Finite Group Databasebeta

← #30609991

Subgroups — #30609991

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462495C212yesmaximal
2#21461082C214yesmaximal, complemented
3#21461067C214yesmaximal
4#21461062C214yesmaximal, complemented
5C22×Q8×Dic61C211yesmaximal, characteristic
6#16840783—22no—
7#16840782C2214yes—
8#16840779—22no—
9#16840778C2214yes—
10#16840765—22no—
11#16840732C2214yes—
12C4×#11758217—22no—
13#16840607C22116yes—
14C2×Q8×Dic61—22no—
15C2×Q8×Dic61—22no—
16C2×Q8×Dic61—22no—
17C2×Q8×Dic61—22no—
18C2×Q8×Dic61C412yes—
19C2×Q8×Dic61C412yes—
20C2×Q8×Dic61C2212yes—
21C2×Q8×Dic61C2212yes—
22C22×C4⋊Dic61C411yescharacteristic
23C22×C4⋊Dic61C2212yes—
24C22×C4×Dic61C412yes—
25C22×C4×Dic61C2211yescharacteristic
26C23×Q8×C61C411yescharacteristic
27C2×#11758217—24no—
28#14519844—24no—
29C4⋊#92420—24no—
30C22⋊#92420—24no—
31C22⋊#92420C2318yes—
32#14519832—24no—
33C4×#92420—24no—
34C23×Dic61C2×C411yescharacteristic
35C23×Dic61C2×C412yes—
36C23×Dic61C2311yescharacteristic
37Q8×Dic61—24no—
38Q8×Dic61—24no—
39Q8×Dic61—24no—
40Q8×Dic61—24no—
41Q8×Dic61—24no—
42Q8×Dic61—24no—
43Q8×Dic61—24no—
44Q8×Dic61—24no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—22no—
47C2×C4⋊Dic61—24no—
48C2×C4⋊Dic61—24no—
49C2×C4⋊Dic61C2×C412yes—
50C2×C4⋊Dic61C2×C412yes—
51C2×C4⋊Dic61C2×C414yes—
52C2×C4⋊Dic61C2314yes—
53C2×C4×Dic61—22no—
54C2×C4×Dic61—22no—
55C2×C4×Dic61—22no—
56C2×C4×Dic61—22no—
57C2×C4×Dic61C2×C414yes—
58C2×C4×Dic61C2×C414yes—
59C2×C4×Dic61C2×C411yescharacteristic
60C2×C4×Dic61C2×C411yescharacteristic
61C2×C4×Dic61D412yes—
62C2×C4×Dic61D412yes—
63C2×C4×Dic61D412yes—
64C2×C4×Dic61D412yes—
65C2×C4×Dic61C2311yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C22×Q8×C61—28no—
68C22×Q8×C61—22no—
69C22×Q8×C61—22no—
70C22×Q8×C61C2×C412yes—
71C22×Q8×C61C2×C412yes—
72C23×C4×C61C2×C412yes—
73C23×C4×C61C2×C411yescharacteristic
74C24×C61C22×C411yescharacteristic
75C2×Q8×C61—216no—
76C2×Q8×C61—216no—
77C2×Q8×C61—24no—
78C2×Q8×C61—24no—
79C2×Q8×C61—28no—
80C2×Q8×C61—24no—
81C2×Q8×C61M1618yescomplemented
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61—28no—
85C22×C4×C61—24no—
86C22×C4×C61C22⋊C412yes—
87C22×C4×C61C22⋊C412yes—
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C22×C414yes—
90C22×C4×C61C22×C411yescharacteristic
91C22×Dic61—24no—
92C22×Dic61—22no—
93C22×Dic61—22no—
94C22×Dic61—42no—
95C22×Dic61—44no—
96C22×Dic61C22⋊C411yescharacteristic
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C22×C411yescharacteristic
101C22×Dic61C22×C411yescharacteristic
102C22×Dic61C22×C414yes—
103C22×Dic61C22×C411yescharacteristic
104C22×Dic61C2×D412yes—
105C22×Dic61C2×D412yes—
106C22×Dic61C2411yescharacteristic
107#11758217—48no—
108C2×#92420—28no—
109C4⋊Dic61—24no—
110C4⋊Dic61—24no—
111C4⋊Dic61—28no—
112C4⋊Dic61—28no—
113C4×Dic61—24no—
