Finite Group Databasebeta

← #29602676

Subgroups — #29602676

Order 15616.

1342 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839630C211yesmaximal, characteristic
2C2×C4×C4⋊Dic61C211yesmaximal, characteristic
3#21444084C212yesmaximal
4#21444076C212yesmaximal
5#21444053C214yesmaximal
6#21444049C212yesmaximal
7#21444046C212yesmaximal
8C2×C61×#376C211yesmaximal, characteristic
9C22×C4⋊Dic61C2212yes—
10C22×C4⋊Dic61C2214yes—
11C22×C4×Dic61C2212yes—
12#16839630—24no—
13C4×C4⋊Dic61—24no—
14C2×#14519684C2211yescharacteristic
15C2×#14519684C2211yescharacteristic
16C2×#14519684C2212yes—
17C2×#14519684C2212yes—
18C2×#14519684C2212yes—
19C2×#14519684C2214yes—
20C2×#14519684C2214yes—
21C2×#14519684C2214yes—
22C22×C42×C61C2212yes—
23C22×C42×C61C2211yescharacteristic
24C61×#376C418yescomplemented
25C2×#95×C61C2212yes—
26C2×#95×C61C2211yescharacteristic
27C2×#95×C61C2211yescharacteristic
28C23×Dic61C2312yes—
29C23×Dic61C2312yes—
30C23×Dic61C2314yes—
31C2×C4⋊Dic61—24no—
32C2×C4⋊Dic61—28no—
33C2×C4⋊Dic61—28no—
34C2×C4⋊Dic61—28no—
35C2×C4⋊Dic61—44no—
36C2×C4×Dic61—24no—
37C2×C4×Dic61—42no—
38C2×C4×Dic61—42no—
39#14519684—24no—
40#14519684—22no—
41#14519684—22no—
42#14519684—24no—
43#14519684—24no—
44#14519684—28no—
45#14519684—28no—
46#14519684—44no—
47C23×C4×C61C2312yes—
48C23×C4×C61C2311yescharacteristic
49C23×C4×C61C2311yescharacteristic
50C23×C4×C61C2311yescharacteristic
51C23×C4×C61C2312yes—
52C2×C42×C61—28no—
53C2×C42×C61C2×C418yes—
54C2×C42×C61C2×C414yes—
55C2×C42×C61D414yes—
56C2×C42×C61Q814yes—
57#95×C61C2×C414yes—
58#95×C61C2×C418yes—
59#95×C61C2×C414yes—
60C24×C61C2411yescharacteristic
61C22×C4×C61—24no—
62C22×C4×C61—24no—
63C22×C4×C61—24no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—22no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—22no—
70C22×C4×C61—24no—
71C22×C4×C61—24no—
72C22×C4×C61—22no—
73C22×C4×C61—22no—
74C22×C4×C61C22×C414yes—
75C22×C4×C61C22×C412yes—
76C22×C4×C61C22×C412yes—
77C22×C4×C61C22×C412yes—
78C22×C4×C61C22×C414yes—
79C22×C4×C61C2×D412yes—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×Q812yes—
83C22×C4×C61C2×Q812yes—
84C22×C4×C61C2×Q812yes—
85C22×C4×C61#4118yes—
86C22×Dic61—24no—
87C22×Dic61—24no—
88C22×Dic61—24no—
89C22×Dic61—24no—
90C22×Dic61—24no—
91C22×Dic61—24no—
92C22×Dic61—22no—
93C22×Dic61—22no—
94C22×Dic61—24no—
95C22×Dic61—24no—
96C22×Dic61—24no—
97C22×Dic61—24no—
98C22×Dic61—22no—
99C22×Dic61—22no—
100C22×Dic61—22no—
101C22×Dic61—22no—
102C22×Dic61#4118yes—
103C42×C61—216no—
104C42×C61C4⋊C4116yescomplemented
105C4⋊Dic61—216no—
106C4⋊Dic61—416no—
107C4×Dic61—48no—
108C23×C61C23×C411yescharacteristic
109C23×C61C22×D411yescharacteristic
110C23×C61C22×Q811yescharacteristic
111C23×C61C2×#4114yes—
112C23×C61C2×#4114yes—
113C23×C61H(5,𝔽2)11yescharacteristic
114C23×C61H(5,𝔽2)11yescharacteristic
