Finite Group Databasebeta

← C2×#80699 #11665014

Subgroups — C2×#80699 #11665014

Order 768.

2098 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5619C212yesmaximal, complemented
2C2×C3×#466C211yesmaximal, characteristic, complemented
3C2×#5151C211yesmaximal, characteristic, complemented
4C2×Dic3×D8C211yesmaximal, characteristic, complemented
5C2×#5115C211yesmaximal, characteristic, complemented
6C2×#5101C211yesmaximal, characteristic, complemented
7C2×#5039C212yesmaximal, complemented
8#80699C2116yesmaximal, direct product factor
9C2×#4878C211yesmaximal, characteristic, complemented
10C2×#4872C211yesmaximal, characteristic, complemented
11C2×#4764C212yesmaximal, complemented
12C2×#4785C212yesmaximal, complemented
13C2×#3176—31nomaximal, Sylow
14C22×C3×D8C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×#121C2212yes—
17C2×#940C2212yes—
18C2×#939—22no—
19C2×#938—22no—
20C2×D4×Dic3C2212yes—
21C22×#213C2212yes—
22C2×#930C2212yes—
23C2×#926C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C22×D24C2211yescharacteristic
26#5619—28no—
27#5619C22116yescomplemented
28C2×#893C2212yes—
29C2×#890C2212yes—
30C2×#886—22no—
31C2×#883—22no—
32C3×#466C2218yescomplemented
33C2×C3×#107C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×#832C2212yes—
36#5151C2218yescomplemented
37Dic3×D8C2218yescomplemented
38#5115C2218yescomplemented
39C2×#821C2212yes—
40C2×#818C2211yescharacteristic
41#5101C2218yescomplemented
42C2×#815C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46#5039C22116yescomplemented
47C2×#809C2212yes—
48C2×#808C2212yes—
49#4878C2218yescomplemented
50#4872C2218yescomplemented
51#4785C22116yescomplemented
52#4764C22116yescomplemented
53C2×#540—32no—
54#3176—316no—
55C2×#499—31no—
56C2×#493—31no—
57C2×#490—32no—
58C2×#469—32no—
59C2×#466—31no—
60C2×#466D311yescharacteristic, complemented
61C2×#442—32no—
62C2×C4×D8—31no—
63C2×C4×D8—31no—
64C2×#405—31no—
65C22×C3×D4—22no—
66C22×C3×D4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C22×#67—22no—
69C22×#67—22no—
70C22×#67—22no—
71C23×Dic3—22no—
72C22×D12C2312yes—
73C22×C4×D3—22no—
74C2×C3×D8—24no—
75C2×C3×D8C2314yes—
76C22×C3×C8—24no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2314yes—
79C22×C3×C8C2311yescharacteristic
80C3×#121—24no—
81C3×#121C2318yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×#217—22no—
85C2×#217D411yescharacteristic
86C2×#217D411yescharacteristic
87#940—24no—
88#940C2318yes—
89#939—216no—
90#938—216no—
91D4×Dic3—24no—
92D4×Dic3C2318yes—
93C2×#213—28no—
94C2×#213C2318yes—
95#930—24no—
96#930C2318yes—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99#926D414yes—
100#926C2314yes—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2312yes—
105C2×Dic3⋊C4—22no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C2×#208—24no—
109C2×#208C2314yes—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C2×D24—24no—
113C2×D24C2314yes—
114#893—24no—
115#893C2318yes—
116#890—24no—
117#890C2318yes—
118#886—216no—
119#883—216no—
120C3×#107C2314yes—
121C3×#102C2318yes—
122#832C2318yes—
123#821C2318yes—
124#818C2314yes—
125#815C2314yes—
126#815C2314yes—
127C8×Dic3C2314yes—
128C8×Dic3C2314yes—
129#809C2318yes—
130#808C2318yes—
131C22×D8—32no—
132C22×D8—31no—
133C22×D8C2×D311yescharacteristic
134C22×M16—31no—
135C23×C8C2×D311yescharacteristic
136#540—316no—
137#540—68no—
138C2×#127—32no—
139C2×#125—32no—
140C2×#123—62no—
141C2×#123—62no—
142C2×#121—32no—
143C2×#121—62no—
144C2×#121—62no—
145C2×#121C2×D312yes—
146C2×C4×D4—32no—
147C2×C4×D4—32no—
148C2×#117—31no—
149#499—38no—
150#493—38no—
151#490—316no—
152#469—316no—
153#466—38no—
154#466C2×D318yescomplemented
155#442—316no—
156C4×D8—38no—
