Finite Group Databasebeta

← C2×#86214 #11703598

Subgroups — C2×#86214 #11703598

Order 768.

11126 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1007C212yesmaximal, complemented
2C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#5816C212yesmaximal, complemented
4#86214C2132yesmaximal, direct product factor
5C2×#5727C214yesmaximal, complemented
6C2×#5726C214yesmaximal, complemented
7C2×#5724C214yesmaximal, complemented
8C2×D3×C4⋊Q8C212yesmaximal, complemented
9C2×#5725C212yesmaximal, complemented
10C2×#5507C218yesmaximal, complemented
11C2×#5477C211yesmaximal, characteristic, complemented
12C2×C4×#235C211yesmaximal, characteristic, complemented
13C2×#3499—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C2×#1007—232no—
16C2×#1007C22132yescomplemented
17C22×#239C2214yes—
18C22×#239C2214yes—
19C22×D3×Q8C2214yes—
20C22×D3×Q8C2214yes—
21C22×#235C2212yes—
22C22×#235C2218yes—
23C22×#235C2211yescharacteristic
24C2×C3×C4⋊Q8C2212yes—
25C2×C3×C4⋊Q8C2212yes—
26C2×C3×C4⋊Q8C2214yes—
27C2×C3×C4⋊Q8C22116yescomplemented
28C22×C3×C4⋊C4C2214yes—
29C22×C3×C42C2211yescharacteristic
30#5816C22132yescomplemented
31C2×#947C2214yes—
32C2×#946C2218yes—
33C2×#944C2214yes—
34C2×#930C2214yes—
35C2×#929C2218yes—
36C2×C4×#67C2214yes—
37C2×#926C2211yescharacteristic
38#5727C22164yescomplemented
39#5726C22164yescomplemented
40#5725C22132yescomplemented
41#5724C22164yescomplemented
42D3×C4⋊Q8C22132yescomplemented
43C2×#896C22116yes—
44#5507C221128yescomplemented
45C2×#895C2218yes—
46C2×#894C22116yes—
47C2×#893C2218yes—
48C2×D3×C4⋊C4C2218yes—
49C2×#888C2218yes—
50#5477C22116yescomplemented
51C2×#875C2212yes—
52C2×#874C2211yescharacteristic
53C4×#235C22116yescomplemented
54C2×C4×D12C2212yes—
55C2×C4×D12C2214yes—
56C2×#872C2212yes—
57C2×C42×D3C2212yes—
58C2×#869C2211yescharacteristic
59C2×C4×Dic6C2212yes—
60C22×2−1+4—32no—
61#3499—332no—
62C2×D4×Q8—34no—
63C2×#547—34no—
64C2×#547—38no—
65C2×#536—32no—
66C2×#536—34no—
67C2×#532—31no—
68C2×#532—32no—
69C22×C4⋊Q8—32no—
70C22×C4⋊Q8D311yescharacteristic, complemented
71C2×C4×#41—31no—
72C22×C3×Q8—216no—
73C22×C3×Q8C2314yes—
74C22×C3×Q8C2314yes—
75C22×C3×Q8C23116yes—
76C23×C3×C4C2314yes—
77C23×C3×C4C2312yes—
78C23×C3×C4C2311yescharacteristic
79C22×#67C2314yes—
80C22×#67C2312yes—
81#1007—2256no—
82C2×#239—232no—
83C2×#239—232no—
84C2×#239C23132yes—
85C2×#239C23132yes—
86C2×D3×Q8—232no—
87C2×D3×Q8—232no—
88C2×D3×Q8C23132yes—
89C2×D3×Q8C23132yes—
90C2×#235—216no—
91C2×#235—264no—
92C2×#235D4116yescomplemented
93C2×#235C23116yes—
94C2×#235C23164yes—
