Finite Group Databasebeta

← C2×#21447253 #29708839

Subgroups — C2×#21447253 #29708839

Order 15616.

1270 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519689C211yesmaximal, characteristic
2C2×#16839619C211yesmaximal, characteristic
3C2×#16839619C211yesmaximal, characteristic
4#21447253C218yesmaximal, direct product factor
5C2×#16839525C211yesmaximal, characteristic, complemented
6C2×#16839524C211yesmaximal, characteristic, complemented
7C2×#16839524C211yesmaximal, characteristic, complemented
8C22×#106×C61C211yesmaximal, characteristic, complemented
9C23×C4×D61C2211yescharacteristic
10C23×#92418C2211yescharacteristic
11C22×C4⋊Dic61C2211yescharacteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×#11758198—21no—
14C22×#11758198—21no—
15C2×#14519689—24no—
16C2×#14519689C2214yes—
17#16839619—22no—
18#16839619—22no—
19#16839619C2214yes—
20#16839619C2214yes—
21C2×#14519684—21no—
22C2×#14519684—21no—
23#16839525C2214yescomplemented
24#16839524C2214yescomplemented
25#16839524C2214yescomplemented
26C23×C8×C61C2211yescharacteristic
27C22×C4⋊C4×C61C2211yescharacteristic
28C22×C4⋊C4×C61C2211yescharacteristic
29C2×#106×C61—24no—
30C2×#106×C61C2214yescomplemented
31C24×D61D411yescharacteristic
32C23×Dic61—21no—
33C23×Dic61—21no—
34C23×Dic61D411yescharacteristic
35C22×C4×D61—22no—
36C22×C4×D61—22no—
37C22×C4×D61—22no—
38C22×C4×D61—22no—
39C22×C4×D61—22no—
40C22×C4×D61—22no—
41C22×C4×D61D412yes—
42C22×C4×D61C2312yes—
43C22×#92418—22no—
44C22×#92418—22no—
45C22×#92418—22no—
46C22×#92418C2312yes—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×#11758198—24no—
56C2×#11758198—24no—
57C2×#11758198—44no—
58C2×#11758198—44no—
59#14519689—28no—
60#14519684—24no—
61#14519684—24no—
62C23×C4×C61—21no—
63C23×C4×C61—21no—
64C23×C4×C61C2311yescharacteristic
65C22×C8×C61—22no—
66C22×C8×C61—22no—
67C22×C8×C61D414yes—
68C22×C8×C61C2312yes—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2312yes—
76C2×C4⋊C4×C61C2312yes—
77#106×C61—28no—
78C24×C61C2×D411yescharacteristic
79C23×D61—22no—
80C23×D61—22no—
81C23×D61—22no—
82C23×D61—22no—
83C23×D61—22no—
84C23×D61—22no—
85C23×D61SD1614yes—
86C23×D61C2×D412yes—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—42no—
90C22×C4×C61—42no—
91C22×C4×C61—42no—
92C22×C4×C61—42no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D412yes—
97C22×C4×C61#4112yes—
98C22×C4×C61#4112yes—
99C22×C4×C61#4112yes—
100C22×C4×C61#4112yes—
101C22×C4×C61#4112yes—
102C22×C4×C61C2411yescharacteristic
103C22×Dic61—22no—
104C22×Dic61—22no—
105C22×Dic61—22no—
106C22×Dic61—22no—
107C22×Dic61—22no—
108C22×Dic61—42no—
109C22×Dic61—42no—
110C22×Dic61—42no—
111C22×Dic61—42no—
112C22×Dic61—42no—
113C22×Dic61—42no—
114C22×Dic61C2×D412yes—
115C2×C4×D61—24no—
116C2×C4×D61—24no—
117C2×C4×D61—24no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—44no—
125C2×C4×D61—44no—
126C2×C4×D61—44no—
127C2×C4×D61—44no—
128C2×C8×C61—28no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C2×C8×C61C2×D414yes—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—42no—
137C4⋊C4×C61—42no—
138#11758198—48no—
139#11758198—48no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—42no—
145C4⋊Dic61—42no—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418—22no—
149C2×#92418—44no—
150C23×C61#12112yes—
151C23×C61#12312yes—
152C23×C61#12312yes—
153C23×C61C2×SD1612yes—
154C23×C61#13512yes—
155C23×C61AGL(1,ℤ/8)12yes—
156C23×C61#13712yes—
157C23×C61C22×D411yescharacteristic
158C22×D61—22no—
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—22no—
162C22×D61—22no—
163C22×D61—22no—
164C22×D61—28no—
165C22×D61—28no—
166C22×D61—28no—
