Finite Group Databasebeta

← C2×#21447780 #29833949

Subgroups — C2×#21447780 #29833949

Order 15616.

1098 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840576C211yesmaximal, characteristic, complemented
2C22×#14519721C211yesmaximal, characteristic, complemented
3C2×#16840029C211yesmaximal, characteristic, complemented
4#21447780C218yesmaximal, direct product factor
5C2×#16839579C211yesmaximal, characteristic, complemented
6C2×#16839569C211yesmaximal, characteristic, complemented
7C2×#16839547C211yesmaximal, characteristic, complemented
8C2×C61×#509C211yesmaximal, characteristic, complemented
9C2×#14519824—21no—
10#16840576—22no—
11#16840576—22no—
12#16840576—22no—
13#16840576C2214yescomplemented
14C2×#14519807—21no—
15C2×#14519806—22no—
16C2×#14519803—21no—
17C2×#14519796—21no—
18C22×Dic244C2211yescharacteristic
19C2×Q16×D61—21no—
20C2×#14519783C2211yescharacteristic
21C2×#14519721—22no—
22C2×#14519721—22no—
23C2×#14519721C2214yescomplemented
24C22×#11758185C2211yescharacteristic
25#16840029—22no—
26#16840029C2214yescomplemented
27C2×#14519713C2211yescharacteristic
28C2×#14519694—21no—
29#16839579C2214yescomplemented
30#16839569C2214yescomplemented
31#16839547C2214yescomplemented
32C2×#135×C61C2211yescharacteristic
33C22×D8×C61C2211yescharacteristic
34C61×#509—22no—
35C61×#509C2214yescomplemented
36C2×SD32×C61—21no—
37C2×D16×C61—21no—
38C2×M32×C61C2211yescharacteristic
39#14519824—24no—
40#14519824—24no—
41#14519824—42no—
42#14519824—42no—
43C2×#11758222—41no—
44C2×#11758222D411yescharacteristic
45C2×#11758224—22no—
46C2×#11758224—42no—
47C22×Dic122—21no—
48C22×Dic122—41no—
49C2×Q8×D61—22no—
50#14519807—24no—
51#14519807—24no—
52#14519806—28no—
53#14519806—28no—
54#14519803—24no—
55#14519803—42no—
56#14519796—24no—
57#14519796—24no—
58C2×Dic244—21no—
59C2×Dic244—21no—
60C2×Dic244—21no—
61C2×Dic244D412yes—
62C2×Dic244D412yes—
63C2×Dic244C2312yes—
64C2×#11758202—22no—
65Q16×D61—24no—
66Q16×D61—24no—
67C2×#11758200—22no—
68C2×#11758200—41no—
69C2×#11758199—41no—
70#14519783—21no—
71#14519783—21no—
72#14519783D412yes—
73#14519783C2312yes—
74C2×#11758194—21no—
75C2×#11758194D411yescharacteristic
76C2×#11758189—21no—
77C2×C8×D61—21no—
78#14519721—28no—
79C2×#11758185—21no—
80C2×#11758185—21no—
81C2×#11758185C2312yes—
82#14519713—21no—
83#14519713C2312yes—
84#14519694—24no—
85C2×#41×C61—21no—
86C22×D4×C61—21no—
87#135×C61—21no—
88#135×C61—21no—
89#135×C61D412yes—
90#135×C61C2312yes—
91C2×Q16×C61D411yescharacteristic
92C2×SD16×C61—21no—
93C2×D8×C61—21no—
94C2×D8×C61—21no—
95C2×D8×C61—21no—
96C2×D8×C61—21no—
97C2×D8×C61—21no—
98C2×D8×C61D411yescharacteristic
99C2×D8×C61C2312yes—
100C22×C8×C61C2311yescharacteristic
101SD32×C61—24no—
102D16×C61—24no—
103M32×C61—21no—
104M32×C61C2312yes—
105C2×C16×C61—21no—
106C24×C61—41no—
107#41×C61—22no—
108#41×C61—22no—
109#41×C61—41no—
110#41×C61—41no—
