Finite Group Databasebeta

← #29678293

Subgroups — #29678293

Order 15616.

1034 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839816C211yesmaximal, characteristic
2C22×#14519687C211yesmaximal, characteristic
3C4×#14519733C211yesmaximal, characteristic
4#21445695C212yesmaximal, complemented
5#21445692C212yesmaximal, complemented
6#21445597C211yesmaximal, characteristic
7#21445593C211yesmaximal, characteristic
8C8×C22⋊Dic61C211yesmaximal, characteristic
9#21445566C212yesmaximal, complemented
10#97×Dic61C212yesmaximal, complemented
11C61×#1752C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×#14519733C412yes—
14C22×C4×Dic61C2211yescharacteristic
15#16839816C418yescomplemented
16C2×#14519706C2212yes—
17C2×C22⋊#92418C2211yescharacteristic
18C2×#14519688C2212yes—
19C2×#14519687C418yescomplemented
20C2×#14519687C2211yescharacteristic
21C2×#14519687C2211yescharacteristic
22C2×#14519687C2212yes—
23C2×#14519687C2211yescharacteristic
24C2×#14519687C2211yescharacteristic
25C2×C8×Dic61C2212yes—
26C2×C8×Dic61C2212yes—
27C4×#11758196C2212yes—
28#16839613C412yes—
29C22⋊C4×Dic61C412yes—
30C4×C22⋊Dic61C2211yescharacteristic
31#16839607C412yes—
32C42×Dic61C2212yes—
33#16839517C2212yes—
34#16839516C2212yes—
35#16839516C2212yes—
36C23×C8×C61C2211yescharacteristic
37C2×C22⋊C8×C61C2211yescharacteristic
38C2×#97×C61C2212yes—
39C4×C22⋊C4×C61C2211yescharacteristic
40C61×#336C2212yes—
41C23×Dic61C2×C411yescharacteristic
42C23×Dic61C2×C411yescharacteristic
43C22×#92418—24no—
44C22×#92418C2×C414yes—
45C22×#92418C2312yes—
46C22×#92418C2311yescharacteristic
47C22×#92418C2311yescharacteristic
48#14519733—28no—
49C2×#11758196C2×C414yes—
50C2×C22⋊Dic61C2×C412yes—
51C2×C4×Dic61C2×C414yes—
52C2×C4×Dic61C2×C414yes—
53C2×C4×Dic61C2×C411yescharacteristic
54C2×C4×Dic61C2×C411yescharacteristic
55C2×C4×Dic61C2×C414yes—
56C2×C4×Dic61C2×C414yes—
57C2×C4×Dic61C2312yes—
58C2×C4×Dic61C2311yescharacteristic
59C2×C4×Dic61C2311yescharacteristic
60#14519706C2×C418yes—
61C22⋊#92418C2×C414yes—
62#14519688—24no—
63#14519687C2×C414yes—
64#14519687C2×C414yes—
65#14519687C2×C414yes—
66#14519687C2×C414yes—
67#14519687C2×C418yes—
68C8×Dic61—24no—
69C8×Dic61—24no—
70#14519684C2×C412yes—
71#14519684C2×C414yes—
72C23×C4×C61C2311yescharacteristic
73C22×C8×C61—24no—
74C22×C8×C61C2×C414yes—
75C22×C8×C61C2312yes—
76C22×C8×C61C2311yescharacteristic
77C22×C8×C61C2311yescharacteristic
78C2×C22⋊C4×C61C2×C412yes—
79C2×C42×C61C2312yes—
80C22⋊C8×C61C2×C414yes—
81#97×C61—24no—
82#95×C61C2×C412yes—
83C24×C61C22×C411yescharacteristic
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61C22×C414yes—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C22×C412yes—
89C22×C4×C61C22×C412yes—
90C22×C4×C61C2411yescharacteristic
91C22×Dic61—24no—
92C22×Dic61—42no—
93C22×Dic61C4214yes—
94C22×Dic61C4214yes—
95C22×Dic61C22×C411yescharacteristic
96C22×Dic61C22×C411yescharacteristic
97C22×Dic61C22×C411yescharacteristic
98C22×Dic61C22×C411yescharacteristic
99C22×Dic61C22×C414yes—
100C2×C8×C61—22no—
101C2×C8×C61—22no—
102C2×C8×C61—22no—
103C2×C8×C61—28no—
104C2×C8×C61C22×C414yes—
105C2×C8×C61C22×C412yes—
