Finite Group Databasebeta

← C2×#21447267 #29708912

Subgroups — C2×#21447267 #29708912

Order 15616.

950 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519708C211yesmaximal, characteristic, complemented
2C2×#16839763C211yesmaximal, characteristic
3C2×#16839729C211yesmaximal, characteristic
4#21447267C218yesmaximal, direct product factor
5C2×#16839531C211yesmaximal, characteristic, complemented
6C2×#16839530C211yesmaximal, characteristic, complemented
7C2×#16839526C211yesmaximal, characteristic, complemented
8C2×C61×#417C211yesmaximal, characteristic
9C22×#11758194C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C2×#14519708—22no—
14C2×#14519708—22no—
15C2×#14519708C2214yescomplemented
16#16839763—22no—
17#16839763C2214yes—
18C2×#14519701—21no—
19C2×#14519698—21no—
20#16839729—22no—
21#16839729C2214yes—
22C2×C22⋊#92418—22no—
23#16839531C2214yescomplemented
24#16839530C2214yescomplemented
25#16839526C2214yescomplemented
26C22×#41×C61C2211yescharacteristic
27C22×M16×C61C2211yescharacteristic
28C22×C4⋊C4×C61C2211yescharacteristic
29C61×#417—22no—
30C61×#417C2214yes—
31C2×#103×C61—21no—
32C2×#102×C61—21no—
33C23×Dic61—21no—
34C2×#11758194—21no—
35C2×#11758194—21no—
36C2×#11758194—22no—
37C2×#11758194—22no—
38C2×#11758194—22no—
39C2×#11758194—22no—
40C2×#11758194C2312yes—
41C2×#11758194C2312yes—
42C22×#92418—22no—
43C22×#92418—22no—
44C22×#92418—21no—
45C22×#92418—21no—
46C22×#92418D412yes—
47C22×#92418D412yes—
48C22×#92418C2312yes—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55#14519708—28no—
56#14519701—24no—
57#14519698—24no—
58C22⋊#92418—28no—
59C2×#41×C61—21no—
60C2×#41×C61—21no—
61C2×#41×C61—22no—
62C2×#41×C61—22no—
63C2×#41×C61—22no—
64C2×#41×C61—22no—
65C2×#41×C61D412yes—
66C2×#41×C61C2312yes—
67C22×Q8×C61D411yescharacteristic
68C22×D4×C61—21no—
69C22×D4×C61D411yescharacteristic
70C23×C4×C61—21no—
71C23×C4×C61—21no—
72C23×C4×C61C2311yescharacteristic
73C2×M16×C61—21no—
74C2×M16×C61—22no—
75C2×M16×C61—22no—
76C2×M16×C61C2312yes—
77C22×C8×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61C2312yes—
84#103×C61—24no—
85#102×C61—24no—
86C24×C61—21no—
87C24×C61#4111yescharacteristic
88#41×C61—24no—
89#41×C61—24no—
90#41×C61—22no—
91#41×C61—22no—
92#41×C61—22no—
93#41×C61—22no—
94#41×C61—42no—
95#41×C61—42no—
96#41×C61—42no—
97#41×C61—42no—
98C2×Q8×C61—21no—
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61C2×D412yes—
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—42no—
108C2×D4×C61—42no—
109C2×D4×C61C2×D412yes—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61—82no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61—41no—
131C22×Dic61—41no—
132C22×Dic61—82no—
133M16×C61—22no—
134M16×C61—22no—
135M16×C61—42no—
136C2×C8×C61—22no—
137C2×C8×C61—42no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—41no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146#11758194—22no—
147#11758194—22no—
148#11758194—22no—
149#11758194—22no—
150#11758194—42no—
151#11758194—42no—
152C2×#92418—24no—
153C2×#92418—24no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—42no—
159C2×#92418—42no—
160C2×#92418C2×D414yes—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—42no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61#12111yescharacteristic
168C23×C61#12111yescharacteristic
169C23×C61#12612yes—
170C23×C61#12612yes—
