Finite Group Databasebeta

← C2×#81089 #11675663

Subgroups — C2×#81089 #11675663

Order 768.

2082 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5730C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic, complemented
4C2×#5080C211yesmaximal, characteristic, complemented
5C2×C4×#213C211yesmaximal, characteristic
6#81089C2116yesmaximal, direct product factor
7C2×#4870C212yesmaximal, complemented
8C2×#4856C212yesmaximal, complemented
9C2×#4846C211yesmaximal, characteristic, complemented
10C2×#4831C211yesmaximal, characteristic, complemented
11C2×#4766C212yesmaximal, complemented
12C2×#4785C212yesmaximal
13C2×#3384—31nomaximal, Sylow
14C23×Dic6—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#946—22no—
18C2×D4×Dic3C2212yes—
19C22×#213C2211yescharacteristic
20C2×#924—21no—
21#5730—24no—
22#5730C2218yescomplemented
23D4×Dic6—24no—
24D4×Dic6C2218yescomplemented
25C2×#897—21no—
26C2×#893C2212yes—
27C2×D3×C4⋊C4—21no—
28C2×#890C2212yes—
29C2×#888C2212yes—
30C2×#878—22no—
31C2×C4×D12C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#476C2218yescomplemented
35C2×C3×#106C2212yes—
36C2×C3×C4⋊C8C2211yescharacteristic
37C2×C3×#102C2212yes—
38C2×#832C2211yescharacteristic
39C2×#821C2212yes—
40C2×#818C2212yes—
41C2×#815C2212yes—
42C2×Dic3⋊C8C2212yes—
43#5080C2218yescomplemented
44C4×#213C2218yes—
45C2×#809C2211yescharacteristic
46C2×#808C2212yes—
47C2×#808C2212yes—
48C2×#807C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4870C22116yescomplemented
51#4856C22116yescomplemented
52#4846C2218yescomplemented
53#4831C2218yescomplemented
54#4785C22116yes—
55#4766C22116yescomplemented
56C2×#554—31no—
57C2×D4×Q8—31no—
58#3384—316no—
59C2×#501—32no—
60C2×#498—31no—
61C2×#478—32no—
62C2×#476—32no—
63C2×#476D311yescharacteristic, complemented
64C2×#474—31no—
65C2×#446—31no—
66C2×#442—32no—
67C2×C4×D8—31no—
68C22×C3×Q8—21no—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×Dic3—22no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—22no—
75C22×Dic6—21no—
76C22×Dic6—22no—
77C22×Dic6—28no—
78C22×Dic6—48no—
79C22×Dic6D411yescharacteristic
80C22×C3×C8C2312yes—
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8C2314yes—
83C3×C4×D4—22no—
84C3×C4×D4C2314yes—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—22no—
91#946—216no—
92D4×Dic3—24no—
93D4×Dic3C2318yes—
94C2×#213Q818yescomplemented
95C2×#213C2314yes—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2312yes—
102C2×C4⋊Dic3C2311yescharacteristic
103#924—28no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4D412yes—
107C2×C4×Dic3C2312yes—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#897—28no—
111#893—24no—
112#893C2318yes—
113D3×C4⋊C4—28no—
114#890—24no—
115#890C2318yes—
116#888—24no—
117#888C2318yes—
118#878—216no—
119C4×D12—22no—
120C4×D12C2314yes—
121#869—22no—