106C2×C8×C61C22×C412yes—
107C2×C8×C61#4114yes—
108C22⋊C4×C61—24no—
109C42×C61—22no—
110C42×C61—22no—
111C22⋊Dic61M1618yes—
112#11758196—28no—
113C4×Dic61—22no—
114C4×Dic61—22no—
115C4×Dic61—28no—
116C4×Dic61—44no—
117C4×Dic61C4214yes—
118C4×Dic61C4214yes—
119C4×Dic61C22×C412yes—
120C4×Dic61C22×C412yes—
121C4×Dic61C22×C412yes—
122C4×Dic61C22×C412yes—
123C2×#92418—22no—
124C2×#92418—22no—
125C2×#92418—22no—
126C2×#92418—28no—
127C2×#92418C22×C412yes—
128C2×#92418C22×C414yes—
129C2×#92418C22×C412yes—
130C2×#92418#4114yes—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61C2×C4212yes—
134C23×C61C2×M1614yes—
135C23×C61C23×C411yescharacteristic
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—24no—
140C2×C4×C61—24no—
141C2×C4×C61—24no—
142C2×C4×C61—24no—
143C2×C4×C61C2×C4212yes—
144C2×C4×C61C23×C411yescharacteristic
145C2×C4×C61C23×C411yescharacteristic
146C2×C4×C61C2×#4112yes—
147C2×C4×C61C2×#4112yes—
148C2×Dic61—24no—
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—24no—
153C2×Dic61—22no—
154C2×Dic61—28no—
155C2×Dic61—44no—
156C2×Dic61C2×C4212yes—
157C2×Dic61C2×C4212yes—
158C2×Dic61C2×C4212yes—
159C2×Dic61C2×C4212yes—
160C2×Dic61C2×M1618yes—
161C8×C61—22no—
162C8×C61—24no—
163C8×C61#11714yescomplemented
164#92418—22no—
165#92418—24no—
166#92418#11714yescomplemented
167#20479—611nomaximal, Sylow
168C22×C61—22no—
169C22×C61—28no—
170C22×C61—22no—
171C22×C61C4×M1614yes—
172C22×C61C22×C4211yescharacteristic
173C22×C61C4×#4112yes—
174C22×C61C22×M1612yes—
175C22×C61#56812yes—
176C4×C61—22no—
177C4×C61—44no—
178C4×C61C2×#11711yescharacteristic
179C4×C61C2×#11711yescharacteristic
180C4×C61C4×#4112yes—
181Dic61—22no—
182Dic61—22no—
183Dic61—22no—
184Dic61—22no—
185Dic61—28no—
186C2×#432—611no—
187C22×#97—611no—
188C4×#117—611no—
189#1771—612no—
190#1771—612no—
191#1752—611no—
192#1752D6111yescharacteristic
193C8×C22⋊C4—611no—
194C8×C22⋊C4—611no—
195#1745—612no—
196C4×#97—612no—
197C2×C61—24no—
198C2×C61C4×#11711yescharacteristic
199C2×C61C2×C4×M1611yescharacteristic
200C2×C61#287311yescharacteristic
201C2×C61#297014yes—
202C23×C8—611no—
203C23×C8C2×D6111yescharacteristic
204C2×#117—612no—
205C22×C42—611no—
206#432—618no—
207C2×C4⋊C8—612no—
208C2×C22⋊C8—611no—
209C2×C22⋊C8C2×D6111yescharacteristic
210C2×#97—618no—
211C2×#97—611no—
212C2×#97—611no—
213C2×#97—611no—
214C2×#97—611no—
215C2×#97—612no—
216C2×#97—612no—
217C2×#97C2×D6112yes—
218C2×C4×C8—612no—
219C2×C4×C8—612no—
220C4×C4⋊C4—612no—
221C4×C4⋊C4—612no—
222C4×C22⋊C4—611no—
223C4×C22⋊C4—612no—
224C4×C22⋊C4C2×D6111yescharacteristic
225#376—612no—
226C43—612no—
227#336—612no—
228#336—612no—
229#336—612no—
230#336C2×D6112yes—
231C61#2047911yescharacteristic, complemented, Sylow
232C23×C4—611no—
233C23×C4—611no—
234C23×C4C22×D6111yescharacteristic
235C22×C8—24no—
236C22×C8—611no—
237C22×C8—614no—
238C22×C8—611no—
239C22×C8—612no—
240C22×C8—1224no—
241C22×C8C4×D6114yes—
242C22×C8C22×D6111yescharacteristic
243C22×C8C22×D6112yes—
244C22×C8C22×D6111yescharacteristic
245#117—1228no—
246C2×C4⋊C4—614no—
247C2×C22⋊C4—612no—
248C2×C22⋊C4C4×D6112yes—
249C2×C42—611no—
