Finite Group Databasebeta

← C2×#83088 #10965955

Subgroups — C2×#83088 #10965955

Order 768.

2594 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#126C211yesmaximal, characteristic
2C22×C4×Dic6C211yesmaximal, characteristic, complemented
3#83088C2116yesmaximal, direct product factor
4C2×#4986C211yesmaximal, characteristic, complemented
5C2×#4979C211yesmaximal, characteristic, complemented
6C2×#4976C212yesmaximal, complemented
7C2×#4978C212yesmaximal, complemented
8C2×#4949C211yesmaximal, characteristic
9C2×#4947C211yesmaximal, characteristic
10C2×#4945C212yesmaximal
11C2×#4944C212yesmaximal
12C2×#4932C211yesmaximal, characteristic, complemented
13C2×#2604—31nomaximal, Sylow
14C23×Dic6C2211yescharacteristic
15C2×C3×#126—28no—
16C2×C3×#126C2218yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×C4⋊Dic3C2211yescharacteristic
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×C4×Dic3C2212yes—
31C22×C4×Dic3C2212yes—
32C2×C4×Dic6—28no—
33C2×C4×Dic6C2218yescomplemented
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41#4986C2218yescomplemented
42#4979C2218yescomplemented
43#4978C22116yescomplemented
44#4976C22116yescomplemented
45#4949C2218yes—
46#4947C2218yes—
47#4945C22116yes—
48#4944C22116yes—
49#4932C2218yescomplemented
50C22×#126D311yescharacteristic
51C22×C4×Q8—31no—
52#2604—316no—
53C2×#400—31no—
54C2×#399—31no—
55C2×#395—31no—
56C2×#393—31no—
57C2×#389—32no—
58C2×#388—32no—
59C2×#388—32no—
60C2×#387—32no—
61C2×#376—31no—
62C25×C3C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C23×Dic3C2312yes—
72C23×Dic3C2312yes—
73C22×Dic6—24no—
74C22×Dic6—28no—
75C22×Dic6C2314yes—
76C3×#126—232no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4C2314yes—
81C2×C3×C4⋊C4C2314yes—
82C2×C3×C4⋊C4C2314yes—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4C2314yes—
87C2×C3×C22⋊C4C2314yes—
88C2×C3×C22⋊C4C2314yes—
89C2×C3×C42D418yes—
90C2×C3×C42C2314yes—
91C2×#217—28no—
92C2×#217—28no—
93C2×#217C2318yes—
94C2×#217C2318yes—
95C2×C4⋊Dic3—24no—
96C2×C4⋊Dic3C2314yes—
97C2×Dic3⋊C4—28no—
98C2×Dic3⋊C4—28no—
99C2×Dic3⋊C4—28no—
100C2×Dic3⋊C4C2318yes—
101C2×Dic3⋊C4C2318yes—
102C2×Dic3⋊C4C2318yes—
103C2×C4×Dic3—28no—
104C2×C4×Dic3—28no—
105C2×C4×Dic3C2318yes—
106C2×C4×Dic3C2318yes—
107C4×Dic6—232no—
108C3⋊#95C2318yes—
109C3⋊#95C2314yes—
110C3⋊#95C2314yes—
111C3⋊#95C2318yes—
112C3⋊#95C2318yes—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C23×Q8—31no—
116C2×#126—28no—
117C2×#126C2×D318yes—
118C2×C4×Q8—38no—
119C2×C4×Q8—68no—
120C22×C4⋊C4—32no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4C2×D311yescharacteristic
125C22×C4⋊C4C2×D311yescharacteristic
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4C2×D311yescharacteristic
130C22×C22⋊C4C2×D311yescharacteristic
131C22×C22⋊C4C2×D311yescharacteristic
132C22×C42—32no—
133C22×C42—32no—
134C22×C42C2×D311yescharacteristic
135#400—38no—
136#399—38no—
137#395—38no—
138#393—38no—
139#389—316no—
140#388—316no—
141#388—316no—
142#387—316no—
143#376—38no—
144C2×#95—31no—
145C2×#95—32no—
146C2×#95—31no—
147C2×#95—32no—
148C2×#95—32no—
