Finite Group Databasebeta

← #30610028

Subgroups — #30610028

Order 15616.

1206 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462684C212yesmaximal, complemented
2#21461076C214yesmaximal, complemented
3#21461068C214yesmaximal, complemented
4#21461062C212yesmaximal, complemented
5#21461052C212yesmaximal, complemented
6C2×#16840244C211yesmaximal, characteristic
7#16840796—28no—
8#16840778C2214yes—
9#16840733C2218yes—
10#16840608C2214yes—
11#16840607C2218yes—
12#16840606C2214yes—
13C22×#11758226C411yescharacteristic
14#16840244—28no—
15C2×Q8×Dic61C2211yescharacteristic
16C2×C4×D244C411yescharacteristic
17C2×C4×D244C2212yes—
18C2×#14519739C412yes—
19C2×#14519739C2211yescharacteristic
20C2×#14519734C412yes—
21C2×#14519734C2211yescharacteristic
22C2×C4⋊C4×D61C411yescharacteristic
23C2×C4⋊C4×D61C2212yes—
24C2×C4×Q8×C61C411yescharacteristic
25C2×#11758213—28no—
26C4⋊#92420—24no—
27#14519834—28no—
28C22⋊#92420C2318yes—
29C4×#92420—24no—
30C2×#11758226—24no—
31C2×#11758226—24no—
32C2×#11758226—24no—
33C22×D244C2×C412yes—
34C22×D244C2×C411yescharacteristic
35C22×C4×D61C2×C412yes—
36C22×C4×D61C2×C411yescharacteristic
37C22×C4×D61C2×C412yes—
38C22×C4×D61C2×C412yes—
39C22×C4×D61C2312yes—
40Q8×Dic61—24no—
41C4×D244—24no—
42C4×D244—28no—
43#14519739—28no—
44#14519739D418yes—
45#14519734—28no—
46#14519734—24no—
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×C4⋊Dic61C2312yes—
49C2×#11758196C2×C412yes—
50C2×#11758196C2×C411yescharacteristic
51C2×#11758196C2311yescharacteristic
52C4⋊C4×D61—28no—
53C4⋊C4×D61—24no—
54C2×#11758198C2×C412yes—
55C2×#11758198C2×C411yescharacteristic
56C2×#11758198C2×C412yes—
57C2×#11758198C2311yescharacteristic
58C2×C4×Dic61C2×C412yes—
59C2×C4×Dic61C2311yescharacteristic
60C22×Q8×C61C2×C411yescharacteristic
61C4×Q8×C61—24no—
62C2×C4⋊C4×C61C2×C412yes—
63C2×C4⋊C4×C61C2×C411yescharacteristic
64C2×C42×C61C2×C412yes—
65C2×C42×C61C2×C411yescharacteristic
66C23×D61C22×C412yes—
67C23×D61C22×C411yescharacteristic
68C2×Q8×C61—22no—
69C2×Q8×C61—22no—
70C2×Q8×C61M1614yescomplemented
71C22×C4×C61C22×C412yes—
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C22×C411yescharacteristic
74C22×C4×C61C22×C412yes—
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61C22×C411yescharacteristic
77C22×Dic61C22×C412yes—
78C22×Dic61C22×C411yescharacteristic
79C22×Dic61C2411yescharacteristic
80#11758226—232no—
81C2×D244—24no—
82C2×D244—24no—
83C2×D244—24no—
84C2×D244—22no—
85C2×D244—22no—
86C2×D244C22⋊C414yes—
87C2×C4×D61—24no—
88C2×C4×D61—22no—
89C2×C4×D61—24no—
90C2×C4×D61—22no—
91C2×C4×D61—24no—
92C2×C4×D61—24no—
93C2×C4×D61—24no—
94C2×C4×D61—82no—
95C2×C4×D61—82no—
96C2×C4×D61—82no—
97C2×C4×D61M1614yescomplemented
98C2×C4×D61C2×D418yes—
99C2×#92420—28no—
100#11758213—88no—
101C4⋊C4×C61—24no—
102C4⋊C4×C61C22⋊C414yes—
103C42×C61—24no—
104C42×C61—22no—
105#11758198—24no—
