Finite Group Databasebeta

← C2×#86243 #11703597

Subgroups — C2×#86243 #11703597

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×#1010C212yesmaximal, complemented
2C2×C3×#532C211yesmaximal, characteristic, complemented
3C2×#5825C212yesmaximal, complemented
4#86243C2132yesmaximal, direct product factor
5C2×D3×C4⋊Q8C212yesmaximal, complemented
6C2×#5716C212yesmaximal, complemented
7C2×#5675C214yesmaximal, complemented
8C2×#5611C218yesmaximal, complemented
9C2×#5551C214yesmaximal, complemented
10C2×D4×Dic6C214yesmaximal, complemented
11C22×#869C211yesmaximal, characteristic, complemented
12C2×C4×#235C211yesmaximal, characteristic, complemented
13C2×#3499—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C2×#1010—232no—
16C2×#1010C22132yescomplemented
17C22×D3×Q8C2214yes—
18C22×#237C2214yes—
19C22×#237C2218yes—
20C22×#235C2214yes—
21C22×#235C2211yescharacteristic
22C23×Dic6C2212yes—
23C23×Dic6C2214yes—
24C3×#532C22116yescomplemented
25C2×C3×C4⋊Q8C2211yescharacteristic
26C2×C3×#127C2211yescharacteristic
27C2×C3×#124C2212yes—
28C2×C3×#121C2214yes—
29C2×C3×C4×D4C2214yes—
30C22×C3×C42C2211yescharacteristic
31#5825C22132yescomplemented
32C2×#944C2214yes—
33C2×#938C2218yes—
34C2×#936C2214yes—
35C2×#935C2218yes—
36C2×D4×Dic3C2218yes—
37C2×C4×#67C2214yes—
38C2×#926C2211yescharacteristic
39C22×C4⋊Dic3C2214yes—
40C2×#924C2218yes—
41D3×C4⋊Q8C22132yescomplemented
42#5716C22132yescomplemented
43#5675C22164yescomplemented
44#5611C221128yescomplemented
45#5551C22164yescomplemented
46D4×Dic6C22164yescomplemented
47C2×D3×C4⋊C4C2218yes—
48C2×#888C2218yes—
49C2×#883C22116yes—
50C2×#878C22116yes—
51C4×#235C22116yescomplemented
52C2×C4×D12C2212yes—
53C2×#872C2212yes—
54C2×C42×D3C2212yes—
55C2×#869C2214yes—
56C2×#869C2212yes—
57C2×#869C2212yes—
58C2×#869C22116yescomplemented
59C2×C4×Dic6C2212yes—
60C22×2−1+4—32no—
61#3499—332no—
62C2×D4×Q8—34no—
63C2×#547—38no—
64C2×#547—34no—
65C2×#536—32no—
66C2×#536—34no—
67C2×#532—32no—
68C2×#532D311yescharacteristic, complemented
69C22×C4⋊Q8—31no—
70C22×C4⋊Q8—32no—
71C2×C4×#41—31no—
72C2×C3×#41—216no—
73C2×C3×#41C23116yes—
74C22×C3×Q8C2312yes—
75C22×C3×D4C2314yes—
76C22×C3×D4C2312yes—
77C22×C3×D4C2314yes—
78C23×C3×C4C2314yes—
79C23×C3×C4C2312yes—
80C23×C3×C4C2311yescharacteristic
81C22×#67C2318yes—
82C22×#67C2312yes—
83C23×Dic3C2318yes—
84C23×Dic3C2314yes—
85#1010—2256no—
86C2×D3×Q8—232no—
87C2×D3×Q8C23132yes—
88C2×#237—264no—
89C2×#237—232no—
90C2×#237C23132yes—
91C2×#237C23164yes—
92C2×#235—232no—
93C2×#235D4116yescomplemented
