Finite Group Databasebeta

← C2×#21459440 #30587102

Subgroups — C2×#21459440 #30587102

Order 15616.

7706 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459440C2132yesmaximal, direct product factor
2C2×#16840491C211yesmaximal, characteristic, complemented
3C2×#16840486C213yesmaximal, complemented
4C2×#16840454C216yesmaximal, complemented
5C2×#16840423C212yesmaximal, complemented
6C2×#16840416C213yesmaximal, complemented
7C2×#16840317C213yesmaximal, complemented
8C2×#16840298C216yesmaximal, complemented
9C2×C4×#11758226C211yesmaximal, characteristic, complemented
10C2×C4×#11758224C213yesmaximal, complemented
11C22×Q8×Dic61C212yesmaximal, complemented
12C2×D4×Q8×C61C211yesmaximal, characteristic, complemented
13C22×#11758226C2211yescharacteristic
14C22×#11758224C2213yes—
15#16840491C22116yescomplemented
16#16840486C22148yescomplemented
17#16840454C22196yescomplemented
18#16840423C22132yescomplemented
19#16840416C22148yescomplemented
20#16840317C22148yescomplemented
21C2×#14519777C2213yes—
22#16840298C22196yescomplemented
23C2×#14519772C22112yes—
24C2×#14519768C2216yes—
25C2×#14519768C2216yes—
26C2×#14519766C2216yes—
27C2×#14519761C22112yes—
28C2×#14519757C2216yes—
29C2×#14519753C2216yes—
30C2×#14519751C2216yes—
31C4×#11758226C22116yescomplemented
32C4×#11758224C22148yescomplemented
33C2×Q8×Dic61C2211yescharacteristic
34C2×Q8×Dic61C2212yes—
35C2×Q8×Dic61C2216yes—
36C2×Q8×Dic61C2216yes—
37C2×Q8×Dic61C22132yescomplemented
38C2×C4×Dic122C2213yes—
39C2×#14519742C2213yes—
40C2×C4×D244C2213yes—
41C2×D4×Dic61C2211yescharacteristic
42C2×#14519739C2213yes—
43C2×C4×C22⋊D61C2216yes—
44C2×#14519737C2216yes—
45C2×#14519735C2213yes—
46C2×#14519735C2213yes—
47C2×#14519735C2216yes—
48C2×#14519733C2216yes—
49C22×C4⋊Dic61C2216yes—
50C22×C4×Dic61C2216yes—
51C2×C42×D61C2213yes—
52C23×Q8×C61C2212yes—
53D4×Q8×C61C22116yescomplemented
54C2×C4⋊Q8×C61C2213yes—
55C2×C22⋊Q8×C61C2216yes—
56C2×C4×Q8×C61C2211yescharacteristic
57C2×C4×D4×C61C2213yes—
58C2×#11758226—28no—
59C2×#11758226C2318yes—
60C2×#11758224—224no—
61C2×#11758224C23124yes—
62C22×Dic122C2313yes—
63C22×D244C2313yes—
64C22×C22⋊D61C2316yes—
65C23×Dic61C2312yes—
66C23×Dic61C2316yes—
67C22×C4×D61C2313yes—
68C22×C4×D61C2313yes—
69C22×C4×D61C2313yes—
70#14519777C23124yes—
71#14519772C23196yes—
72#14519768C23148yes—
73#14519768C23148yes—
74#14519766C23148yes—
75#14519761C23196yes—
76#14519757C23148yes—
77#14519753C23148yes—
78#14519751C23148yes—
79Q8×Dic61C23148yes—
80Q8×Dic61C2318yes—
81Q8×Dic61C23116yes—
82Q8×Dic61C23148yes—
83C4×Dic122C23124yes—
84#14519742C23124yes—
85C4×D244C23124yes—
86D4×Dic61C2318yes—
