Finite Group Databasebeta

← C2×#21456370 #30549668

Subgroups — C2×#21456370 #30549668

Order 15616.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#125×D61C211yesmaximal, characteristic
3#21456370C2116yesmaximal, direct product factor
4C2×#16840172C211yesmaximal, characteristic, complemented
5C2×#16840171C211yesmaximal, characteristic, complemented
6C2×#16840168C211yesmaximal, characteristic, complemented
7C2×#16840163C212yesmaximal, complemented
8C2×#16840162C212yesmaximal, complemented
9C2×#16839803C212yesmaximal
10C2×#16839801C212yesmaximal
11C2×#16839714C211yesmaximal, characteristic
12C2×C61×#501C211yesmaximal, characteristic, complemented
13C22×#11758189C2212yes—
14#16840317—24no—
15#16840317C2218yes—
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519768C2211yescharacteristic
20C2×#14519768C2212yes—
21C2×#14519767—21no—
22C2×#14519767—21no—
23#125×D61—24no—
24#125×D61C2218yes—
25C2×#14519757—21no—
26C2×#14519753—21no—
27C2×Q8×Dic61—21no—
28C2×#14519742—21no—
29C2×#14519735C2212yes—
30C2×C4⋊C4×D61—21no—
31C2×C4⋊C4×D61—21no—
32C2×C4⋊C4×D61C2211yescharacteristic
33C2×C4⋊C4×D61C2211yescharacteristic
34C2×C42×D61C2211yescharacteristic
35#16840172C2218yescomplemented
36#16840171C2218yescomplemented
37#16840168C2218yescomplemented
38#16840163C22116yescomplemented
39#16840162C22116yescomplemented
40#16839803C22116yes—
41#16839801C22116yes—
42C2×#14519711C2212yes—
43C2×#14519710C2212yes—
44C2×#14519710C2212yes—
45C2×#14519709C2212yes—
46C2×#14519708C2212yes—
47C2×#14519708C2212yes—
48#16839714C2218yes—
49C2×#14519688C2211yescharacteristic
50C2×C8×Dic61C2212yes—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#501C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×#106×C61C2212yes—
56C2×#97×C61C2211yescharacteristic
57C22×Dic122—21no—
58C22×C4×D61—21no—
59C22×C4×D61—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C2×#11758189Q8116yescomplemented
64C2×#11758189C2318yes—
65C22×#92418C2312yes—
66#14519777—22no—
67#14519777C2314yes—
68#14519769—22no—
69#14519769C2314yes—
70#14519768—22no—
71#14519768—22no—
72#14519768—24no—
73#14519768C2314yes—
74#14519768C2314yes—
75#14519768C2318yes—
76#14519767—28no—
77#14519767—28no—
78#14519757—28no—
79#14519753—28no—
80Q8×Dic61—28no—
81#14519742—28no—
82#14519735—24no—
83#14519735C2318yes—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—22no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2312yes—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—22no—
98C4⋊C4×D61—28no—
99C4⋊C4×D61—28no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61—22no—
102C4⋊C4×D61C2314yes—
103C4⋊C4×D61C2314yes—
104C2×#11758198—22no—
105C2×C4×Dic61—22no—
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C42×D61D414yes—
110C42×D61C2314yes—
111#14519711C2318yes—
112#14519710C2318yes—
113#14519710C2318yes—
114#14519709C2318yes—
115#14519708C2318yes—
116#14519708C2318yes—
117#14519688C2314yes—
118C8×Dic61C2318yes—
119C22×Q8×C61—21no—
120C22×C8×C61C2312yes—
121C4⋊Q8×C61—22no—
122C4⋊Q8×C61C2314yes—
123#125×C61—22no—
124#125×C61C2314yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2312yes—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132#107×C61C2318yes—
133#106×C61C2318yes—
134#97×C61C2314yes—
135C23×D61C2×D411yescharacteristic
136C2×Q8×C61—24no—
137C2×Q8×C61—44no—
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—42no—
153C2×C4×D61—42no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61C2×D414yes—
157C2×C4×D61C2×D414yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×Q818yes—
160C2×C4×D61C2412yes—
161C2×Dic122—24no—
162C2×Dic122—44no—
163C2×C8×C61C2×Q818yes—
164C2×C8×C61C2414yes—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61C2412yes—