94C2×#235C23132yes—
95C2×#235C2318yes—
96C22×D12C2311yescharacteristic
97C22×C4×D3C2318yes—
98C22×C4×D3C2314yes—
99C22×C4×D3C2312yes—
100C22×Dic6—216no—
101C22×Dic6—232no—
102C22×Dic6C2314yes—
103C22×Dic6C2314yes—
104C22×Dic6C2318yes—
105C22×Dic6C23116yes—
106C22×Dic6C23132yes—
107C22×Dic6C2311yescharacteristic
108C3×C4⋊Q8C2318yes—
109C3×#127C2318yes—
110C3×#124C23116yes—
111C3×#121C23132yes—
112C3×C4×D4C23132yes—
113C2×C3×C4⋊C4C2314yes—
114C2×C3×C22⋊C4C2318yes—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2318yes—
119C2×#217C23116yes—
120C2×#217C2312yes—
121#944C23132yes—
122#938C23164yes—
123#936C23132yes—
124#935C23164yes—
125D4×Dic3C23164yes—
126C4×#67C23132yes—
127C2×(C2×D3)⋊C4C2314yes—
128#926C2318yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2318yes—
131C2×C4⋊Dic3C2318yes—
132C2×C4⋊Dic3C23132yes—
133#924C23164yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C23116yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2312yes—
138D3×C4⋊C4C23164yes—
139#888C23164yes—
140#883C231128yes—
141#878C231128yes—
142C4×D12C23116yes—
143#872C23116yes—
144C42×D3C23116yes—
145#869C23116yes—
146#869C23116yes—
147#869C23132yes—
148C4×Dic6C23116yes—
149C2×2−1+4—332no—
150C2×2−1+4—632no—
151C22×#41—34no—
152C22×#41—38no—
153C22×#41—31no—
154C22×#41—34no—
155C22×#41C2×D312yes—
156C23×Q8—34no—
157C23×Q8—32no—
158C23×Q8—34no—
159D4×Q8—364no—
160#547—3128no—
161#547—364no—
162#536—332no—
163#536—364no—
164#532—332no—
165#532C2×D3116yescomplemented
166C2×C4⋊Q8—34no—
167C2×C4⋊Q8—332no—
168C2×C4⋊Q8—38no—
169C2×C4⋊Q8—32no—
170C2×C4⋊Q8—32no—
171C2×C4⋊Q8—34no—
172C2×C4⋊Q8—316no—
173C2×C4⋊Q8C2×D311yescharacteristic
174C2×#127C2×D311yescharacteristic
175C2×#124—34no—
176C2×#124C2×D312yes—
177C2×#123—38no—
178C2×#123—316no—
179C2×C22⋊Q8—38no—
180C2×C22⋊Q8—316no—
181C2×#121—38no—
182C2×#121C2×D314yes—
183C4×#41—316no—
184C2×C4×Q8—32no—
185C2×C4×D4—38no—
186C2×C4×D4—34no—
187C2×C4×D4—32no—
188C2×C4×D4C2×D314yes—
189C2×#117—31no—
190C2×#117—32no—
191C22×C4⋊C4—34no—
192C22×C4⋊C4—38no—
193C22×C42—32no—
194C22×C42C2×D311yescharacteristic
195C24×C3C2414yes—
196C24×C3C2411yescharacteristic
197C23×D3C2412yes—
198C3×#41—264no—
199C2×C3×Q8—28no—
200C2×C3×Q8C2418yes—
201C2×C3×D4—216no—
202C2×C3×D4—28no—
203C2×C3×D4—216no—
204C2×C3×D4C24116yes—
205C2×C3×D4C24116yes—
206C2×C3×D4C2418yes—
207C22×C3×C4—216no—
208C22×C3×C4—28no—
209C22×C3×C4C2×D418yes—
210C22×C3×C4C2414yes—
211C22×C3×C4C2412yes—
212C22×C3×C4C24116yes—
213C22×C3×C4C2418yes—
