Finite Group Databasebeta

← C2×#21458211 #30558474

Subgroups — C2×#21458211 #30558474

Order 15616.

1942 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C212yesmaximal
2#21458211C2116yesmaximal, direct product factor
3C2×#16840099C214yesmaximal, complemented
4C2×#16840054C212yesmaximal, complemented
5C2×#16840004C212yesmaximal, complemented
6C2×C4×Dic244C211yesmaximal, characteristic, complemented
7C2×C8×D244C211yesmaximal, characteristic
8C2×#107×D61C212yesmaximal, complemented
9C2×C61×#500C211yesmaximal, characteristic, complemented
10C22×Dic244C2211yescharacteristic
11C22×C8×D61C2212yes—
12#16840473—28no—
13#16840473C22116yes—
14C2×#14519768—22no—
15C2×#14519757C2214yes—
16C2×#14519753—24no—
17C2×C4×Dic122C2212yes—
18C2×C4×D244C2211yescharacteristic
19C2×#14519739—22no—
20C2×C4⋊C4×D61C2214yes—
21#16840099C22132yescomplemented
22#16840054C22116yescomplemented
23#16840004C22116yescomplemented
24C4×Dic244C2218yescomplemented
25C8×D244C2218yes—
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2214yes—
28#107×D61C22116yescomplemented
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519702C2212yes—
31C2×#14519700C2214yes—
32C2×#14519689C2212yes—
33C2×C4⋊Q8×C61C2212yes—
34C61×#500C2218yescomplemented
35C2×#107×C61C2212yes—
36C2×#107×C61C2212yes—
37C2×C4×C8×C61C2211yescharacteristic
38C22×Dic122C2312yes—
39C22×D244D411yescharacteristic
40C22×C4×D61—24no—
41C22×C4×D61C2312yes—
42C2×Dic244—24no—
43C2×Dic244C2314yes—
44C22×#92418C2312yes—
45C2×C8×D61—28no—
46C2×C8×D61C2318yes—
47#14519768—216no—
48#14519757—28no—
49#14519757C23116yes—
50#14519753—232no—
51C4×Dic122—24no—
52C4×Dic122C2318yes—
53C4×D244D414yes—
54C4×D244C2314yes—
55#14519739—216no—
56C2×C4⋊Dic61—24no—
57C2×C4⋊Dic61D411yescharacteristic
58C2×C4⋊Dic61C2312yes—
59C2×#11758196—22no—
60C2×#11758196—24no—
61C4⋊C4×D61—28no—
62C4⋊C4×D61C23116yes—
63C2×#11758198—24no—
64C2×#11758198D412yes—
65C2×C4×Dic61—22no—
66#14519711C2314yes—
67#14519710C23116yes—
68C4⋊#92418C2314yes—
69#14519702C2318yes—
70#14519700C23116yes—
71#14519689C2318yes—
72C22×Q8×C61—22no—
73C22×C8×C61C2311yescharacteristic
74C22×C8×C61C2311yescharacteristic
75C4⋊Q8×C61—24no—
76C4⋊Q8×C61C2318yes—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61C2314yes—
79C2×C42×C61C2311yescharacteristic
80#107×C61C2318yes—
81#107×C61C2318yes—
82C4×C8×C61C2314yes—
83C23×D61C2×D412yes—
84C2×Q8×C61—28no—
85C2×Q8×C61—84no—
86C22×C4×C61—24no—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2411yescharacteristic
90C22×Dic61—22no—
91C22×Dic61—22no—
92C22×Dic61C2×D412yes—
93C2×D244Q1618yescomplemented
94C2×D244C2×D414yes—
95C2×C4×D61—24no—
96C2×C4×D61—24no—
97C2×C4×D61—216no—
98C2×C4×D61—88no—
99C2×C4×D61C2×D414yes—
100C2×C4×D61C2414yes—
101C2×Dic122—24no—
102C2×Dic122—24no—
103C2×Dic122—22no—
104C2×Dic122C2414yes—
105C2×C8×C61—22no—
106C2×C8×C61#4114yes—
107C2×C8×C61C2412yes—
108C2×C8×C61C2412yes—
109C4⋊C4×C61—28no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61C2418yes—
112C42×C61C2×D412yes—
113C42×C61C2412yes—
114#11758198—216no—
