Finite Group Databasebeta

← C2×#86773 #11703441

Subgroups — C2×#86773 #11703441

Order 768.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic, complemented
2C2×#5834C212yesmaximal, complemented
3C2×#5827C212yesmaximal, complemented
4#86773C2132yesmaximal, direct product factor
5C2×#5718C213yesmaximal, complemented
6C2×#5716C213yesmaximal, complemented
7C2×#5618C216yesmaximal, complemented
8C2×#5619C216yesmaximal, complemented
9C2×#5570C211yesmaximal, characteristic, complemented
10C2×C4×#239C211yesmaximal, characteristic, complemented
11C2×#5557C213yesmaximal, complemented
12C2×#5555C213yesmaximal, complemented
13C2×#3517—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C22×#239C2211yescharacteristic
16C22×#237C2216yes—
17C3×#550C22116yescomplemented
18C2×C3×#127C2213yes—
19C2×C3×#121C2216yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2213yes—
22#5834C22132yescomplemented
23#5827C22132yescomplemented
24C2×Q8×Dic3C2211yescharacteristic
25C2×Q8×Dic3C2212yes—
26C2×#940C2216yes—
27C2×#939C22112yes—
28C2×#938C2213yes—
29C2×#938C2216yes—
30C2×#938C2216yes—
31C2×#936C2213yes—
32C2×D4×Dic3C2216yes—
33C2×D4×Dic3C2216yes—
34C2×D4×Dic3C2216yes—
35C2×#930C2216yes—
36C2×C4×#67C2216yes—
37C2×#926C2216yes—
38C22×C4⋊Dic3C2216yes—
39#5718C22148yescomplemented
40#5716C22148yescomplemented
41#5619C22196yescomplemented
42#5618C22196yescomplemented
43C4×#239C22116yescomplemented
44#5570C22116yescomplemented
45#5557C22148yescomplemented
46#5555C22148yescomplemented
47C2×#897C2213yes—
48C2×#893C2213yes—
49C2×#892C2213yes—
50C2×#890C2211yescharacteristic
51C2×#890C2216yes—
52C2×#886C22112yes—
53C2×#883C2216yes—
54C2×#879C2216yes—
55C2×C4×D12C2213yes—
56C2×C42×D3C2213yes—
57C2×#869C2213yes—
58C2×C4×Dic6C2213yes—
59#3517—332no—
60C2×#557—31no—
61C2×#553—36no—
62C2×#550—32no—
63C2×#550—33no—
64C2×#550D311yescharacteristic, complemented
65C2×#547—33no—
66C2×#540—36no—
67C2×#532—33no—
68C2×#529—33no—
69C2×#520—32no—
70C2×C4×#41—31no—
71C2×C3×#41—216no—
72C2×C3×#41C23116yes—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4C2313yes—
75C22×C3×D4C2316yes—
76C22×C3×D4C2316yes—
77C23×C3×C4C2316yes—
78C22×#67C23112yes—
79C22×#67C2316yes—
80C23×Dic3C23112yes—
81C2×#239—28no—
82C2×#239C2318yes—
83C2×#237—424no—
84C2×#237C23148yes—
85C22×D12C2313yes—
86C22×C4×D3C2313yes—
87C22×C4×D3C2313yes—
88C22×C4×D3C2313yes—
89C22×Dic6C2313yes—
90C3×#127C23124yes—
91C3×#121C23148yes—
92C3×C4×Q8C2318yes—
93C3×C4×D4C23124yes—
94C2×C3×C4⋊C4C2313yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4C2316yes—
97C2×C3×C42C2313yes—
98C2×#217C2316yes—
99C2×#217C2316yes—
100C2×#217C2316yes—
101Q8×Dic3C23116yes—
