Finite Group Databasebeta

← C2×#21462365 #30624903

Subgroups — C2×#21462365 #30624903

Order 15616.

2198 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462365C2116yesmaximal, direct product factor
2C2×#16840713C214yesmaximal
3C2×#16840711C214yesmaximal
4C2×#16840711C214yesmaximal
5C2×#16840706C212yesmaximal
6C2×#125×D61C211yesmaximal, characteristic
7C22×C4⋊#6923C2212yes—
8C22×C4⋊#6923C2214yes—
9C22×C4⋊#6923C2212yes—
10C22×C4×#6923C2214yes—
11#16840713C22132yes—
12#16840711C22132yes—
13#16840711C22132yes—
14#16840706C22116yes—
15C2×(C2×#6923)⋊C4C2214yes—
16C2×(C2×#6923)⋊C4C2214yes—
17C2×(C2×#6923)⋊C4C2214yes—
18C2×(C2×#6923)⋊C4C2214yes—
19C2×#14519769C411yescharacteristic, complemented
20C2×#14519768C412yescomplemented
21C2×#14519767C414yescomplemented
22#125×D61C418yescomplemented
23#125×D61C2218yes—
24C2×C4⋊C4×D61C2214yes—
25C2×C4⋊C4×D61C2212yes—
26C2×C42×D61C2211yescharacteristic
27C2×#125×C61C411yescharacteristic, complemented
28C23×#6923C2314yes—
29C23×#6923C2314yes—
30C2×C4⋊#6923—48no—
31C2×C4⋊#6923—44no—
32C2×C4⋊#6923—44no—
33C2×C4⋊#6923C2318yes—
34C2×C4⋊#6923C2318yes—
35C2×C4⋊#6923C23116yes—
36C2×C4×#6923—48no—
37C2×C4×#6923C23116yes—
38(C2×#6923)⋊C4C23116yes—
39(C2×#6923)⋊C4C23116yes—
40(C2×#6923)⋊C4C23116yes—
41(C2×#6923)⋊C4C23116yes—
42C22×C4×D61C2312yes—
43C22×C4×D61C2311yescharacteristic
44C22×C4×D61C2314yes—
45#14519769C2×C418yescomplemented
46#14519768C2×C4116yescomplemented
47#14519767C2×C4132yescomplemented
48C2×C4⋊Dic61C2×C414yes—
49C2×C4⋊Dic61C2×C412yes—
50C2×#11758196C2×C414yes—
51C2×#11758196C2×C414yes—
52C2×#11758196C2×C414yes—
53C4⋊C4×D61C2×C418yes—
54C4⋊C4×D61C2×C4116yes—
55C4⋊C4×D61C2318yes—
56C4⋊C4×D61C23116yes—
57C2×C4×Dic61C2×C412yes—
58C2×C4×Dic61C2×C411yescharacteristic
59C42×D61C2×C414yes—
60C42×D61C2314yes—
61#125×C61C2×C418yescomplemented
62C2×C4⋊C4×C61C2×C414yes—
63C2×C4⋊C4×C61C2×C412yes—
64C2×C42×C61C2×C411yescharacteristic
65C23×D61C2411yescharacteristic
66C22×#6923—28no—
67C22×#6923—28no—
68C22×#6923—28no—
69C22×#6923—28no—
70C22×#6923—28no—
71C22×#6923—28no—
72C22×#6923C2418yes—
73C22×#6923C2418yes—
74C22×C4×C61C22×C412yes—
75C22×C4×C61C22×C411yescharacteristic
76C22×C4×C61C22×C414yes—
77C22×Dic61C22×C411yescharacteristic
78C22×Dic61C22×C412yes—
79C22×Dic61C22×C414yes—
80C2×C4×D61—28no—
81C2×C4×D61—24no—
82C2×C4×D61—22no—
83C2×C4×D61—22no—
84C2×C4×D61—24no—
85C2×C4×D61—28no—
86C2×C4×D61C22×C412yes—
87C2×C4×D61C22×C414yes—
88C2×C4×D61C22×C418yes—
89C2×C4×D61C2412yes—
90C2×C4×D61C2418yes—
91C2×C4×D61C2414yes—
92C4⋊#6923—416no—
93C4⋊#6923—48no—
94C4⋊#6923—48no—
95C4×#6923—416no—
96C4⋊C4×C61C22×C4116yes—
97C4⋊C4×C61C22×C418yes—
98C42×C61C22×C414yes—
99C4⋊Dic61C22×C4116yes—
100C4⋊Dic61C22×C418yes—
101#11758196C22×C4116yes—
102#11758196C22×C4116yes—