122#869C2314yes—
123C4×Dic6D414yes—
124C4×Dic6C2314yes—
125C3×#106C2318yes—
126C3×C4⋊C8C2314yes—
127C3×#102C2318yes—
128#832C2314yes—
129#821C2318yes—
130#818C2318yes—
131#815C2318yes—
132Dic3⋊C8C2318yes—
133#809C2314yes—
134#808C2318yes—
135#808C2318yes—
136#807C2314yes—
137C4×C3⋊C8C2314yes—
138C23×Q8—61no—
139C22×D8—31no—
140#554—38no—
141#554—64no—
142D4×Q8—38no—
143D4×Q8—64no—
144C2×C4⋊Q8—32no—
145C2×C4⋊Q8—31no—
146C2×C4⋊Q8C2×D311yescharacteristic
147C2×#125—32no—
148C2×C22⋊Q8—62no—
149C2×C22⋊Q8—61no—
150C2×C22⋊Q8—61no—
151C2×C22⋊Q8—62no—
152C2×C4×Q8—31no—
153C2×C4×D4—32no—
154C2×C4×D4—32no—
155C2×C4×D4—31no—
156C2×C4×D4C2×D311yescharacteristic
157C22×C4⋊C4—61no—
158#501—316no—
159#498—38no—
160#478—316no—
161#476—316no—
162#476C2×D318yescomplemented
163#474—38no—
164#446—38no—
165#442—316no—
166C4×D8—38no—
167C2×#107—32no—
168C2×#107—31no—
169C2×#106—32no—
170C2×#106—32no—
171C2×#106C2×D312yes—
172C2×C4⋊C8—32no—
173C2×C4⋊C8C2×D311yescharacteristic
174C2×#102—31no—
175C2×#102—32no—
176C2×#102—31no—
177C2×#102C2×D312yes—
178C2×#97—32no—
179C2×C4×C8—31no—
180C24×C3—21no—
181C23×D3—21no—
182C2×C3×Q8—24no—
183C2×C3×Q8—82no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4C2×Q814yes—
187C2×C3×D4C2412yes—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—44no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—28no—
198C22×Dic3—48no—
199C22×Dic3C2×D412yes—
200C22×Dic3C2×D412yes—
201C2×D12—21no—
202C2×D12—22no—
203C2×D12C2×Q814yescomplemented
204C2×D12C2412yes—
205C2×C4×D3—24no—
206C2×C4×D3—28no—
207C2×C4×D3—44no—
208C2×C4×D3—84no—
209C2×Dic6—24no—
210C2×Dic6—24no—
211C2×Dic6—28no—
212C2×Dic6—432no—
213C2×Dic6—48no—
214C2×Dic6—44no—
215C2×Dic6C2×D414yes—
216C2×C3×C8—24no—
217C2×C3×C8C2414yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4C2412yes—
222C3×C4⋊C4C2414yes—
223C3×C22⋊C4—24no—
224C3×C42C2×D412yes—
225C3×C42C2412yes—
226#217—28no—
227#213C2×Q8116yescomplemented
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—28no—
230C4⋊Dic3—22no—
231C4⋊Dic3—24no—
232C4⋊Dic3—21no—
233C4⋊Dic3—28no—
234C4⋊Dic3C2×D414yes—
235C4⋊Dic3C2412yes—
236C4⋊Dic3C2414yes—
237Dic3⋊C4—28no—
238Dic3⋊C4—28no—
239Dic3⋊C4C2×D418yes—
240C4×Dic3C2×D414yes—
241C4×Dic3C2414yes—
242C2×C3⋊C8—22no—
243C2×C3⋊C8C2×Q814yescomplemented
244C2×C3⋊C8C2412yes—
245C2×C3⋊C8C2412yes—
246C22×Q8—21no—
247C22×Q8—31no—
248C22×Q8—62no—
249C22×Q8—68no—
250C22×Q8—61no—
251C22×Q8—128no—
252C22×D4—31no—
253C22×D4C22×D311yescharacteristic
254C23×C4—21no—
255C23×C4—62no—
256C23×C4—62no—
257C23×C4—61no—
258C2×D8—38no—
259C2×D8—34no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8C22×D312yes—
263C4⋊Q8—22no—
264C4⋊Q8—38no—
265C4⋊Q8—34no—
266C4⋊Q8—64no—
267C4⋊Q8—62no—
268C4⋊Q8C22×D314yes—
269#125—38no—
270#125—64no—
271C22⋊Q8—616no—
272C22⋊Q8—68no—
273C22⋊Q8—68no—
274C22⋊Q8—616no—