87#14519739C23124yes—
88C4×C22⋊D61C23148yes—
89#14519737C23148yes—
90#14519735C23124yes—
91#14519735C23124yes—
92#14519735C23148yes—
93#14519733C23148yes—
94C2×C4⋊Dic61C23148yes—
95C2×C4⋊Dic61C2313yes—
96C2×C4⋊Dic61C2316yes—
97C2×C4⋊Dic61C23112yes—
98C2×C4⋊Dic61C2311yescharacteristic
99C2×C4⋊Dic61C2316yes—
100C2×#11758196C2316yes—
101C2×#11758196C2316yes—
102C2×C22⋊Dic61C2312yes—
103C2×#11758198C2316yes—
104C2×#11758198C2316yes—
105C2×#11758198C23112yes—
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2313yes—
108C2×C4×Dic61C2313yes—
109C2×C4×Dic61C2316yes—
110C2×C4×Dic61C2316yes—
111C2×C4×Dic61C2316yes—
112C2×C4×Dic61C23148yes—
113C42×D61C23124yes—
114C22×Q8×C61—216no—
115C22×Q8×C61C2316yes—
116C22×Q8×C61C23116yes—
117C22×Q8×C61C2311yescharacteristic
118C22×D4×C61C2311yescharacteristic
119C23×C4×C61C2316yes—
120C4⋊Q8×C61C23124yes—
121C22⋊Q8×C61C23148yes—
122C4×Q8×C61C2318yes—
123C4×D4×C61C23124yes—
124C2×C4⋊C4×C61C2313yes—
125C2×C4⋊C4×C61C2316yes—
126C2×C4⋊C4×C61C2313yes—
127C2×C22⋊C4×C61C2316yes—
128C2×C42×C61C2313yes—
129C24×C61C2412yes—
130C23×D61C2413yes—
131C2×Q8×C61—264no—
132C2×Q8×C61—224no—
133C2×Q8×C61#4118yescomplemented
134C2×Q8×C61C24124yes—
135C2×Q8×C61C2414yes—
136C2×D4×C61#4118yescomplemented
137C2×D4×C61C2414yes—
138C22×C4×C61—224no—
139C22×C4×C61C2413yes—
140C22×C4×C61C2416yes—
141C22×C4×C61C24124yes—
142C22×C4×C61C2411yescharacteristic
143C22×C4×C61C2413yes—
144C2×C22⋊D61—224no—
145C2×C22⋊D61C24124yes—
146C22×Dic61—224no—
147C22×Dic61—28no—
148C22×Dic61C2411yescharacteristic
149C22×Dic61C2413yes—
150C22×Dic61C2416yes—
151C22×Dic61C24124yes—
152C22×Dic61C2412yes—
153C22×Dic61C2418yes—
154C22×Dic61C2411yescharacteristic
155#11758226—232no—
156#11758224—296no—
157C2×D244—212no—
158C2×D244C24112yes—
159C2×C4×D61—212no—
160C2×C4×D61—212no—
161C2×C4×D61—212no—
162C2×C4×D61C24112yes—
163C2×C4×D61C24112yes—
164C2×C4×D61C24112yes—
165C2×Dic122—212no—
166C2×Dic122C24112yes—
167C4⋊C4×C61C24112yes—
168C4⋊C4×C61C24112yes—
169C4⋊C4×C61C24124yes—
170C22⋊C4×C61C24124yes—
171C42×C61C24112yes—
172C22⋊Dic61C2418yes—
173#11758198C24124yes—
174#11758198C24124yes—
175#11758198C24148yes—
176C4⋊Dic61C24112yes—
177C4⋊Dic61C24124yes—
178C4⋊Dic61C24148yes—
179C4⋊Dic61C2414yes—
180C4⋊Dic61C24124yes—
181#11758196C24124yes—
182#11758196C24124yes—
183C4×Dic61C2414yes—
184C4×Dic61C24112yes—
185C4×Dic61C24112yes—
186C4×Dic61C24124yes—
187C4×Dic61C24124yes—
188C4×Dic61C24124yes—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61C2×#4118yes—
192C23×C61C2514yes—
193C23×C61C2511yescharacteristic
194C22×D61—26no—
195C22×D61—26no—
196C22×D61—26no—
