Finite Group Databasebeta

← C2×#21462101 #30623966

Subgroups — C2×#21462101 #30623966

Order 15616.

1230 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 224 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
1C2×#16840741C212yesmaximal
2#21462101C218yesmaximal, direct product factor
3C2×#16840668C214yesmaximal, complemented
4C2×#16840336C211yesmaximal, characteristic
5C2×#14519856—22no—
6#16840741—24no—
7#16840741C2218yes—
8C2×#14519836C2214yes—
9C2×#14519835—22no—
10#16840668C22116yescomplemented
11C22×#11758222C2211yescharacteristic
12#16840336—24no—
13#16840336C414yescomplemented
14#16840336C2214yes—
15C2×#14519775C412yescomplemented
16C2×#14519766C2212yes—
17C2×#14519760—22no—
18C2×#14519753—22no—
19C2×#14519736C411yescharacteristic, complemented
20C23×Q8×C61C411yescharacteristic, complemented
21#14519856—28no—
22#14519856—44no—
23C2×C4⋊#6923—42no—
24C2×C22⋊#6923—24no—
25C2×C4×#6923—42no—
26#14519836—24no—
27#14519836C2318yes—
28#14519835—28no—
29C2×#11758222—42no—
30C2×#11758222—42no—
31C2×#11758222C2×C412yes—
32C2×#11758222D414yes—
33C2×#11758222C2312yes—
34C22×Dic122C2×C411yescharacteristic
35C22×D244C2311yescharacteristic
36C22×C22⋊D61D412yes—
37C22×C4×D61D412yes—
38#14519775—24no—
39#14519775C2×C418yescomplemented
40#14519766—24no—
41#14519766C2×C414yes—
42#14519766C2314yes—
43#14519760—28no—
44#14519760—28no—
45#14519753—28no—
46#14519753—28no—
47#14519736—22no—
48#14519736C2×C414yescomplemented
49C2×C4⋊Dic61—21no—
50C2×#11758196—24no—
51C2×C22⋊Dic61—21no—
52C2×#11758198—24no—
53C2×C4×Dic61C2×C412yes—
54C22×Q8×C61—21no—
55C22×Q8×C61—22no—
56C22×Q8×C61—42no—
57C22×Q8×C61—42no—
58C22×Q8×C61C2×C412yes—
59C22×Q8×C61C2×C414yescomplemented
60C23×C4×C61—21no—
61C23×C4×C61C2×C411yescharacteristic
62C24×C61C22⋊C411yescharacteristic
63C23×D61C2×D412yes—
64C22×#6923—44no—
65C2×Q8×C61—22no—
66C2×Q8×C61—24no—
67C2×Q8×C61—48no—
68C2×Q8×C61—48no—
69C2×Q8×C61—42no—
70C2×Q8×C61—42no—
71C2×Q8×C61—42no—
72C2×Q8×C61—42no—
73C2×Q8×C61C22×C414yes—
74C22×C4×C61—22no—
75C22×C4×C61—22no—
76C22×C4×C61—22no—
77C22×C4×C61—24no—
78C22×C4×C61—42no—
79C22×C4×C61—42no—
80C22×C4×C61C22⋊C412yes—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C22×C412yes—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—42no—
86C2×C22⋊D61C22⋊C414yes—
87C2×C22⋊D61C2×D414yes—
88C22×Dic61—22no—
89C22×Dic61C22⋊C412yes—
90#11758222—44no—
91#11758222—44no—
92#11758222—44no—
93#11758222—44no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244C22×C411yescharacteristic
97C2×D244C2×D412yes—
98C2×D244C2411yescharacteristic
99C2×C4×D61—22no—
100C2×C4×D61—22no—
101C2×C4×D61—24no—
102C2×C4×D61C22⋊C414yes—
103C2×C4×D61C2×D414yes—
104C2×Dic122—41no—
105C2×Dic122—41no—
106C2×Dic122C22⋊C412yes—
107C2×Dic122C22×C412yes—
108C22⋊#6923—28no—
109C22⋊#6923—44no—
110C4⋊#6923—48no—
111C4×#6923—48no—
112C22⋊Dic61—24no—
113#11758198—28no—
114#11758198—28no—
115C4⋊Dic61—24no—
116#11758196—216no—
117C4×Dic61—22no—
118C4×Dic61C22×C414yes—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—42no—
122C23×C61C2×C22⋊C411yescharacteristic
123C23×C61C2×C22⋊C412yes—
124C22×D61—22no—
125C22×D61—22no—
126C22×D61—44no—
127C22×D61C2×C22⋊C412yes—
128C22×D61C22×D412yes—
129C2×#6923—48no—
130C2×#6923—88no—
131Q8×C61—44no—
132Q8×C61—44no—
133Q8×C61—48no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—24no—
137C2×C4×C61—24no—
138C2×C4×C61—24no—
139C2×C4×C61—48no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61#9912yes—
143C2×C4×C61#9912yes—
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C2×C4×C61C2×C22⋊C412yes—
146C2×C4×C61C23×C411yescharacteristic
147C22⋊D61—44no—
148C22⋊D61—44no—
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—24no—
152C2×Dic61—44no—
153C2×Dic61C2×C22⋊C414yes—
154D244—41no—
155D244—42no—
156D244—41no—
157C4×D61—24no—
158C4×D61—24no—
159C4×D61—24no—
160C4×D61—24no—
161Dic122—42no—
162Dic122—42no—
163C2×#2129—611nomaximal, Sylow
164C22×C61—22no—
165C22×C61—22no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61C2×#9911yescharacteristic
170C22×C61C2×#9911yescharacteristic
171C22×C61C22×C22⋊C411yescharacteristic
172C2×D61—22no—
173C2×D61—22no—
174C2×D61—44no—
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—82no—
178#6923—88no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—44no—
182C4×C61C2×#9914yes—
183Dic61—24no—
184Dic61—44no—