103#11758196C22×C4116yes—
104C4×Dic61C22×C418yes—
105C4×Dic61C22×C414yes—
106C23×C61C23×C411yescharacteristic
107C22×D61C23×C411yescharacteristic
108C22×D61H(5,𝔽2)14yes—
109C22×D612−1+412yes—
110C22×D612−1+412yes—
111C22×D612−1+414yes—
112C22×D61C2511yescharacteristic
113C2×#6923—28no—
114C2×#6923—28no—
115C2×#6923—28no—
116C2×#6923—28no—
117C2×#6923—28no—
118C2×#6923—28no—
119C2×#6923—416no—
120C2×#6923—416no—
121C2×C4×C61—22no—
122C2×C4×C61—28no—
123C2×C4×C61—24no—
124C2×C4×C61C23×C414yes—
125C2×C4×C61C23×C412yes—
126C2×C4×C61C23×C418yes—
127C2×Dic61—22no—
128C2×Dic61—28no—
129C2×Dic61—24no—
130C2×Dic61C23×C412yes—
131C2×Dic61C23×C414yes—
132C2×Dic61C23×C418yes—
133C4×D61—24no—
134C4×D61—216no—
135C4×D61—28no—
136C4×D61—28no—
137C4×D61—24no—
138C4×D61—22no—
139C4×D61—22no—
140C4×D61—28no—
141C4×D61—24no—
142C2×#2382—611nomaximal, Sylow
143C22×C61#51912yes—
144C22×C61#52014yes—
145C22×C61C24×C411yescharacteristic
146C2×D61#51912yes—
147C2×D61#52014yes—
148C2×D61#56214yes—
149C2×D61#56318yes—
150C2×D61#56414yes—
151C2×D61C2×H(5,𝔽2)12yes—
152C2×D61C2×2−1+411yescharacteristic
153C2×D61C2×2−1+411yescharacteristic
154C2×D61C2×2−1+412yes—
155#6923—48no—
156#6923—48no—
157C4×C61—22no—
158C4×C61—28no—
159C4×C61—24no—
160Dic61—22no—
161Dic61—28no—
162Dic61—24no—
163C22×#125—611no—
164#2382—6116no—
165C2×#389—614no—
166C2×#387—614no—
167C2×#387—614no—
168C2×#382—612no—
169C2×C61#238214yescomplemented
170C2×C61C2×#51911yescharacteristic
171C2×C61C2×#52012yes—
172D61#238214yescomplemented
173D61C2×#56211yescharacteristic
174D61C2×#56312yes—
175D61C2×#56411yescharacteristic
176C2×#125—618no—
177C2×#125—618no—
178C2×#125—614no—
179C2×#125—612no—
180C2×#125—611no—
181C2×#125#692311yescharacteristic, complemented
182C22×C4⋊C4—612no—
183C22×C4⋊C4—614no—
184C22×C4⋊C4—612no—
185C22×C4⋊C4—612no—
186C22×C4⋊C4—614no—
187C22×C42—611no—
188C22×C42—614no—
189#389—6132no—
190#387—6132no—
191#387—6132no—
192#382—6116no—
193C2×#95—614no—
194C2×#95—614no—
195C2×#95—614no—
196C2×#95—614no—
197C61C2×#238211yescharacteristic, complemented, Sylow
198C23×C4—612no—
199C23×C4—614no—
200C23×C4—611no—
201C23×C4—614no—
202C23×C4—614no—
203#125—6132no—
204#125—6116no—
205#125—618no—
206#125C2×#692318yescomplemented
207C2×C4⋊C4—614no—
208C2×C4⋊C4—614no—
209C2×C4⋊C4—614no—
210C2×C4⋊C4—614no—
211C2×C4⋊C4—612no—
212C2×C4⋊C4—618no—
213C2×C4⋊C4—6116no—
214C2×C4⋊C4—618no—
215C2×C4⋊C4—6116no—
216C2×C4⋊C4—618no—
217C2×C4⋊C4—618no—
218C2×C4⋊C4—6116no—
219C2×C4⋊C4—2448no—
220C2×C4⋊C4—2444no—
221C2×C4⋊C4—2444no—
222C2×C4⋊C4C2×#692312yes—
223C2×C4⋊C4C2×#692314yes—
224C2×C42—612no—
225C2×C42—611no—
226C2×C42—614no—
227C2×C42—614no—
228C2×C42—6116no—
229C2×C42—2448no—
230C2×C42C2×#692311yescharacteristic
231#95—6116no—
232#95—6116no—
233#95—6116no—
234#95—6116no—