197C22×D61C2516yes—
198Q8×C61—248no—
199Q8×C61C2×#41116yescomplemented
200D4×C61C2×#41116yescomplemented
201C2×C4×C61—212no—
202C2×C4×C61—26no—
203C2×C4×C61—248no—
204C2×C4×C61C2×#41112yes—
205C2×C4×C61C2×#4114yes—
206C2×C4×C61C25112yes—
207C2×C4×C61C2516yes—
208C2×C4×C61C2516yes—
209C2×C4×C61C2512yes—
210C22⋊D61—248no—
211C2×Dic61—212no—
212C2×Dic61—22no—
213C2×Dic61—26no—
214C2×Dic61—248no—
215C2×Dic61—24no—
216C2×Dic61—216no—
217C2×Dic612−1+414yes—
218C2×Dic61C2514yes—
219C2×Dic61C2512yes—
220C2×Dic61C2516yes—
221C2×Dic61C25112yes—
222C2×Dic61C2512yes—
223D244—224no—
224C4×D61—224no—
225C4×D61—224no—
226C4×D61—224no—
227Dic122—224no—
228C2×#3482—611nomaximal, Sylow
229C22×C61—24no—
230C22×C61—24no—
231C22×C61—216no—
232C22×C61C22×#4112yes—
233C22×C61C22×#4118yes—
234C22×C61C22×#4112yes—
235C22×C61C2×2−1+412yes—
236C22×C61C2611yescharacteristic
237C2×D61—26no—
238C2×D61—26no—
239C2×D61—26no—
240C2×D61—412no—
241C4×C61—212no—
242C4×C61—26no—
243C4×C61C22×#41112yes—
244C4×C61C22×#4114yes—
245Dic61—212no—
246Dic61—22no—
247Dic61—26no—
248Dic61—24no—
249Dic61C2×2−1+414yes—
250#3482—6132no—
251C2×#557—611no—
252C2×#556—613no—
253C2×#552—616no—
254C2×D4×Q8D6111yescharacteristic, complemented
255C2×#548—612no—
256C2×#547—613no—
257C2×#533—613no—
258C2×#529—616no—
259C2×C4×#41—611no—
260C2×C4×#41—613no—
261C22×C4×Q8—612no—
262C2×C61—28no—
263C2×C61#348214yescomplemented
264C2×C61C23×#4111yescharacteristic
265C2×C61C23×#4111yescharacteristic
266C2×C61C22×2−1+411yescharacteristic
267D61—46no—
268C22×#41—611no—
269C22×#41—613no—
270C23×Q8C2×D6112yes—
271#557—6116no—
272#556—6148no—
273#552—6196no—
274D4×Q8C2×D61116yescomplemented
275#548—6132no—
276#547—6148no—
277#533—6148no—
278C2×C4⋊Q8—613no—
279C2×C4⋊Q8C2×D6113yes—
280#529—6196no—
281C2×#126—6112no—
282C2×#125—616no—
283C2×#125—616no—
284C2×#124—616no—
285C2×#123—6112no—
286C2×C22⋊Q8—616no—
287C2×C22⋊Q8—616no—
288C2×C22⋊Q8C2×D6116yes—
289C2×#121—616no—
290C4×#41—6116no—
291C4×#41—6148no—
292C2×C4×Q8—611no—
293C2×C4×Q8—616no—
294C2×C4×Q8—612no—
295C2×C4×Q8—616no—
296C2×C4×Q8—6132no—
297C2×C4×Q8—613no—
298C2×C4×Q8—613no—
299C2×C4×Q8C2×D6111yescharacteristic
300C2×C4×D4—611no—
301C2×C4×D4—616no—
302C2×C4×D4—613no—
303C2×C4×D4—616no—
304C2×C4×D4—613no—
305C2×C4×D4C2×D6113yes—
306C2×#117—613no—
307C2×#117—613no—
308C2×#117—616no—
309C2×#117—616no—
310C22×C4⋊C4—616no—
311C22×C42—616no—
312C22×C42—613no—
313C61C2×#348211yescharacteristic, complemented, Sylow
314C2×#41—618no—
315C2×#41—6124no—
316C2×#41—1228no—
317C2×#41—12224no—
318C22×Q8—216no—
319C22×Q8—613no—
320C22×Q8C22×D6116yes—
321C22×Q8C22×D61116yes—
