Finite Group Databasebeta

← C2×#21457020 #30552122

Subgroups — C2×#21457020 #30552122

Order 15616.

1770 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#125×D61C211yesmaximal, characteristic
3#21457020C2116yesmaximal, direct product factor
4C2×#16840172C211yesmaximal, characteristic, complemented
5C2×#16840171C211yesmaximal, characteristic, complemented
6C2×#16840130C211yesmaximal, characteristic, complemented
7C2×#16840096C212yesmaximal
8C2×#16840091C212yesmaximal
9C2×#16840071C212yesmaximal, complemented
10C2×#16840067C212yesmaximal, complemented
11C2×#16839817C211yesmaximal, characteristic
12C2×C61×#491C211yesmaximal, characteristic, complemented
13#16840317—24no—
14#16840317C2218yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519768—21no—
18C2×#14519768C2211yescharacteristic
19C2×#14519768C2211yescharacteristic
20C2×#14519767—21no—
21C2×#14519767—21no—
22#125×D61—24no—
23#125×D61C2218yes—
24C2×#14519757—21no—
25C2×#14519753C2212yes—
26C2×Q8×Dic61C2212yes—
27C2×#14519742—21no—
28C2×#14519735—21no—
29C2×C4⋊C4×D61—21no—
30C2×C4⋊C4×D61—21no—
31C2×C4⋊C4×D61C2211yescharacteristic
32C2×C4⋊C4×D61C2211yescharacteristic
33C2×C42×D61C2211yescharacteristic
34#16840172C2218yescomplemented
35#16840171C2218yescomplemented
36#16840130C2218yescomplemented
37#16840096C22116yes—
38#16840091C22116yes—
39#16840071C22116yescomplemented
40#16840067C22116yescomplemented
41#16839817C2218yes—
42C2×#14519711C2212yes—
43C2×#14519710C2212yes—
44C2×#14519709C2212yes—
45C2×#14519708C2212yes—
46C2×#14519706C2212yes—
47C2×#14519701C2212yes—
48C2×#14519701C2212yes—
49C2×#14519689C2212yes—
50C2×#14519688C2211yescharacteristic
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#491C2218yescomplemented
54C2×#103×C61C2212yes—
55C2×#103×C61C2212yes—
56C2×#97×C61C2211yescharacteristic
57C22×Dic122—21no—
58C22×C4×D61—21no—
59C22×C4×D61—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2312yes—
64#14519777—22no—
65#14519777C2314yes—
66#14519769—22no—
67#14519769C2314yes—
68#14519768—22no—
69#14519768—22no—
70#14519768—28no—
71#14519768C2314yes—
72#14519768C2314yes—
73#14519767—28no—
74#14519767—28no—
75#14519757—28no—
76#14519753—24no—
77#14519753C2318yes—
78Q8×Dic61—24no—
79Q8×Dic61C2318yes—
80#14519742—28no—
81#14519735—28no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—22no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2312yes—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—22no—
96C4⋊C4×D61—28no—
97C4⋊C4×D61—28no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61—22no—
100C4⋊C4×D61C2314yes—
101C4⋊C4×D61C2314yes—
102C2×#11758198—22no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C42×D61D414yes—
108C42×D61C2314yes—
109#14519711C2318yes—
110#14519710C2318yes—
111#14519709C2318yes—
112#14519708C2318yes—
113#14519706C2318yes—
114#14519701C2318yes—
115#14519701C2318yes—
116#14519689C2318yes—
117#14519688C2314yes—
118C22×Q8×C61C2312yes—
119C22×C8×C61C2312yes—
120C4⋊Q8×C61—22no—
121C4⋊Q8×C61C2314yes—
122#125×C61—22no—
123#125×C61C2314yes—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131#103×C61C2318yes—
132#103×C61C2318yes—
133#97×C61C2314yes—
134C23×D61C2×D411yescharacteristic
135C2×Q8×C61—22no—
136C2×Q8×C61—28no—
137C2×Q8×C61C2414yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×C4×D61C2×D414yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D414yes—
160C2×C4×D61C2412yes—
161C2×Dic122—24no—
162C2×Dic122—44no—
163C2×C8×C61—24no—
164C2×C8×C61C2414yes—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2412yes—
172C4⋊C4×C61C2412yes—
173C42×C61C2×D412yes—
174C42×C61C2412yes—
