Finite Group Databasebeta

← #30614580

Subgroups — #30614580

Order 15616.

1314 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840733C212yesmaximal
2#21461560C218yesmaximal, complemented
3#21461545C212yesmaximal, complemented
4C4×C22⋊#92420C212yesmaximal, complemented
5C2×#16839609C211yesmaximal, characteristic
6C22×#11758213C2214yes—
7#16840733C4116yescomplemented
8C2×#14519834C2214yes—
9C2×C22⋊#92420C2214yes—
10C2×#14519832C2214yescomplemented
11C2×C4×#92420C2214yescomplemented
12#16840628C2214yescomplemented
13#16840628C2214yescomplemented
14C23×C4×D61C2211yescharacteristic
15C22×#11758198C411yescharacteristic
16C22×#11758198C411yescharacteristic
17C22×C4×Dic61C2211yescharacteristic
18C22×C4×Dic61C2211yescharacteristic
19#16839609C414yes—
20#16839609C2214yes—
21C2×#14519684C411yescharacteristic
22C2×#95×C61C411yescharacteristic
23C2×#11758213—44no—
24C2×#11758213—44no—
25C2×#11758213C2×C4116yescomplemented
26C22×#92420C2318yes—
27#14519834C2×C4116yescomplemented
28C22⋊#92420C2×C4116yes—
29#14519832—44no—
30C4×#92420—44no—
31C24×D61C2×C411yescharacteristic
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61C2311yescharacteristic
35C22×C4×D61—42no—
36C22×C4×D61—42no—
37C22×C4×D61C2×C412yes—
38C22×C4×D61C2×C418yes—
39C22×C4×D61C2312yes—
40C2×#11758198—24no—
41C2×#11758198—24no—
42C2×#11758198C2×C414yes—
43C2×#11758198C2×C414yes—
44C2×C4×Dic61—42no—
45C2×C4×Dic61—42no—
46C2×C4×Dic61C2×C412yes—
47C2×C4×Dic61C2×C412yes—
48C2×C4×Dic61C2312yes—
49C2×C4×Dic61C2312yes—
50#14519684C2×C414yes—
51C23×C4×C61C2×C411yescharacteristic
52C23×C4×C61C2×C411yescharacteristic
53C23×C4×C61C2×C411yescharacteristic
54#95×C61C2×C414yes—
55C24×C61C22×C411yescharacteristic
56C23×D61—22no—
57C23×D61—22no—
58C23×D61—41no—
59C23×D61—41no—
60C23×D61—41no—
61C23×D61—41no—
62C23×D61C4214yes—
63C23×D61C22×C412yes—
64C22×C4×C61—22no—
65C22×C4×C61—41no—
66C22×C4×C61—41no—
67C22×C4×C61—41no—
68C22×C4×C61—41no—
69C22×C4×C61—41no—
70C22×C4×C61—41no—
71C22×C4×C61C4214yes—
72C22×C4×C61M1614yescomplemented
73C22×C4×C61C22×C412yes—
74C22×C4×C61C22×C412yes—
75C22×C4×C61C22×C412yes—
76C22×Dic61—22no—
77C22×Dic61—22no—
78C22×Dic61—22no—
79C22×Dic61—41no—
80C22×Dic61—41no—
81C22×Dic61—41no—
82C22×Dic61—41no—
83C22×Dic61M1614yescomplemented
84C22×Dic61C22×C412yes—
85C22×Dic61C22×C411yescharacteristic
86C22×Dic61C22×C414yes—
87C22×Dic61C22×C412yes—
88C22×Dic61C2411yescharacteristic
89C2×C4×D61—24no—
90C2×C4×D61—24no—
91C2×C4×D61—416no—
92C2×C4×D61—42no—
93C2×C4×D61—42no—
94C2×C4×D61C4218yes—
95C2×C4×D61C22×C414yes—
96C2×C4×D61C22×C414yes—
97C2×#92420—28no—
98C2×#92420—28no—
99C2×#92420C22×C4116yes—
100#11758213—416no—
101#11758198—28no—
102#11758198—28no—
103C4×Dic61—24no—
104C4×Dic61—24no—
105C4×Dic61—42no—
106C4×Dic61—42no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61C2×C4212yes—
112C23×C61C2×M1612yes—
113C23×C61C2×M1612yes—
114C23×C61C23×C411yescharacteristic
115C22×D61—22no—
116C22×D61—22no—
117C22×D61—22no—
118C22×D61—22no—
