Finite Group Databasebeta

← #29575013

Subgroups — #29575013

Order 15616.

800 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448437C211yesmaximal, characteristic
2#21445754C212yesmaximal
3C2×#16839488C211yesmaximal, characteristic
4#21443698C212yesmaximal
5C2×C61×#395C211yesmaximal, characteristic
6C2×#14519736—21no—
7C2×#14519701—24no—
8C2×C22⋊#92418C2212yes—
9#16839661—24no—
10#16839622—21no—
11#16839622C2212yes—
12#16839488—22no—
13C23×Q8×C61C2211yescharacteristic
14C22×C4⋊C4×C61—21no—
15C22×C4⋊C4×C61C2211yescharacteristic
16C61×#395—22no—
17C61×#395C414yescomplemented
18C2×#95×C61—21no—
19C2×#95×C61C2211yescharacteristic
20C22×#92418—22no—
21#14519736—42no—
22#14519736—42no—
23C2×C4⋊Dic61—22no—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61D414yes—
26C2×C22⋊Dic61—22no—
27C2×C4×Dic61—22no—
28#14519701—48no—
29C22⋊#92418—22no—
30C22⋊#92418—22no—
31#14519684—22no—
32#14519684—22no—
33C22×Q8×C61—21no—
34C22×Q8×C61—22no—
35C22×Q8×C61—22no—
36C22×Q8×C61—44no—
37C22×Q8×C61C2×C412yes—
38C23×C4×C61—21no—
39C23×C4×C61—21no—
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42C23×C4×C61C2311yescharacteristic
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61—24no—
46C2×C4⋊C4×C61—44no—
47C2×C4⋊C4×C61C2×C412yes—
48C2×C4⋊C4×C61D412yes—
49#95×C61—24no—
50#95×C61—21no—
51#95×C61C2×C412yes—
52C24×C61C2×D411yescharacteristic
53C2×Q8×C61—28no—
54C2×Q8×C61—24no—
55C2×Q8×C61—44no—
56C2×Q8×C61—44no—
57C2×Q8×C61—44no—
58C2×Q8×C61—48no—
59C2×Q8×C61—42no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—24no—
65C22×C4×C61—24no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—41no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61—41no—
77C22×C4×C61—42no—
78C22×C4×C61C22⋊C412yes—
79C22×C4×C61C22⋊C412yes—
80C22×C4×C61SD1612yes—
81C22×C4×C61Q1612yes—
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61#4112yes—
86C22×Dic61—24no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61—48no—
90C22⋊Dic61—42no—
91C22⋊Dic61—42no—
92C4⋊Dic61—44no—
93C4⋊Dic61—44no—
94C4⋊Dic61—82no—
95C4×Dic61—44no—
96C2×#92418—44no—
97C2×#92418—42no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61C2×C22⋊C411yescharacteristic
101C23×C61#12011yescharacteristic
102C23×C61#12311yescharacteristic
103C23×C61C2×SD1611yescharacteristic
104C23×C61C2×Q1611yescharacteristic
105C23×C61AGL(1,ℤ/8)11yescharacteristic
106C23×C61#13711yescharacteristic
107Q8×C61—28no—
108Q8×C61—48no—
109Q8×C61—44no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—24no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—24no—
118C2×C4×C61—24no—
119C2×C4×C61—22no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—44no—
125C2×C4×C61—44no—
126C2×C4×C61—42no—
127C2×C4×C61—44no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—44no—
132C2×C4×C61—42no—
133C2×C4×C61—81no—
134C2×C4×C61#10314yescomplemented
135C2×C4×C61C4×D412yes—
136C2×C4×C61#12112yes—
137C2×Dic61—44no—
138C2×Dic61—44no—
139C2×Dic61—44no—
140#92418—82no—
141#9015—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#38611yescharacteristic
151C22×C61C2×#10311yescharacteristic
152C22×C61#41711yescharacteristic
153C22×C61#45011yescharacteristic
154C22×C61#45111yescharacteristic
155C22×C61#45612yes—
156C22×C61B(4,𝔽2)12yes—
157C22×C61#48311yescharacteristic
158C22×C61#48411yescharacteristic
159C4×C61—24no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61—44no—
163C4×C61—44no—
164C4×C61—42no—
165Dic61—84no—
166C2×#395D6111yescharacteristic
167#2316—611no—
