Finite Group Databasebeta

← C2×#21458461 #30559577

Subgroups — C2×#21458461 #30559577

Order 15616.

2058 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840491C211yesmaximal, characteristic, complemented
2#21458461C2116yesmaximal, direct product factor
3C2×#16840114C213yesmaximal, complemented
4C2×#16840044C213yesmaximal, complemented
5C2×#16839967C213yesmaximal
6C2×Q8×#92418C211yesmaximal, characteristic, complemented
7C2×C4×#11758200C213yesmaximal
8C2×D4×Q8×C61C211yesmaximal, characteristic
9C22×#11758200C2213yes—
10#16840491—24no—
11#16840491C2218yescomplemented
12C2×#14519777C2213yes—
13C2×#14519768—23no—
14C2×Q8×Dic61—21no—
15C2×C4×Dic122C2213yes—
16#16840114C22124yescomplemented
17#16840044C22124yescomplemented
18#16839967C22124yes—
19Q8×#92418C2218yescomplemented
20C4×#11758200C22124yes—
21C2×#14519708C2213yes—
22C2×C4⋊#92418C2213yes—
23C2×#14519700C2213yes—
24C2×#14519700C2216yes—
25C2×#14519698C2211yescharacteristic
26C2×C4×#92418C2213yes—
27C23×Q8×C61—21no—
28D4×Q8×C61—24no—
29D4×Q8×C61C2218yes—
30C2×C4⋊Q8×C61C2213yes—
31C2×C22⋊Q8×C61—23no—
32C2×C4×Q8×C61C2211yescharacteristic
33C2×C4×D4×C61C2213yes—
34C22×Dic122C2313yes—
35C2×#11758200—212no—
36C2×#11758200C23112yes—
37C22×#92418C2311yescharacteristic
38C22×#92418C2313yes—
39#14519777—26no—
40#14519777C23112yes—
41#14519768—224no—
42Q8×Dic61—28no—
43C4×Dic122—26no—
44C4×Dic122C23112yes—
45C2×C4⋊Dic61—23no—
46C2×C4⋊Dic61—23no—
47C2×C4⋊Dic61C2311yescharacteristic
48C2×#11758196—23no—
49C2×C4×Dic61—23no—
50#14519708C23112yes—
51C4⋊#92418C23112yes—
52#14519700C23124yes—
53#14519700C23112yes—
54#14519698C2314yes—
55C4×#92418C23112yes—
56C22×Q8×C61—28no—
57C22×Q8×C61—23no—
58C22×Q8×C61—48no—
59C22×Q8×C61D411yescharacteristic
60C22×D4×C61C2311yescharacteristic
61C23×C4×C61—23no—
62C4⋊Q8×C61—26no—
63C4⋊Q8×C61C23112yes—
64C22⋊Q8×C61—224no—
65C4×Q8×C61D414yes—
66C4×Q8×C61C2314yes—
67C4×D4×C61—26no—
68C4×D4×C61C23112yes—
69C2×C4⋊C4×C61—23no—
70C2×C4⋊C4×C61D413yes—
71C2×C4⋊C4×C61C2313yes—
72C2×C22⋊C4×C61—23no—
73C2×C42×C61C2313yes—
74C24×C61—21no—
75C2×Q8×C61—212no—
76C2×Q8×C61—432no—
77C2×Q8×C61—412no—
78C2×Q8×C61SD1618yescomplemented
79C2×Q8×C61C2×D414yes—
80C2×D4×C61—21no—
81C2×D4×C61—22no—
82C2×D4×C61#4114yes—
83C2×D4×C61C2412yes—
84C22×C4×C61—23no—
85C22×C4×C61—212no—
86C22×C4×C61—412no—
87C22×C4×C61C2×D413yes—
88C22×C4×C61C2×D413yes—
89C22×C4×C61C2411yescharacteristic
90C22×Dic61—23no—
91C22×Dic61—21no—
92C2×Dic122—26no—
93C2×Dic122—26no—
94C2×Dic122—23no—
95C2×Dic122C2416yes—
96C4⋊C4×C61—212no—
97C4⋊C4×C61—23no—
98C4⋊C4×C61C2×D4112yes—
99C4⋊C4×C61C2416yes—
100C22⋊C4×C61—212no—
101C42×C61C2×D416yes—
102C42×C61C2416yes—
103#11758200—224no—
104C4⋊Dic61—212no—
105C4⋊Dic61—212no—
106C4⋊Dic61—21no—
107C4⋊Dic61C2412yes—
