Finite Group Databasebeta

← #30597238

Subgroups — #30597238

Order 15616.

994 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462263C212yesmaximal
2#21460718C214yesmaximal
3#21448198C211yesmaximal, characteristic
4C2×#14519856—22no—
5#16840705—22no—
6#16840704—22no—
7#16840704C414yescomplemented
8#16840703C2214yes—
9C4×C22⋊#6923—22no—
10C4×C22⋊#6923C414yescomplemented
11C23×C4×D61C2211yescharacteristic
12C2×#14519736C411yescharacteristic, complemented
13#16839623C412yescomplemented
14#16839619—22no—
15#16839617C2212yes—
16C4×#11758198—22no—
17C2×#117×C61C411yescharacteristic, complemented
18#14519856—44no—
19#14519856—44no—
20C2×C4⋊#6923—24no—
21C2×C22⋊#6923—24no—
22C2×C22⋊#6923—24no—
23C2×C22⋊#6923C2×C418yes—
24C2×C4×#6923—24no—
25(C2×#6923)⋊C4—24no—
26(C2×#6923)⋊C4—24no—
27C24×D61C2311yescharacteristic
28C23×Dic61C2×C411yescharacteristic
29C22×C4×D61—42no—
30C22×C4×D61—42no—
31C22×C4×D61C2×C412yes—
32C22×C4×D61C2×C412yes—
33C22×C4×D61D412yes—
34C22×C4×D61D412yes—
35C22×C4×D61D412yes—
36C22×C4×D61Q812yes—
37#14519736—42no—
38C2×C4⋊Dic61—21no—
39C2×C22⋊Dic61C2×C412yes—
40C2×#11758198—24no—
41C2×C4×Dic61—21no—
42#14519684—24no—
43C23×C4×C61C2×C411yescharacteristic
44#117×C61—42no—
45C2×C4⋊C4×C61—21no—
46C2×C22⋊C4×C61C2×C412yes—
47C2×C42×C61—21no—
48C24×C61C22×C411yescharacteristic
49C23×D61—44no—
50C23×D61—41no—
51C23×D61—41no—
52C23×D61C22×C412yes—
53C23×D61C2×D411yescharacteristic
54C23×D61C2×D412yes—
55C23×D61C2×Q811yescharacteristic
56C22×#6923—28no—
57C22×C4×C61—22no—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61C4212yes—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C4⋊C412yes—
64C22×Dic61—22no—
65C22×Dic61—41no—
66C22×Dic61—41no—
67C22×Dic61C4211yescharacteristic
68C22×Dic61C4211yescharacteristic
69C22×Dic61C22⋊C412yes—
70C22×Dic61C4⋊C412yes—
71C2×C4×D61—24no—
72C2×C4×D61—24no—
73C2×C4×D61—44no—
74C2×C4×D61—44no—
75C2×C4×D61—44no—
76C2×C4×D61—44no—
77C2×C4×D61—42no—
78C2×C4×D61—42no—
79C2×C4×D61C22⋊C412yes—
80C2×C4×D61C22⋊C412yes—
81C2×C4×D61C22⋊C412yes—
82C2×C4×D61C22⋊C412yes—
83C2×C4×D61C4⋊C412yes—
84C2×C4×D61C4⋊C412yes—
85C2×C4×D61C4⋊C412yes—
86C2×C4×D61C4⋊C412yes—
87C22⋊#6923—48no—
88C22⋊#6923—44no—
89C22⋊#6923—44no—
90C22⋊#6923—48no—
91C4⋊#6923—48no—
92C4×#6923—48no—
93C4⋊C4×C61—42no—
94C22⋊C4×C61—42no—
95C42×C61—42no—
96C22⋊Dic61—42no—
97#11758198—48no—
98C4⋊Dic61—42no—
99C4×Dic61—42no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—42no—
103C23×C61C2×C4211yescharacteristic
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61C2×C4⋊C411yescharacteristic
106C22×D61—22no—
107C22×D61—22no—
108C22×D61—24no—
109C22×D61—24no—
110C22×D61—24no—
111C22×D61—42no—
112C22×D61—41no—
113C22×D61—41no—
114C22×D61—42no—
115C22×D61—42no—
116C22×D61—42no—
117C22×D61—48no—
118C22×D61—48no—
119C22×D61C2×C22⋊C412yes—
120C22×D61C2×C4⋊C412yes—
121C2×#6923—48no—
122C2×#6923—48no—
123C2×#6923—48no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—24no—
127C2×C4×C61—24no—
128C2×C4×C61—44no—
129C2×C4×C61—42no—
130C2×C4×C61#9512yes—
131C2×C4×C61#9512yes—
132C2×C4×C61#9912yes—
133C2×C4×C61#9912yes—
134C2×Dic61—22no—
135C2×Dic61—22no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—42no—
139C2×Dic61—44no—
140C2×Dic61—42no—
141C2×Dic61#9512yes—
142C2×Dic61#9512yes—
143C4×D61—24no—
144C4×D61—24no—
145C4×D61—24no—
146C4×D61—24no—
147C4×D61—42no—
148C4×D61—42no—
149C4×D61—42no—
150C4×D61—42no—
151#8102—611nomaximal, Sylow
152C22×C61—22no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—22no—
156C22×C61—22no—
157C22×C61—42no—
158C22×C61—42no—
159C22×C61C2×#9511yescharacteristic
160C22×C61C2×#9911yescharacteristic
161C22×C61C2×#9911yescharacteristic
162C2×D61—22no—
163C2×D61—22no—
164C2×D61—22no—
165C2×D61—22no—
166C2×D61—22no—
167C2×D61—44no—
168C2×D61—44no—
169C2×D61—44no—
170C2×D61—48no—
171C2×D61—44no—
172C2×D61#41014yes—
173#6923—88no—
174C4×C61—22no—
175C4×C61—82no—
176C4×C61#34214yes—
177Dic61—22no—
178Dic61—22no—
179Dic61—82no—
180#2280—611no—
