Finite Group Databasebeta

← C2×#21456340 #30549626

Subgroups — C2×#21456340 #30549626

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×#16840318C211yesmaximal, characteristic
2C2×#16840317C211yesmaximal, characteristic
3#21456340C2116yesmaximal, direct product factor
4C2×#16840172C211yesmaximal, characteristic, complemented
5C2×#16840171C211yesmaximal, characteristic, complemented
6C2×#16840170C212yesmaximal, complemented
7C2×#16840169C212yesmaximal, complemented
8C2×#16840168C211yesmaximal, characteristic, complemented
9C2×#16839799C212yesmaximal
10C2×#16839797C212yesmaximal
11C2×#97×D61C211yesmaximal, characteristic
12C2×C61×#501C211yesmaximal, characteristic, complemented
13C22×C8×D61C2212yes—
14#16840318—24no—
15#16840318C2218yes—
16#16840317—24no—
17#16840317C2218yes—
18C2×#14519777C2211yescharacteristic
19C2×#14519776C2211yescharacteristic
20C2×#14519776C2211yescharacteristic
21C2×#14519769C2211yescharacteristic
22C2×#14519768C2212yes—
23C2×#14519757—21no—
24C2×#14519755—21no—
25C2×#14519755—21no—
26C2×#14519753—21no—
27C2×Q8×Dic61—21no—
28C2×#14519742—21no—
29C2×#14519742—21no—
30C2×#14519742—21no—
31C2×#14519735C2211yescharacteristic
32C2×#14519735C2212yes—
33C2×#14519735C2211yescharacteristic
34C2×C42×D61C2211yescharacteristic
35#16840172C2218yescomplemented
36#16840171C2218yescomplemented
37#16840170C22116yescomplemented
38#16840169C22116yescomplemented
39#16840168C2218yescomplemented
40#16839799C22116yes—
41#16839797C22116yes—
42C2×#14519711C2212yes—
43C2×#14519710C2212yes—
44C2×#14519710C2212yes—
45C2×#14519709C2212yes—
46C2×#14519708C2212yes—
47C2×#14519708C2212yes—
48C2×#14519688C2211yescharacteristic
49C2×#14519687C2212yes—
50#97×D61C2218yes—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#501C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×#106×C61C2212yes—
56C2×#97×C61C2211yescharacteristic
57C22×Dic122—21no—
58C22×Dic122—21no—
59C22×Dic122—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2312yes—
64C2×C8×D61Q8116yescomplemented
65C2×C8×D61C2318yes—
66#14519777—22no—
67#14519777C2314yes—
68#14519776—22no—
69#14519776—22no—
70#14519776C2314yes—
71#14519776C2314yes—
72#14519769—22no—
73#14519769C2314yes—
74#14519768—24no—
75#14519768C2318yes—
76#14519757—28no—
77#14519755—28no—
78#14519755—28no—
79#14519753—28no—
80Q8×Dic61—28no—
81#14519742—28no—
82#14519742—28no—
83#14519742—28no—
84#14519735—24no—
85#14519735—22no—
86#14519735—22no—
87#14519735C2314yes—
88#14519735C2318yes—
89#14519735C2314yes—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—22no—
92C2×C4⋊Dic61C2312yes—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—22no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758198—22no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
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—
118#14519687C2318yes—
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—42no—
151C2×C4×D61—42no—
152C2×C4×D61C2×D414yes—
153C2×C4×D61C2×D414yes—
154C2×C4×D61C2×D412yes—
155C2×C4×D61C2×Q818yes—
156C2×C4×D61C2412yes—
157C2×Dic122—24no—
158C2×Dic122—24no—
159C2×Dic122—24no—
160C2×Dic122—44no—
161C2×Dic122—44no—
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—
177#11758198—24no—
178#11758198—24no—
179C4⋊Dic61—24no—
180C4⋊Dic61—22no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—28no—
184C4⋊Dic61C2412yes—
185C4⋊Dic61C2414yes—
186C4⋊Dic61C2412yes—
187#11758196—24no—
188#11758196—24no—
189#11758196—28no—
190#11758196—24no—
191#11758196—24no—
192C4×Dic61—24no—
193C4×Dic61—24no—
194C4×Dic61—28no—
195C4×Dic61C2×D412yes—
196C4×Dic61C2×D414yes—
197C4×Dic61C2×D414yes—
198C4×Dic61C2412yes—
199C2×#92418C2×Q818yes—
200C2×#92418C2414yes—
201C8×D61C2×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—22no—
224C2×Dic61—22no—
225C2×Dic61—24no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—44no—
229C2×Dic61C4⋊Q814yes—
230C2×Dic61C22×D412yes—
231C2×Dic61C22×D412yes—
232C2×Dic61C22×D412yes—
233C4×D61—44no—
234C4×D61—44no—
235C4×D61C4⋊Q818yes—
236C4×D61C22×Q818yes—
237Dic122—48no—
238Dic122—48no—
239Dic122—48no—
240C8×C61C22×Q818yes—
241#92418C22×Q818yes—
242C2×#3162—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—42no—
261Dic61—42no—
262Dic61—44no—
263Dic61C2×C4⋊Q814yes—
264C2×#533—611no—
265C2×#533—611no—
266#3162—6116no—
267C2×#501—612no—
268C2×#501—612no—
269C2×#501—611no—
270C2×#501—611no—
271C2×#501—611no—
272C2×#501D6111yescharacteristic, complemented
273C2×#427—612no—
274C2×#427—612no—
275C22×#97—611no—
276C2×C61#316214yescomplemented
277C2×C61C22×C4⋊Q811yescharacteristic