167C22×D61—44no—
168C22×D61—44no—
169C22×D61—44no—
170C22×D61—44no—
171C22×D61C2×SD1614yes—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—84no—
179C2×C4×C61—84no—
180C2×C4×C61#12112yes—
181C2×C4×C61#12314yes—
182C2×C4×C61#12312yes—
183C2×C4×C61#12312yes—
184C2×C4×C61#12414yes—
185C2×C4×C61#12614yes—
186C2×C4×C61#12614yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—44no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—84no—
204C2×Dic61—84no—
205C4×D61—42no—
206C4×D61—42no—
207C4×D61—42no—
208C4×D61—42no—
209C4×D61—48no—
210C8×C61—24no—
211#92418—42no—
212C2×#2077—611nomaximal, Sylow
213C22×C61#39714yes—
214C22×C61#46514yes—
215C22×C61#46814yes—
216C22×C61#48114yes—
217C22×C61#48214yes—
218C22×C61#48314yes—
219C22×C61#48514yes—
220C22×C61C2×#12111yescharacteristic
221C22×C61C2×#12311yescharacteristic
222C22×C61C2×#12311yescharacteristic
223C22×C61C22×SD1611yescharacteristic
224C22×C61C2×#13511yescharacteristic
225C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
226C22×C61C2×#13711yescharacteristic
227C2×D61—24no—
228C2×D61—24no—
229C2×D61—24no—
230C2×D61—42no—
231C2×D61—42no—
232C2×D61—416no—
233C2×D61—42no—
234C2×D61—42no—
235C4×C61—82no—
236C4×C61—82no—
237C4×C61#39714yes—
238C4×C61C2×#12311yescharacteristic
239C4×C61C2×#12411yescharacteristic
240C4×C61C2×#12611yescharacteristic
241C4×C61C2×#12611yescharacteristic
242Dic61—42no—
243Dic61—82no—
244Dic61—82no—
245C22×#106D6111yescharacteristic, complemented
246C22×C22⋊C8—611no—
247C2×#380—611no—
248C2×#380—611no—
249#2077—618no—
250C2×#340—611no—
251C2×#340—611no—
252C2×#340—611no—
253C2×C61#207718yescomplemented
254C2×C61C2×#39711yescharacteristic
255C2×C61C2×#46511yescharacteristic
256C2×C61C2×#46811yescharacteristic
257C2×C61C2×#48311yescharacteristic
258C2×C61C2×#48211yescharacteristic
259C2×C61C2×#48511yescharacteristic
260C2×C61C2×#48111yescharacteristic
261D61—44no—
262C24×C4—611no—
263C23×C8—611no—
264C23×C8C2×D6111yescharacteristic
265C22×C4⋊C4—611no—
266C22×C4⋊C4—611no—
267C22×C4⋊C4C2×D6111yescharacteristic
268C22×C4⋊C4C2×D6111yescharacteristic
269C22×C22⋊C4—1221no—
270C22×C22⋊C4—1221no—
271C2×#106—24no—
272C2×#106C2×D6114yescomplemented
273C2×C22⋊C8—614no—
274C2×C22⋊C8—1224no—
275#380—614no—
276#380—614no—
277#380—1222no—
278#380—1222no—
279C2×#95—1221no—
280C2×#95—1221no—
281#340—614no—
282#340—614no—
283#340—614no—
284C61C2×#207711yescharacteristic, complemented, Sylow
285C25—611no—
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—611no—
289C23×C4—612no—
290C23×C4—612no—
291C23×C4—1222no—
292C23×C4—1222no—
293C23×C4—1222no—
294C23×C4—1222no—
295C23×C4—1222no—
296C23×C4—1222no—
297C23×C4—1221no—
298C23×C4—1221no—
299C23×C4C22×D6111yescharacteristic
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—612no—
303C22×C8—1222no—
304C22×C8—1222no—
305C22×C8—1222no—
306C22×C8C22⋊D6114yes—
307C22×C8C22×D6112yes—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—612no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4C22×D6112yes—
323C2×C4⋊C4C22×D6112yes—
324C2×C22⋊C4—1224no—
325C2×C22⋊C4—1224no—
326C2×C22⋊C4—2444no—
327C2×C22⋊C4—2444no—
328#106—28no—
329C22⋊C8—1228no—
330#95—1224no—
331#95—1224no—
332C24—612no—
333C24—614no—
334C24—1222no—
335C24—1222no—
336C24—1222no—
337C24—1222no—
338C24—1222no—
339C24—1222no—
340C24D4×D6111yescharacteristic
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—612no—
350C22×C4—1222no—
351C22×C4—1222no—
352C22×C4—1222no—
353C22×C4—1222no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—1222no—
359C22×C4—1224no—
360C22×C4—1224no—
361C22×C4—1224no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4—2442no—