95C2×#235C2318yes—
96C22×D12C2312yes—
97C22×D12C2318yes—
98C22×D12C2314yes—
99C22×D12C2311yescharacteristic
100C22×C4×D3C2314yes—
101C22×C4×D3C2318yes—
102C22×C4×D3C2318yes—
103C22×C4×D3C2314yes—
104C22×C4×D3C2312yes—
105C22×Dic6C2312yes—
106C22×Dic6C2318yes—
107C22×Dic6C2311yescharacteristic
108C3×C4⋊Q8C23116yes—
109C3×C4⋊Q8C23116yes—
110C3×C4⋊Q8C23132yes—
111C2×C3×C4⋊C4C2318yes—
112C2×C3×C4⋊C4C2318yes—
113C2×C3×C4⋊C4C23132yes—
114C2×C3×C42C2312yes—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2318yes—
118C2×#217C2312yes—
119#947C23132yes—
120#946C23164yes—
121#944C23132yes—
122#930C23132yes—
123#929C23164yes—
124C4×#67C23132yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2318yes—
127C2×(C2×D3)⋊C4C23116yes—
128#926C2318yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C23116yes—
133C2×C4×Dic3C2314yes—
134C2×C4×Dic3C2312yes—
135#896C231128yes—
136#895C23164yes—
137#894C231128yes—
138#893C23164yes—
139D3×C4⋊C4C23164yes—
140#888C23164yes—
141#875C23116yes—
142#874C2318yes—
143C4×D12C23116yes—
144C4×D12C23132yes—
145#872C23116yes—
146C42×D3C23116yes—
147#869C2318yes—
148C4×Dic6C23116yes—
149C2×2−1+4—332no—
150C2×2−1+4—632no—
151C22×#41—34no—
152C22×#41—34no—
153C22×#41—31no—
154C22×#41—32no—
155C22×#41—38no—
156C23×Q8—34no—
157C23×Q8—34no—
158C23×Q8C2×D312yes—
159D4×Q8—364no—
160#547—364no—
161#547—3128no—
162#536—332no—
163#536—364no—
164#532—332no—
165#532—316no—
166C2×C4⋊Q8—34no—
167C2×C4⋊Q8—332no—
168C2×C4⋊Q8—38no—
169C2×C4⋊Q8—31no—
170C2×C4⋊Q8C2×D312yes—
171C2×C4⋊Q8C2×D312yes—
172C2×C4⋊Q8C2×D314yes—
173C2×C4⋊Q8C2×D3116yescomplemented
174C2×#127—31no—
175C2×#124—34no—
176C2×#124—32no—
177C2×#123—38no—
178C2×#123—316no—
179C2×C22⋊Q8—38no—
180C2×C22⋊Q8—316no—
181C2×#121—34no—
182C2×#121—38no—
183C4×#41—316no—
184C2×C4×Q8—32no—
185C2×C4×D4—34no—
186C2×C4×D4—38no—
187C2×C4×D4—32no—
188C2×C4×D4—34no—
189C2×#117—31no—
190C2×#117—32no—
191C22×C4⋊C4—38no—
192C22×C4⋊C4C2×D314yes—
193C22×C42—32no—
194C22×C42C2×D311yescharacteristic
195C24×C3C2411yescharacteristic
196C23×D3C2414yes—
197C23×D3C2412yes—
198C2×C3×Q8—216no—
199C2×C3×Q8—216no—
200C2×C3×Q8—264no—
201C2×C3×Q8C24116yes—
202C2×C3×Q8C24116yes—
203C22×C3×C4—216no—
204C22×C3×C4—28no—
205C22×C3×C4C2×D418yes—
206C22×C3×C4C2412yes—
207C22×C3×C4C2418yes—
208C22×C3×C4C2418yes—
209C22×C3×C4C24116yes—
210C22×C3×C4C2411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×C3×C4C2414yes—
213C22×C3×C4C2412yes—
214C2×#67—216no—
215C2×#67C2×D4116yes—
216C2×#67C24116yes—
217C2×#67C2418yes—
218C22×Dic3C2414yes—