171C23×C61C2×#4111yescharacteristic
172Q8×C61—22no—
173Q8×C61—22no—
174Q8×C61—42no—
175D4×C61—22no—
176D4×C61—22no—
177D4×C61—42no—
178D4×C61—42no—
179D4×C61—44no—
180D4×C61—42no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—44no—
186C2×C4×C61—42no—
187C2×C4×C61—82no—
188C2×C4×C61—82no—
189C2×C4×C61—82no—
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#12312yes—
193C2×C4×C61#12312yes—
194C2×C4×C61#12312yes—
195C2×C4×C61#12312yes—
196C2×C4×C61#12312yes—
197C2×C4×C61#12312yes—
198C2×C4×C61#12412yes—
199C2×C4×C61#12412yes—
200C2×C4×C61#12612yes—
201C2×C4×C61#12612yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×Dic61—42no—
209C2×Dic61—82no—
210C2×Dic61—82no—
211C2×Dic61—82no—
212C8×C61—42no—
213#92418—24no—
214#92418—42no—
215#92418—42no—
216C2×#2079—611nomaximal, Sylow
217C22×C61—24no—
218C22×C61—24no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—84no—
227C22×C61#39712yes—
228C22×C61#39712yes—
229C22×C61#47112yes—
230C22×C61#47112yes—
231C22×C61C2×#12111yescharacteristic
232C22×C61C2×#12611yescharacteristic
233C22×C61C2×#12611yescharacteristic
234C4×C61—42no—
235C4×C61—82no—
236C4×C61#39712yes—
237C4×C61#39712yes—
238C4×C61C2×#12311yescharacteristic
239C4×C61C2×#12311yescharacteristic
240C4×C61C2×#12311yescharacteristic
241C4×C61C2×#12411yescharacteristic
242Dic61—82no—
243C22×#106—611no—
244C2×#417—611no—
245C2×#417D6111yescharacteristic
246C2×#408—611no—
247#2079—618no—
248C2×#343—611no—
249C2×#343—611no—
250C2×#341—611no—
251C2×C61—21no—
252C2×C61—21no—
253C2×C61—22no—
254C2×C61—82no—
255C2×C61#207914yescomplemented
256C2×C61C2×#39711yescharacteristic
257C2×C61C2×#47111yescharacteristic
258C2×C61C2×#47111yescharacteristic
259C22×#41C2×D6111yescharacteristic
260C22×M16—611no—
261C22×M16—611no—
262C22×M16C2×D6111yescharacteristic
263C23×C8—611no—
264C22×C4⋊C4—611no—
265C22×C4⋊C4C2×D6111yescharacteristic
266C2×#106—614no—
267C2×#106—1222no—
268C2×#106—1222no—
269#417—22no—
270#417—614no—
271#417—1222no—
272#417C2×D6114yes—
273C2×#103—21no—
274C2×#103—1221no—
275C2×#102—21no—
276C2×#102—1221no—
277#408—614no—
278#408—1222no—
279C2×C22⋊C8—1222no—
280#343—614no—
281#343—614no—
282#341—614no—
283C61C2×#207911yescharacteristic, complemented, Sylow
284C2×#41—21no—
285C2×#41—21no—
286C2×#41—22no—
287C2×#41—22no—
288C2×#41—22no—
289C2×#41—22no—
290C2×#41C22⋊D6112yes—
291C2×#41C22×D6112yes—
292C22×Q8C22⋊D6111yescharacteristic
293C22×D4—21no—
294C22×D4C22⋊D6111yescharacteristic
295C23×C4—21no—
296C23×C4—21no—
297C23×C4—1221no—
298C23×C4C22×D6111yescharacteristic
299C2×M16—21no—
300C2×M16—22no—
301C2×M16—22no—
302C2×M16—612no—
303C2×M16—612no—
304C2×M16—1221no—
305C2×M16—1221no—
306C2×M16—1222no—
307C2×M16—1222no—
308C2×M16—1222no—
309C2×M16—1222no—
310C2×M16C22×D6112yes—
311C22×C8—21no—
312C22×C8—612no—
313C22×C8—612no—
314C22×C8—612no—
315C22×C8—1222no—
316C22×C8—1222no—
317C22×C8—1221no—
318C22×C8—1221no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—612no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4C22×D6112yes—
331#106—1228no—
332#103—24no—
333#103—1224no—
334#102—24no—
335#102—1224no—
336C22⋊C8—1228no—
337C24—21no—
338C24#1175822411yescharacteristic
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—24no—
344#41—24no—
345#41—42no—
346#41—42no—
347#41—42no—