149C2×#95—32no—
150C2×#95—32no—
151C24×C3—22no—
152C24×C3—24no—
153C24×C3—24no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3C2412yes—
157C24×C3C2411yescharacteristic
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2412yes—
173C22×C3×C4C2412yes—
174C22×C3×C4C2412yes—
175C22×C3×C4C2412yes—
176C22×C3×C4C2412yes—
177C22×C3×C4C2412yes—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3#4118yes—
189C22×Dic3#4118yes—
190C22×Dic3C2414yes—
191C22×Dic3C2414yes—
192C22×Dic3C2414yes—
193C22×Dic3C2414yes—
194C2×Dic6—232no—
195C2×Dic6—28no—
196C2×Dic6—216no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—28no—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—28no—
202C3×C22⋊C4—28no—
203C3×C42C2×D4116yes—
204#217—216no—
205#217—216no—
206C4⋊Dic3—28no—
207Dic3⋊C4—216no—
208Dic3⋊C4—216no—
209Dic3⋊C4—216no—
210C4×Dic3—216no—
211C4×Dic3—216no—
212C25C22×D311yescharacteristic
213C22×Q8—34no—
214C22×Q8—68no—
215C22×Q8—64no—
216C23×C4—32no—
217C23×C4—32no—
218C23×C4—32no—
219C23×C4—32no—
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226#126—232no—
227C4×Q8—632no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—38no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—38no—
234C2×C4⋊C4—38no—
235C2×C4⋊C4—68no—
236C2×C4⋊C4—68no—
237C2×C4⋊C4—68no—
238C2×C4⋊C4—64no—
239C2×C4⋊C4C22×D314yes—
240C2×C4⋊C4C22×D314yes—
241C2×C4⋊C4C22×D314yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—38no—
246C2×C22⋊C4—38no—
247C2×C22⋊C4—68no—
248C2×C22⋊C4—68no—
249C2×C22⋊C4C22×D314yes—
250C2×C22⋊C4C22×D314yes—
251C2×C22⋊C4C22×D314yes—
252C2×C42—38no—
253C2×C42—38no—
254C2×C42—68no—
255C2×C42—68no—
256C2×C42#6718yes—
257C2×C42C22×D314yes—
258#95—38no—
259#95—34no—
260#95—34no—
261#95—38no—
262#95—38no—
263#95—38no—
264#95—38no—
265C23×C3—22no—
266C23×C3—24no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—44no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—48no—
274C23×C3—48no—
275C23×C3C22×D412yes—
276C23×C3C2×#4114yes—
277C23×C3C2×#4114yes—
278C23×C32−1+412yes—
279C23×C32−1+412yes—
280C23×C3C2511yescharacteristic
281C2×C3×C4—28no—
282C2×C3×C4—28no—
283C2×C3×C4—28no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4C22×D414yes—
299C2×C3×C4C22×D414yes—
300C2×C3×C4C22×D414yes—
301C2×Dic3—216no—
302C2×Dic3—216no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—48no—
314C2×Dic3—48no—
315C2×Dic3C2×#4118yes—
316C2×Dic3C2×#4118yes—
317Dic6—232no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24C23×D312yes—
324C24C23×D311yescharacteristic
325C2×Q8—616no—
326C2×Q8—632no—
327C2×Q8—68no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—38no—
340C22×C4—38no—
341C22×C4—34no—
342C22×C4—34no—
343C22×C4—34no—
344C22×C4—34no—
345C22×C4—64no—
346C22×C4—64no—
347C22×C4—64no—
348C22×C4—64no—
349C22×C4—64no—
350C22×C4—64no—
351C22×C4—64no—
352C22×C4—64no—
353C22×C4—64no—
354C22×C4—64no—
355C22×C4C2×#6714yes—
356C22×C4C2×#6714yes—
357C22×C4C2×#6714yes—
358C22×C4C23×D312yes—
359C22×C4C23×D312yes—
360C22×C4C23×D312yes—
361C22×C4C23×D312yes—
362C22×C4C23×D312yes—
