Finite Group Databasebeta

← C2×#83630 #11675664

Subgroups — C2×#83630 #11675664

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×C3×#554C211yesmaximal, characteristic, complemented
2C2×Q8×D12C211yesmaximal, characteristic, complemented
3#83630C2116yesmaximal, direct product factor
4C2×#5084C211yesmaximal, characteristic, complemented
5C2×#5080C211yesmaximal, characteristic, complemented
6C2×#5066C212yesmaximal, complemented
7C2×#5064C212yesmaximal, complemented
8C2×#5024C211yesmaximal, characteristic, complemented
9C2×#5008C212yesmaximal, complemented
10C2×#5007C211yesmaximal, characteristic, complemented
11C2×#5014C212yesmaximal
12C2×C4×#213C211yesmaximal, characteristic
13C2×#3384—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C3×#554—24no—
16C3×#554C2218yescomplemented
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2212yes—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8—22no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2212yes—
24C22×C3×C4⋊C4—21no—
25C22×#213C2211yescharacteristic
26Q8×D12—24no—
27Q8×D12C2218yescomplemented
28C2×#897—21no—
29C2×#897—22no—
30C2×C4×D12C2211yescharacteristic
31C2×C4×D12C2212yes—
32C2×#869C2211yescharacteristic
33C2×#869C2212yes—
34C2×#832C2211yescharacteristic
35C2×#832C2212yes—
36#5084C2218yescomplemented
37#5080C2218yescomplemented
38#5066C22116yescomplemented
39#5064C22116yescomplemented
40#5024C2218yescomplemented
41#5014C22116yes—
42C4×#213C2218yes—
43#5008C22116yescomplemented
44#5007C2218yescomplemented
45C2×#809C2212yes—
46C2×#809C2211yescharacteristic
47C2×#808C2212yes—
48C2×#808C2212yes—
49C2×#808C2212yes—
50C2×#807C2211yescharacteristic
51C2×#807C2212yes—
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2212yes—
54C2×#803C2212yes—
55C2×C4×C3⋊C8C2211yescharacteristic
56C2×#554D311yescharacteristic, complemented
57C2×D4×Q8—31no—
58#3384—316no—
59C2×#501—32no—
60C2×#498—31no—
61C2×#478—32no—
62C2×#476—31no—
63C2×#476—32no—
64C2×#474—31no—
65C2×#446—31no—
66C2×#442—32no—
67C2×C4×D8—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4—22no—
73C2×D3×Q8—28no—
74C2×D3×Q8—48no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—22no—
77C22×C4×D3—21no—
78C22×Dic6—22no—
79C22×Dic6—21no—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8C2314yes—
82C3×#125—24no—
83C3×#125C2318yes—
84C3×C22⋊Q8—28no—
85C3×C22⋊Q8—216no—
86C3×C4×Q8D414yes—
87C3×C4×Q8C2314yes—
88C3×C4×D4—24no—
89C3×C4×D4—22no—
90C3×C4×D4C2314yes—
91C3×C4×D4C2318yes—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—28no—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D412yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2312yes—
104C2×#213Q818yescomplemented
105C2×#213C2314yes—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—22no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—22no—
110C2×C4⋊Dic3C2312yes—
111C2×C4⋊Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2311yescharacteristic
115#897—28no—
116#897—216no—
117C4×D12—24no—
118C4×D12—22no—
119C4×D12C2314yes—
120C4×D12C2318yes—
121#869—24no—
122#869—22no—
123#869C2314yes—
124#869C2318yes—
125#832C2318yes—
126#832C2314yes—
127#809C2314yes—
128#809C2318yes—
129#808C2318yes—
130#808C2318yes—
131#808C2318yes—
132#807C2318yes—
133#807C2314yes—
134C4⋊(C3⋊C8)C2314yes—
135C4⋊(C3⋊C8)C2318yes—
136#803C2318yes—
137C4×C3⋊C8C2314yes—
138C23×Q8—61no—
139C22×D8—31no—
140#554—24no—
141#554C2×D318yescomplemented
142D4×Q8—38no—
143D4×Q8—64no—
144C2×C4⋊Q8—31no—
145C2×C4⋊Q8—32no—
146C2×C4⋊Q8C2×D311yescharacteristic
147C2×#125C2×D312yes—
148C2×C22⋊Q8—21no—
149C2×C22⋊Q8—22no—
150C2×C22⋊Q8—61no—
151C2×C22⋊Q8—62no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—32no—
154C2×C4×D4—31no—
155C2×C4×D4C2×D311yescharacteristic
