Finite Group Databasebeta

← C2×#21444904 #29655897

Subgroups — C2×#21444904 #29655897

Order 15616.

1190 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic, complemented
2C2×#16840135C211yesmaximal, characteristic, complemented
3C2×#16839958C211yesmaximal, characteristic, complemented
4C2×#16839957C211yesmaximal, characteristic, complemented
5#21444904C218yesmaximal, direct product factor
6C2×#16839510C211yesmaximal, characteristic, complemented
7C2×#16839491C211yesmaximal, characteristic, complemented
8C2×C61×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×#11758190—22no—
10C22×D488—21no—
11C22×#11758199—21no—
12#16840477—22no—
13#16840477—22no—
14#16840477—22no—
15#16840477C2214yescomplemented
16C2×#14519777C2211yescharacteristic
17C2×#14519774C2211yescharacteristic
18C2×#14519763—22no—
19C2×#14519751—21no—
20C2×#14519748—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519736—21no—
23C2×#14519735—21no—
24#16840135—22no—
25#16840135C2214yescomplemented
26#16839958—22no—
27#16839958C2214yescomplemented
28#16839957—22no—
29#16839957C2214yescomplemented
30C2×C4⋊#92418C2211yescharacteristic
31C2×#14519699—21no—
32C2×#14519697—21no—
33#16839510C2214yescomplemented
34#16839491C2214yescomplemented
35C2×C4×D4×C61C2211yescharacteristic
36C2×C4⋊C8×C61C2211yescharacteristic
37C2×C4×C8×C61C2211yescharacteristic
38C61×D4⋊C8C2214yescomplemented
39C22×Dic122D412yes—
40C22×D244—21no—
41C22×D244—21no—
42C22×D244—21no—
43C22×D244D411yescharacteristic
44C22×C22⋊D61—22no—
45C22×C4×D61—21no—
46C2×#11758190—28no—
47C2×#11758190—48no—
48C2×D488—24no—
49C2×D488—44no—
50C2×#11758199—24no—
51C2×#11758199—44no—
52C22×#92418—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777D412yes—
56#14519777C2312yes—
57#14519774—21no—
58#14519774—21no—
59#14519774—21no—
60#14519774C2312yes—
61#14519763—28no—
62#14519763—28no—
63#14519751—24no—
64#14519751—24no—
65#14519748—24no—
66#14519748—24no—
67C4×D244—21no—
68C4×D244—21no—
69C4×D244D412yes—
70C4×D244C2312yes—
71#14519736—24no—
72#14519736—24no—
73#14519735—24no—
74#14519735—24no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×C22⋊Dic61—22no—
79C2×#11758198—22no—
80C2×#11758198—21no—
81C2×C4×Dic61—22no—
82C4⋊#92418—21no—
83C4⋊#92418C2312yes—
84#14519699—24no—
85#14519697—24no—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C22×C8×C61—21no—
89C22×C8×C61—21no—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61D412yes—
93C4×D4×C61C2312yes—
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C22⋊C4×C61—21no—
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—21no—
98C4⋊C8×C61C2312yes—
99C4×C8×C61—21no—
100C4×C8×C61C2312yes—
101C24×C61—21no—
102C23×D61—21no—
103C23×D61—41no—
104C2×D4×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61C2×D412yes—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C2×C22⋊D61—28no—
117C2×C22⋊D61—84no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—21no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—44no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244—42no—
130C2×D244—42no—
131C2×D244—41no—
132C2×D244—41no—
133C2×D244—41no—
134C2×D244C2×D412yes—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×Dic122—22no—
140C2×Dic122—44no—
141C2×Dic122C2×D414yes—
142C2×C8×C61—24no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—42no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2411yescharacteristic
154C22⋊Dic61—28no—
155#11758199—48no—
156#11758198—28no—
157#11758198—22no—
158#11758198—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61C2×D412yes—
165C4×Dic61—28no—
166C2×#92418—22no—
167C2×#92418—42no—
168D488—48no—
169#11758190—416no—
