Finite Group Databasebeta

← #29575733

Subgroups — #29575733

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454158C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic
3C2×#16839498C212yesmaximal
4#21443698C212yesmaximal
5C2×C4×C4⋊C4×C61C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7C2×C22⋊#92418C2212yes—
8#16839668C412yescomplemented
9#16839660C412yescomplemented
10#16839646—22no—
11#16839635—21no—
12#16839625—21no—
13#16839622C2211yescharacteristic
14#16839517—22no—
15#16839498—24no—
16C22×C4⋊C4×C61—21no—
17C22×C42×C61—21no—
18C22×C42×C61C2211yescharacteristic
19C4×C4⋊C4×C61—24no—
20C2×#95×C61C2212yes—
21C22×Dic122C2×C411yescharacteristic
22C22×#92418—22no—
23#14519756—22no—
24#14519756—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2×C412yes—
27C2×#11758196—22no—
28C2×#11758196—21no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—21no—
31C22⋊#92418—22no—
32C22⋊#92418—22no—
33#14519684—22no—
34#14519684—22no—
35#14519684—22no—
36#14519684—21no—
37C23×C4×C61—22no—
38C23×C4×C61D411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C23×C4×C61C2311yescharacteristic
41C2×C4⋊C4×C61—24no—
42C2×C4⋊C4×C61—28no—
43C2×C42×C61—22no—
44C2×C42×C61—22no—
45C2×C42×C61—24no—
46C2×C42×C61—42no—
47C2×C42×C61—42no—
48C2×C42×C61C2×C412yes—
49C2×C42×C61D412yes—
50#95×C61—22no—
51#95×C61D414yes—
52C24×C61C2×D411yescharacteristic
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—24no—
60C22×C4×C61—28no—
61C22×C4×C61—24no—
62C22×C4×C61—42no—
63C22×C4×C61—44no—
64C22×C4×C61—42no—
65C22×C4×C61—42no—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22⋊C412yes—
68C22×C4×C61SD1612yes—
69C22×C4×C61Q1612yes—
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D412yes—
73C22×C4×C61#4112yes—
74C22×Dic61—21no—
75C22×Dic61—22no—
76C22×Dic61—21no—
77C2×Dic122—22no—
78C2×Dic122—42no—
79C4⋊C4×C61—216no—
80C42×C61—24no—
81C42×C61—24no—
82C42×C61—48no—
83C22⋊Dic61—41no—
84C22⋊Dic61—41no—
85C22⋊Dic61—41no—
86C22⋊Dic61—41no—
87C4⋊Dic61—22no—
88C4⋊Dic61—42no—
89#11758196—44no—
90#11758196—42no—
91C2×#92418—44no—
92C2×#92418—42no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61C2×C22⋊C411yescharacteristic
96C23×C61#12011yescharacteristic
97C23×C61#12311yescharacteristic
98C23×C61C2×SD1611yescharacteristic
99C23×C61C2×Q1611yescharacteristic
100C23×C61AGL(1,ℤ/8)11yescharacteristic
101C23×C61#13711yescharacteristic
102C2×C4×C61—24no—
103C2×C4×C61—24no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—28no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—44no—
118C2×C4×C61—48no—
119C2×C4×C61—48no—
120C2×C4×C61—44no—
121C2×C4×C61#10314yes—
122C2×C4×C61#10414yes—
123C2×C4×C61C4×D412yes—
124C2×C4×C61#12112yes—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132Dic122—82no—
133#92418—82no—
134#9144—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—22no—
142C22×C61—22no—
143C22×C61#38611yescharacteristic
144C22×C61#41012yes—
145C22×C61C2×#10311yescharacteristic
146C22×C61#41711yescharacteristic
147C22×C61C2×#10412yes—
148C22×C61#45011yescharacteristic
149C22×C61#45111yescharacteristic
150C22×C61#48311yescharacteristic
151C22×C61#48411yescharacteristic
152C4×C61—24no—
153C4×C61—24no—
154C4×C61—22no—
155C4×C61—22no—
156C4×C61—48no—
157Dic61—81no—
158Dic61—81no—
159Dic61—82no—
160#2855—611no—
161C2×C4×C4⋊C4D6111yescharacteristic