322C22×Q8C22×D6111yescharacteristic
323C22×D4—616no—
324C22×D4—613no—
325C22×D4C22×D6111yescharacteristic
326C23×C4—616no—
327C23×C4—612no—
328C23×C4—613no—
329C23×C4—613no—
330C23×C4—613no—
331C23×C4C22×D6116yes—
332C4⋊Q8—6124no—
333C4⋊Q8C22×D61124yes—
334#126—6196no—
335#125—6148no—
336#125—6148no—
337#124—6148no—
338#123—6196no—
339C22⋊Q8—6148no—
340C22⋊Q8—6148no—
341C22⋊Q8C22×D61148yes—
342#121—6148no—
343C4×Q8—618no—
344C4×Q8—6148no—
345C4×Q8—6116no—
346C4×Q8—6148no—
347C4×Q8—6124no—
348C4×Q8—6124no—
349C4×Q8C22×D6118yes—
350C4×D4—618no—
351C4×D4—6148no—
352C4×D4—6124no—
353C4×D4—6148no—
354C4×D4—6124no—
355C4×D4C22×D61124yes—
356#117—6124no—
357#117—6124no—
358#117—6148no—
359#117—6148no—
360C2×C4⋊C4—613no—
361C2×C4⋊C4—616no—
362C2×C4⋊C4—6112no—
363C2×C4⋊C4—6148no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—616no—
366C2×C4⋊C4—616no—
367C2×C4⋊C4—616no—
368C2×C4⋊C4C22×D6113yes—
369C2×C4⋊C4C22×D6113yes—
370C2×C4⋊C4C22×D6116yes—
371C2×C22⋊C4—612no—
372C2×C22⋊C4—616no—
373C2×C22⋊C4—616no—
374C2×C22⋊C4—6112no—
375C2×C22⋊C4C22×D6116yes—
376C2×C42—613no—
377C2×C42—616no—
378C2×C42—616no—
379C2×C42—616no—
380C2×C42—613no—
381C2×C42—611no—
382C2×C42—6148no—
383C2×C42—6124no—
384C2×C42C22×D6113yes—
385C24—613no—
386C24C23×D6112yes—
387#41—12232no—
388#41—12296no—
389C2×Q8—264no—
390C2×Q8—224no—
391C2×Q8—6112no—
392C2×Q8—12212no—
393C2×Q8#1175822418yescomplemented
394C2×Q8C23×D61124yes—
395C2×Q8C23×D6114yes—
396C2×D4—6124no—
397C2×D4—6112no—
398C2×D4—12224no—
399C2×D4—12212no—
400C2×D4#1175822618yescomplemented
401C2×D4C23×D6114yes—
402C22×C4—224no—
403C22×C4—611no—
404C22×C4—618no—
405C22×C4—6124no—
406C22×C4—612no—
407C22×C4—611no—
408C22×C4—613no—
409C22×C4—616no—
410C22×C4—6112no—
411C22×C4—6112no—
412C22×C4—6112no—
413C22×C4—12212no—
414C22×C4—1228no—
415C22×C4—12224no—
416C22×C4—12212no—
417C22×C4—12212no—
418C22×C4C23×D61124yes—
419C22×C4C23×D6116yes—
420C22×C4C23×D6113yes—
421C22×C4C23×D6113yes—
422C22×C4C23×D6111yescharacteristic
423C4⋊C4—6112no—
424C4⋊C4—6124no—
425C4⋊C4—6148no—
426C4⋊C4—6124no—
427C4⋊C4—6124no—
428C4⋊C4—614no—
429C4⋊C4—6124no—
430C4⋊C4C23×D61112yes—
431C4⋊C4C23×D61112yes—
432C4⋊C4C23×D61124yes—
433C22⋊C4—618no—
434C22⋊C4—6124no—
435C22⋊C4—6124no—
436C22⋊C4—6148no—
437C22⋊C4C23×D61124yes—
438C42—6112no—
439C42—614no—
440C42—6112no—
441C42—6124no—
442C42—6124no—
443C42—6124no—
444C42C23×D61112yes—
445C23—24no—
446C23—24no—
447C23—616no—
448C23—1226no—
449C23—1226no—
450C23—1226no—
451C23C2×#1175822618yes—
452C23C24×D6114yes—
453C23C24×D6111yescharacteristic
454Q8—248no—
455Q8—12224no—
456Q8C2×#11758224116yescomplemented