175#11758198—28no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—24no—
179C4⋊Dic61—22no—
180C4⋊Dic61—24no—
181C4⋊Dic61—24no—
182C4⋊Dic61—28no—
183C4⋊Dic61C2412yes—
184C4⋊Dic61C2412yes—
185C4⋊Dic61C2414yes—
186#11758196—24no—
187#11758196—24no—
188#11758196—24no—
189#11758196—24no—
190#11758196—24no—
191#11758196—24no—
192#11758196—28no—
193C4×Dic61—28no—
194C4×Dic61C2×D412yes—
195C4×Dic61C2×D414yes—
196C4×Dic61C2×D414yes—
197C4×Dic61C2412yes—
198C2×#92418—24no—
199C2×#92418C2414yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—22no—
203C22×D61#13714yes—
204C22×D61C22×D412yes—
205Q8×C61—28no—
206Q8×C61—44no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—44no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61C22×D412yes—
216C2×C4×C61C22×D412yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C612−1+414yes—
219C2×C4×C61C2511yescharacteristic
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—42no—
227C2×Dic61—44no—
228C2×Dic61—42no—
229C2×Dic61C22×D412yes—
230C2×Dic61C22×D412yes—
231C2×Dic61C22×D412yes—
232C4×D61—28no—
233C4×D61—24no—
234C4×D61—24no—
235C4×D61—44no—
236C4×D61—48no—
237C4×D61—48no—
238Dic122—48no—
239C8×C61—24no—
240#92418—24no—
241C2×#3236—611nomaximal, Sylow
242C22×C61#53612yes—
243C22×C61C2×#13712yes—
244C22×C61#57512yes—
245C22×C61C23×D411yescharacteristic
246C2×D61—22no—
247C2×D61—28no—
248C2×D61—22no—
249C2×D61C2×#13714yes—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—44no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61#53612yes—
256C4×C61C2×2−1+412yes—
257Dic61—22no—
258Dic61—22no—
259Dic61—22no—
260Dic61—42no—
261Dic61—44no—
262Dic61—42no—
263C2×#533—611no—
264C22×#125—611no—
265#3236—6116no—
266C2×#501—611no—
267C2×#501—611no—
268C2×#491—611no—
269C2×#491D6111yescharacteristic, complemented
270C2×#484—612no—
271C2×#483—612no—
272C2×#479—612no—
273C2×#479—612no—
274C2×#432—611no—
275C2×C61#323614yescomplemented
276C2×C61C2×#53611yescharacteristic
277C2×C61C22×#13711yescharacteristic
278C2×C61C2×#57511yescharacteristic
279D61—24no—
280#533—618no—
281#533—1224no—
282C2×C4⋊Q8—611no—
283C2×C4⋊Q8C2×D6111yescharacteristic
284C2×#125—618no—
285C2×#125—611no—
286C2×#125—611no—
287C2×#125—611no—
288C2×#125—1221no—
289C2×#125—1221no—
290C2×#125—1221no—
291C2×#125—1224no—
292C2×#125C2×D6111yescharacteristic
293C2×C22⋊Q8—612no—
294C2×C22⋊Q8—1221no—
295C2×C4×Q8—612no—
296C2×C4×Q8—1221no—
297C2×#117—1221no—
298C22×C4⋊C4—611no—
299C22×C4⋊C4—611no—
300C22×C4⋊C4—1221no—
301C22×C4⋊C4—1221no—
302C22×C42—611no—
303#501—618no—
304#501—618no—
305#491—618no—
306#491C2×D6118yescomplemented
307#484—6116no—
308#483—6116no—
309#479—6116no—
310#479—6116no—
311#432—618no—
312C2×#107—612no—
313C2×#107—612no—
314C2×#106—612no—
315C2×#106—612no—
316C2×C4⋊C8—612no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103C2×D6112yes—
320C2×#103C2×D6112yes—
321C2×C22⋊C8—612no—
322C2×#97—611no—
323C2×#97C2×D6111yescharacteristic
324C61C2×#323611yescharacteristic, complemented, Sylow
325C22×Q8—1221no—
326C22×Q8C22×D6112yes—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—1221no—
331C23×C4—1221no—
332C22×C8—612no—
333C22×C8C22×D6112yes—
334C4⋊Q8—22no—
335C4⋊Q8—614no—
336C4⋊Q8—1222no—
337C4⋊Q8C22×D6114yes—
338#125—22no—
339#125—614no—
340#125—614no—
341#125—614no—
342#125—1228no—
343#125—1228no—
344#125—1222no—
345#125—1222no—
346#125—1228no—
347#125—1222no—
348#125C22×D6114yes—
349C22⋊Q8—618no—
350C22⋊Q8—1228no—
351C22⋊Q8—1224no—
352C4×Q8—618no—
353C4×Q8—1228no—
354C4×Q8—1224no—
355#117—1228no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—614no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1228no—