275C4×Q8—34no—
276C4×Q8—34no—
277C4×D4—22no—
278C4×D4—38no—
279C4×D4—38no—
280C4×D4—34no—
281C4×D4—64no—
282C4×D4—64no—
283C4×D4—62no—
284C4×D4C22×D314yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—68no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4C22×D312yes—
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—61no—
301C2×C42—32no—
302C2×C42C22×D311yescharacteristic
303#107—38no—
304#107—34no—
305#106—38no—
306#106—38no—
307#106C22×D318yes—
308C4⋊C8—38no—
309C4⋊C8C22×D314yes—
310#102—34no—
311#102—38no—
312#102—34no—
313#102C22×D318yes—
314#97—38no—
315C4×C8—34no—
316C23×C3—24no—
317C23×C3—22no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3C22×D411yescharacteristic
321C22×D3—24no—
322C22×D3—22no—
323C22×D3—42no—
324C22×D3—42no—
325C3×Q8—84no—
326C3×D4—24no—
327C3×D4—22no—
328C3×D4C22×Q814yes—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4—48no—
335C2×C3×C4AGL(1,ℤ/8)14yes—
336C2×C3×C4C22×D412yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D412yes—
339C2×C3×C4C22×Q812yes—
340C2×C3×C42−1+412yes—
341C2×C3×C4C2511yescharacteristic
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—416no—
347C2×Dic3—44no—
348C2×Dic3—42no—
349C2×Dic3C22×D414yes—
350C2×Dic3C22×D414yes—
351D12—24no—
352D12—22no—
353D12C22×Q814yescomplemented
354C4×D3—48no—
355C4×D3—88no—
356Dic6—28no—
357Dic6—416no—
358Dic6—48no—
359C3×C8—24no—
360C3⋊C8—22no—
361C3⋊C8C22×Q814yescomplemented
362C24—21no—
363C24—61no—
364C2×Q8—24no—
365C2×Q8—34no—
366C2×Q8—68no—
367C2×Q8—64no—
368C2×Q8—64no—
369C2×Q8—82no—
370C2×Q8—128no—
371C2×Q8—1232no—
372C2×Q8—124no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—34no—
376C2×D4—32no—
377C2×D4—61no—
378C2×D4—62no—
379C2×D4D3×Q814yes—
380C2×D4C23×D312yes—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—32no—
385C22×C4—32no—
386C22×C4—44no—
387C22×C4—68no—
388C22×C4—68no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—64no—
392C22×C4—128no—
393C22×C4—124no—
394C22×C4—244no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4C23×D311yescharacteristic
398D8—316no—
399C2×C8—24no—
400C2×C8—34no—
401C2×C8—32no—
402C2×C8—32no—
403C2×C8—62no—
404C2×C8C23×D314yes—
405C4⋊C4—24no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—34no—
409C4⋊C4—38no—
410C4⋊C4—34no—
411C4⋊C4—32no—
412C4⋊C4—68no—
413C4⋊C4—68no—
414C4⋊C4—68no—
415C4⋊C4—61no—
416C4⋊C4—62no—
417C4⋊C4—64no—
418C4⋊C4C23×D312yes—
419C4⋊C4C23×D314yes—
420C22⋊C4—24no—
421C22⋊C4—68no—
422C22⋊C4—68no—
423C22⋊C4—64no—
424C42—34no—
425C42—34no—
426C42D3×D412yes—
427C42C23×D312yes—
428C22×C3—24no—
429C22×C3—48no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3D4×Q812yes—
433C22×C3C2×AGL(1,ℤ/8)12yes—
434C22×C3#57712yes—
435C22×C3C23×D411yescharacteristic
436C2×D3—24no—
437C2×D3—48no—
438C2×D3—42no—
439C2×D3—42no—
440C3×C4—22no—
441C3×C4—42no—