185C2×#536—611no—
186C2×#410—612no—
187#2129—618no—
188C2×#354—614no—
189C2×C61—21no—
190C2×C61—21no—
191C2×C61—42no—
192C2×C61#212912yescomplemented
193C2×C61C22×#9911yescharacteristic
194D61—42no—
195D61—42no—
196C22×#41—611no—
197C23×Q8#692311yescharacteristic, complemented
198#536—614no—
199#536—614no—
200#536—1224no—
201C2×C4⋊Q8—612no—
202C2×#124—612no—
203C2×#123—1222no—
204C2×C22⋊Q8—1222no—
205C2×#117—611no—
206C2×#117—1222no—
207#410—618no—
208#410—1224no—
209C2×#99—614no—
210C2×#99—1222no—
211#354—6116no—
212C61C2×#212911yescharacteristic, complemented, Sylow
213C2×#41—614no—
214C2×#41—612no—
215C2×#41—612no—
216C2×#41—2442no—
217C2×#41—2442no—
218C22×Q8—22no—
219C22×Q8—21no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8—611no—
223C22×Q8C2×#692312yes—
224C22×Q8C2×#692314yescomplemented
225C22×D4—612no—
226C22×D4—611no—
227C23×C4—21no—
228C23×C4—612no—
229C23×C4C2×#692311yescharacteristic
230C4⋊Q8—618no—
231C4⋊Q8—1224no—
232#124—614no—
233#124—614no—
234#124—1224no—
235#123—1228no—
236#123—1228no—
237C22⋊Q8—1228no—
238C22⋊Q8—1228no—
239#117—614no—
240#117—1222no—
241#117—1228no—
242#117—2444no—
243C2×C4⋊C4—1224no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—2442no—
246C2×C22⋊C4—1224no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1224no—
249C2×C42—612no—
250C2×C42—2442no—
251#99—618no—
252#99—1228no—
253#99—1224no—
254C24—612no—
255C24C22⋊#692311yescharacteristic
256#41—2444no—
257#41—2444no—
258#41—2444no—
259#41—2444no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—48no—
263C2×Q8—48no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—612no—
269C2×Q8—612no—
270C2×Q8—2441no—
271C2×Q8—2441no—
272C2×Q8C22×#692314yes—
273C2×D4—614no—
274C2×D4—614no—
275C2×D4—612no—
276C2×D4—611no—
277C2×D4—611no—
278C2×D4—2441no—
279C2×D4—2441no—
280C2×D4—2442no—
281C2×D4—2442no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—612no—
289C22×C4—614no—
290C22×C4—614no—
291C22×C4—1222no—
292C22×C4—1222no—
293C22×C4—1224no—
294C22×C4—1222no—
295C22×C4—2444no—
296C22×C4C22⋊#692312yes—
297C22×C4C22⋊#692311yescharacteristic
298C22×C4C22×#692311yescharacteristic
299C22×C4C22×#692312yes—
300C4⋊C4—12216no—
301C4⋊C4—1224no—
302C4⋊C4—2448no—
303C22⋊C4—1228no—
304C22⋊C4—1228no—
305C22⋊C4—1224no—
306C22⋊C4—1228no—
307C22⋊C4—2444no—
308C42—614no—
309C42—1222no—
310C42—2448no—
311C23—21no—
312C23—21no—
313C23—42no—
314C23—612no—
315C23—612no—
316C23—1222no—
317C23—1222no—
318C23—2444no—
319C23C2×C22⋊#692311yescharacteristic
320C23C2×C22⋊#692312yes—
321Q8—44no—
322Q8—44no—
323Q8—48no—
324Q8—2442no—
325Q8—2442no—
326D4—2442no—
327D4—2441no—
328D4—2441no—
329D4—2444no—
330D4—2444no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—48no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—614no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—1224no—
343C2×C4—1224no—
344C2×C4—1224no—
345C2×C4—1224no—
346C2×C4—1224no—
347C2×C4—4888no—
348C2×C4—2448no—
349C2×C4—2444no—
350C2×C4#1451983512yes—
351C2×C4#1451983512yes—
352C2×C4C2×C22⋊#692311yescharacteristic
353C2×C4C2×C22⋊#692312yes—
354C2×C4C23×#692311yescharacteristic
355C22—22no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—1222no—
361C22—1222no—
362C22—4882no—
363C22—2442no—
364C22—2442no—
365C22—2444no—
366C22C2×#1451983511yescharacteristic
367C22C2×#1451983511yescharacteristic
368C22C22×C22⋊#692311yescharacteristic
369C4—21no—
370C4—21no—
371C4—44no—
372C4—1224no—
373C4—4888no—
374C4—2444no—
375C4C2×#1451983514yes—
376C2—21no—
377C2—21no—
378C2—42no—
379C2—2442no—
380C2—2442no—
381C2#2146210112yesdirect product factor
382C2C22×#1451983511yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC23×Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC2×D244C24
hypercenterC23×Q8#6923
hypocenterC61C2×#2129
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #21462101C2direct (N × H)
3. C2×#16840668C2semidirect
10. #16840668C22semidirect
13. #16840336C4semidirect
15. C2×#14519775C4semidirect
19. C2×#14519736C4semidirect
20. C23×Q8×C61C4semidirect
39. #14519775C2×C4semidirect
48. #14519736C2×C4semidirect
59. C22×Q8×C61C2×C4semidirect
192. C2×C61#2129semidirect
197. C23×Q8#6923semidirect
212. C61C2×#2129semidirect
224. C22×Q8C2×#6923semidirect
381. C2#21462101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407412
#214621012
C2×#168406682
C2×#168403362
C2×#212961