108#11758196—212no—
109C4×Dic61—212no—
110C2×#92418—26no—
111C2×#92418—22no—
112C2×#92418C2412yes—
113C2×#92418C2416yes—
114C23×C61—24no—
115C23×C61—22no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61C22×D411yescharacteristic
119Q8×C61—424no—
120Q8×C61C2×SD16116yescomplemented
121D4×C61—24no—
122D4×C61—22no—
123D4×C61C2×#4114yes—
124C2×C4×C61—26no—
125C2×C4×C61—26no—
126C2×C4×C61—46no—
127C2×C4×C61—424no—
128C2×C4×C61C2×SD16112yes—
129C2×C4×C61C22×D416yes—
130C2×C4×C61C22×D411yescharacteristic
131C2×C4×C61C22×D416yes—
132C2×C4×C61C2×#4112yes—
133C2×C4×C612−1+412yes—
134C2×C4×C61C2511yescharacteristic
135C2×Dic61—22no—
136C2×Dic61—26no—
137C2×Dic61—46no—
138C2×Dic61—42no—
139Dic122—26no—
140Dic122—26no—
141Dic122—46no—
142#92418—26no—
143#92418—22no—
144C2×#3359—611nomaximal, Sylow
145C22×C61—24no—
146C22×C61—48no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61#54812yes—
150C22×C61C22×SD1612yes—
151C22×C61#57812yes—
152C22×C61C23×D411yescharacteristic
153C4×C61—26no—
154C4×C61—46no—
155C4×C61#54812yes—
156C4×C61C22×SD16112yes—
157C4×C61C22×#4111yescharacteristic
158C4×C61C2×2−1+411yescharacteristic
159Dic61—46no—
160Dic61—42no—
161C2×#557—611no—
162C2×D4×Q8D6111yescharacteristic
163#3359—6116no—
164C2×#487—613no—
165C2×#475—613no—
166C2×#461—613no—
167C2×C8×Q8—611no—
168C2×C4×SD16—613no—
169C2×C61—44no—
170C2×C61#335914yescomplemented
171C2×C61C2×#54811yescharacteristic
172C2×C61C23×SD1611yescharacteristic
173C2×C61C2×#57811yescharacteristic
174C23×Q8—21no—
175C22×SD16—613no—
176#557—618no—
177#557—1224no—
178D4×Q8—24no—
179D4×Q8C2×D6118yes—
180C2×C4⋊Q8—613no—
181C2×C4⋊Q8C2×D6113yes—
182C2×#125—1223no—
183C2×C22⋊Q8—23no—
184C2×C4×Q8—613no—
185C2×C4×Q8—1221no—
186C2×C4×Q8C2×D6111yescharacteristic
187C2×C4×D4C2×D6113yes—
188#487—6124no—
189#475—6124no—
190#461—6124no—
191C8×Q8—618no—
192C4×SD16—6124no—
193C2×#106—613no—
194C2×C4⋊C8—613no—
195C2×#103—616no—
196C2×#103—613no—
197C2×#102—611no—
198C2×C4×C8—613no—
199C61C2×#335911yescharacteristic, complemented, Sylow
200C22×Q8—28no—
201C22×Q8—23no—
202C22×Q8—48no—
203C22×Q8—613no—
204C22×Q8C22⋊D6111yescharacteristic
205C22×D4C22×D6111yescharacteristic
206C23×C4—23no—
207C2×SD16—6112no—
208C2×SD16—12212no—
209C22×C8—613no—
210C22×C8—611no—
211C4⋊Q8—26no—
212C4⋊Q8—6112no—
213C4⋊Q8—1226no—
214C4⋊Q8C22×D61112yes—
215#125—12224no—
216C22⋊Q8—224no—
217C4×Q8—6112no—
218C4×Q8—1228no—
219C4×Q8—1226no—
220C4×Q8C22⋊D6114yes—
221C4×Q8C22×D6114yes—
222C4×D4—26no—
223C4×D4C22×D61112yes—
224C2×C4⋊C4—23no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—1223no—
227C2×C4⋊C4—1223no—
228C2×C4⋊C4—1223no—
229C2×C4⋊C4C22⋊D6113yes—
230C2×C4⋊C4C22×D6113yes—
231C2×C22⋊C4—23no—
232C2×C42—1223no—
233C2×C42C22×D6113yes—
234#106—6112no—
235C4⋊C8—6112no—
