosocleC23×C4×C61C23
hypercenter#1797D61
hypocenterC61#9654
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
187. C61#9654semidirect

Maximal subgroups

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

subgroupindex
C2×#168396602
#214455812
C2×#168394872
C61×#17972
#965461