348#41—42no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8C2×C22⋊D6112yes—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4C2×C22⋊D6112yes—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—82no—
370C22×C4—1222no—
371C22×C4—4882no—
372C22×C4—2441no—
373C22×C4—2441no—
374C22×C4#1175822212yes—
375C22×C4#1175822212yes—
376C22×C4D4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4#1175822412yes—
379C22×C4#1175822412yes—
380C22×C4#1175822411yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383C22×C4C23×D6111yescharacteristic
384M16—22no—
385M16—22no—
386M16—42no—
387M16—1222no—
388M16—1222no—
389M16—1222no—
390M16—1222no—
391M16—2442no—
392M16—2442no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—614no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1224no—
399C2×C8—1224no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—2442no—
403C2×C8—2442no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—2441no—
411C4⋊C4—2441no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23#1451974911yescharacteristic
419C23#1451974911yescharacteristic
420C23#1451977012yes—
421C23#1451977012yes—
422C23C2×#1175822411yescharacteristic
423Q8—22no—
424Q8—22no—
425Q8—42no—
426D4—22no—
427D4—22no—
428D4—44no—
429D4—42no—
430D4—42no—
431D4—42no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—4882no—
442C2×C4—4882no—
443C2×C4—4882no—
444C2×C4—2442no—
445C2×C4#1451974911yescharacteristic
446C2×C4#1451974911yescharacteristic
447C2×C4#1451975812yes—
448C2×C4#1451975812yes—
449C2×C4#1451975912yes—
450C2×C4#1451975912yes—
451C2×C4#1451976012yes—
452C2×C4#1451976012yes—
453C2×C4#1451976312yes—
454C2×C4#1451976312yes—
455C2×C4#1451977012yes—
456C2×C4#1451977012yes—
457C2×C4C2×D4×D6111yescharacteristic
458C2×C4C22×C22⋊D6111yescharacteristic
459C2×C4C2×#1175822411yescharacteristic
460C2×C4C2×#1175822411yescharacteristic
461C2×C4C2×#1175822211yescharacteristic
462C2×C4C2×#1175822211yescharacteristic
463C8—42no—
464C8—1224no—
465C8—2442no—
466C8—2442no—
467C22—24no—
468C22—24no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—84no—
477C22#1683969012yes—
478C22#1683969012yes—
479C22#1684002312yes—
480C22#1684002712yes—
481C22C2×#1451974911yescharacteristic
482C22C2×#1451977011yescharacteristic
483C22C2×#1451977011yescharacteristic
484C4—42no—
485C4—82no—
486C4—4882no—
487C4#1683969012yes—
488C4#1683969012yes—
489C4C2×#1451975811yescharacteristic
490C4C2×#1451975911yescharacteristic
491C4C2×#1451976011yescharacteristic
492C4C2×#1451976311yescharacteristic
493C2—22no—
494C2—21no—
495C2—21no—
496C2—82no—
497C2#2144726714yesdirect product factor
498C2C2×#1683969011yescharacteristic
499C2C2×#1684002311yescharacteristic
500C2C2×#1684002711yescharacteristic
subgroupG/H
centerC23#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61#121
prime radicalC22×#41×C61C22
hypercenterC2×#417D61
hypocenterC61C2×#2079
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C22×#14519708C2semidirect
4. #21447267C2direct (N × H)
5. C2×#16839531C2semidirect
6. C2×#16839530C2semidirect
7. C2×#16839526C2semidirect
15. C2×#14519708C22semidirect
23. #16839531C22semidirect
24. #16839530C22semidirect
25. #16839526C22semidirect
255. C2×C61#2079semidirect
283. C61C2×#2079semidirect
497. C2#21447267direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197082
C2×#168397632
C2×#168397292
#214472672
C2×#168395312
C2×#168395302
C2×#168395262
C2×C61×#4172
C2×#207961