157C4×D8—38no—
158C2×#107—31no—
159C2×#107C2×D311yescharacteristic
160C2×#106—32no—
161C2×#102—32no—
162C2×#102—32no—
163C2×#102—32no—
164C2×#102C2×D312yes—
165#405—38no—
166C2×#97—31no—
167C2×#97—31no—
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C24×C3—22no—
171C24×C3C2×D411yescharacteristic
172C23×D3—22no—
173C2×C3×D4—28no—
174C2×C3×D4—22no—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—84no—
178C2×C3×D4C2414yes—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2412yes—
185C22×C3×C4C2411yescharacteristic
186C2×#67—28no—
187C2×#67—28no—
188C2×#67—28no—
189C2×#67—84no—
190C2×#67—84no—
191C2×#67—84no—
192C22×Dic3—22no—
193C22×Dic3—28no—
194C22×Dic3—84no—
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×D12—24no—
200C2×D12—24no—
201C2×D12—22no—
202C2×D12C2414yes—
203C2×C4×D3—28no—
204C2×C4×D3—84no—
205C3×D8—28no—
206C2×C3×C8—22no—
207C2×C3×C8—28no—
208C2×C3×C8#4114yes—
209C2×C3×C8C2412yes—
210C2×C3×C8C2412yes—
211C3×C4⋊C4—22no—
212C3×C4⋊C4C2414yes—
213C3×C22⋊C4—28no—
214#217—28no—
215#217C2×D414yes—
216#217C2×D414yes—
217#213—216no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—28no—
220C4⋊Dic3—22no—
221C4⋊Dic3—28no—
222C4⋊Dic3C2×D414yes—
223C4⋊Dic3C2×D414yes—
224C4⋊Dic3C2414yes—
225Dic3⋊C4—28no—
226C4×Dic3C2×D412yes—
227C4×Dic3C2×D412yes—
228C4×Dic3C2412yes—
229C4×Dic3C2412yes—
230#208—28no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8#4114yes—
233C2×C3⋊C8C2412yes—
234C2×C3⋊C8C2412yes—
235D24—28no—
236C22×D4—22no—
237C22×D4—32no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—62no—
241C22×D4C22×D312yes—
242C23×C4—62no—
243C23×C4—62no—
244C23×C4C22×D311yescharacteristic
245C2×D8—24no—
246C2×D8—38no—
247C2×D8—34no—
248C2×D8—68no—
249C2×D8—64no—
250C2×D8C22×D314yes—
251C2×M16—34no—
252C2×M16—64no—
253C22×C8—24no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8C22×D311yescharacteristic
257C22×C8C22×D314yes—
258C22×C8C22×D311yescharacteristic
259#127—38no—
260#127—64no—
261#125—38no—
262#125—64no—
263#123—616no—
264#123—616no—
265#121—24no—
266#121—38no—
267#121—616no—
268#121—616no—
269#121—64no—
270#121C22×D318yes—
271C4×D4—38no—
272C4×D4—38no—
273C4×D4—64no—
274C4×D4—64no—
275#117—34no—
276#117—34no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4C22×D312yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—31no—
286C2×C22⋊C4—62no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C42—31no—
290C2×C42—31no—
291#107—34no—
292#107C22×D314yes—
293#106—38no—
294#102—38no—
295#102—38no—
296#102—38no—
297#102C22×D318yes—
298#97—34no—
299#97—34no—
300C4×C8—34no—
301C4×C8—34no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—24no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—44no—
309C23×C3C22×D412yes—
310C23×C3C22×D411yescharacteristic
311C22×D3—24no—
312C22×D3—44no—
313C22×D3—44no—
314C22×D3—44no—
315C3×D4—24no—
316C3×D4—24no—
317C3×D4—44no—
318C3×D4—88no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—44no—
325C2×C3×C4C22×D412yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4112yes—
328C2×C3×C4C2×#4112yes—
329C2×C3×C4C2511yescharacteristic
330#67—88no—
331#67—88no—
332#67—88no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—44no—
339C2×Dic3—88no—
340C2×Dic3C22×D412yes—
341C2×Dic3C22×D412yes—
342C2×Dic3C22×D412yes—
343C2×Dic3C22×D412yes—
344D12—24no—
345D12—24no—
346D12—44no—
347C4×D3—88no—
348C3×C8—22no—
349C3×C8C2×#4114yes—
350C3⋊C8—22no—
351C3⋊C8C2×#4114yes—
352C24—22no—
353C24—62no—
354C24D3×D411yescharacteristic
355C2×D4—28no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—34no—
360C2×D4—68no—
361C2×D4—68no—
362C2×D4—68no—
363C2×D4—62no—
364C2×D4—64no—
365C2×D4—64no—
366C2×D4—84no—
367C2×D4—244no—