168#1787—612no—
169C2×#324—611no—
170#1294—612no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61#160412yes—
176C2×C61#160512yes—
177C2×C61#187211yescharacteristic
178C2×C61#188611yescharacteristic
179C2×C61#188911yescharacteristic
180C23×Q8C2×D6111yescharacteristic
181C2×#117—1221no—
182C22×C4⋊C4—21no—
183C22×C4⋊C4C2×D6111yescharacteristic
184C2×#103—1224no—
185C2×C22⋊C8—612no—
186#395—22no—
187#395Dic6114yescomplemented
188#389—1224no—
189#380—612no—
190#380—1221no—
191C2×#95—21no—
192C2×#95C2×D6111yescharacteristic
193#324—1222no—
194C61#901511yescharacteristic, complemented, Sylow
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—44no—
199C22×Q8C2×Dic6112yes—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4C22⋊D6111yescharacteristic
203C23×C4C22⋊D6111yescharacteristic
204C23×C4C22×D6111yescharacteristic
205C22×C8—1222no—
206#117—2442no—
207#117—2442no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—614no—
213C2×C4⋊C4—1222no—
214C2×C4⋊C4—1222no—
215C2×C4⋊C4C2×Dic6112yes—
216C2×C4⋊C4C22⋊D6112yes—
217C2×C22⋊C4—1222no—
218C2×C42—1222no—
219#103—2448no—
220C22⋊C8—1222no—
221C22⋊C8—1222no—
222#95—24no—
223#95—21no—
224#95—1222no—
225#95—1222no—
226#95C2×Dic6112yes—
227C24C2×C22⋊D6111yescharacteristic
228C2×Q8—28no—
229C2×Q8—24no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—48no—
234C2×Q8—42no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—1224no—
254C22×C4#1175820112yes—
255C22×C4#1175820212yes—
256C22×C4C22⋊Dic6112yes—
257C22×C4C22⋊Dic6112yes—
258C22×C4D4×D6112yes—
259C22×C4#1175822412yes—
260C22×C4C22×Dic6111yescharacteristic
261C22×C4C2×C22⋊D6111yescharacteristic
262C2×C8—2444no—
263C2×C8—2442no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—48no—
267C4⋊C4—4882no—
268C4⋊C4—2444no—
269C4⋊C4—2444no—
270C22⋊C4—2442no—
271C22⋊C4—2442no—
272C42—2444no—
273C23—22no—
274C23—22no—
275C23C2×C22⋊Dic6111yescharacteristic
276C23#1451974511yescharacteristic
277C23#1451975811yescharacteristic
278C23C2×#1175820111yescharacteristic
279C23C2×#1175820211yescharacteristic
280C23#1451980211yescharacteristic
281C23#1451980711yescharacteristic
282Q8—28no—
283Q8—48no—
284Q8—44no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—81no—
309C2×C4—2444no—
310C2×C4—2444no—
311C2×C4—2444no—
312C2×C4#1451970114yescomplemented
313C2×C4D4×Dic6112yes—
314C2×C4#1451974912yes—
315C8—4882no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#1683964211yescharacteristic
325C22C2×#1451970111yescharacteristic
326C22#1683976311yescharacteristic
327C22#1683991211yescharacteristic
328C22#1683991711yescharacteristic
329C22#1683994412yes—
330C22#1683994812yes—
331C22#1684009011yescharacteristic
332C22#1684009511yescharacteristic
333C4—22no—
334C4—22no—
335C4—24no—
336C4—44no—
337C4—44no—
338C4—42no—
339C4—4884no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#2144478712yes—
345C2#2144479212yes—
346C2#2144612711yescharacteristic
347C2#2144621311yescharacteristic
348C2#2144622811yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#395C2
socleC23×C61#120
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#395D61
hypocenterC61#9015
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
17. C61×#395C4semidirect
134. C2×C4×C61#103semidirect
187. #395Dic61semidirect
194. C61#9015semidirect
312. C2×C4#14519701semidirect

Maximal subgroups

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

subgroupindex
#214484372
#214457542
C2×#168394882
#214436982
C2×C61×#3952
#901561