442C3×C4—44no—
443C3×C4D4×Q812yes—
444C3×C4C2×AGL(1,ℤ/8)14yes—
445C3×C4C23×Q811yescharacteristic
446C3×C4C2×2−1+411yescharacteristic
447Dic3—24no—
448Dic3—24no—
449Dic3—44no—
450Dic3—42no—
451C23—24no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23—64no—
456C23—62no—
457C23—122no—
458C23—122no—
459C23C2×D3×D411yescharacteristic
460Q8—68no—
461Q8—84no—
462Q8—1216no—
463Q8—128no—
464D4—24no—
465D4—22no—
466D4—34no—
467D4—64no—
468D4—62no—
469D4C2×D3×Q814yes—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—34no—
474C2×C4—34no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—62no—
481C2×C4—64no—
482C2×C4—1216no—
483C2×C4—124no—
484C2×C4—122no—
485C2×C4—128no—
486C2×C4—248no—
487C2×C4#91414yes—
488C2×C4C2×D3×D412yes—
489C2×C4C2×D3×D411yescharacteristic
490C2×C4C2×D3×D412yes—
491C2×C4C2×D3×Q812yes—
492C2×C4#101012yes—
493C2×C4C24×D311yescharacteristic
494C8—24no—
495C8—34no—
496C8—62no—
497C2×C3—44no—
498C2×C3#338414yescomplemented
499C2×C3C2×D4×Q811yescharacteristic
500C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
501C2×C3C2×#57711yescharacteristic
502D3—44no—
503C22—24no—
504C22—42no—
505C22—42no—
506C22—48no—
507C22—64no—
508C22—122no—
509C22—122no—
510C22—128no—
511C22#563212yes—
512C22C2×#91412yes—
513C22#577512yes—
514C22C22×D3×D411yescharacteristic
515C4—22no—
516C4—42no—
517C4—44no—
518C4—64no—
519C4—64no—
520C4—124no—
521C4—122no—
522C4#563212yes—
523C4C2×#91414yes—
524C4C22×D3×Q811yescharacteristic
525C4C2×#101011yescharacteristic
526C3C2×#338411yescharacteristic, complemented, Sylow
527C2—44no—
528C2—124no—
529C2#8108914yesdirect product factor
530C2C2×#563211yescharacteristic
531C2C22×#91411yescharacteristic
532C2C2×#577511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#476C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#476D3
hypocenterC3C2×#3384
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#5730C2semidirect
2. C2×D4×Dic6C2semidirect
3. C2×C3×#476C2semidirect
4. C2×#5080C2semidirect
6. #81089C2direct (N × H)
7. C2×#4870C2semidirect
8. C2×#4856C2semidirect
9. C2×#4846C2semidirect
10. C2×#4831C2semidirect
11. C2×#4766C2semidirect
22. #5730C22semidirect
24. D4×Dic6C22semidirect
34. C3×#476C22semidirect
43. #5080C22semidirect
50. #4870C22semidirect
51. #4856C22semidirect
52. #4846C22semidirect
53. #4831C22semidirect
55. #4766C22semidirect
63. C2×#476D3semidirect
94. C2×#213Q8semidirect
162. #476C2×D3semidirect
203. C2×D12C2×Q8semidirect
227. #213C2×Q8semidirect
243. C2×C3⋊C8C2×Q8semidirect
353. D12C22×Q8semidirect
361. C3⋊C8C22×Q8semidirect
498. C2×C3#3384semidirect
526. C3C2×#3384semidirect
529. C2#81089direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57302
C2×D4×Dic62
C2×C3×#4762
C2×#50802
C2×C4×#2132
#810892
C2×#48702
C2×#48562
C2×#48462
C2×#48312
C2×#47662
C2×#47852
C2×#33843