106#11758198—22no—
107#11758198—22no—
108#11758198C22⋊C418yes—
109C4⋊Dic61—22no—
110C4⋊Dic61—24no—
111#11758196—24no—
112#11758196—22no—
113#11758196—22no—
114C4×Dic61—24no—
115C4×Dic61C2×D414yes—
116C23×C61C23×C411yescharacteristic
117C22×D61—21no—
118C22×D61—22no—
119C22×D61—22no—
120C22×D61—21no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61C2×C22⋊C414yes—
126C22×D61C2×M1612yes—
127Q8×C61—28no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—42no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61C2×C22⋊C414yes—
136C2×C4×C61C2×C22⋊C412yes—
137C2×C4×C61C2×M1614yes—
138C2×C4×C61C2×M1612yes—
139C2×C4×C61C2×M1612yes—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—22no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—42no—
146C2×Dic61C2×C22⋊C412yes—
147C2×Dic61C2×M1612yes—
148C2×Dic61C22×D412yes—
149D244—216no—
150D244—28no—
151C4×D61—216no—
152C4×D61—28no—
153C4×D61—48no—
154C4×D61—84no—
155C4×D61—84no—
156#92420—48no—
157#17370—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61C22×C22⋊C411yescharacteristic
161C22×C61C22×M1611yescharacteristic
162C22×C61C22×M1611yescharacteristic
163C2×D61—24no—
164C2×D61—22no—
165C2×D61—24no—
166C2×D61—24no—
167C2×D61—44no—
168C2×D61—44no—
169C2×D61—42no—
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—42no—
173C4×C61—24no—
174C4×C61—24no—
175C4×C61—41no—
176C4×C61—41no—
177C4×C61#40714yes—
178Dic61—22no—
179Dic61—22no—
180Dic61—22no—
181Dic61—42no—
182C2×#520—611no—
183#2975—612no—
184#1465—614no—
185#1463—614no—
186#1461—612no—
187#1458—612no—
188C2×C61—21no—
189C2×C61—21no—
190C2×C61C2×#40711yescharacteristic
191C2×C61#288311yescharacteristic
192C2×C61#288911yescharacteristic
193D61—44no—
194D61—42no—
195C22×#41—611no—
196#520—1228no—
197C2×C4×Q8—611no—
198C2×C4×Q8#692311yescharacteristic
199C2×C4×D4—612no—
200C2×C4×D4—611no—
201C2×C4×D4—611no—
202C2×C4×D4—612no—
203C2×#117—612no—
204C2×#117—611no—
205C22×C4⋊C4—612no—
206C22×C4⋊C4—611no—
207#436—1228no—
208#431—614no—
209#407—618no—
210#324—618no—
211#324—614no—
212#323—614no—
213C61#1737011yescharacteristic, complemented, Sylow
214C2×#41—1224no—
215C2×#41—1224no—
216C2×#41—1224no—
217C22×Q8C2×#692311yescharacteristic
218C22×D4—611no—
219C22×D4—612no—
220C23×C4—611no—
221C23×C4—612no—
222C23×C4—612no—
223C23×C4—612no—
224C23×C4—612no—
225C2×M16—1228no—
226C4×Q8—24no—
227C4×Q8—1224no—
228C4×D4—618no—
229C4×D4—1228no—
230C4×D4—1224no—
231C4×D4—1228no—
232#117—1224no—
233#117—1228no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—612no—
236C2×C4⋊C4—612no—
237C2×C4⋊C4—611no—
238C2×C4⋊C4—611no—
239C2×C4⋊C4—1228no—
240C2×C4⋊C4—1224no—
241C2×C4⋊C4C2×#692312yes—
242C2×C4⋊C4C2×#692311yescharacteristic
243C2×C22⋊C4—612no—
244C2×C22⋊C4—611no—
245C2×C22⋊C4—611no—
246C2×C22⋊C4—612no—
247C2×C42—611no—
248C2×C42—612no—
249C2×C42C2×#692312yes—
250C2×C42C2×#692311yescharacteristic