214C22×C3×C4C2411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×C3×C4C2414yes—
217C22×C3×C4C2412yes—
218C2×#67—232no—
219C2×#67C2×D4116yes—
220C2×#67C24132yes—
221C2×#67C2418yes—
222C22×Dic3—216no—
223C22×Dic3—232no—
224C22×Dic3C24132yes—
225C22×Dic3C2414yes—
226C22×Dic3C2418yes—
227C22×Dic3C24116yes—
228C22×Dic3C2412yes—
229D3×Q8—2128no—
230#237—2256no—
231#237—2128no—
232#235—2128no—
233#235C2×D4164yescomplemented
234C2×D12C2×D418yes—
235C2×D12C2414yes—
236C2×C4×D3—232no—
237C2×C4×D3—216no—
238C2×C4×D3C2×D4116yes—
239C2×C4×D3C24116yes—
240C2×C4×D3C24132yes—
241C2×C4×D3C2418yes—
242C2×Dic6—216no—
243C2×Dic6—216no—
244C2×Dic6—264no—
245C2×Dic6—232no—
246C2×Dic6—2128no—
247C2×Dic6C2×D418yes—
248C2×Dic6C24116yes—
249C2×Dic6C24116yes—
250C2×Dic6C24132yes—
251C2×Dic6C2414yes—
252C3×C4⋊C4C24116yes—
253C3×C22⋊C4C24132yes—
254C3×C42C2418yes—
255C3×C42C2414yes—
256C3×C42C2414yes—
257#217C24164yes—
258#217C2418yes—
259(C2×D3)⋊C4C24116yes—
260C4⋊Dic3C2418yes—
261C4⋊Dic3C24132yes—
262C4⋊Dic3C24132yes—
263Dic3⋊C4C24116yes—
264Dic3⋊C4C24164yes—
265C4×Dic3C24116yes—
266C4×Dic3C2418yes—
2672−1+4—6256no—
268C2×#41—216no—
269C2×#41—332no—
270C2×#41—364no—
271C2×#41—316no—
272C2×#41—38no—
273C2×#41—332no—
274C2×#41—632no—
275C2×#41—632no—
276C2×#41—664no—
277C2×#41C22×D3116yes—
278C22×Q8—332no—
279C22×Q8—31no—
280C22×Q8—38no—
281C22×Q8—34no—
282C22×Q8—34no—
283C22×Q8—316no—
284C22×Q8—332no—
285C22×Q8—632no—
286C22×Q8—632no—
287C22×Q8—616no—
288C22×Q8C22×D312yes—
289C22×D4—32no—
290C22×D4—38no—
291C22×D4—31no—
292C22×D4C22×D314yes—
293C22×D4C22×D314yes—
294C22×D4C22×D312yes—
295C23×C4—32no—
296C23×C4—34no—
297C23×C4—38no—
298C23×C4—34no—
299C23×C4—38no—
300C23×C4C22×D312yes—
301C23×C4C22×D314yes—
302C23×C4C22×D311yescharacteristic
303C4⋊Q8—332no—
304C4⋊Q8—364no—
305C4⋊Q8—316no—
306C4⋊Q8—316no—
307C4⋊Q8—332no—
308C4⋊Q8C22×D318yes—
309#127C22×D318yes—
310#124—332no—
311#124C22×D3116yes—
312#123—364no—
313#123—3128no—
314C22⋊Q8—364no—
315C22⋊Q8—3128no—
316#121—364no—
317#121C22×D3132yes—
318C4×Q8—316no—
319C4×D4—364no—
320C4×D4—332no—
321C4×D4—316no—
322C4×D4C22×D3132yes—
323#117—38no—
324#117—316no—
325C2×C4⋊C4—38no—
326C2×C4⋊C4—316no—
327C2×C4⋊C4—38no—
328C2×C4⋊C4—332no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—364no—
332C2×C4⋊C4C22×D314yes—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—316no—
335C2×C22⋊C4—34no—
336C2×C22⋊C4C22×D318yes—
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×C3—28no—
346C23×C3—216no—
347C23×C3C22×D414yes—
348C23×C3C2518yes—