172C4⋊C4×C61C2414yes—
173C4⋊C4×C61C2412yes—
174C42×C61C2×D412yes—
175C42×C61C2412yes—
176#11758198—28no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—24no—
180C4⋊Dic61—22no—
181C4⋊Dic61—24no—
182C4⋊Dic61—24no—
183C4⋊Dic61—28no—
184C4⋊Dic61C2412yes—
185C4⋊Dic61C2414yes—
186C4⋊Dic61C2412yes—
187#11758196—24no—
188#11758196—24no—
189#11758196—24no—
190#11758196—24no—
191#11758196—24no—
192#11758196—24no—
193#11758196—28no—
194C4×Dic61—28no—
195C4×Dic61C2×D412yes—
196C4×Dic61C2×D414yes—
197C4×Dic61C2×D414yes—
198C4×Dic61C2412yes—
199C2×#92418C2×Q818yes—
200C2×#92418C2414yes—
201#11758189C2×Q8132yescomplemented
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—22no—
205C22×D61C4⋊Q814yes—
206C22×D61C22×D412yes—
207Q8×C61—48no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—44no—
216C2×C4×C61C22×D412yes—
217C2×C4×C61C22×D412yes—
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×Q814yes—
220C2×C4×C61C2511yescharacteristic
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—44no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61C4⋊Q814yes—
230C2×Dic61C22×D412yes—
231C2×Dic61C22×D412yes—
232C2×Dic61C22×D412yes—
233C4×D61—44no—
234C4×D61—44no—
235C4×D61—48no—
236C4×D61—48no—
237C4×D61C4⋊Q818yes—
238C4×D61C22×Q818yes—
239Dic122—48no—
240C8×C61C22×Q818yes—
241#92418C22×Q818yes—
242C2×#3166—611nomaximal, Sylow
243C22×C61C2×C4⋊Q812yes—
244C22×C61#57512yes—
245C22×C61#57512yes—
246C22×C61C23×D411yescharacteristic
247C2×D61—28no—
248C2×D61—22no—
249C2×D61—22no—
250C2×D61C2×C4⋊Q814yes—
251C4×C61—22no—
252C4×C61—22no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61—44no—
256C4×C61C2×C4⋊Q812yes—
257C4×C61C23×Q812yes—
258Dic61—22no—
259Dic61—22no—
260Dic61—44no—
261Dic61—42no—
262Dic61—42no—
263Dic61C2×C4⋊Q814yes—
264C2×#533—611no—
265C22×#125—611no—
266#3166—6116no—
267C2×#501—611no—
268C2×#501—611no—
269C2×#501—611no—
270C2×#501D6111yescharacteristic, complemented
271C2×#499—612no—
272C2×#499—612no—
273C2×#428—612no—
274C2×#428—612no—
275C2×C4×M16—611no—
276C2×C61#316614yescomplemented
277C2×C61C22×C4⋊Q811yescharacteristic
278C2×C61C2×#57511yescharacteristic
279C2×C61C2×#57511yescharacteristic
280D61—24no—
281C22×M16—612no—
282#533—618no—
283#533—1224no—
284C2×C4⋊Q8—611no—
285C2×C4⋊Q8C2×D6111yescharacteristic
286C2×#125—618no—
287C2×#125—611no—
288C2×#125—611no—
289C2×#125—612no—
290C2×#125—611no—
291C2×#125—1221no—
292C2×#125—1221no—
293C2×#125—1224no—
294C2×#125C2×D6111yescharacteristic
295C2×C22⋊Q8—1221no—
296C2×C22⋊Q8—1221no—
297C2×C4×Q8—1221no—
298C2×C4×Q8—1221no—
299C2×#117—612no—
300C22×C4⋊C4—611no—
301C22×C4⋊C4—611no—
302C22×C4⋊C4—1221no—
303C22×C4⋊C4—1221no—
304C22×C42—611no—
305#501—618no—
306#501—618no—
307#501—618no—
308#501C2×D6118yescomplemented
309#499—6116no—
310#499—6116no—
311#428—6116no—
312#428—6116no—
313C2×#107—612no—
314C2×#107—612no—
315C2×#107—612no—
316C2×#107C2×D6112yes—
317C2×#106—612no—
318C2×#106—612no—
319C2×#106—612no—
320C2×#106C2×D6112yes—
321C4×M16—618no—
322C2×#97—611no—
323C2×#97C2×D6111yescharacteristic
324C2×C4×C8—612no—
325C61C2×#316611yescharacteristic, complemented, Sylow
326C22×Q8—21no—
327C22×Q8—1221no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—1221no—
332C23×C4—1221no—
333C2×M16—6116no—
334C2×M16—618no—
335C22×C8—612no—
336C22×C8C22×D6112yes—
337C4⋊Q8—22no—
338C4⋊Q8—614no—
339C4⋊Q8—1222no—
340C4⋊Q8C22×D6114yes—
341#125—22no—
342#125—614no—
343#125—614no—
344#125—618no—
345#125—614no—
346#125—1228no—
347#125—1228no—
348#125—1222no—
349#125—1222no—
350#125—1224no—
351#125—1222no—
352#125C22×D6114yes—
353C22⋊Q8—1228no—
354C22⋊Q8—1228no—
355C4×Q8—1228no—
356C4×Q8—1228no—
357#117—618no—