181#2280—612no—
182#1304—614no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—41no—
186C2×C61—41no—
187C2×C61C2×#34211yescharacteristic
188C2×C61#174012yes—
189D61—24no—
190D61—42no—
191C24×C4—611no—
192C2×#117—611no—
193C2×#117—1222no—
194C2×#117#692311yescharacteristic, complemented
195#381—612no—
196#381—1222no—
197#380—614no—
198#380—1222no—
199#380—1222no—
200#379—614no—
201#379—612no—
202C4×C22⋊C4—614no—
203C4×C22⋊C4—1222no—
204C4×C22⋊C4—1222no—
205C61#810211yescharacteristic, complemented, Sylow
206C25—611no—
207C23×C4—611no—
208C23×C4—612no—
209C23×C4—612no—
210C23×C4—612no—
211C23×C4—612no—
212C23×C4—612no—
213C23×C4—612no—
214C23×C4—2442no—
215C23×C4—2442no—
216C23×C4C2×#692311yescharacteristic
217#117—42no—
218#117—2444no—
219#117—2444no—
220#117—2442no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—1224no—
223C2×C4⋊C4—1221no—
224C2×C22⋊C4—618no—
225C2×C22⋊C4—612no—
226C2×C22⋊C4—1224no—
227C2×C22⋊C4—1224no—
228C2×C22⋊C4—1224no—
229C2×C22⋊C4C2×#692312yes—
230C2×C42—21no—
231C2×C42—1224no—
232C2×C42—1221no—
233#95—1224no—
234#95—1224no—
235#95—1224no—
236C24—612no—
237C24—611no—
238C24—612no—
239C24—611no—
240C24—2441no—
241C24—2441no—
242C24—2444no—
243C24C22×#692311yescharacteristic
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—612no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—612no—
251C22×C4—612no—
252C22×C4—612no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—612no—
256C22×C4—612no—
257C22×C4—612no—
258C22×C4—612no—
259C22×C4—1224no—
260C22×C4—1224no—
261C22×C4—1228no—
262C22×C4—1222no—
263C22×C4—2444no—
264C22×C4—2444no—
265C22×C4—2444no—
266C22×C4—2444no—
267C22×C4—2442no—
268C22×C4—2442no—
269C22×C4—2441no—
270C22×C4—2441no—
271C22×C4C4×#692312yes—
272C22×C4C4⋊#692312yes—
273C22×C4C22⋊#692311yescharacteristic
274C22×C4C22⋊#692311yescharacteristic
275C4⋊C4—42no—
276C4⋊C4—2448no—
277C4⋊C4—2442no—
278C22⋊C4—42no—
279C22⋊C4—2448no—
280C22⋊C4—2444no—
281C22⋊C4—2444no—
282C22⋊C4—2448no—
283C22⋊C4—2442no—
284C22⋊C4—2448no—
285C42—42no—
286C42—2442no—
287C42—2448no—
288C23—21no—
289C23—21no—
290C23—42no—
291C23—612no—
292C23—612no—
293C23—1224no—
294C23—1224no—
295C23—1224no—
296C23—1222no—
297C23—1222no—
298C23—2442no—
299C23—2441no—
300C23—2441no—
301C23—2442no—
302C23—2442no—
303C23—2442no—
304C23—2448no—
305C23—2448no—
306C23C2×C4×#692311yescharacteristic
307C23C2×C22⋊#692311yescharacteristic
308C23C2×C4⋊#692311yescharacteristic
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—612no—
316C2×C4—612no—
317C2×C4—1222no—
318C2×C4—1222no—
319C2×C4—1224no—
320C2×C4—1224no—
321C2×C4—1224no—
322C2×C4—1224no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2442no—
330C2×C4—2444no—
331C2×C4—2448no—
332C2×C4—2448no—
333C2×C4—2448no—
334C2×C4—2442no—
335C2×C4(C2×#6923)⋊C412yes—
336C2×C4(C2×#6923)⋊C412yes—
337C2×C4#1451983512yes—
338C2×C4#1451983612yes—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—614no—
347C22—1222no—
348C22—1222no—
349C22—1222no—
350C22—1222no—
351C22—1222no—
352C22—2444no—
353C22—2444no—
354C22—2444no—
355C22—2448no—
356C22—2444no—
357C22C2×(C2×#6923)⋊C411yescharacteristic
358C22C2×#1451983511yescharacteristic
359C22C2×#1451983611yescharacteristic
360C4—22no—
361C4—82no—
362C4—1222no—
363C4—1222no—
364C4—4882no—
365C4—4888no—
366C4#1684064014yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—1224no—
372C2—2442no—
373C2C2×#1684064011yescharacteristic
374C2#2146151712yes—
subgroupG/H
centerC22C2×(C2×#6923)⋊C4
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#95
prime radical, cosocleC24×D61C23
hypercenterC2×#117#6923
hypocenterC61#8102
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
7. #16840704C4semidirect
10. C4×C22⋊#6923C4semidirect
12. C2×#14519736C4semidirect
13. #16839623C4semidirect
17. C2×#117×C61C4semidirect
194. C2×#117#6923semidirect
205. C61#8102semidirect

Maximal subgroups

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

subgroupindex
#214622632
#214607182
#214481982
#810261