119C22×D61—22no—
120C22×D61—22no—
121C22×D61—28no—
122C22×D61—44no—
123C22×D61—44no—
124C22×D61—44no—
125C22×D61—44no—
126C22×D61—44no—
127C22×D61—44no—
128C22×D61C2×C4214yes—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—44no—
137C2×C4×C61—44no—
138C2×C4×C61—44no—
139C2×C4×C61C2×C4214yes—
140C2×C4×C61C2×M1614yes—
141C2×Dic61—22no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—22no—
145C2×Dic61—28no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—44no—
150C2×Dic61—44no—
151C2×Dic61C2×C4212yes—
152C2×Dic61C2×M1614yes—
153C2×Dic61C23×C412yes—
154C2×Dic61H(5,𝔽2)12yes—
155C2×Dic61H(5,𝔽2)12yes—
156C4×D61—28no—
157C4×D61—28no—
158C4×D61—48no—
159#92420—216no—
160#20524—611nomaximal, Sylow
161C22×C61—22no—
162C22×C61—22no—
163C22×C61—22no—
164C22×C61—22no—
165C22×C61—22no—
166C22×C61—22no—
167C22×C61C4×M1614yes—
168C22×C61C22×C4211yescharacteristic
169C22×C61#51811yescharacteristic
170C22×C61#51811yescharacteristic
171C22×C61C22×M1611yescharacteristic
172C22×C61C22×M1611yescharacteristic
173C22×C61#56811yescharacteristic
174C22×C61#56811yescharacteristic
175C2×D61—24no—
176C2×D61—24no—
177C2×D61—24no—
178C2×D61—416no—
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—42no—
183C4×C61—24no—
184C4×C61—24no—
185C4×C61—42no—
186Dic61—24no—
187Dic61—22no—
188Dic61—42no—
189Dic61#51814yes—
190C2×#407—612no—
191C2×C4×C22⋊C4—611no—
192#1753—618no—
193#1749—612no—
194C4×C22⋊C8—612no—
195C2×C61—24no—
196C2×C61#227311yescharacteristic
197C2×C61C2×C4×M1611yescharacteristic
198C2×C61#287311yescharacteristic
199C2×C61#297311yescharacteristic
200C2×C61#297311yescharacteristic
201C2×C61#297511yescharacteristic
202C2×C61#297511yescharacteristic
203D61—44no—
204C24×C4—611no—
205C22×M16—614no—
206C22×C22⋊C4—611no—
207C22×C22⋊C4—611no—
208C22×C42—611no—
209C22×C42—611no—
210#407—6116no—
211C2×C22⋊C8—614no—
212C2×C22⋊C8—614no—
213C2×#97—614no—
214C2×C4×C8—614no—
215C4×C22⋊C4—614no—
216C4×C22⋊C4—614no—
217C2×#95—611no—
218C2×#95#692311yescharacteristic
219#336—614no—
220#336—614no—
221C61#2052411yescharacteristic, complemented, Sylow
222C25—611no—
223C23×C4—611no—
224C23×C4—611no—
225C23×C4—612no—
226C23×C4—618no—
227C23×C4—612no—
228C23×C4—611no—
229C23×C4—2442no—
230C23×C4—2442no—
231C23×C4C2×#692311yescharacteristic
232C23×C4C2×#692311yescharacteristic
233C23×C4C2×#692311yescharacteristic
234C2×M16—6116no—
235C2×M16—2444no—
236C2×M16—2444no—
237C22×C8—618no—
238C2×C22⋊C4—614no—
239C2×C22⋊C4—614no—
240C2×C22⋊C4—1224no—
241C2×C22⋊C4—1224no—
242C2×C42—612no—
243C2×C42—612no—
244C2×C42—612no—
245C2×C42—612no—
246C2×C42—2442no—
247C2×C42—2442no—
248C22⋊C8—6116no—
249C22⋊C8—6116no—
250#97—2444no—
251C4×C8—2444no—
252#95—614no—
253#95C2×#692314yes—
254C24—612no—
255C24—614no—
256C24—1222no—
257C24—1222no—
258C24—2441no—
259C24—2441no—
260C24—2441no—
261C24—2441no—
262C24C22×#692311yescharacteristic
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—611no—
271C22×C4—614no—
272C22×C4—612no—
273C22×C4—611no—