170C23×C61—22no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61#12011yescharacteristic
175C22×D61—22no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—82no—
181C22×D61—82no—
182C22×D61—82no—
183D4×C61—22no—
184D4×C61—22no—
185D4×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—44no—
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#13512yes—
196C2×C4×C61#13512yes—
197C2×C4×C61AGL(1,ℤ/8)12yes—
198C2×C4×C61AGL(1,ℤ/8)12yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C22⋊D61—88no—
203C2×Dic61—22no—
204C2×Dic61—24no—
205C2×Dic61—42no—
206C2×Dic61—44no—
207D244—22no—
208D244—22no—
209D244—44no—
210D244—42no—
211D244—42no—
212D244—42no—
213D244—42no—
214D244—42no—
215D244—82no—
216D244—82no—
217D244—82no—
218C4×D61—42no—
219C4×D61—42no—
220C4×D61—44no—
221Dic122—44no—
222Dic122—82no—
223C8×C61—24no—
224C8×C61—42no—
225#92418—42no—
226C2×#1624—611nomaximal, Sylow
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—84no—
231C22×C61#44912yes—
232C22×C61#44912yes—
233C22×C61#45312yes—
234C22×C61C2×#12011yescharacteristic
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—84no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—164no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—42no—
246C4×C61#44912yes—
247C4×C61#44912yes—
248C4×C61C2×#13511yescharacteristic
249C4×C61C2×#13511yescharacteristic
250C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
252Dic61—42no—
253Dic61—44no—
254C2×#555—611no—
255C2×#492—611no—
256C2×#459—611no—
257C2×#459—611no—
258#1624—618no—
259C2×#333—611no—
260C2×D4⋊C8—611no—
261C2×D4⋊C8D6111yescharacteristic, complemented
262C2×C61—82no—
263C2×C61#162414yescomplemented
264C2×C61C2×#44911yescharacteristic
265C2×C61C2×#44911yescharacteristic
266C2×C61C2×#45311yescharacteristic
267D61—82no—
268D61—162no—
269C22×SD16—1222no—
270C22×D8—1221no—
271C22×D8—1221no—
272#555—614no—
273#555—1222no—
274#555—1222no—
275#555—1222no—
276C2×C4⋊Q8—611no—
277C2×#127—611no—
278C2×#124—1222no—
279C2×#121—1221no—
280C2×#121—1221no—
281C2×C4×D4—611no—
282C2×C4×D4C2×D6111yescharacteristic
283C2×#117—1221no—
284C2×#117—1221no—
285#492—614no—
286#492—1222no—
287#459—614no—
288#459—614no—
289#459—1222no—
290#459—1222no—
291C2×C4⋊C8—611no—
292C2×C4⋊C8C2×D6111yescharacteristic
293C2×#102—1221no—
294C2×#102—1221no—
295C2×C4×C8C2×D6111yescharacteristic
296#333—614no—
297D4⋊C8—614no—
298D4⋊C8C2×D6114yescomplemented
299C61C2×#162411yescharacteristic, complemented, Sylow
300C22×Q8—612no—
301C22×D4—611no—
302C22×D4—1222no—
303C22×D4—1221no—
304C22×D4—1221no—
305C22×D4—1221no—
306C22×D4D24411yescharacteristic
307C23×C4—21no—
308C23×C4—1221no—
309C2×SD16—1228no—
310C2×SD16—2448no—
311C2×D8—1224no—
312C2×D8—1224no—
313C2×D8—2444no—
314C2×D8—2444no—
315C22×C8—21no—
316C22×C8—21no—
317C22×C8—1221no—
318C4⋊Q8—612no—
319C4⋊Q8—612no—
320C4⋊Q8—1221no—
321C4⋊Q8—1221no—
322#127—612no—
323#127—1221no—
324#127—1221no—
325#127—1221no—
326#124—1228no—
327#124—1228no—
328#121—1224no—
329#121—1224no—
330#121—1224no—
331#121—1224no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—612no—
335C4×D4—612no—
336C4×D4—1221no—
337C4×D4—1221no—
338C4×D4D24412yes—
339C4×D4C22×D6112yes—
340#117—1224no—
341#117—1224no—
342#117—1224no—
343#117—1224no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4D24411yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1222no—
351C2×C22⋊C4—1221no—
352C2×C42—1222no—
353C2×C42C22×D6111yescharacteristic
354C4⋊C8—21no—
355C4⋊C8—612no—
356C4⋊C8—1221no—
357C4⋊C8C22×D6112yes—
358#102—1224no—
359#102—1224no—
360C4×C8—21no—
361C4×C8C22×D6112yes—
362C24—21no—
363C24—1221no—
364C24—2441no—
365C2×Q8—614no—
366C2×Q8—1222no—
367C2×Q8—2444no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—612no—