114C4×Dic61—24no—
115C4×Dic61—24no—
116C4×Dic61—24no—
117C4×Dic61C22⋊C412yes—
118C4×Dic61C22⋊C412yes—
119C4×Dic61C22⋊C412yes—
120C4×Dic61C22⋊C412yes—
121C4×Dic61C2×D412yes—
122C4×Dic61C2×D412yes—
123C4×Dic61C2×D412yes—
124C4×Dic61C2×D412yes—
125C23×C61—22no—
126C23×C61—22no—
127C23×C61—22no—
128C23×C61C2×C22⋊C411yescharacteristic
129C23×C61C2×C22⋊C411yescharacteristic
130C23×C61C23×C411yescharacteristic
131Q8×C61—216no—
132Q8×C61—216no—
133C2×C4×C61—28no—
134C2×C4×C61—28no—
135C2×C4×C61—24no—
136C2×C4×C61—22no—
137C2×C4×C61—24no—
138C2×C4×C61—24no—
139C2×C4×C61—24no—
140C2×C4×C61C2×C22⋊C412yes—
141C2×C4×C61C2×C22⋊C412yes—
142C2×C4×C61C2×M1618yes—
143C2×C4×C61C2×M1614yes—
144C2×Dic61—24no—
145C2×Dic61—22no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—22no—
150C2×Dic61—24no—
151C2×Dic61—41no—
152C2×Dic61—42no—
153C2×Dic61—41no—
154C2×Dic61—44no—
155C2×Dic61—44no—
156C2×Dic61—44no—
157C2×Dic61#10114yes—
158C2×Dic61C2×C22⋊C411yescharacteristic
159C2×Dic61C2×C22⋊C411yescharacteristic
160C2×Dic61C2×C22⋊C412yes—
161C2×Dic61C2×C22⋊C411yescharacteristic
162C2×Dic61C2×C22⋊C411yescharacteristic
163C2×Dic61C22×D412yes—
164#92420—48no—
165#17365—611nomaximal, Sylow
166C22×C61—28no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61#41012yes—
174C22×C61C2×#10112yes—
175C22×C61C22×C22⋊C411yescharacteristic
176C22×C61C22×M1612yes—
177C4×C61—22no—
178C4×C61—28no—
179C4×C61#40714yes—
180Dic61—22no—
181Dic61—22no—
182Dic61—21no—
183Dic61—21no—
184Dic61—44no—
185Dic61C2×#10112yes—
186Dic61C2×#10112yes—
187C22×C4×Q8—611no—
188#2920—612no—
189#1467—614no—
190#1463—614no—
191#1461—614no—
192C2×C61—22no—
193C2×C61—22no—
194C2×C61—22no—
195C2×C61C2×#40711yescharacteristic
196C2×C61C2×#41011yescharacteristic
197C2×C61C22×#10111yescharacteristic
198C23×Q8#692311yescharacteristic
199C2×C4×Q8—612no—
200C2×C4×Q8—612no—
201C2×C4×Q8—612no—
202C2×C4×Q8—612no—
203C2×C4×Q8—1222no—
204C2×C4×Q8—1222no—
205C2×C4×Q8—1222no—
206C2×C4×Q8—1222no—
207C22×C4⋊C4—612no—
208C22×C4⋊C4—611no—
209C22×C42—611no—
210C22×C42—612no—
211#432—614no—
212#432—1222no—
213#431—614no—
214#431—1222no—
215#423—1222no—
216#407—614no—
217C4×M16—1222no—
218#324—6116no—
219C61#1736511yescharacteristic, complemented, Sylow
220C22×Q8—28no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8C2×#692312yes—
224C22×Q8C2×#692312yes—
225C23×C4—611no—
226C23×C4—611no—
227C23×C4—612no—
228C23×C4C2×#692312yes—
229C23×C4C2×#692311yescharacteristic
230C2×M16—1224no—
231C4×Q8—1224no—
232C4×Q8—1224no—
233C4×Q8—1224no—
234C4×Q8—1224no—
235C4×Q8—1224no—
236C4×Q8—1224no—
237C4×Q8—1224no—
238C4×Q8—1224no—
239C2×C4⋊C4—614no—
240C2×C4⋊C4—614no—
241C2×C4⋊C4—612no—
242C2×C4⋊C4—612no—
243C2×C4⋊C4—1224no—
244C2×C4⋊C4—1224no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1222no—
247C2×C42—612no—
248C2×C42—612no—