250C2×C42—611no—
251C2×C42—614no—
252C2×C42—614no—
253C2×C42—611no—
254C2×C42—611no—
255C2×C42—614no—
256C2×C42—612no—
257C2×C42—614no—
258C2×C42C22×D6112yes—
259C4⋊C8—618no—
260C22⋊C8—614no—
261C22⋊C8C4×D6114yes—
262#97—24no—
263#97—614no—
264#97—614no—
265#97—614no—
266#97—614no—
267#97—618no—
268#97—1224no—
269C4×C8—1224no—
270C4×C8—1224no—
271#95—614no—
272#95—612no—
273#95C4×D6112yes—
274C24C2×C4×D6111yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—611no—
278C22×C4—614no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—614no—
283C22×C4—614no—
284C22×C4—1224no—
285C22×C4—2442no—
286C22×C4C2×C4×D6114yes—
287C22×C4C2×C4×D6112yes—
288C22×C4C2×C4×D6112yes—
289C22×C4C2×C4×D6111yescharacteristic
290C22×C4C23×D6111yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—28no—
295C2×C8—612no—
296C2×C8—614no—
297C2×C8—612no—
298C2×C8—614no—
299C2×C8—1228no—
300C2×C8—1222no—
301C2×C8—1222no—
302C2×C8—1222no—
303C2×C8C2×C4×D6112yes—
304C2×C8C2×C4×D6114yes—
305C2×C8C2×C4×D6112yes—
306C2×C8#1175822414yes—
307C4⋊C4—1228no—
308C22⋊C4—24no—
309C22⋊C4—618no—
310C42—22no—
311C42—22no—
312C42—612no—
313C42—612no—
314C42—612no—
315C42—612no—
316C42—614no—
317C42—614no—
318C42—1228no—
319C42—1222no—
320C42—1222no—
321C42—2444no—
322C23—22no—
323C23—22no—
324C23C42×D6112yes—
325C23M16×D6114yes—
326C23C22×C4×D6111yescharacteristic
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—612no—
335C2×C4—612no—
336C2×C4—612no—
337C2×C4—612no—
338C2×C4—618no—
339C2×C4—1228no—
340C2×C4—1224no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—1224no—
345C2×C4—1222no—
346C2×C4—2444no—
347C2×C4C42×D6112yes—
348C2×C4C22×C4×D6111yescharacteristic
349C2×C4C22×C4×D6111yescharacteristic
350C2×C4#41×D6112yes—
351C2×C4C2×#1175822412yes—
352C8—22no—
353C8—24no—
354C8—614no—
355C8—1222no—
356C8—1224no—
357C8#1451973314yescomplemented
358C22—22no—
359C22—22no—
360C22—28no—
361C22#1683971414yes—
362C22C2×C42×D6111yescharacteristic
363C22C4×#1175822412yes—
364C22C2×M16×D6112yes—
365C22#1684052712yes—
366C4—22no—
367C4—44no—
368C4—1222no—
369C4—1222no—
370C4—1222no—
371C4—1222no—
372C4—1228no—
373C4#117×D6111yescharacteristic
374C4C2×#1451973311yescharacteristic
375C4C4×#1175822412yes—
376C2—24no—
377C2C4×#1451973311yescharacteristic
378C2C2×#1683971411yescharacteristic
379C2#2145423511yescharacteristic
380C2#2145482914yes—
subgroupG/H
center, FrattiniC22×C4C23×D61
commutatorC22×C61C22×C42
FittingC61×#1752C2
socleC23×C61C23×C4
prime radicalC24×C61C22×C4
cosocleC22×C4×C61C24
hypercenter#1752D61
hypocenterC61#20479
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
4. #21445695C2semidirect
5. #21445692C2semidirect
9. #21445566C2semidirect
10. #97×Dic61C2semidirect
15. #16839816C4semidirect
19. C2×#14519687C4semidirect
163. C8×C61#117semidirect
166. #92418#117semidirect
231. C61#20479semidirect
357. C8#14519733semidirect

Maximal subgroups

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

subgroupindex
C2×#168398162
C22×#145196872
C4×#145197332
#214456952
#214456922
#214455972
#214455932
C8×C22⋊Dic612
#214455662
#97×Dic612
C61×#17522
#2047961