102Q8×Dic3C2318yes—
103#940C23148yes—
104#939C23196yes—
105#938C23148yes—
106#938C23148yes—
107#938C23124yes—
108#936C23124yes—
109D4×Dic3C23148yes—
110D4×Dic3C23148yes—
111D4×Dic3C23148yes—
112#930C23148yes—
113C4×#67C23148yes—
114C2×(C2×D3)⋊C4C2316yes—
115C2×(C2×D3)⋊C4C2316yes—
116#926C23148yes—
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2316yes—
119C2×C4⋊Dic3C2316yes—
120C2×C4⋊Dic3C2313yes—
121C2×C4⋊Dic3C2316yes—
122C2×C4⋊Dic3C2312yes—
123C2×C4⋊Dic3C23148yes—
124C2×Dic3⋊C4C2316yes—
125C2×Dic3⋊C4C2312yes—
126C2×C4×Dic3C2316yes—
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2313yes—
129C2×C4×Dic3C2313yes—
130#897C23124yes—
131#893C23124yes—
132#892C23124yes—
133#890C2318yes—
134#890C23148yes—
135#886C23196yes—
136#883C23148yes—
137#879C23148yes—
138C4×D12C23124yes—
139C42×D3C23124yes—
140#869C23124yes—
141C4×Dic6C23124yes—
142C22×#41—31no—
143C22×#41—36no—
144C22×#41C2×D312yes—
145#557—316no—
146#553—396no—
147#550—332no—
148#550—348no—
149#550C2×D3116yescomplemented
150#547—348no—
151#540—396no—
152#532—348no—
153C2×C4⋊Q8—33no—
154C2×#127—36no—
155C2×#127C2×D313yes—
156#529—348no—
157C2×#126—36no—
158C2×#125—31no—
159C2×#125—36no—
160C2×#124—33no—
161C2×#123—36no—
162C2×#123—312no—
163C2×C22⋊Q8—33no—
164C2×#121—33no—
165C2×#121—36no—
166C2×#121—36no—
167C2×#121—312no—
168C2×#121—36no—
169C2×#121C2×D316yes—
170#520—332no—
171C4×#41—316no—
172C2×C4×Q8—31no—
173C2×C4×Q8—32no—
174C2×C4×Q8—33no—
175C2×C4×Q8C2×D311yescharacteristic
176C2×C4×D4—36no—
177C2×C4×D4—36no—
178C2×C4×D4—36no—
179C2×C4×D4—36no—
180C2×C4×D4—33no—
181C2×C4×D4—33no—
182C2×C4×D4C2×D313yes—
183C2×#117—36no—
184C2×#117—33no—
185C22×C4⋊C4—36no—
186C22×C42—33no—
187C24×C3C2416yes—
188C23×D3C2413yes—
189C3×#41—264no—
190C2×C3×Q8#4118yescomplemented
191C2×C3×Q8C2414yes—
192C2×C3×D4—224no—
193C2×C3×D4—212no—
194C2×C3×D4—412no—
195C2×C3×D4C24124yes—
196C2×C3×D4C24112yes—
197C2×C3×D4C24124yes—
198C22×C3×C4—224no—
199C22×C3×C4C2413yes—
200C22×C3×C4C2412yes—
201C22×C3×C4C24124yes—
202C22×C3×C4C2413yes—
203C22×C3×C4C2411yescharacteristic
204C2×#67—424no—
205C2×#67—412no—
206C2×#67C24124yes—
207C2×#67C24148yes—
208C22×Dic3—424no—
209C22×Dic3C2411yescharacteristic
210C22×Dic3C2411yescharacteristic
211C22×Dic3C2412yes—
212C22×Dic3C2413yes—
213C22×Dic3C2416yes—
214C22×Dic3C24148yes—
215#239—232no—
216#237—496no—
217C2×D12—212no—
218C2×D12C24112yes—
219C2×C4×D3—212no—
220C2×C4×D3—212no—
221C2×C4×D3—46no—
222C2×C4×D3C24112yes—
223C2×C4×D3C24112yes—
224C2×C4×D3C24112yes—
225C2×Dic6—212no—
226C2×Dic6C24112yes—
227C3×C4⋊C4C24112yes—
228C3×C4⋊C4C2414yes—
229C3×C22⋊C4C24124yes—
230C3×C42C24112yes—