236#103—6112no—
237#103—6124no—
238#102—614no—
239C4×C8—6112no—
240C24—21no—
241C2×Q8—212no—
242C2×Q8—432no—
243C2×Q8—412no—
244C2×Q8—616no—
245C2×Q8—1223no—
246C2×Q8—1226no—
247C2×Q8—1226no—
248C2×Q8#1175820018yescomplemented
249C2×Q8C2×C22⋊D6114yes—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4#1175822614yes—
253C2×D4C23×D6112yes—
254C22×C4—23no—
255C22×C4—212no—
256C22×C4—412no—
257C22×C4—1221no—
258C22×C4—1223no—
259C22×C4C2×C22⋊D6113yes—
260C22×C4C2×C22⋊D6113yes—
261C22×C4C23×D6111yescharacteristic
262SD16—12224no—
263C2×C8—612no—
264C2×C8—616no—
265C2×C8—1226no—
266C2×C8—1222no—
267C4⋊C4—212no—
268C4⋊C4—23no—
269C4⋊C4—612no—
270C4⋊C4—1221no—
271C4⋊C4—12212no—
272C4⋊C4—12212no—
273C4⋊C4—12212no—
274C4⋊C4C2×C22⋊D61112yes—
275C4⋊C4C23×D6116yes—
276C22⋊C4—212no—
277C42—12212no—
278C42C2×C22⋊D6116yes—
279C42C23×D6116yes—
280C23—22no—
281C23—24no—
282C23—42no—
283C23—42no—
284C23C22×C22⋊D6111yescharacteristic
285Q8—424no—
286Q8—1226no—
287Q8—1226no—
288Q8—2446no—
289Q8C2×#11758200116yescomplemented
290D4—22no—
291D4—24no—
292D4C2×#1175822614yes—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—46no—
296C2×C4—424no—
297C2×C4—1222no—
298C2×C4—1226no—
299C2×C4—2442no—
300C2×C4—2446no—
301C2×C4C2×#11758200112yes—
302C2×C4C22×C22⋊D6116yes—
303C2×C4C22×C22⋊D6116yes—
304C2×C4C22×C22⋊D6111yescharacteristic
305C2×C4C2×#1175822612yes—
306C2×C4#1451982512yes—
307C2×C4C24×D6111yescharacteristic
308C8—1226no—
309C8—1222no—
310C22—24no—
311C22—48no—
312C22—42no—
313C22—42no—
314C22#1684042312yes—
315C22C22×#1175820012yes—
316C22#1684057712yes—
317C22C23×C22⋊D6111yescharacteristic
318C4—26no—
319C4—46no—
320C4—2446no—
321C4—2442no—
322C4#1684042312yes—
323C4C22×#11758200112yes—
324C4C22×#1175822611yescharacteristic
325C4C2×#1451982511yescharacteristic
326C2—44no—
327C2#2145846114yesdirect product factor
328C2C2×#1684042311yescharacteristic
329C2C23×#1175820011yescharacteristic
330C2C2×#1684057711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×D4×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×D4×Q8D61
hypocenterC61C2×#3359
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840491C2semidirect
2. #21458461C2direct (N × H)
3. C2×#16840114C2semidirect
4. C2×#16840044C2semidirect
6. C2×Q8×#92418C2semidirect
11. #16840491C22semidirect
16. #16840114C22semidirect
17. #16840044C22semidirect
19. Q8×#92418C22semidirect
78. C2×Q8×C61SD16semidirect
120. Q8×C61C2×SD16semidirect
170. C2×C61#3359semidirect
199. C61C2×#3359semidirect
248. C2×Q8#11758200semidirect
289. Q8C2×#11758200semidirect
327. C2#21458461direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404912
#214584612
C2×#168401142
C2×#168400442
C2×#168399672
C2×Q8×#924182
C2×C4×#117582002
C2×D4×Q8×C612
C2×#335961