111C2×Q8×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—21no—
123C2×C22⋊D61—42no—
124C22×Dic61—42no—
125C22×Dic61—41no—
126Q8×D61—28no—
127Q8×D61—44no—
128#11758224—28no—
129#11758224—44no—
130#11758224—48no—
131#11758224—48no—
132#11758222—21no—
133#11758222—21no—
134#11758222—44no—
135#11758222—44no—
136#11758222D812yes—
137#11758222C2×D412yes—
138C2×D244D811yescharacteristic
139C2×C4×D61—21no—
140C2×C4×D61—42no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—44no—
145C2×Dic122—42no—
146C2×Dic122—41no—
147C2×Dic122—41no—
148C2×Dic122—41no—
149C2×Dic122—44no—
150C2×Dic122D811yescharacteristic
151Q16×C61—21no—
152Q16×C61C2×D412yes—
153SD16×C61—22no—
154SD16×C61—22no—
155D8×C61—21no—
156D8×C61—22no—
157D8×C61—22no—
158D8×C61—41no—
159D8×C61—41no—
160D8×C61C2×D412yes—
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2411yescharacteristic
167#11758202—28no—
168#11758200—28no—
169#11758200—44no—
170#11758199—44no—
171#11758194—21no—
172#11758194—22no—
173#11758194—41no—
174#11758194C2×D412yes—
175C2×#92418—21no—
176C2×#92418—41no—
177Dic244—22no—
178Dic244—22no—
179Dic244—22no—
180Dic244C2×D414yes—
181#11758189—22no—
182#11758189—22no—
183C8×D61—22no—
184C8×D61—22no—
185C16×C61—22no—
186#11758185—22no—
187C23×C61—41no—
188C23×C61—42no—
189C23×C61—81no—
190C23×C61—81no—
191C23×C61—162no—
192C23×C61AGL(1,ℤ/8)11yescharacteristic
193C22×D61—21no—
194Q8×C61—22no—
195Q8×C61—41no—
196D4×C61—22no—
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—41no—
202D4×C61—81no—
203D4×C61—81no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61C2×D811yescharacteristic
210C2×C4×C61C2×D811yescharacteristic
211C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C22⋊D61—22no—
214C22⋊D61—22no—
215C22⋊D61—48no—
216C2×Dic61—21no—
217C2×Dic61—48no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221D244—21no—
222D244C2×D812yes—
223C4×D61—22no—
224C4×D61—22no—
225C4×D61—48no—
226Dic122—21no—
227Dic122—24no—
228Dic122—48no—
229Dic122—42no—
230Dic122—42no—
231Dic122—42no—
232Dic122C2×D812yes—
233C8×C61#12011yescharacteristic
234C8×C61#12011yescharacteristic
235C8×C61C22×D411yescharacteristic
236C8×C61C22×D411yescharacteristic
237#92418—22no—
238#92418—42no—
239C2×#2193—611nomaximal, Sylow
240C22×C61—22no—
241C22×C61—42no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61—162no—
245C22×C61—162no—
246C22×C61—162no—
247C22×C61#44711yescharacteristic
248C22×C61#44711yescharacteristic
249C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C2×D61—22no—
251C2×D61—42no—
252C4×C61—42no—
253C4×C61#44711yescharacteristic
254C4×C61#44711yescharacteristic
255C4×C61C2×#12011yescharacteristic
256C4×C61C22×D811yescharacteristic
257Dic61—22no—
258Dic61—44no—
259C2×#578—611no—
260C2×#509D6111yescharacteristic, complemented
261C22×SD32—611no—
262C2×#472—611no—
263#2193—618no—
264C2×#362—611no—
265C2×#359—611no—