162#1817—611no—
163C2×#328—612no—
164#1294—612no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61#150312yes—
170C2×C61#150912yes—
171C2×C61#187211yescharacteristic
172C2×C61#189611yescharacteristic
173C2×C61#189911yescharacteristic
174C2×C22⋊Q8—611no—
175C22×C4⋊C4—21no—
176C22×C42—21no—
177C22×C42C2×D6111yescharacteristic
178C2×C22⋊C8—612no—
179#391—612no—
180#389—612no—
181#387—1222no—
182#385—1221no—
183#381—1221no—
184#380—611no—
185C4×C4⋊C4—24no—
186C2×#95C2×D6112yes—
187#336—1222no—
188#328—1224no—
189C61#914411yescharacteristic, complemented, Sylow
190C22×Q8—611no—
191C23×C4—22no—
192C23×C4C22⋊D6111yescharacteristic
193C23×C4C22⋊D6111yescharacteristic
194C23×C4C22×D6111yescharacteristic
195C22×C8—1222no—
196C22⋊Q8—1222no—
197C22⋊Q8—1222no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—611no—
201C2×C4⋊C4—612no—
202C2×C4⋊C4—1222no—
203C2×C4⋊C4—1221no—
204C2×C22⋊C4—1221no—
205C2×C22⋊C4—1221no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C4×D6112yes—
212C2×C42D24412yes—
213C22⋊C8—1222no—
214C22⋊C8—1222no—
215#95—22no—
216#95—1222no—
217#95—1222no—
218#95—1222no—
219#95—1221no—
220#95C22⋊D6114yes—
221C24C2×C22⋊D6111yescharacteristic
222C2×Q8—1222no—
223C2×Q8—2442no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—1222no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4#1175819812yes—
241C22×C4#1175819812yes—
242C22×C4#1175820012yes—
243C22×C4#1175820212yes—
244C22×C4C2×C4×D6111yescharacteristic
245C22×C4C2×D24411yescharacteristic
246C22×C4#1175822212yes—
247C22×C4C2×C22⋊D6112yes—
248C2×C8—2444no—
249C2×C8—2442no—
250C4⋊C4—216no—
251C4⋊C4—1222no—
252C4⋊C4—2442no—
253C4⋊C4—2444no—
254C4⋊C4—2442no—
255C22⋊C4—2441no—
256C22⋊C4—2441no—
257C22⋊C4—2441no—
258C22⋊C4—2441no—
259C42—24no—
260C42—24no—
261C42—48no—
262C23—22no—
263C23—22no—
264C23C2×#1175819811yescharacteristic
265C23#1451974611yescharacteristic
266C23#1451976011yescharacteristic
267C23C2×#1175820011yescharacteristic
268C23C2×#1175820211yescharacteristic
269C23#1451980311yescharacteristic
270C23#1451980811yescharacteristic
271Q8—4882no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—28no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4#1451970014yes—
299C2×C4#1451970514yes—
300C2×C4C4×C22⋊D6112yes—
301C2×C4#1451975112yes—
302C8—4882no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22#1683964111yescharacteristic
312C22#1683973512yes—
313C22C2×#1451970011yescharacteristic
314C22#1683976111yescharacteristic
315C22C2×#1451970512yes—
316C22#1683991311yescharacteristic
317C22#1683991811yescharacteristic
318C22#1684009211yescharacteristic
319C22#1684009711yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—48no—
325C4—4881no—
326C4—4881no—
327C4—4882no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2144431712yes—
333C2#2144434212yes—
334C2#2144612611yescharacteristic
335C2#2144626211yescharacteristic
336C2#2144627711yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C4×C4⋊C4×C61C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×C4×C4⋊C4D61
hypocenterC61#9144
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #16839668C4semidirect
9. #16839660C4semidirect
189. C61#9144semidirect

Maximal subgroups

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

subgroupindex
#214541582
#214458872
C2×#168394982
#214436982
C2×C4×C4⋊C4×C612
#914461