115C23×C612−1+411yescharacteristic
116C23×C612−1+411yescharacteristic
117C2×C4×C61—24no—
118C2×C4×C61—24no—
119C2×C4×C61—24no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—24no—
123C2×C4×C61—24no—
124C2×C4×C61—24no—
125C2×C4×C61—24no—
126C2×C4×C61—24no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—28no—
132C2×C4×C61—28no—
133C2×C4×C61—24no—
134C2×C4×C61—24no—
135C2×C4×C61—24no—
136C2×C4×C61—44no—
137C2×C4×C61—44no—
138C2×C4×C61C2×C4⋊C414yes—
139C2×C4×C61C2×C4⋊C414yes—
140C2×C4×C61C2×C4⋊C414yes—
141C2×C4×C61#11718yes—
142C2×Dic61—28no—
143C2×Dic61—28no—
144C2×Dic61—28no—
145C2×Dic61—44no—
146C2×Dic61—44no—
147C2×Dic61—44no—
148C2×Dic61—44no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—42no—
154C2×Dic61—42no—
155C2×Dic61—44no—
156C2×Dic61—44no—
157C2×Dic61—44no—
158C2×Dic61—44no—
159#13952—611nomaximal, Sylow
160C22×C61C22×C4⋊C411yescharacteristic
161C22×C61C2×#11712yes—
162C22×C61C4×#4112yes—
163C22×C61#52011yescharacteristic
164C22×C61#52011yescharacteristic
165C22×C61#52512yes—
166C22×C61#53212yes—
167C22×C61#53312yes—
168C22×C61#53312yes—
169C22×C61#53511yescharacteristic
170C22×C61#53611yescharacteristic
171C22×C61#54411yescharacteristic
172C22×C61#54411yescharacteristic
173C22×C61#55114yes—
174C22×C61#55214yes—
175C22×C61#55314yes—
176C22×C61#55614yes—
177C4×C61—24no—
178C4×C61—28no—
179C4×C61—24no—
180C4×C61—24no—
181C4×C61—44no—
182Dic61—48no—
183Dic61—82no—
184Dic61—82no—
185C2×#383—611no—
186C2×C4×C4⋊C4—611no—
187C2×#376D6111yescharacteristic
188#1444—612no—
189#1442—612no—
190#1436—614no—
191C4⋊#95—612no—
192#1434—612no—
193C2×C61#228511yescharacteristic
194C2×C61#229611yescharacteristic
195C2×C61#231711yescharacteristic
196C2×C61#234312yes—
197C2×C61#234712yes—
198C2×C61#260312yes—
199C2×C61#260412yes—
200C2×C61#260512yes—
201C2×C61#260612yes—
202C22×C4⋊C4—612no—
203C22×C4⋊C4—614no—
204C22×C42—612no—
205C22×C42C2×D6112yes—
206C22×C42C2×D6111yescharacteristic
207#383—1224no—
208C4×C4⋊C4—1224no—
209#376Dic6118yescomplemented
210C2×#95—611no—
211C2×#95—612no—
212C2×#95—614no—
213C2×#95—614no—
214C2×#95—611no—
215C2×#95—612no—
216C2×#95—612no—
217C2×#95—614no—
218C2×#95C2×D6112yes—
219C2×#95C2×D6111yescharacteristic
220C2×#95C2×D6111yescharacteristic
221C61#1395211yescharacteristic, complemented, Sylow
222C23×C4—614no—
223C23×C4—612no—
224C23×C4—612no—
225C23×C4C22×D6112yes—
226C23×C4C22×D6111yescharacteristic
227C23×C4C22×D6111yescharacteristic
228C23×C4C22×D6111yescharacteristic
229C23×C4C22×D6112yes—
230C2×C4⋊C4—1228no—
231C2×C4⋊C4—1224no—
232C2×C4⋊C4—1228no—
233C2×C4⋊C4—1228no—
234C2×C4⋊C4—2444no—
235C2×C42—28no—
236C2×C42—1224no—
237C2×C42—2442no—
238C2×C42—2442no—
239C2×C42Dic12214yes—
240C2×C42D24414yes—
241C2×C42C2×Dic6118yes—
242C2×C42C2×Dic6114yes—
243#95—1228no—
244#95—1224no—
245#95—1222no—
246#95—1222no—
247#95—1224no—