266C2×#350—611no—
267C2×C61—21no—
268C2×C61—21no—
269C2×C61—82no—
270C2×C61—162no—
271C2×C61#219312yescomplemented
272C2×C61C2×#44711yescharacteristic
273D61—42no—
274C2×2−1+4—1221no—
275#578—614no—
276#578—1222no—
277#578—1222no—
278#578—1222no—
279C2×#137—1221no—
280C2×#137—1222no—
281C2×AGL(1,ℤ/8)—1221no—
282C2×#135—1221no—
283C2×#135C2×D6111yescharacteristic
284C22×Q16—611no—
285C22×Q16—1221no—
286C22×D8C2×D6111yescharacteristic
287C2×#131—611no—
288#509—22no—
289#509C2×D6114yescomplemented
290C2×SD32—21no—
291C2×SD32—614no—
292C2×SD32—1222no—
293C2×SD32—1222no—
294C2×D16—21no—
295C2×M32C2×D6111yescharacteristic
296C22×C16—611no—
297#472—614no—
298#472—1222no—
299C2×#108—611no—
300C2×#100—1221no—
301#362—614no—
302#359—614no—
303#350—614no—
304C61C2×#219311yescharacteristic, complemented, Sylow
3052−1+4—1224no—
3062−1+4—1224no—
3072−1+4—2442no—
3082−1+4—2442no—
309C2×#41—21no—
310C2×#41—611no—
311C2×#41—1222no—
312C2×#41—2441no—
313C2×#41—2442no—
314C22×Q8—1222no—
315C22×Q8—1221no—
316C22×Q8—2441no—
317C22×D4—21no—
318#137—1224no—
319#137—1228no—
320#137—1228no—
321#137—1224no—
322AGL(1,ℤ/8)—1224no—
323AGL(1,ℤ/8)—2442no—
324#135—21no—
325#135—21no—
326#135—1224no—
327#135—1224no—
328#135C22⋊D6112yes—
329#135C22×D6112yes—
330C2×Q16—612no—
331C2×Q16—612no—
332C2×Q16—612no—
333C2×Q16—1224no—
334C2×Q16—1222no—
335C2×Q16—1224no—
336C2×Q16—1221no—
337C2×Q16—1221no—
338C2×Q16—1221no—
339C2×Q16C22⋊D6111yescharacteristic
340C2×SD16—21no—
341C2×SD16—1222no—
342C2×SD16—2441no—
343C2×D8—21no—
344C2×D8—21no—
345C2×D8—21no—
346C2×D8—21no—
347C2×D8—21no—
348C2×D8—2441no—
349C2×D8C22⋊D6111yescharacteristic
350C2×D8C22×D6112yes—
351#131—612no—
352#131—612no—
353#131—1221no—
354#131—1221no—
355C2×M16—611no—
356C2×M16—1221no—
357C2×M16—1221no—
358C22×C8—1221no—
359C22×C8C22×D6111yescharacteristic
360SD32—24no—
361SD32—1228no—
362D16—24no—
363M32—21no—
364M32C22×D6112yes—
365C2×C16—21no—
366C2×C16—612no—
367C2×C16—1221no—
368C2×C16—1221no—
369#108—612no—
370#108—1221no—
371#100—1224no—
372C24—41no—
373#41—22no—
374#41—22no—
375#41—41no—
376#41—41no—
377#41—612no—
378#41—612no—
379#41—1221no—
380#41—1221no—
381#41—1228no—
382#41—2444no—
383#41—2444no—
384#41—2444no—
385#41—2448no—
386#41—2448no—
387C2×Q8—21no—
388C2×Q8—611no—
389C2×Q8—1228no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—1222no—
393C2×Q8—2442no—
394C2×Q8—2444no—
395C2×Q8—2444no—
396C2×Q8—2444no—
397C2×Q8—2441no—
398C2×Q8—2441no—
399C2×Q8—2441no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—611no—
409C2×D4—1221no—
410C2×D4—2442no—
411C22×C4—41no—
412C22×C4—1221no—
413C22×C4—2442no—
414C22×C4—2442no—
415C22×C4—2441no—
416C22×C4D4×D6111yescharacteristic
417Q16—21no—
418Q16—614no—
419Q16—1228no—
420Q16—1222no—
421Q16—1222no—
422Q16—1222no—
423Q16C2×C22⋊D6112yes—
424SD16—22no—
425SD16—22no—
426SD16—1228no—
427SD16—2444no—
428D8—22no—