278C2×C61C2×#57511yescharacteristic
279C2×C61C2×#57511yescharacteristic
280D61—24no—
281C23×C8—612no—
282#533—618no—
283#533—618no—
284#533—1224no—
285#533—1224no—
286C2×C4⋊Q8—611no—
287C2×C4⋊Q8—611no—
288C2×C4⋊Q8—611no—
289C2×C4⋊Q8C2×D6111yescharacteristic
290C2×#125—612no—
291C2×#125—611no—
292C2×#125C2×D6111yescharacteristic
293C2×C22⋊Q8—1221no—
294C2×C22⋊Q8—1221no—
295C2×C22⋊Q8—1221no—
296C2×C22⋊Q8—1221no—
297C2×C4×Q8—1221no—
298C2×C4×Q8—1221no—
299C2×C4×Q8—1221no—
300C2×C4×Q8—1221no—
301C2×#117—612no—
302C2×#117—611no—
303C2×#117—611no—
304C22×C42—611no—
305#501—618no—
306#501—618no—
307#501—6116no—
308#501—6116no—
309#501—618no—
310#501C2×D6118yescomplemented
311#427—6116no—
312#427—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—
321C2×#97—612no—
322C2×#97—611no—
323C2×#97—618no—
324C2×#97C2×D6111yescharacteristic
325C61C2×#316211yescharacteristic, complemented, Sylow
326C22×Q8—21no—
327C22×Q8—1221no—
328C22×Q8—1221no—
329C22×Q8—1221no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—611no—
333C22×C8—612no—
334C22×C8—6116no—
335C22×C8—618no—
336C22×C8C22×D6112yes—
337C4⋊Q8—22no—
338C4⋊Q8—614no—
339C4⋊Q8—614no—
340C4⋊Q8—614no—
341C4⋊Q8—1222no—
342C4⋊Q8—1222no—
343C4⋊Q8—1222no—
344C4⋊Q8C22×D6114yes—
345#125—22no—
346#125—618no—
347#125—614no—
348#125—1224no—
349#125—1222no—
350#125C22×D6114yes—
351C22⋊Q8—1228no—
352C22⋊Q8—1228no—
353C22⋊Q8—1228no—
354C22⋊Q8—1228no—
355C4×Q8—1228no—
356C4×Q8—1228no—
357C4×Q8—1228no—
358C4×Q8—1228no—
359#117—614no—
360#117—618no—
361#117—614no—
362#117—1222no—
363#117—1222no—
364#117—1224no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4C22×D6112yes—
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1222no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—614no—
388C2×C42—614no—
389C2×C42—1221no—
390C2×C42—1221no—
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—618no—
402#97—614no—
403#97C22×D6114yes—
404C24—611no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—1224no—
408C2×Q8—1224no—
409C2×Q8—1224no—
410C2×Q8—2444no—
411C2×Q8—2444no—
412C2×Q8—2444no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—611no—
417C22×C4—612no—
418C22×C4—618no—
419C22×C4—612no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—614no—
423C22×C4—614no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4D4×D6111yescharacteristic
430C22×C4D4×D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432C2×C8—618no—
433C2×C8—614no—
434C2×C8—6132no—
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—614no—
445C4⋊C4—612no—
446C4⋊C4—1228no—
447C4⋊C4—1224no—
448C4⋊C4—1224no—
449C4⋊C4—1228no—
450C4⋊C4—1222no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1224no—
454C4⋊C4—1224no—
455C4⋊C4—1224no—
456C4⋊C4C23×D6112yes—
457C4⋊C4C23×D6114yes—
458C4⋊C4C23×D6112yes—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C22⋊C4—1228no—
462C42—614no—
463C42—614no—
464C42—612no—
465C42—612no—
466C42—1224no—
467C42—1224no—
468C42—1228no—
469C42D4×D6112yes—
470C42C23×D6112yes—
471C23—614no—
472C23—612no—
473C23—1222no—
474C23—1222no—
475C23C2×D4×D6111yescharacteristic
476Q8—48no—
477Q8—2448no—
478Q8—2448no—
479Q8—2448no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—614no—
489C2×C4—612no—
490C2×C4—618no—
491C2×C4—618no—
492C2×C4—612no—
493C2×C4—612no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1224no—
499C2×C4—2444no—
500C2×C4—2444no—
501C2×C4—2442no—
502C2×C4—2444no—
503C2×C4—2442no—
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—44no—
522C4—42no—
523C4—42no—
524C4—614no—
525C4—1222no—
526C4—1222no—
527C4—2442no—
528C4—2444no—
529C4—2442no—
530C4C4⋊Q8×D6112yes—
531C4C22×Q8×D6112yes—
532C2—1224no—
533C2#2145634014yesdirect 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×#3162
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. #21456340C2direct (N × H)
4. C2×#16840172C2semidirect
5. C2×#16840171C2semidirect
6. C2×#16840170C2semidirect
7. C2×#16840169C2semidirect
8. C2×#16840168C2semidirect
12. C2×C61×#501C2semidirect
35. #16840172C22semidirect
36. #16840171C22semidirect
37. #16840170C22semidirect
38. #16840169C22semidirect
39. #16840168C22semidirect
53. C61×#501C22semidirect
64. C2×C8×D61Q8semidirect
201. C8×D61C2×Q8semidirect
272. C2×#501D61semidirect
276. C2×C61#3162semidirect
310. #501C2×D61semidirect
325. C61C2×#3162semidirect
533. C2#21456340direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403182
C2×#168403172
#214563402
C2×#168401722
C2×#168401712
C2×#168401702
C2×#168401692
C2×#168401682
C2×#168397992
C2×#168397972
C2×#97×D612
C2×C61×#5012
C2×#316261