457D4—12248no—
458D4—12224no—
459D4C2×#11758226116yescomplemented
460C2×C4—212no—
461C2×C4—26no—
462C2×C4—248no—
463C2×C4—614no—
464C2×C4—612no—
465C2×C4—614no—
466C2×C4—612no—
467C2×C4—616no—
468C2×C4—6112no—
469C2×C4—12224no—
470C2×C4—12216no—
471C2×C4—1222no—
472C2×C4—1226no—
473C2×C4—12212no—
474C2×C4—12248no—
475C2×C4—1224no—
476C2×C4—12224no—
477C2×C4—12224no—
478C2×C4C2×#11758224112yes—
479C2×C4C2×#1175822614yes—
480C2×C4C24×D61112yes—
481C2×C4C24×D6116yes—
482C2×C4C24×D6116yes—
483C2×C4C24×D6112yes—
484C22—24no—
485C22—24no—
486C22—216no—
487C22—1226no—
488C22—1226no—
489C22—1226no—
490C22—24412no—
491C22C22×#1175822412yes—
492C22C22×#1175822618yes—
493C22C22×#1175822612yes—
494C222−1+4×D6112yes—
495C22C25×D6111yescharacteristic
496C4—212no—
497C4—26no—
498C4—614no—
499C4—1222no—
500C4—1226no—
501C4—12212no—
502C4—1224no—
503C4C22×#11758224112yes—
504C4C22×#1175822614yes—
505C2—28no—
506C2—2446no—
507C2#2145944014yesdirect product factor
508C2C23×#1175822411yescharacteristic
509C2C23×#1175822611yescharacteristic
510C2C2×2−1+4×D6111yescharacteristic
subgroupG/H
centerC23C24×D61
commutator, cosocleC22×C61C26
FrattiniC22C25×D61
FittingC2×D4×Q8×C61C2
socleC23×C61C25
hypercenterC2×D4×Q8D61
hypocenterC61C2×#3482
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #21459440C2direct (N × H)
2. C2×#16840491C2semidirect
3. C2×#16840486C2semidirect
4. C2×#16840454C2semidirect
5. C2×#16840423C2semidirect
6. C2×#16840416C2semidirect
7. C2×#16840317C2semidirect
8. C2×#16840298C2semidirect
9. C2×C4×#11758226C2semidirect
10. C2×C4×#11758224C2semidirect
11. C22×Q8×Dic61C2semidirect
12. C2×D4×Q8×C61C2semidirect
15. #16840491C22semidirect
16. #16840486C22semidirect
17. #16840454C22semidirect
18. #16840423C22semidirect
19. #16840416C22semidirect
20. #16840317C22semidirect
22. #16840298C22semidirect
31. C4×#11758226C22semidirect
32. C4×#11758224C22semidirect
37. C2×Q8×Dic61C22semidirect
53. D4×Q8×C61C22semidirect
133. C2×Q8×C61#41semidirect
136. C2×D4×C61#41semidirect
199. Q8×C61C2×#41semidirect
200. D4×C61C2×#41semidirect
254. C2×D4×Q8D61semidirect
263. C2×C61#3482semidirect
274. D4×Q8C2×D61semidirect
313. C61C2×#3482semidirect
393. C2×Q8#11758224semidirect
400. C2×D4#11758226semidirect
456. Q8C2×#11758224semidirect
459. D4C2×#11758226semidirect
507. C2#21459440direct (N × H)

Maximal subgroups

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

subgroupindex
#214594402
C2×#168404912
C2×#168404862
C2×#168404542
C2×#168404232
C2×#168404162
C2×#168403172
C2×#168402982
C2×C4×#117582262
C2×C4×#117582242
C22×Q8×Dic612
C2×D4×Q8×C612
C2×#348261