274C22×C4—614no—
275C22×C4—614no—
276C22×C4—614no—
277C22×C4—618no—
278C22×C4—612no—
279C22×C4—1224no—
280C22×C4—1224no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—2442no—
285C22×C4—2442no—
286C22×C4—24416no—
287C22×C4—2441no—
288C22×C4—2441no—
289C22×C4—2441no—
290C22×C4—2441no—
291C22×C4#1175821314yescomplemented
292C22×C4C4×#692314yes—
293C22×C4C22×#692312yes—
294C22×C4C22×#692312yes—
295C22×C4C22×#692312yes—
296M16—24416no—
297C2×C8—6116no—
298C2×C8—1228no—
299C2×C8—1228no—
300C22⋊C4—1228no—
301C22⋊C4—1228no—
302C42—1224no—
303C42—1224no—
304C42—2442no—
305C42—2442no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—614no—
311C23—1228no—
312C23—1222no—
313C23—1222no—
314C23—1222no—
315C23—1222no—
316C23—1222no—
317C23—1222no—
318C23—2444no—
319C23—2444no—
320C23—2444no—
321C23—2444no—
322C23—2444no—
323C23—2444no—
324C23C2×C4×#692312yes—
325C23D61⋊M1612yes—
326C23C2×#1175821312yes—
327C23C23×#692311yescharacteristic
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—612no—
339C2×C4—612no—
340C2×C4—612no—
341C2×C4—612no—
342C2×C4—614no—
343C2×C4—1228no—
344C2×C4—1228no—
345C2×C4—1228no—
346C2×C4—1222no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1222no—
350C2×C4—1222no—
351C2×C4—1222no—
352C2×C4—1222no—
353C2×C4—2448no—
354C2×C4—2444no—
355C2×C4—2444no—
356C2×C4C2×C4×#692314yes—
357C2×C4C2×#1175821314yes—
358C8—12216no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—1224no—
366C22—1224no—
367C22—1224no—
368C22—24416no—
369C22—2442no—
370C22—2442no—
371C22—2442no—
372C22—2442no—
373C22C4×#1175821314yes—
374C22C22×C4×#692311yescharacteristic
375C22#1684082911yescharacteristic
376C22#1684082911yescharacteristic
377C22C2×D61⋊M1611yescharacteristic
378C22C22×#1175821311yescharacteristic
379C22#1684084011yescharacteristic
380C22#1684084211yescharacteristic
381C4—24no—
382C4—24no—
383C4—42no—
384C4—614no—
385C4—1224no—
386C4—1222no—
387C4—2442no—
388C2—24no—
389C2—2444no—
390C2#2146225511yescharacteristic
391C2C2×C4×#1175821311yescharacteristic
392C2#2146238511yescharacteristic
393C2#2146267211yescharacteristic
394C2#2146267411yescharacteristic
395C2#2146268311yescharacteristic
396C2#2146268411yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutatorC22×C61C22×C42
FittingC2×#95×C61C4
socleC23×C61C23×C4
prime radicalC24×D61C2×C4
cosocleC22×Dic61C24
hypercenterC2×#95#6923
hypocenterC61#20524
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. #21461560C2semidirect
3. #21461545C2semidirect
4. C4×C22⋊#92420C2semidirect
7. #16840733C4semidirect
10. C2×#14519832C22semidirect
11. C2×C4×#92420C22semidirect
12. #16840628C22semidirect
13. #16840628C22semidirect
25. C2×#11758213C2×C4semidirect
27. #14519834C2×C4semidirect
72. C22×C4×C61M16semidirect
83. C22×Dic61M16semidirect
221. C61#20524semidirect
291. C22×C4#11758213semidirect

Maximal subgroups

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

subgroupindex
C2×#168407332
#214615602
#214615452
C4×C22⋊#924202
C2×#168396092
#2052461