115#11758198C2×D418yes—
116C4⋊Dic61—22no—
117C4⋊Dic61—216no—
118C4⋊Dic61C2×D414yes—
119C4⋊Dic61C2414yes—
120#11758196—28no—
121#11758196—216no—
122C4×Dic61—28no—
123C2×#92418—24no—
124C2×#92418C2414yes—
125Dic244—28no—
126C8×D61—216no—
127C23×C61C22×D411yescharacteristic
128C22×D61—24no—
129C22×D61—24no—
130C22×D61C2×Q1618yes—
131C22×D61C22×D414yes—
132Q8×C61—88no—
133C2×C4×C61—22no—
134C2×C4×C61—28no—
135C2×C4×C61—48no—
136C2×C4×C61C2×Q1614yes—
137C2×C4×C61C22×D412yes—
138C2×C4×C61C22×D411yescharacteristic
139C2×C4×C61C22×D412yes—
140C2×C4×C61C2×#4112yes—
141C2×C4×C612−1+412yes—
142C2×C4×C61C2511yescharacteristic
143C2×Dic61—24no—
144C2×Dic61—24no—
145C2×Dic61—24no—
146C2×Dic61—44no—
147C2×Dic61—44no—
148C2×Dic61C22×D414yes—
149D244C2×Q16116yescomplemented
150C4×D61—28no—
151C4×D61—24no—
152C4×D61—24no—
153C4×D61—816no—
154Dic122—24no—
155Dic122—24no—
156Dic122—44no—
157C8×C61—22no—
158C8×C61C2×#4114yes—
159#92418—24no—
160C2×#3338—611nomaximal, Sylow
161C22×C61#54712yes—
162C22×C61C22×Q1612yes—
163C22×C61#57612yes—
164C22×C61C23×D411yescharacteristic
165C2×D61—216no—
166C2×D61—24no—
167C2×D61—24no—
168C2×D61C22×Q1618yes—
169C4×C61—22no—
170C4×C61—48no—
171C4×C61#54712yes—
172C4×C61C22×Q1614yes—
173C4×C61C22×#4111yescharacteristic
174C4×C61C2×2−1+411yescharacteristic
175Dic61—24no—
176Dic61—44no—
177Dic61—44no—
178C2×#554—612no—
179#3338—6116no—
180C2×#500D6111yescharacteristic, complemented
181C2×#484—614no—
182C2×#477—612no—
183C2×#467—612no—
184C2×C4×Q16—611no—
185C2×C8×D4—611no—
186C22×#107—612no—
187C2×C61#333814yescomplemented
188C2×C61C2×#54711yescharacteristic
189C2×C61C23×Q1611yescharacteristic
190C2×C61C2×#57611yescharacteristic
191D61—28no—
192C22×Q16—611no—
193C23×C8—612no—
194#554—6116no—
195#554—1228no—
196C2×C4⋊Q8C2×D6112yes—
197C2×#125—1222no—
198C2×C22⋊Q8—614no—
199C2×C22⋊Q8—1224no—
200C2×C4×Q8—612no—
201C2×C4×D4—611no—
202C2×C4×D4—1222no—
203C22×C4⋊C4—614no—
204#500C2×D6118yescomplemented
205#484—6132no—
206#477—6116no—
207#467—6116no—
208C4×Q16—618no—
209C8×D4—618no—
210C2×#107—611no—
211C2×#107—614no—
212C2×#107—6116no—
213C2×#107C2×D6112yes—
214C2×#107C2×D6112yes—
215C2×C4⋊C8—611no—
216C2×#103—612no—
217C2×#103—614no—
218C2×C22⋊C8—612no—
219C2×C4×C8C2×D6111yescharacteristic
220C61C2×#333811yescharacteristic, complemented, Sylow
221C22×Q8—22no—
222C22×Q8—612no—
223C22×D4—611no—
224C23×C4—612no—
225C23×C4—1224no—
226C2×Q16—614no—
227C2×Q16—1224no—
228C22×C8—612no—
229C22×C8—618no—
230C22×C8—1228no—
231C22×C8C22×D6111yescharacteristic
232C22×C8C22×D6111yescharacteristic
233C4⋊Q8—24no—
234C4⋊Q8C22×D6118yes—
235#125—12216no—
236C22⋊Q8—6116no—
237C22⋊Q8—12232no—
238C22⋊Q8—1228no—
239C4×Q8—618no—
240C4×Q8—1224no—
241C4×D4—614no—
242C4×D4—614no—
243C4×D4—12216no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—612no—
247C2×C4⋊C4—6116no—
248C2×C4⋊C4—1228no—
249C2×C4⋊C4—1224no—
250C2×C4⋊C4—1224no—
251C2×C4⋊C4—1222no—
252C2×C4⋊C4C22×D6114yes—
253C2×C22⋊C4—612no—
254C2×C22⋊C4—1224no—
255C2×C42—1222no—
256C2×C42C22×D6111yescharacteristic
257#107—614no—