156C2×C4×D4C2×D312yes—
157C22×C4⋊C4—21no—
158#501—316no—
159#498—38no—
160#478—316no—
161#476—38no—
162#476—316no—
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×#106—32no—
172C2×C4⋊C8—32no—
173C2×C4⋊C8—31no—
174C2×#102—31no—
175C2×#102—32no—
176C2×#102—31no—
177C2×#102—32no—
178C2×#97—32no—
179C2×C4×C8—31no—
180C24×C3—21no—
181C23×D3—21no—
182C2×C3×Q8—24no—
183C2×C3×Q8—24no—
184C2×C3×Q8—82no—
185C2×C3×Q8C2×D414yes—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4C2×Q814yescomplemented
189C2×C3×D4C2412yes—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—24no—
193C22×C3×C4—28no—
194C22×C3×C4—44no—
195C22×C3×C4—84no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—22no—
203D3×Q8—432no—
204C2×D12—21no—
205C2×D12—22no—
206C2×D12C2×Q814yes—
207C2×D12C2412yes—
208C2×C4×D3—28no—
209C2×C4×D3—24no—
210C2×C4×D3—48no—
211C2×C4×D3—44no—
212C2×Dic6—28no—
213C2×Dic6—24no—
214C2×Dic6—48no—
215C2×Dic6—44no—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—28no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4C2×D414yes—
222C3×C4⋊C4C2×D418yes—
223C3×C4⋊C4C2412yes—
224C3×C4⋊C4C2414yes—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—28no—
227C3×C42C2×D414yes—
228C3×C42C2×D412yes—
229C3×C42C2412yes—
230C3×C42C2414yes—
231#213C2×Q8116yescomplemented
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—28no—
234C4⋊Dic3—21no—
235C4⋊Dic3—22no—
236C4⋊Dic3—24no—
237C4⋊Dic3—28no—
238C4⋊Dic3C2414yes—
239C4⋊Dic3C2412yes—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—24no—
242C2×C3⋊C8C2×Q814yescomplemented
243C2×C3⋊C8C2412yes—
244C2×C3⋊C8C2414yes—
245C2×C3⋊C8C2412yes—
246C22×Q8—21no—
247C22×Q8—68no—
248C22×Q8—62no—
249C22×Q8—61no—
250C22×Q8—128no—
251C22×Q8#6711yescharacteristic
252C22×D4—31no—
253C22×D4C22×D311yescharacteristic
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—62no—
257C23×C4—61no—
258C2×D8—38no—
259C2×D8—34no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8—32no—
263C4⋊Q8—22no—
264C4⋊Q8—34no—
265C4⋊Q8—38no—
266C4⋊Q8—64no—
267C4⋊Q8—62no—
268C4⋊Q8C22×D314yes—
269#125—24no—
270#125C22×D318yes—
271C22⋊Q8—28no—
272C22⋊Q8—216no—
273C22⋊Q8—68no—
274C22⋊Q8—616no—
275C4×Q8#6714yes—
276C4×Q8C22×D314yes—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—34no—
280C4×D4—38no—
281C4×D4—64no—
282C4×D4—62no—
283C4×D4C22×D314yes—
284C4×D4C22×D318yes—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4#6711yescharacteristic
294C2×C4⋊C4#6712yes—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D312yes—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—62no—
301C2×C42C22×D312yes—
302C2×C42C22×D311yescharacteristic
303#107—34no—
304#107—38no—
305#106—38no—
306#106—38no—
307#106—38no—
308C4⋊C8—34no—
309C4⋊C8—38no—
310#102—34no—
311#102—38no—
312#102—34no—
313#102—38no—
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—28no—
326C3×Q8—84no—
327C3×D4—24no—
328C3×D4—22no—
329C3×D4C22×Q814yescomplemented
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—48no—
338C2×C3×C4—88no—
339C2×C3×C4AGL(1,ℤ/8)14yes—
340C2×C3×C4C22×D412yes—
341C2×C3×C4C22×D414yes—
342C2×C3×C4C22×D414yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D412yes—
345C2×C3×C4C22×Q812yes—
346C2×C3×C42−1+412yes—
347C2×C3×C4C2511yescharacteristic
348C2×Dic3—22no—
349C2×Dic3—24no—
350C2×Dic3—42no—
351C2×Dic3—44no—
352D12—24no—
353D12—22no—
354D12C22×Q814yes—
355C4×D3—416no—
356C4×D3—48no—
357Dic6—416no—
358Dic6—48no—
359C3⋊C8—22no—
360C3⋊C8—24no—
361C3⋊C8C22×Q814yescomplemented
362C24—21no—
363C24—61no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—68no—
367C2×Q8—64no—
368C2×Q8—82no—
369C2×Q8—128no—
370C2×Q8—1232no—
371C2×Q8—124no—
372C2×Q8C2×#6714yes—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—34no—