372C2×D4—1228no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—1222no—
376C2×D4—1221no—
377C2×D4—1222no—
378C2×D4—1222no—
379C2×D4—4884no—
380C2×D4—2444no—
381C2×D4—2441no—
382C2×D4—2441no—
383C2×D4—2441no—
384C2×D4—2442no—
385C2×D4—2442no—
386C2×D4—2442no—
387C2×D4—2442no—
388C2×D4C2×D24412yes—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—1221no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4C2×D24411yescharacteristic
401C22×C4D4×D6111yescharacteristic
402C22×C4D4×D6111yescharacteristic
403SD16—24416no—
404D8—2448no—
405D8—2448no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—42no—
410C2×C8—1222no—
411C2×C8—2442no—
412C4⋊C4—21no—
413C4⋊C4—612no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1221no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4C2×D24412yes—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—1228no—
423C22⋊C4—1228no—
424C22⋊C4—1222no—
425C22⋊C4—1222no—
426C42—1228no—
427C42C2×D24411yescharacteristic
428C42D4×D6111yescharacteristic
429C42D4×D6111yescharacteristic
430C42C23×D6111yescharacteristic
431C23—22no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—1222no—
436C23—4882no—
437C23—4882no—
438C23—4882no—
439C23—2442no—
440C23—2442no—
441C23—2442no—
442C23—2442no—
443C23#1451974711yescharacteristic
444Q8—4882no—
445Q8—2444no—
446D4—22no—
447D4—22no—
448D4—42no—
449D4—1222no—
450D4—1222no—
451D4—4882no—
452D4—4888no—
453D4—4882no—
454D4—4882no—
455D4—2444no—
456D4—2442no—
457D4—2442no—
458D4—2442no—
459D4—2442no—
460D4—2442no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—1222no—
468C2×C4—1224no—
469C2×C4—2442no—
470C2×C4—2444no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2444no—
474C2×C4#1451974711yescharacteristic
475C2×C4#1451974711yescharacteristic
476C2×C4#1451974711yescharacteristic
477C2×C4#1451979812yes—
478C2×C4#1451979912yes—
479C2×C4#1451980112yes—
480C2×C4#1451980512yes—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4C22×D24411yescharacteristic
484C8—24no—
485C8—42no—
486C8—2442no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—84no—
491C22—9764no—
492C22—4882no—
493C22—4882no—
494C22—4882no—
495C22—4884no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22#1683990812yes—
500C22#1683990912yes—
501C22#1683993112yes—
502C22C2×#1451974711yescharacteristic
503C4—21no—
504C4—21no—
505C4—42no—
506C4—2444no—
507C4—2442no—
508C4#1683990812yes—
509C4#1683990912yes—
510C4C2×#1451979811yescharacteristic
511C4C2×#1451979911yescharacteristic
512C4C2×#1451980111yescharacteristic
513C4C2×#1451980511yescharacteristic
514C2—82no—
515C2—9762no—
516C2—4882no—
517C2#2144490414yesdirect product factor
518C2C2×#1683990811yescharacteristic
519C2C2×#1683990911yescharacteristic
520C2C2×#1683993111yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×D4⋊C8C2
socleC23×C61#120
hypercenterC2×D4⋊C8D61
hypocenterC61C2×#1624
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#16840477C2semidirect
2. C2×#16840135C2semidirect
3. C2×#16839958C2semidirect
4. C2×#16839957C2semidirect
5. #21444904C2direct (N × H)
6. C2×#16839510C2semidirect
7. C2×#16839491C2semidirect
8. C2×C61×D4⋊C8C2semidirect
15. #16840477C22semidirect
25. #16840135C22semidirect
27. #16839958C22semidirect
29. #16839957C22semidirect
33. #16839510C22semidirect
34. #16839491C22semidirect
38. C61×D4⋊C8C22semidirect
261. C2×D4⋊C8D61semidirect
263. C2×C61#1624semidirect
298. D4⋊C8C2×D61semidirect
299. C61C2×#1624semidirect
517. C2#21444904direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404772
C2×#168401352
C2×#168399582
C2×#168399572
#214449042
C2×#168395102
C2×#168394912
C2×C61×D4⋊C82
C2×#162461