258#107—6116no—
259#107C22×D6118yes—
260#107C22×D6118yes—
261C4⋊C8—614no—
262#103—618no—
263#103—6116no—
264C22⋊C8—618no—
265C4×C8C22×D6114yes—
266C24—612no—
267C2×Q8—28no—
268C2×Q8—84no—
269C2×Q8—614no—
270C2×Q8—1222no—
271C2×Q8—1224no—
272C2×Q8—1224no—
273C2×D4—614no—
274C2×D4—618no—
275C22×C4—24no—
276C22×C4—612no—
277C22×C4—614no—
278C22×C4—614no—
279C22×C4—12216no—
280C22×C4—1224no—
281C22×C4—1224no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—4888no—
285C22×C4D4×D6111yescharacteristic
286C22×C4D4×D6111yescharacteristic
287C22×C4C23×D6111yescharacteristic
288Q16—1228no—
289C2×C8—22no—
290C2×C8—614no—
291C2×C8—1224no—
292C2×C8—12216no—
293C2×C8#1175822614yes—
294C2×C8C23×D6112yes—
295C2×C8C23×D6112yes—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—614no—
299C4⋊C4—614no—
300C4⋊C4—12216no—
301C4⋊C4—12216no—
302C4⋊C4—1222no—
303C4⋊C4—1228no—
304C4⋊C4C23×D6118yes—
305C22⋊C4—618no—
306C22⋊C4—12216no—
307C42—1228no—
308C42D4×D6112yes—
309C42C23×D6112yes—
310C23—618no—
311C23—614no—
312C23—1224no—
313C23—1224no—
314C23C2×D4×D6111yescharacteristic
315Q8—88no—
316Q8—1224no—
317Q8—1224no—
318Q8—2444no—
319D4—6116no—
320C2×C4—22no—
321C2×C4—28no—
322C2×C4—48no—
323C2×C4—614no—
324C2×C4—1224no—
325C2×C4—1224no—
326C2×C4—1224no—
327C2×C4—1228no—
328C2×C4—1224no—
329C2×C4—1224no—
330C2×C4—48816no—
331C2×C4—2444no—
332C2×C4—2444no—
333C2×C4Q16×D6114yes—
334C2×C4C2×D4×D6112yes—
335C2×C4C2×D4×D6112yes—
336C2×C4C2×D4×D6111yescharacteristic
337C2×C4C2×#1175822612yes—
338C2×C4#1451982512yes—
339C2×C4C24×D6111yescharacteristic
340C8—22no—
341C8—1224no—
342C8C2×#1175822614yes—
343C22—618no—
344C22—12216no—
345C22—1224no—
346C22—1224no—
347C22#1684041512yes—
348C22C2×Q16×D6112yes—
349C22#1684056312yes—
350C22C22×D4×D6111yescharacteristic
351C4—22no—
352C4—48no—
353C4—1224no—
354C4—2444no—
355C4—2444no—
356C4#1684041512yes—
357C4C2×Q16×D6114yes—
358C4C22×#1175822611yescharacteristic
359C4C2×#1451982511yescharacteristic
360C2—1228no—
361C2#2145821114yesdirect product factor
362C2C2×#1684041511yescharacteristic
363C2C22×Q16×D6111yescharacteristic
364C2C2×#1684056311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#500C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenterC2×#500D61
hypocenterC61C2×#3338
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
2. #21458211C2direct (N × H)
3. C2×#16840099C2semidirect
4. C2×#16840054C2semidirect
5. C2×#16840004C2semidirect
6. C2×C4×Dic244C2semidirect
8. C2×#107×D61C2semidirect
9. C2×C61×#500C2semidirect
21. #16840099C22semidirect
22. #16840054C22semidirect
23. #16840004C22semidirect
24. C4×Dic244C22semidirect
28. #107×D61C22semidirect
34. C61×#500C22semidirect
93. C2×D244Q16semidirect
149. D244C2×Q16semidirect
180. C2×#500D61semidirect
187. C2×C61#3338semidirect
204. #500C2×D61semidirect
220. C61C2×#3338semidirect
361. C2#21458211direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404732
#214582112
C2×#168400992
C2×#168400542
C2×#168400042
C2×C4×Dic2442
C2×C8×D2442
C2×#107×D612
C2×C61×#5002
C2×#333861