358#117—1224no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—614no—
366C2×C4⋊C4—614no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4—1228no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6112yes—
385C2×C22⋊C4—1222no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—614no—
390C2×C42—614no—
391C2×C42—1222no—
392C2×C42C22×D6111yescharacteristic
393#107—618no—
394#107—618no—
395#107—618no—
396#107C22×D6118yes—
397#106—618no—
398#106—618no—
399#106—618no—
400#106C22×D6118yes—
401#97—614no—
402#97C22×D6114yes—
403C4×C8—618no—
404C24—611no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—1224no—
408C2×Q8—2444no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—611no—
413C22×C4—612no—
414C22×C4—618no—
415C22×C4—612no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—614no—
419C22×C4—614no—
420C22×C4—1224no—
421C22×C4—1224no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4—2444no—
428C22×C4—2444no—
429C22×C4D4×D6111yescharacteristic
430C22×C4D4×D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432M16—6132no—
433C2×C8—618no—
434C2×C8—614no—
435C2×C8Q8×D6118yes—
436C2×C8C23×D6114yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—612no—
444C4⋊C4—612no—
445C4⋊C4—614no—
446C4⋊C4—1224no—
447C4⋊C4—1224no—
448C4⋊C4—1224no—
449C4⋊C4—1224no—
450C4⋊C4—1224no—
451C4⋊C4—1224no—
452C4⋊C4—1228no—
453C4⋊C4—1228no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1222no—
457C4⋊C4—1224no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4C23×D6112yes—
461C4⋊C4C23×D6114yes—
462C4⋊C4C23×D6112yes—
463C22⋊C4—1228no—
464C42—614no—
465C42—614no—
466C42—612no—
467C42—612no—
468C42—1228no—
469C42D4×D6112yes—
470C42C23×D6112yes—
471C23—614no—
472C23—612no—
473C23—1222no—
474C23—1222no—
475C23C2×D4×D6111yescharacteristic
476Q8—48no—
477Q8—2448no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—614no—
487C2×C4—612no—
488C2×C4—618no—
489C2×C4—618no—
490C2×C4—612no—
491C2×C4—612no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1224no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—2444no—
498C2×C4—2444no—
499C2×C4—2448no—
500C2×C4—2448no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2444no—
504C2×C4C2×D4×D6112yes—
505C2×C4C2×D4×D6112yes—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C2×Q8×D6114yes—
508C2×C4C24×D6111yescharacteristic
509C8—618no—
510C8C2×Q8×D6118yes—
511C22—614no—
512C22—1222no—
513C22—1222no—
514C22—1228no—
515C22C4⋊Q8×D6112yes—
516C22#1684055812yes—
517C22#1684056112yes—
518C22C22×D4×D6111yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—614no—
525C4—1222no—
526C4—1222no—
527C4—2442no—
528C4—2442no—
529C4—2444no—
530C4C4⋊Q8×D6112yes—
531C4C22×Q8×D6112yes—
532C2—1224no—
533C2#2145637014yesdirect product factor
534C2C2×C4⋊Q8×D6111yescharacteristic
535C2C2×#1684055811yescharacteristic
536C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#501D61
hypocenterC61C2×#3166
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #21456370C2direct (N × H)
4. C2×#16840172C2semidirect
5. C2×#16840171C2semidirect
6. C2×#16840168C2semidirect
7. C2×#16840163C2semidirect
8. C2×#16840162C2semidirect
12. C2×C61×#501C2semidirect
35. #16840172C22semidirect
36. #16840171C22semidirect
37. #16840168C22semidirect
38. #16840163C22semidirect
39. #16840162C22semidirect
53. C61×#501C22semidirect
63. C2×#11758189Q8semidirect
201. #11758189C2×Q8semidirect
270. C2×#501D61semidirect
276. C2×C61#3166semidirect
308. #501C2×D61semidirect
325. C61C2×#3166semidirect
533. C2#21456370direct (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
#214563702
C2×#168401722
C2×#168401712
C2×#168401682
C2×#168401632
C2×#168401622
C2×#168398032
C2×#168398012
C2×#168397142
C2×C61×#5012
C2×#316661