370C22×C4—2444no—
371C22×C4—2444no—
372C22×C4—2444no—
373C22×C4—2444no—
374C22×C4#1175822212yes—
375C22×C4#1175822212yes—
376C22×C4D4×D6111yescharacteristic
377C22×C4#1175822412yes—
378C22×C4#1175822612yes—
379C22×C4#1175822612yes—
380C22×C4C2×C22⋊D6112yes—
381C22×C4C23×D6111yescharacteristic
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—28no—
385C2×C8—1222no—
386C2×C8—1222no—
387C2×C8—1222no—
388C2×C8—2444no—
389C2×C8C2×C22⋊D6114yes—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—1222no—
397C4⋊C4—1222no—
398C4⋊C4—1222no—
399C4⋊C4—1222no—
400C4⋊C4—2442no—
401C4⋊C4—2442no—
402C22⋊C4—2448no—
403C22⋊C4—2448no—
404C23—614no—
405C23—1228no—
406C23—1222no—
407C23—1222no—
408C23—1222no—
409C23—1222no—
410C23—1222no—
411C23—1222no—
412C23—1228no—
413C23—1228no—
414C23—2444no—
415C23—2444no—
416C23—2444no—
417C23—2444no—
418C23#1451974812yes—
419C23#1451976212yes—
420C23#1451976212yes—
421C23SD16×D6112yes—
422C23#1451979812yes—
423C23#1451980412yes—
424C23#1451980612yes—
425C23C2×D4×D6111yescharacteristic
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—1222no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—4884no—
438C2×C4—4884no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4—2444no—
446C2×C4—2448no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4#1451974812yes—
452C2×C4#1451976014yes—
453C2×C4#1451976212yes—
454C2×C4#1451976212yes—
455C2×C4#1451976614yes—
456C2×C4#1451977214yes—
457C2×C4#1451977214yes—
458C2×C4C22×C22⋊D6111yescharacteristic
459C2×C4C2×#1175822411yescharacteristic
460C2×C4C2×#1175822211yescharacteristic
461C2×C4C2×#1175822211yescharacteristic
462C2×C4C2×#1175822611yescharacteristic
463C2×C4C2×#1175822611yescharacteristic
464C8—24no—
465C8—2442no—
466C22—1224no—
467C22—1224no—
468C22—1224no—
469C22—2442no—
470C22—2442no—
471C22—24416no—
472C22—2442no—
473C22—2442no—
474C22#1683968814yes—
475C22#1683998814yes—
476C22#1684000514yes—
477C22#1684008214yes—
478C22#1684008814yes—
479C22#1684009414yes—
480C22#1684010614yes—
481C22C2×#1451974811yescharacteristic
482C22C2×#1451976211yescharacteristic
483C22C2×#1451976211yescharacteristic
484C22C2×SD16×D6111yescharacteristic
485C22C2×#1451979811yescharacteristic
486C22C2×#1451980411yescharacteristic
487C22C2×#1451980611yescharacteristic
488C4—82no—
489C4—82no—
490C4—4882no—
491C4—4882no—
492C4—2442no—
493C4#1683968814yes—
494C4C2×#1451976011yescharacteristic
495C4C2×#1451976611yescharacteristic
496C4C2×#1451977211yescharacteristic
497C4C2×#1451977211yescharacteristic
498C2—2444no—
499C2#2144725318yesdirect product factor
500C2C2×#1683968811yescharacteristic
501C2C2×#1683998811yescharacteristic
502C2C2×#1684000511yescharacteristic
503C2C2×#1684009411yescharacteristic
504C2C2×#1684008811yescharacteristic
505C2C2×#1684010611yescharacteristic
506C2C2×#1684008211yescharacteristic
subgroupG/H
centerC24D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#106×C61C2
socleC24×C61C2×D4
prime radicalC24×D61D4
hypercenterC22×#106D61
hypocenterC61C2×#2077
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
4. #21447253C2direct (N × H)
5. C2×#16839525C2semidirect
6. C2×#16839524C2semidirect
7. C2×#16839524C2semidirect
8. C22×#106×C61C2semidirect
23. #16839525C22semidirect
24. #16839524C22semidirect
25. #16839524C22semidirect
30. C2×#106×C61C22semidirect
245. C22×#106D61semidirect
253. C2×C61#2077semidirect
272. C2×#106C2×D61semidirect
284. C61C2×#2077semidirect
499. C2#21447253direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145196892
C2×#168396192
#214472532
C2×#168395252
C2×#168395242
C22×#106×C612
C2×#207761