219C22×Dic3C2414yes—
220C22×Dic3C2412yes—
221#239—2128no—
222#239—2128no—
223D3×Q8—2128no—
224D3×Q8—2128no—
225#235—264no—
226#235—2256no—
227#235C2×D4164yescomplemented
228C2×D12—232no—
229C2×D12—28no—
230C2×D12—216no—
231C2×D12C2×D418yes—
232C2×D12C24116yes—
233C2×D12C2418yes—
234C2×D12C24132yes—
235C2×D12C2414yes—
236C2×C4×D3—232no—
237C2×C4×D3—216no—
238C2×C4×D3—216no—
239C2×C4×D3—232no—
240C2×C4×D3C2×D4116yes—
241C2×C4×D3C24116yes—
242C2×C4×D3C24116yes—
243C2×C4×D3C24132yes—
244C2×C4×D3C24132yes—
245C2×C4×D3C2418yes—
246C2×Dic6—28no—
247C2×Dic6—232no—
248C2×Dic6C2×D418yes—
249C2×Dic6C2418yes—
250C2×Dic6C24132yes—
251C2×Dic6C2414yes—
252C3×C4⋊C4C24132yes—
253C3×C4⋊C4C24132yes—
254C3×C42C2418yes—
255C3×C42C2414yes—
256C3×C42C2414yes—
257#217C2418yes—
258(C2×D3)⋊C4C24116yes—
259(C2×D3)⋊C4C24132yes—
260(C2×D3)⋊C4C24164yes—
261C4⋊Dic3C2418yes—
262C4⋊Dic3C24116yes—
263Dic3⋊C4C24116yes—
264Dic3⋊C4C24164yes—
265C4×Dic3C24116yes—
266C4×Dic3C2418yes—
2672−1+4—6256no—
268C2×#41—332no—
269C2×#41—332no—
270C2×#41—316no—
271C2×#41—38no—
272C2×#41—316no—
273C2×#41—364no—
274C2×#41—664no—
275C2×#41—632no—
276C2×#41—632no—
277C2×#41—616no—
278C22×Q8—216no—
279C22×Q8—332no—
280C22×Q8—332no—
281C22×Q8—31no—
282C22×Q8—38no—
283C22×Q8—32no—
284C22×Q8—632no—
285C22×Q8—632no—
286C22×Q8C22×D314yes—
287C22×Q8C22×D314yes—
288C22×Q8C22×D3116yes—
289C22×D4—32no—
290C22×D4—34no—
291C22×D4—31no—
292C22×D4—34no—
293C22×D4—38no—
294C22×D4—32no—
295C23×C4—32no—
296C23×C4—34no—
297C23×C4—38no—
298C23×C4—38no—
299C23×C4—34no—
300C23×C4C22×D312yes—
301C23×C4C22×D314yes—
302C23×C4C22×D311yescharacteristic
303C4⋊Q8—332no—
304C4⋊Q8—364no—
305C4⋊Q8—38no—
306C4⋊Q8C22×D3116yes—
307C4⋊Q8C22×D3116yes—
308C4⋊Q8C22×D3132yes—
309#127—38no—
310#124—332no—
311#124—316no—
312#123—364no—
313#123—3128no—
314C22⋊Q8—364no—
315C22⋊Q8—3128no—
316#121—332no—
317#121—364no—
318C4×Q8—316no—
319C4×D4—332no—
320C4×D4—364no—
321C4×D4—316no—
322C4×D4—332no—
323#117—38no—
324#117—316no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—316no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—364no—
330C2×C4⋊C4C22×D318yes—
331C2×C4⋊C4C22×D318yes—
332C2×C4⋊C4C22×D3132yes—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4—316no—
336C2×C22⋊C4—38no—
337C2×C42—32no—
338C2×C42—34no—
339C2×C42—316no—
340C2×C42C22×D311yescharacteristic
341C2×C42C22×D312yes—
342C2×C42C22×D318yes—
343C2×C42C22×D311yescharacteristic
344C23×C3—24no—
345C23×C3C22×D414yes—
346C23×C3C2512yes—
347C23×C3C2511yescharacteristic
348C22×D3—28no—