429D8—22no—
430D8—21no—
431D8—41no—
432D8—41no—
433D8—2444no—
434D8C2×C22⋊D6112yes—
435M16—612no—
436M16—1221no—
437M16—1222no—
438M16—1222no—
439M16—1222no—
440M16—2441no—
441C2×C8—1221no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—2441no—
445C2×C8D4×D6111yescharacteristic
446C2×C8D4×D6111yescharacteristic
447C2×C8D4×D6111yescharacteristic
448C2×C8C2×C22⋊D6111yescharacteristic
449C2×C8C2×C22⋊D6111yescharacteristic
450C2×C8C23×D6111yescharacteristic
451C16—22no—
452C16—1222no—
453C23—41no—
454C23—42no—
455C23—81no—
456C23—81no—
457C23—162no—
458C23—1221no—
459C23#1451980111yescharacteristic
460Q8—22no—
461Q8—41no—
462Q8—612no—
463Q8—1221no—
464Q8—1224no—
465Q8—2448no—
466Q8—2442no—
467Q8—2442no—
468Q8—2442no—
469D4—22no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—41no—
475D4—81no—
476D4—81no—
477D4—612no—
478D4—1222no—
479D4—1222no—
480D4—1221no—
481D4—2448no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—1221no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—2448no—
489C2×C4—2448no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4#1451974511yescharacteristic
494C2×C4#1451974511yescharacteristic
495C2×C4D8×D6111yescharacteristic
496C2×C4D8×D6111yescharacteristic
497C2×C4#1451980111yescharacteristic
498C2×C4C2×D4×D6111yescharacteristic
499C8—1222no—
500C8—2442no—
501C8#1451974511yescharacteristic
502C8#1451974511yescharacteristic
503C8C2×D4×D6111yescharacteristic
504C8C22×C22⋊D6111yescharacteristic
505C22—22no—
506C22—42no—
507C22—82no—
508C22—82no—
509C22—162no—
510C22—162no—
511C22—162no—
512C22—1222no—
513C22—2442no—
514C22#1683989411yescharacteristic
515C22#1683989411yescharacteristic
516C22C2×#1451980111yescharacteristic
517C4—42no—
518C4—1222no—
519C4—2444no—
520C4#1683989411yescharacteristic
521C4#1683989411yescharacteristic
522C4C2×#1451974511yescharacteristic
523C4C2×D8×D6111yescharacteristic
524C2—21no—
525C2—21no—
526C2—82no—
527C2—162no—
528C2—2442no—
529C2#2144778012yesdirect product factor
530C2C2×#1683989411yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61#447
hypercenterC2×#509D61
hypocenterC61C2×#2193
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#16840576C2semidirect
2. C22×#14519721C2semidirect
3. C2×#16840029C2semidirect
4. #21447780C2direct (N × H)
5. C2×#16839579C2semidirect
6. C2×#16839569C2semidirect
7. C2×#16839547C2semidirect
8. C2×C61×#509C2semidirect
13. #16840576C22semidirect
23. C2×#14519721C22semidirect
26. #16840029C22semidirect
29. #16839579C22semidirect
30. #16839569C22semidirect
31. #16839547C22semidirect
35. C61×#509C22semidirect
260. C2×#509D61semidirect
271. C2×C61#2193semidirect
289. #509C2×D61semidirect
304. C61C2×#2193semidirect
529. C2#21447780direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168405762
C22×#145197212
C2×#168400292
#214477802
C2×#168395792
C2×#168395692
C2×#168395472
C2×C61×#5092
C2×#219361