363C22×C4C23×D312yes—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—616no—
368C4⋊C4—616no—
369C4⋊C4—616no—
370C4⋊C4—68no—
371C22⋊C4—28no—
372C22⋊C4—28no—
373C22⋊C4—28no—
374C22⋊C4—616no—
375C22⋊C4—616no—
376C42—616no—
377C42—616no—
378C42C2×#67116yes—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—44no—
383C22×C3—44no—
384C22×C3—88no—
385C22×C3#52514yes—
386C22×C3#53214yes—
387C22×C3#53614yes—
388C22×C3#55218yes—
389C22×C3#55618yes—
390C22×C3C23×D411yescharacteristic
391C22×C3C22×#4112yes—
392C22×C3C22×#4112yes—
393C22×C3C2×2−1+411yescharacteristic
394C22×C3C2×2−1+411yescharacteristic
395C3×C4—24no—
396C3×C4—24no—
397C3×C4—24no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—42no—
401Dic3—28no—
402Dic3—28no—
403Dic3—44no—
404Dic3—44no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—48no—
411C23—48no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23D3×#4114yes—
416C23D3×#4114yes—
417C23#101012yes—
418C23#101012yes—
419C23C22×#6712yes—
420C23C24×D311yescharacteristic
421Q8—632no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—38no—
437C2×C4—38no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—64no—
442C2×C4—64no—
443C2×C4—64no—
444C2×C4—616no—
445C2×C4—64no—
446C2×C4—616no—
447C2×C4—64no—
448C2×C4—64no—
449C2×C4—64no—
450C2×C4—64no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—128no—
454C2×C4—128no—
455C2×C4C22×#6714yes—
456C2×C4C22×#6714yes—
457C2×C4C22×#6714yes—
458C2×C3—82no—
459C2×C3#260418yescomplemented
460C2×C3C2×#52511yescharacteristic
461C2×C3C2×#53211yescharacteristic
462C2×C3C2×#53611yescharacteristic
463C2×C3C2×#55212yes—
464C2×C3C2×#55612yes—
465C22—44no—
466C22—44no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—88no—
471C22#570218yes—
472C22#570818yes—
473C22#582514yes—
474C22#582814yes—
475C22#583214yes—
476C22C2×D3×#4112yes—
477C22C2×D3×#4112yes—
478C22C2×#101011yescharacteristic
479C22C2×#101011yescharacteristic
480C22C23×#6711yescharacteristic
481C4—24no—
482C4—24no—
483C4—24no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—68no—
488C4—68no—
489C4—124no—
490C4—124no—
491C3C2×#260411yescharacteristic, complemented, Sylow
492C2—82no—
493C2#8308818yesdirect product factor
494C2C2×#570212yes—
495C2C2×#570812yes—
496C2C2×#583211yescharacteristic
497C2C2×#582511yescharacteristic
498C2C2×#582811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×#126C2
socleC24×C3C24
prime radicalC25×C3C23
hypercenterC22×#126D3
hypocenterC3C2×#2604
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic6C2semidirect
3. #83088C2direct (N × H)
4. C2×#4986C2semidirect
5. C2×#4979C2semidirect
6. C2×#4976C2semidirect
7. C2×#4978C2semidirect
12. C2×#4932C2semidirect
33. C2×C4×Dic6C22semidirect
41. #4986C22semidirect
42. #4979C22semidirect
43. #4978C22semidirect
44. #4976C22semidirect
49. #4932C22semidirect
459. C2×C3#2604semidirect
491. C3C2×#2604semidirect
493. C2#83088direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1262
C22×C4×Dic62
#830882
C2×#49862
C2×#49792
C2×#49762
C2×#49782
C2×#49492
C2×#49472
C2×#49452
C2×#49442
C2×#49322
C2×#26043