251C4⋊C8—1224no—
252C22⋊C8—618no—
253C22⋊C8—1228no—
254C4×C8—1224no—
255C24—612no—
256C24—611no—
257#41—12232no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8#1175821314yescomplemented
261C2×D4—614no—
262C2×D4—1224no—
263C2×D4—1224no—
264C2×D4—1222no—
265C2×D4—1224no—
266C2×D4—1222no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—611no—
270C22×C4—614no—
271C22×C4—618no—
272C22×C4—612no—
273C22×C4—1224no—
274C22×C4—1224no—
275C22×C4—1224no—
276C22×C4—1222no—
277C22×C4—1224no—
278C22×C4—1222no—
279C22×C4—1224no—
280C22×C4—4882no—
281C22×C4—4882no—
282C22×C4—4882no—
283C22×C4C22×#692312yes—
284C22×C4C22×#692311yescharacteristic
285C22×C4C22×#692311yescharacteristic
286C22×C4C22×#692312yes—
287C22×C4C22×#692311yescharacteristic
288M16—4888no—
289C2×C8—1228no—
290C4⋊C4—24no—
291C4⋊C4—1222no—
292C4⋊C4—1224no—
293C4⋊C4—1222no—
294C4⋊C4—1224no—
295C4⋊C4—1222no—
296C4⋊C4C22⋊#692314yes—
297C22⋊C4—618no—
298C22⋊C4—1222no—
299C22⋊C4—1222no—
300C22⋊C4—1224no—
301C42—24no—
302C42—22no—
303C42—614no—
304C42—1224no—
305C23—614no—
306C23—612no—
307C23—1221no—
308C23—1222no—
309C23—1222no—
310C23—1221no—
311C23—2442no—
312C23—2442no—
313C23—2441no—
314C23—2441no—
315C23C23×#692311yescharacteristic
316Q8—28no—
317D4—12216no—
318D4—1228no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—612no—
327C2×C4—612no—
328C2×C4—612no—
329C2×C4—12216no—
330C2×C4—1221no—
331C2×C4—1221no—
332C2×C4—1228no—
333C2×C4—1222no—
334C2×C4—4884no—
335C2×C4—4884no—
336C2×C4—2448no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2442no—
340C2×C4C2×C22⋊#692314yes—
341C2×C4C2×C22⋊#692312yes—
342C2×C4D61⋊M1612yes—
343C2×C4C2×#1175821314yes—
344C2×C4C2×#1175821312yes—
345C8—2448no—
346C22—21no—
347C22—21no—
348C22—1224no—
349C22—1222no—
350C22—1224no—
351C22—1224no—
352C22—2442no—
353C22—2444no—
354C22—2444no—
355C22—2442no—
356C22—2441no—
357C22—2441no—
358C22C22×C22⋊#692311yescharacteristic
359C22C2×D61⋊M1611yescharacteristic
360C22C22×#1175821311yescharacteristic
361C4—24no—
362C4—24no—
363C4—41no—
364C4—41no—
365C4—1222no—
366C4—1222no—
367C4—1222no—
368C4—2442no—
369C4#1684073314yes—
370C2—21no—
371C2—21no—
372C2—2444no—
373C2—2442no—
374C2C2×#1684073311yescharacteristic
375C2#2146241911yescharacteristic
376C2#2146243211yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758226C4
cosocleC22×Dic61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#17370
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21462684C2semidirect
2. #21461076C2semidirect
3. #21461068C2semidirect
4. #21461062C2semidirect
5. #21461052C2semidirect
70. C2×Q8×C61M16semidirect
97. C2×C4×D61M16semidirect
213. C61#17370semidirect
260. C2×Q8#11758213semidirect

Maximal subgroups

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

subgroupindex
#214626842
#214610762
#214610682
#214610622
#214610522
C2×#168402442
#1737061