376C2×D4—32no—
377C2×D4—62no—
378C2×D4—61no—
379C2×D4D3×Q814yescomplemented
380C2×D4C23×D312yes—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—44no—
386C22×C4—62no—
387C22×C4—61no—
388C22×C4—68no—
389C22×C4—64no—
390C22×C4—84no—
391C22×C4—128no—
392C22×C4—124no—
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6712yes—
395C22×C4C2×#6712yes—
396C22×C4C2×#6711yescharacteristic
397C22×C4C23×D311yescharacteristic
398D8—316no—
399C2×C8—34no—
400C2×C8—32no—
401C2×C8—32no—
402C2×C8—34no—
403C2×C8—62no—
404C2×C8—64no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—28no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—32no—
411C4⋊C4—34no—
412C4⋊C4—62no—
413C4⋊C4—61no—
414C4⋊C4—64no—
415C4⋊C4—68no—
416C4⋊C4C2×#6714yes—
417C4⋊C4C2×#6718yes—
418C4⋊C4C23×D314yes—
419C4⋊C4C23×D312yes—
420C22⋊C4—24no—
421C22⋊C4—28no—
422C22⋊C4—64no—
423C22⋊C4—68no—
424C42C2×#6714yes—
425C42C2×#6712yes—
426C42C23×D312yes—
427C42C23×D314yes—
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—24no—
441C3×C4—24no—
442C3×C4—22no—
443C3×C4—42no—
444C3×C4—44no—
445C3×C4D4×Q812yes—
446C3×C4C2×AGL(1,ℤ/8)14yes—
447C3×C4C23×Q811yescharacteristic
448C3×C4C2×2−1+411yescharacteristic
449Dic3—42no—
450Dic3—44no—
451C23—22no—
452C23—24no—
453C23—42no—
454C23—42no—
455C23—62no—
456C23—64no—
457C23—122no—
458C23—122no—
459C23C22×#6711yescharacteristic
460Q8—28no—
461Q8—84no—
462Q8—1216no—
463Q8—128no—
464D4—22no—
465D4—24no—
466D4—34no—
467D4—62no—
468D4—64no—
469D4C2×D3×Q814yescomplemented
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—64no—
479C2×C4—62no—
480C2×C4—88no—
481C2×C4—1216no—
482C2×C4—124no—
483C2×C4—122no—
484C2×C4—128no—
485C2×C4#94914yes—
486C2×C4C2×D3×Q812yes—
487C2×C4#100712yes—
488C2×C4C22×#6712yes—
489C2×C4C22×#6714yes—
490C2×C4C22×#6714yes—
491C2×C4C22×#6711yescharacteristic
492C2×C4C22×#6712yes—
493C2×C4C24×D311yescharacteristic
494C8—34no—
495C8—62no—
496C8—64no—
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—48no—
505C22—42no—
506C22—42no—
507C22—64no—
508C22—128no—
509C22—122no—
510C22—122no—
511C22Q8×#6712yes—
512C22C2×#94912yes—
513C22#583612yes—
514C22C23×#6711yescharacteristic
515C4—24no—
516C4—24no—
517C4—22no—
518C4—42no—
519C4—44no—
520C4—124no—
521C4—122no—
522C4Q8×#6712yes—
523C4C2×#94914yes—
524C4C22×D3×Q811yescharacteristic
525C4C2×#100711yescharacteristic
526C3C2×#338411yescharacteristic, complemented, Sylow
527C2—44no—
528C2—124no—
529C2#8363014yesdirect product factor
530C2C2×Q8×#6711yescharacteristic
531C2C22×#94911yescharacteristic
532C2C2×#583611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#554C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#554D3
hypocenterC3C2×#3384
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C3×#554C2semidirect
2. C2×Q8×D12C2semidirect
3. #83630C2direct (N × H)
4. C2×#5084C2semidirect
5. C2×#5080C2semidirect
6. C2×#5066C2semidirect
7. C2×#5064C2semidirect
8. C2×#5024C2semidirect
9. C2×#5008C2semidirect
10. C2×#5007C2semidirect
16. C3×#554C22semidirect
27. Q8×D12C22semidirect
36. #5084C22semidirect
37. #5080C22semidirect
38. #5066C22semidirect
39. #5064C22semidirect
40. #5024C22semidirect
43. #5008C22semidirect
44. #5007C22semidirect
56. C2×#554D3semidirect
104. C2×#213Q8semidirect
141. #554C2×D3semidirect
188. C2×C3×D4C2×Q8semidirect
231. #213C2×Q8semidirect
242. C2×C3⋊C8C2×Q8semidirect
329. C3×D4C22×Q8semidirect
361. C3⋊C8C22×Q8semidirect
379. C2×D4D3×Q8semidirect
469. D4C2×D3×Q8semidirect
498. C2×C3#3384semidirect
526. C3C2×#3384semidirect
529. C2#83630direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×Q8×D122
#836302
C2×#50842
C2×#50802
C2×#50662
C2×#50642
C2×#50242
C2×#50082
C2×#50072
C2×#50142
C2×C4×#2132
C2×#33843