235C24—611no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—614no—
239C22×C4—612no—
240C22×C4—614no—
241C22×C4—618no—
242C22×C4—614no—
243C22×C4—612no—
244C22×C4—618no—
245C22×C4—618no—
246C22×C4—618no—
247C22×C4—1228no—
248C22×C4—1224no—
249C22×C4—1222no—
250C22×C4—1222no—
251C22×C4—1228no—
252C22×C4—1224no—
253C22×C4—1228no—
254C22×C4—1228no—
255C22×C4—1228no—
256C22×C4—1228no—
257C22×C4—1228no—
258C22×C4—1228no—
259C22×C4C22×#692311yescharacteristic
260C22×C4C22×#692312yes—
261C22×C4C22×#692314yes—
262C4⋊C4—6116no—
263C4⋊C4—6116no—
264C4⋊C4—6116no—
265C4⋊C4—6116no—
266C4⋊C4—618no—
267C4⋊C4—24416no—
268C4⋊C4—2448no—
269C4⋊C4—2448no—
270C4⋊C4C22×#692318yes—
271C4⋊C4C22×#6923116yes—
272C42—618no—
273C42—614no—
274C42—24416no—
275C42C22×#692314yes—
276C23—612no—
277C23—612no—
278C23—614no—
279C23—614no—
280C23—611no—
281C23—611no—
282C23C23×#692311yescharacteristic
283C2×C4—22no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—612no—
287C2×C4—614no—
288C2×C4—618no—
289C2×C4—1228no—
290C2×C4—1224no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1224no—
294C2×C4—1228no—
295C2×C4—1224no—
296C2×C4—1228no—
297C2×C4—12216no—
298C2×C4—1222no—
299C2×C4—1224no—
300C2×C4—1228no—
301C2×C4—1228no—
302C2×C4—1228no—
303C2×C4—1228no—
304C2×C4—1228no—
305C2×C4—1228no—
306C2×C4—1228no—
307C2×C4—24416no—
308C2×C4—24416no—
309C2×C4C23×#692312yes—
310C2×C4C23×#692314yes—
311C2×C4C23×#692318yes—
312C22—611no—
313C22—611no—
314C22—612no—
315C22—612no—
316C22—612no—
317C22—614no—
318C22—614no—
319C22—618no—
320C22—614no—
321C22#1684083012yes—
322C22#1684083114yes—
323C22C24×#692311yescharacteristic
324C4—22no—
325C4—28no—
326C4—24no—
327C4—1222no—
328C4—1224no—
329C4—1228no—
330C4—2448no—
331C4—2448no—
332C2—611no—
333C2—612no—
334C2—614no—
335C2—611no—
336C2#2146236514yesdirect product factor
337C2C2×#1684083011yescharacteristic
338C2C2×#1684083112yes—
subgroupG/H
centerC23C23×#6923
commutatorC22×C61C24×C4
FrattiniC22C24×#6923
FittingC2×#125×C61C4
socleC23×C61C23×C4
prime radicalC23×D61C24
cosocleC22×D61C25
hypercenterC2×#125#6923
hypocenterC61C2×#2382
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #21462365C2direct (N × H)
19. C2×#14519769C4semidirect
20. C2×#14519768C4semidirect
21. C2×#14519767C4semidirect
22. #125×D61C4semidirect
27. C2×#125×C61C4semidirect
45. #14519769C2×C4semidirect
46. #14519768C2×C4semidirect
47. #14519767C2×C4semidirect
61. #125×C61C2×C4semidirect
169. C2×C61#2382semidirect
172. D61#2382semidirect
181. C2×#125#6923semidirect
197. C61C2×#2382semidirect
206. #125C2×#6923semidirect
336. C2#21462365direct (N × H)

Maximal subgroups

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

subgroupindex
#214623652
C2×#168407132
C2×#168407112
C2×#168407062
C2×#125×D612
C2×#238261