368C2×D4—244no—
369C2×D4—244no—
370C2×D4C23×D314yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—68no—
379C22×C4—68no—
380C22×C4—62no—
381C22×C4—244no—
382C22×C4—244no—
383C22×C4D3×D412yes—
384C22×C4D3×D411yescharacteristic
385C22×C4C23×D312yes—
386C22×C4C23×D311yescharacteristic
387D8—28no—
388D8—616no—
389D8—68no—
390M16—68no—
391C2×C8—22no—
392C2×C8—28no—
393C2×C8—34no—
394C2×C8—32no—
395C2×C8—32no—
396C2×C8—62no—
397C2×C8#23714yes—
398C2×C8C23×D312yes—
399C2×C8C23×D312yes—
400C4⋊C4—22no—
401C4⋊C4—34no—
402C4⋊C4—34no—
403C4⋊C4—34no—
404C4⋊C4—68no—
405C4⋊C4—68no—
406C4⋊C4—62no—
407C4⋊C4C23×D314yes—
408C22⋊C4—28no—
409C22⋊C4—34no—
410C22⋊C4—34no—
411C22⋊C4—68no—
412C22⋊C4—68no—
413C22⋊C4—68no—
414C42—32no—
415C42—32no—
416C42—32no—
417C42—32no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—88no—
426C22×C3#53212yes—
427C22×C3#57612yes—
428C22×C3#57612yes—
429C22×C3C23×D411yescharacteristic
430C2×D3—44no—
431C2×D3—44no—
432C2×D3—44no—
433C2×D3—88no—
434C3×C4—22no—
435C3×C4—44no—
436C3×C4#53212yes—
437C3×C4C22×#4111yescharacteristic
438C3×C4C22×#4111yescharacteristic
439Dic3—22no—
440Dic3—22no—
441Dic3—22no—
442Dic3—22no—
443Dic3—44no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—64no—
452C23—124no—
453C23—124no—
454C23—124no—
455C23C2×D3×D412yes—
456C23C2×D3×D411yescharacteristic
457D4—24no—
458D4—24no—
459D4—44no—
460D4—64no—
461D4—64no—
462D4—88no—
463D4—124no—
464D4—248no—
465D4—248no—
466D4—248no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—32no—
473C2×C4—32no—
474C2×C4—32no—
475C2×C4—32no—
476C2×C4—44no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—64no—
482C2×C4—124no—
483C2×C4—248no—
484C2×C4—248no—
485C2×C4C2×D3×D412yes—
486C2×C4C2×D3×D411yescharacteristic
487C2×C4C2×#23712yes—
488C2×C4D3×#4112yes—
489C2×C4C24×D311yescharacteristic
490C8—22no—
491C8—34no—
492C8—62no—
493C8C2×#23714yes—
494C2×C3—42no—
495C2×C3—84no—
496C2×C3#317614yescomplemented
497C2×C3C2×#53211yescharacteristic
498C2×C3C2×#57611yescharacteristic
499C2×C3C2×#57611yescharacteristic
500D3—84no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—88no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—248no—
513C22#559612yes—
514C22#575712yes—
515C22#576912yes—
516C22C22×D3×D411yescharacteristic
517C4—22no—
518C4—44no—
519C4—62no—
520C4—62no—
521C4—62no—
522C4—62no—
523C4—124no—
524C4#559612yes—
525C4C22×#23711yescharacteristic
526C4C2×D3×#4111yescharacteristic
527C3C2×#317611yescharacteristic, complemented, Sylow
528C2—42no—
529C2—84no—
530C2—244no—
531C2#8069914yesdirect product factor
532C2C2×#559611yescharacteristic
533C2C2×#575711yescharacteristic
534C2C2×#576911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#466C2
socleC23×C3C22×D4
hypercenterC2×#466D3
hypocenterC3C2×#3176
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×#5619C2semidirect
2. C2×C3×#466C2semidirect
3. C2×#5151C2semidirect
4. C2×Dic3×D8C2semidirect
5. C2×#5115C2semidirect
6. C2×#5101C2semidirect
7. C2×#5039C2semidirect
8. #80699C2direct (N × H)
9. C2×#4878C2semidirect
10. C2×#4872C2semidirect
11. C2×#4764C2semidirect
12. C2×#4785C2semidirect
27. #5619C22semidirect
32. C3×#466C22semidirect
36. #5151C22semidirect
37. Dic3×D8C22semidirect
38. #5115C22semidirect
41. #5101C22semidirect
46. #5039C22semidirect
49. #4878C22semidirect
50. #4872C22semidirect
51. #4785C22semidirect
52. #4764C22semidirect
60. C2×#466D3semidirect
154. #466C2×D3semidirect
496. C2×C3#3176semidirect
527. C3C2×#3176semidirect
531. C2#80699direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56192
C2×C3×#4662
C2×#51512
C2×Dic3×D82
C2×#51152
C2×#51012
C2×#50392
#806992
C2×#48782
C2×#48722
C2×#47642
C2×#47852
C2×#31763