349C22×D3—28no—
350C22×D3—216no—
351C22×D3C22×D418yes—
352C22×D3C2518yes—
353C22×D3C2514yes—
354C3×Q8—232no—
355C3×Q8—232no—
356C2×C3×C4—216no—
357C2×C3×C4—232no—
358C2×C3×C4—24no—
359C2×C3×C4—216no—
360C2×C3×C4C22×D414yes—
361C2×C3×C4C22×D414yes—
362C2×C3×C4C22×D4116yes—
363C2×C3×C42−1+418yescomplemented
364C2×C3×C4C2514yes—
365C2×C3×C4C25116yes—
366C2×C3×C4C2512yes—
367C2×C3×C4C2512yes—
368C2×C3×C4C2514yes—
369#67—232no—
370#67C22×D4132yes—
371C2×Dic3—28no—
372C2×Dic3—28no—
373C2×Dic3C22×D418yes—
374C2×Dic3C2518yes—
375C2×Dic3C2518yes—
376C2×Dic3C2514yes—
377D12—264no—
378D12—216no—
379D12—232no—
380D12C22×D4116yes—
381C4×D3—264no—
382C4×D3—232no—
383C4×D3—232no—
384C4×D3—264no—
385C4×D3C22×D4132yes—
386Dic6—216no—
387Dic6—264no—
388Dic6C22×D4116yes—
389C24—32no—
390C24—34no—
391C24C23×D311yescharacteristic
392#41—364no—
393#41—6256no—
394#41—6128no—
395#41—6128no—
396#41—664no—
397C2×Q8—216no—
398C2×Q8—216no—
399C2×Q8—264no—
400C2×Q8—38no—
401C2×Q8—34no—
402C2×Q8—332no—
403C2×Q8—38no—
404C2×Q8—632no—
405C2×Q8—6128no—
406C2×Q8—6128no—
407C2×Q8—68no—
408C2×Q8C23×D3116yes—
409C2×Q8C23×D3116yes—
410C2×D4—316no—
411C2×D4—38no—
412C2×D4—316no—
413C2×D4—38no—
414C2×D4—34no—
415C2×D4—316no—
416C2×D4—332no—
417C2×D4—38no—
418C2×D4—632no—
419C2×D4—616no—
420C2×D4—68no—
421C2×D4—616no—
422C22×C4—216no—
423C22×C4—28no—
424C22×C4—32no—
425C22×C4—316no—
426C22×C4—38no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—316no—
430C22×C4—332no—
431C22×C4—332no—
432C22×C4—316no—
433C22×C4—632no—
434C22×C4—632no—
435C22×C4—616no—
436C22×C4—616no—
437C22×C4D3×D418yes—
438C22×C4C23×D318yes—
439C22×C4C23×D3116yes—
440C22×C4C23×D312yes—
441C22×C4C23×D318yes—
442C22×C4C23×D314yes—
443C22×C4C23×D311yescharacteristic
444C22×C4C23×D311yescharacteristic
445C22×C4C23×D312yes—
446C4⋊C4—316no—
447C4⋊C4—364no—
448C4⋊C4—316no—
449C4⋊C4—38no—
450C4⋊C4C23×D3132yes—
451C4⋊C4C23×D3132yes—
452C22⋊C4—38no—
453C22⋊C4—316no—
454C22⋊C4—364no—
455C22⋊C4—332no—
456C42—38no—
457C42—316no—
458C42C23×D314yes—
459C42C23×D314yes—
460C42C23×D318yes—
461C22×C3—24no—
462C22×C3—28no—
463C22×C3C23×D414yes—
464C22×C3C23×D412yes—
465C22×C3C2×2−1+414yes—
466C22×C3C2611yescharacteristic
467C2×D3—28no—
468C2×D3—216no—
469C2×D3—28no—
470C2×D3—216no—
471C2×D3—416no—
472C2×D3C23×D418yes—
473C3×C4—216no—
474C3×C4—24no—
475C3×C4C23×D414yes—
476C3×C4C23×D414yes—
477C3×C4C2×2−1+418yescomplemented
478Dic3—28no—
479Dic3—28no—
480Dic3C23×D418yes—
481C23—24no—
482C23—38no—
483C23—34no—
484C23—38no—
485C23—68no—
486C23—68no—
487C23—616no—