349C23×C3C2512yes—
350C23×C3C2511yescharacteristic
351C22×D3—28no—
352C22×D3C22×D418yes—
353C22×D3C2514yes—
354C3×Q8—216no—
355C3×D4—232no—
356C3×D4—216no—
357C3×D4—232no—
358C2×C3×C4—28no—
359C2×C3×C4—24no—
360C2×C3×C4—232no—
361C2×C3×C4—216no—
362C2×C3×C4C22×D414yes—
363C2×C3×C4C22×D414yes—
364C2×C3×C4C22×D4116yes—
365C2×C3×C42−1+418yescomplemented
366C2×C3×C4C2518yes—
367C2×C3×C4C2514yes—
368C2×C3×C4C2512yes—
369C2×C3×C4C2512yes—
370C2×C3×C4C2514yes—
371#67—264no—
372#67C22×D4132yes—
373C2×Dic3—216no—
374C2×Dic3—232no—
375C2×Dic3—28no—
376C2×Dic3—264no—
377C2×Dic3C22×D418yes—
378C2×Dic3C2518yes—
379C2×Dic3C25116yes—
380C2×Dic3C2514yes—
381D12C22×D4116yes—
382C4×D3—264no—
383C4×D3—232no—
384C4×D3C22×D4132yes—
385Dic6—232no—
386Dic6—232no—
387Dic6—264no—
388Dic6C22×D4116yes—
389C24—32no—
390C24C23×D314yes—
391C24C23×D311yescharacteristic
392#41—264no—
393#41—364no—
394#41—6128no—
395#41—6128no—
396#41—6256no—
397C2×Q8—28no—
398C2×Q8—38no—
399C2×Q8—34no—
400C2×Q8—332no—
401C2×Q8—316no—
402C2×Q8—316no—
403C2×Q8—632no—
404C2×Q8—6128no—
405C2×Q8—6128no—
406C2×Q8—616no—
407C2×Q8—616no—
408C2×Q8—664no—
409C2×Q8C23×D318yes—
410C2×D4—216no—
411C2×D4—28no—
412C2×D4—216no—
413C2×D4—316no—
414C2×D4—38no—
415C2×D4—332no—
416C2×D4—38no—
417C2×D4—34no—
418C2×D4—632no—
419C2×D4C23×D3116yes—
420C2×D4C23×D3116yes—
421C2×D4C23×D318yes—
422C22×C4—216no—
423C22×C4—28no—
424C22×C4—32no—
425C22×C4—316no—
426C22×C4—38no—
427C22×C4—316no—
428C22×C4—332no—
429C22×C4—34no—
430C22×C4—38no—
431C22×C4—316no—
432C22×C4—332no—
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×D314yes—
442C22×C4C23×D314yes—
443C22×C4C23×D311yescharacteristic
444C22×C4C23×D311yescharacteristic
445C22×C4C23×D312yes—
446C4⋊C4—332no—
447C4⋊C4—364no—
448C4⋊C4—332no—
449C4⋊C4—316no—
450C4⋊C4—38no—
451C4⋊C4C23×D3116yes—
452C22⋊C4—38no—
453C22⋊C4—364no—
454C22⋊C4—316no—
455C22⋊C4C23×D3132yes—
456C42—38no—
457C42—316no—
458C42C23×D314yes—
459C42C23×D314yes—
460C42C23×D318yes—
461C22×C3—24no—
462C22×C3—28no—
463C22×C3—28no—
464C22×C3—216no—
465C22×C3—416no—
466C22×C3C23×D414yes—
467C22×C3C23×D412yes—
468C22×C3C2×2−1+414yes—
469C22×C3C2611yescharacteristic
470C2×D3—28no—
471C2×D3—216no—
472C2×D3C23×D418yes—
473C3×C4—28no—
474C3×C4—24no—
475C3×C4C23×D414yes—
476C3×C4C23×D414yes—
477C3×C4C2×2−1+418yescomplemented
478Dic3—216no—
479Dic3—28no—
480Dic3C23×D418yes—
481C23—24no—
482C23—28no—
483C23—216no—
484C23—38no—
485C23—34no—
486C23—68no—
487C23C2×D3×D414yes—
488C23C24×D318yes—
489C23C24×D312yes—