248#95—1224no—
249#95—1228no—
250#95—2444no—
251#95C2×Dic6114yes—
252#95C2×Dic6118yes—
253#95C2×Dic6114yes—
254C24C23×D6111yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—618no—
269C22×C4—1224no—
270C22×C4—1224no—
271C22×C4—1224no—
272C22×C4—1224no—
273C22×C4—1224no—
274C22×C4—1224no—
275C22×C4—1222no—
276C22×C4—1222no—
277C22×C4—1224no—
278C22×C4—1224no—
279C22×C4—1224no—
280C22×C4—1224no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4C2×Dic12212yes—
286C22×C4C2×Dic12212yes—
287C22×C4C2×Dic12212yes—
288C22×C4C2×D24412yes—
289C22×C4C2×D24412yes—
290C22×C4C2×D24412yes—
291C22×C4#1175822218yes—
292C22×C4C22×Dic6114yes—
293C22×C4C22×Dic6112yes—
294C22×C4C22×Dic6112yes—
295C22×C4C22×Dic6112yes—
296C22×C4C22×Dic6114yes—
297C4⋊C4—12216no—
298C4⋊C4—24416no—
299C42—216no—
300C42—2448no—
301C42C4⋊Dic61116yescomplemented
302C23C23×Dic6111yescharacteristic
303C23C22×D24411yescharacteristic
304C23C22×Dic12211yescharacteristic
305C23#41×D6114yes—
306C23C2×#1175822214yes—
307C23#1451982311yescharacteristic
308C23#1451982311yescharacteristic
309C23#1451982411yescharacteristic
310C23#1451982411yescharacteristic
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—1228no—
333C2×C4—1228no—
334C2×C4—1228no—
335C2×C4—2444no—
336C2×C4—2444no—
337C2×C4—2442no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2442no—
343C2×C4—2444no—
344C2×C4—2444no—
345C2×C4—2444no—
346C2×C4—2444no—
347C2×C4—2444no—
348C2×C4—2444no—
349C2×C4C2×C4⋊Dic6114yes—
350C2×C4C2×C4⋊Dic6114yes—
351C2×C4C2×C4⋊Dic6114yes—
352C2×C4#1451973618yes—
353C22C22×C4⋊Dic6111yescharacteristic
354C22C2×#1451973612yes—
355C22#41×Dic6112yes—
356C22#1684024511yescharacteristic
357C22#1684024511yescharacteristic
358C22#1684027412yes—
359C22#1684031212yes—
360C22#1684031512yes—
361C22#1684031912yes—
362C22#1684033111yescharacteristic
363C22#1684033811yescharacteristic
364C22#1684039011yescharacteristic
365C22#1684039011yescharacteristic
366C22#1684043614yes—
367C22#1684044514yes—
368C22#1684045614yes—
369C22#1684048014yes—
370C4—24no—
371C4—24no—
372C4—28no—
373C4—24no—
374C4—44no—
375C4—4882no—
376C4—4882no—
377C4—2448no—
378C2#2144822611yescharacteristic
379C2#2144830311yescharacteristic
380C2#2144844211yescharacteristic
381C2#2144867912yes—
382C2#2144871212yes—
383C2#2145144612yes—
384C2#2145145712yes—
385C2#2145146712yes—
386C2#2145147612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#376C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×#376D61
hypocenterC61#13952
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
24. C61×#376C4semidirect
104. C42×C61C4⋊C4semidirect
209. #376Dic61semidirect
221. C61#13952semidirect
301. C42C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396302
C2×C4×C4⋊Dic612
#214440842
#214440762
#214440532
#214440492
#214440462
C2×C61×#3762
#1395261