231#217C24124yes—
232#217C24124yes—
233#217C24124yes—
234(C2×D3)⋊C4C24124yes—
235(C2×D3)⋊C4C24124yes—
236C4⋊Dic3C2414yes—
237C4⋊Dic3C24124yes—
238C4⋊Dic3C24124yes—
239C4⋊Dic3C24112yes—
240C4⋊Dic3C24124yes—
241C4⋊Dic3C2418yes—
242Dic3⋊C4C24124yes—
243Dic3⋊C4C2418yes—
244C4×Dic3C24124yes—
245C4×Dic3C2414yes—
246C4×Dic3C24112yes—
247C4×Dic3C24112yes—
248C2×#41—216no—
249C2×#41—38no—
250C2×#41—348no—
251C2×#41—68no—
252C2×#41—1224no—
253C2×#41C22×D3116yes—
254C22×Q8—33no—
255C22×Q8C22×D311yescharacteristic
256C22×D4—36no—
257C22×D4—312no—
258C22×D4—33no—
259C22×D4C22×D316yes—
260C22×D4C22×D313yes—
261C22×D4C22×D316yes—
262C23×C4—312no—
263C23×C4—33no—
264C23×C4—33no—
265C23×C4—33no—
266C23×C4C22×D316yes—
267C4⋊Q8—324no—
268#127—348no—
269#127C22×D3124yes—
270#126—348no—
271#125—38no—
272#125—348no—
273#124—324no—
274#123—348no—
275#123—396no—
276C22⋊Q8—324no—
277#121—324no—
278#121—348no—
279#121—348no—
280#121—396no—
281#121—348no—
282#121C22×D3148yes—
283C4×Q8—38no—
284C4×Q8—316no—
285C4×Q8—324no—
286C4×Q8C22×D318yes—
287C4×D4—348no—
288C4×D4—348no—
289C4×D4—348no—
290C4×D4—348no—
291C4×D4—324no—
292C4×D4—324no—
293C4×D4C22×D3124yes—
294#117—348no—
295#117—324no—
296C2×C4⋊C4—33no—
297C2×C4⋊C4—36no—
298C2×C4⋊C4—36no—
299C2×C4⋊C4—348no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—36no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—36no—
305C2×C4⋊C4C22×D313yes—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—36no—
308C2×C22⋊C4—36no—
309C2×C22⋊C4—36no—
310C2×C22⋊C4—36no—
311C2×C22⋊C4—36no—
312C2×C22⋊C4C22×D316yes—
313C2×C42—33no—
314C2×C42—33no—
315C2×C42—36no—
316C2×C42—31no—
317C2×C42—324no—
318C2×C42C22×D313yes—
319C23×C3—212no—
320C23×C3—212no—
321C23×C3—212no—
322C23×C3C25112yes—
323C23×C3C2511yescharacteristic
324C22×D3—26no—
325C22×D3—26no—
326C22×D3—26no—
327C22×D3C2516yes—
328C3×Q8C2×#41116yescomplemented
329C3×D4—248no—
330C3×D4—224no—
331C3×D4—424no—
332C2×C3×C4—24no—
333C2×C3×C4—26no—
334C2×C3×C4—248no—
335C2×C3×C4C2×#41112yes—
336C2×C3×C4H(5,𝔽2)14yes—
337C2×C3×C4C2516yes—
338C2×C3×C4C2514yes—
339C2×C3×C4C2516yes—
340C2×C3×C4C2512yes—
341#67—448no—
342#67—424no—
343C2×Dic3—22no—
344C2×Dic3—212no—
345C2×Dic3—26no—
346C2×Dic3—22no—
347C2×Dic3—24no—
348C2×Dic3—448no—
349C2×Dic3C2512yes—
350C2×Dic3C2512yes—
351C2×Dic3C2514yes—
352C2×Dic3C2516yes—
353C2×Dic3C25112yes—
354D12—224no—
355C4×D3—224no—
356C4×D3—224no—
357C4×D3—412no—
358Dic6—224no—
359C24—33no—
360C24C23×D316yes—
361#41—264no—
362#41—632no—
363#41—1296no—
364C2×Q8—312no—
365C2×Q8—612no—
366C2×Q8#23718yescomplemented
367C2×Q8C23×D314yes—
368C2×D4—224no—
369C2×D4—212no—
370C2×D4—324no—
371C2×D4—348no—