368C2×C4⋊C4—1228no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—1222no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—614no—
387C2×C42—614no—
388C2×C42—1222no—
389C2×C42C22×D6111yescharacteristic
390#107—618no—
391#107—618no—
392#106—618no—
393#106—618no—
394C4⋊C8—618no—
395#103—618no—
396#103—618no—
397#103C22×D6118yes—
398#103C22×D6118yes—
399C22⋊C8—618no—
400#97—614no—
401#97C22×D6114yes—
402C24—611no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—1224no—
406C2×Q8—2444no—
407C2×Q8C23×D6114yes—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—614no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—614no—
418C22×C4—1224no—
419C22×C4—1224no—
420C22×C4—1224no—
421C22×C4—1224no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—2442no—
426C22×C4—2444no—
427C22×C4—2444no—
428C22×C4D4×D6111yescharacteristic
429C22×C4D4×D6111yescharacteristic
430C22×C4C23×D6111yescharacteristic
431C2×C8—24no—
432C2×C8—614no—
433C2×C8—1224no—
434C2×C8C23×D6114yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—612no—
442C4⋊C4—612no—
443C4⋊C4—614no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1224no—
447C4⋊C4—1224no—
448C4⋊C4—1224no—
449C4⋊C4—1224no—
450C4⋊C4—1228no—
451C4⋊C4—1228no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1222no—
455C4⋊C4—1224no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4C23×D6112yes—
459C4⋊C4C23×D6112yes—
460C22⋊C4—1228no—
461C42—614no—
462C42—612no—
463C42—614no—
464C42—612no—
465C42—1228no—
466C42D4×D6112yes—
467C42C23×D6112yes—
468C23—614no—
469C23—612no—
470C23—1222no—
471C23—1222no—
472C23C2×D4×D6111yescharacteristic
473Q8—28no—
474Q8—44no—
475Q8—2448no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—612no—
485C2×C4—612no—
486C2×C4—612no—
487C2×C4—1228no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1224no—
492C2×C4—1224no—
493C2×C4—1224no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—2444no—
497C2×C4—2448no—
498C2×C4—2448no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2444no—
502C2×C4C2×D4×D6112yes—
503C2×C4C2×D4×D6112yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4#1451982414yes—
506C2×C4C24×D6111yescharacteristic
507C8—24no—
508C8—1224no—
509C22—614no—
510C22—1222no—
511C22—1228no—
512C22—1222no—
513C22#1684033212yes—
514C22#137×D6112yes—
515C22#1684056112yes—
516C22C22×D4×D6111yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—44no—
522C4—1222no—
523C4—1222no—
524C4—1222no—
525C4—2442no—
526C4—2442no—
527C4—2444no—
528C4#1684033212yes—
529C4C2×#1451982412yes—
530C2—1224no—
531C2#2145702014yesdirect product factor
532C2C2×#1684033211yescharacteristic
533C2C2×#137×D6111yescharacteristic
534C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#491C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#491D61
hypocenterC61C2×#3236
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
3. #21457020C2direct (N × H)
4. C2×#16840172C2semidirect
5. C2×#16840171C2semidirect
6. C2×#16840130C2semidirect
9. C2×#16840071C2semidirect
10. C2×#16840067C2semidirect
12. C2×C61×#491C2semidirect
34. #16840172C22semidirect
35. #16840171C22semidirect
36. #16840130C22semidirect
39. #16840071C22semidirect
40. #16840067C22semidirect
53. C61×#491C22semidirect
269. C2×#491D61semidirect
275. C2×C61#3236semidirect
306. #491C2×D61semidirect
324. C61C2×#3236semidirect
531. C2#21457020direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#125×D612
#214570202
C2×#168401722
C2×#168401712
C2×#168401302
C2×#168400962
C2×#168400912
C2×#168400712
C2×#168400672
C2×#168398172
C2×C61×#4912
C2×#323661