488C23C2×D3×D414yes—
489C23C24×D312yes—
490C23C24×D311yescharacteristic
491Q8—232no—
492Q8—232no—
493Q8—316no—
494Q8—664no—
495Q8—616no—
496D4—332no—
497D4—316no—
498D4—664no—
499D4—632no—
500D4—616no—
501D4—632no—
502C2×C4—216no—
503C2×C4—232no—
504C2×C4—24no—
505C2×C4—216no—
506C2×C4—38no—
507C2×C4—34no—
508C2×C4—332no—
509C2×C4—38no—
510C2×C4—38no—
511C2×C4—664no—
512C2×C4—664no—
513C2×C4—68no—
514C2×C4—68no—
515C2×C4—632no—
516C2×C4—632no—
517C2×C4C2×D3×D414yes—
518C2×C4C2×D3×D414yes—
519C2×C4C2×D3×D4116yes—
520C2×C4#100718yescomplemented
521C2×C4C24×D314yes—
522C2×C4C24×D3116yes—
523C2×C4C24×D312yes—
524C2×C4C24×D312yes—
525C2×C4C24×D314yes—
526C2×C3—24no—
527C2×C3#349914yescomplemented
528C2×C3C24×D411yescharacteristic
529C2×C3C22×2−1+412yes—
530D3—28no—
531D3—48no—
532C22—24no—
533C22—28no—
534C22—38no—
535C22—68no—
536C22—616no—
537C22—68no—
538C22—616no—
539C22—1216no—
540C22C22×D3×D414yes—
541C22C22×D3×D412yes—
542C22C2×#100714yes—
543C22C25×D311yescharacteristic
544C4—216no—
545C4—24no—
546C4—38no—
547C4—68no—
548C4—68no—
549C4C22×D3×D414yes—
550C4C22×D3×D414yes—
551C4C2×#100718yescomplemented
552C3C2×#349911yescharacteristic, complemented, Sylow
553C2—24no—
554C2—68no—
555C2—128no—
556C2#8621414yesdirect product factor
557C2C23×D3×D411yescharacteristic
558C2C22×#100712yes—
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×C4⋊Q8C2
socleC23×C3C25
hypercenterC22×C4⋊Q8D3
hypocenterC3C2×#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#1007C2semidirect
2. C22×C3×C4⋊Q8C2semidirect
3. C2×#5816C2semidirect
4. #86214C2direct (N × H)
5. C2×#5727C2semidirect
6. C2×#5726C2semidirect
7. C2×#5724C2semidirect
8. C2×D3×C4⋊Q8C2semidirect
9. C2×#5725C2semidirect
10. C2×#5507C2semidirect
11. C2×#5477C2semidirect
12. C2×C4×#235C2semidirect
16. C2×#1007C22semidirect
27. C2×C3×C4⋊Q8C22semidirect
30. #5816C22semidirect
38. #5727C22semidirect
39. #5726C22semidirect
40. #5725C22semidirect
41. #5724C22semidirect
42. D3×C4⋊Q8C22semidirect
44. #5507C22semidirect
50. #5477C22semidirect
53. C4×#235C22semidirect
70. C22×C4⋊Q8D3semidirect
92. C2×#235D4semidirect
173. C2×C4⋊Q8C2×D3semidirect
227. #235C2×D4semidirect
363. C2×C3×C42−1+4semidirect
477. C3×C4C2×2−1+4semidirect
520. C2×C4#1007semidirect
527. C2×C3#3499semidirect
551. C4C2×#1007semidirect
552. C3C2×#3499semidirect
556. C2#86214direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#10072
C22×C3×C4⋊Q82
C2×#58162
#862142
C2×#57272
C2×#57262
C2×#57242
C2×D3×C4⋊Q82
C2×#57252
C2×#55072
C2×#54772
C2×C4×#2352
C2×#34993