249C2×C42—612no—
250C2×C42—612no—
251C2×C42—611no—
252C2×C42—611no—
253C2×C42—611no—
254C2×C42—611no—
255C2×C42—614no—
256C2×C42—614no—
257C2×C42—1222no—
258C2×C42—1222no—
259C2×C42—1222no—
260C2×C42—1222no—
261C4⋊C8—1224no—
262C4⋊C8—1224no—
263C22⋊C8—618no—
264C22⋊C8—1224no—
265#97—1224no—
266C4×C8—1224no—
267C24C22×#692311yescharacteristic
268C2×Q8—216no—
269C2×Q8—216no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—28no—
273C2×Q8—24no—
274C2×Q8#1175821718yescomplemented
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—612no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—612no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—614no—
290C22×C4—1224no—
291C22×C4—1222no—
292C22×C4—1222no—
293C22×C4—2442no—
294C22×C4—2444no—
295C22×C4C22⋊#692312yes—
296C22×C4C22⋊#692312yes—
297C22×C4C22×#692311yescharacteristic
298C22×C4C22×#692314yes—
299C22×C4C22×#692311yescharacteristic
300M16—2448no—
301C2×C8—1228no—
302C4⋊C4—1228no—
303C4⋊C4—1228no—
304C4⋊C4—1224no—
305C4⋊C4—1224no—
306C42—612no—
307C42—612no—
308C42—612no—
309C42—612no—
310C42—612no—
311C42—612no—
312C42—612no—
313C42—612no—
314C42—1224no—
315C42—1224no—
316C42—1224no—
317C42—1224no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23C2×C22⋊#692311yescharacteristic
322C23C2×C22⋊#692311yescharacteristic
323C23C23×#692311yescharacteristic
324Q8—216no—
325Q8—216no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—614no—
334C2×C4—611no—
335C2×C4—611no—
336C2×C4—611no—
337C2×C4—611no—
338C2×C4—612no—
339C2×C4—612no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—1224no—
345C2×C4—1222no—
346C2×C4—1224no—
347C2×C4—2441no—
348C2×C4—2442no—
349C2×C4—2441no—
350C2×C4—2444no—
351C2×C4—2444no—
352C2×C4—2444no—
353C2×C4C2×C22⋊#692312yes—
354C2×C4C2×C22⋊#692312yes—
355C2×C4C2×#1175821718yes—
356C2×C4C2×#1175821714yes—
357C8—2448no—
358C22—28no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22#1684074312yes—
366C22D61⋊#10112yes—
367C22C22×C22⋊#692311yescharacteristic
368C22C22×#1175821712yes—
369C4—22no—
370C4—28no—
371C4—612no—
372C4—612no—
373C4—1222no—
374C4—1222no—
375C4—1221no—
376C4—1221no—
377C4—2444no—
378C4#1684073214yes—
379C2—22no—
380C2—22no—
381C2—22no—
382C2C2×#1684073211yescharacteristic
383C2C2×#1684074311yescharacteristic
384C2C2×D61⋊#10111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC23×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C22×C4
cosocleC22×Dic61C24
hypercenterC23×Q8#6923
hypocenterC61#17365
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #21461082C2semidirect
4. #21461062C2semidirect
81. C2×Q8×C61M16semidirect
219. C61#17365semidirect
274. C2×Q8#11758217semidirect

Maximal subgroups

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

subgroupindex
#214624952
#214610822
#214610672
#214610622
C22×Q8×Dic612
#1736561