490C23C24×D311yescharacteristic
491Q8—216no—
492Q8—316no—
493Q8—664no—
494Q8—632no—
495Q8—632no—
496D4—232no—
497D4—216no—
498D4—232no—
499D4—332no—
500D4—316no—
501D4—664no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—232no—
505C2×C4—216no—
506C2×C4—38no—
507C2×C4—34no—
508C2×C4—332no—
509C2×C4—38no—
510C2×C4—316no—
511C2×C4—664no—
512C2×C4—664no—
513C2×C4—616no—
514C2×C4—632no—
515C2×C4—68no—
516C2×C4—632no—
517C2×C4C2×D3×D414yes—
518C2×C4C2×D3×D414yes—
519C2×C4C2×D3×D4116yes—
520C2×C4#101018yescomplemented
521C2×C4C24×D314yes—
522C2×C4C24×D318yes—
523C2×C4C24×D312yes—
524C2×C4C24×D312yes—
525C2×C4C24×D314yes—
526C2×C3—24no—
527C2×C3—48no—
528C2×C3#349914yescomplemented
529C2×C3C24×D411yescharacteristic
530C2×C3C22×2−1+412yes—
531D3—28no—
532C22—24no—
533C22—28no—
534C22—28no—
535C22—216no—
536C22—38no—
537C22—416no—
538C22—68no—
539C22—616no—
540C22C22×D3×D414yes—
541C22C22×D3×D412yes—
542C22C2×#101014yes—
543C22C25×D311yescharacteristic
544C4—28no—
545C4—24no—
546C4—38no—
547C4—616no—
548C4—68no—
549C4C22×D3×D414yes—
550C4C22×D3×D414yes—
551C4C2×#101018yescomplemented
552C3C2×#349911yescharacteristic, complemented, Sylow
553C2—24no—
554C2—48no—
555C2—68no—
556C2#8624314yesdirect product factor
557C2C23×D3×D411yescharacteristic
558C2C22×#101012yes—
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×#532C2
socleC23×C3C25
hypercenterC2×#532D3
hypocenterC3C2×#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C22×#1010C2semidirect
2. C2×C3×#532C2semidirect
3. C2×#5825C2semidirect
4. #86243C2direct (N × H)
5. C2×D3×C4⋊Q8C2semidirect
6. C2×#5716C2semidirect
7. C2×#5675C2semidirect
8. C2×#5611C2semidirect
9. C2×#5551C2semidirect
10. C2×D4×Dic6C2semidirect
11. C22×#869C2semidirect
12. C2×C4×#235C2semidirect
16. C2×#1010C22semidirect
24. C3×#532C22semidirect
31. #5825C22semidirect
41. D3×C4⋊Q8C22semidirect
42. #5716C22semidirect
43. #5675C22semidirect
44. #5611C22semidirect
45. #5551C22semidirect
46. D4×Dic6C22semidirect
51. C4×#235C22semidirect
58. C2×#869C22semidirect
68. C2×#532D3semidirect
93. C2×#235D4semidirect
165. #532C2×D3semidirect
233. #235C2×D4semidirect
365. C2×C3×C42−1+4semidirect
477. C3×C4C2×2−1+4semidirect
520. C2×C4#1010semidirect
528. C2×C3#3499semidirect
551. C4C2×#1010semidirect
552. C3C2×#3499semidirect
556. C2#86243direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#10102
C2×C3×#5322
C2×#58252
#862432
C2×D3×C4⋊Q82
C2×#57162
C2×#56752
C2×#56112
C2×#55512
C2×D4×Dic62
C22×#8692
C2×C4×#2352
C2×#34993