372C2×D4—312no—
373C2×D4—412no—
374C2×D4—612no—
375C2×D4—1224no—
376C2×D4—1212no—
377C2×D4C23×D3124yes—
378C2×D4C23×D3112yes—
379C2×D4C23×D3124yes—
380C22×C4—224no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—32no—
384C22×C4—33no—
385C22×C4—36no—
386C22×C4—348no—
387C22×C4—312no—
388C22×C4—312no—
389C22×C4—312no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—126no—
393C22×C4—1224no—
394C22×C4C23×D3124yes—
395C22×C4C23×D313yes—
396C22×C4C23×D312yes—
397C22×C4C23×D313yes—
398C22×C4C23×D311yescharacteristic
399C4⋊C4—312no—
400C4⋊C4—324no—
401C4⋊C4—324no—
402C4⋊C4—38no—
403C4⋊C4—38no—
404C4⋊C4—324no—
405C4⋊C4—34no—
406C4⋊C4—324no—
407C4⋊C4C23×D3112yes—
408C4⋊C4C23×D314yes—
409C22⋊C4—324no—
410C22⋊C4—324no—
411C22⋊C4—324no—
412C22⋊C4—324no—
413C22⋊C4—324no—
414C22⋊C4C23×D3124yes—
415C42—34no—
416C42—312no—
417C42—312no—
418C42—324no—
419C42C23×D3112yes—
420C22×C3—212no—
421C22×C3—212no—
422C22×C3—212no—
423C22×C3—424no—
424C22×C3C22×#4112yes—
425C22×C3C2×H(5,𝔽2)12yes—
426C22×C3#58512yes—
427C22×C3C2611yescharacteristic
428C2×D3—26no—
429C2×D3—26no—
430C2×D3—26no—
431C2×D3—412no—
432C3×C4—24no—
433C3×C4—26no—
434C3×C4C22×#41112yes—
435C3×C4C2×H(5,𝔽2)14yes—
436Dic3—22no—
437Dic3—212no—
438Dic3—26no—
439Dic3—22no—
440Dic3—24no—
441C23—212no—
442C23—212no—
443C23—212no—
444C23—36no—
445C23—66no—
446C23—66no—
447C23—66no—
448C23C24×D3112yes—
449C23C24×D311yescharacteristic
450Q8—624no—
451Q8C2×#237116yescomplemented
452D4—248no—
453D4—224no—
454D4—424no—
455D4—624no—
456D4—1248no—
457D4—1224no—
458C2×C4—24no—
459C2×C4—26no—
460C2×C4—248no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—34no—
464C2×C4—36no—
465C2×C4—312no—
466C2×C4—62no—
467C2×C4—66no—
468C2×C4—612no—
469C2×C4—62no—
470C2×C4—64no—
471C2×C4—624no—
472C2×C4—624no—
473C2×C4—1212no—
474C2×C4—1248no—
475C2×C4C2×#237112yes—
476C2×C4#100914yes—
477C2×C4C24×D316yes—
478C2×C4C24×D314yes—
479C2×C4C24×D316yes—
480C2×C4C24×D312yes—
481C2×C3—412no—
482C2×C3#351714yescomplemented
483C2×C3C23×#4111yescharacteristic
484C2×C3C22×H(5,𝔽2)11yescharacteristic
485C2×C3C2×#58511yescharacteristic
486D3—46no—
487C22—212no—
488C22—212no—
489C22—212no—
490C22—424no—
491C22—66no—
492C22—66no—
493C22—66no—
494C22—1212no—
495C22C22×#23712yes—
496C22C2×#100912yes—
497C22#596612yes—
498C22C25×D311yescharacteristic
499C4—24no—
500C4—26no—
501C4—62no—
502C4—66no—
503C4—612no—
504C4—62no—
505C4—64no—
506C4C22×#237112yes—
507C4C2×#100914yes—
508C3C2×#351711yescharacteristic, complemented, Sylow
509C2—412no—
510C2—126no—
511C2#8677314yesdirect product factor
512C2C23×#23711yescharacteristic
513C2C22×#100911yescharacteristic
514C2C2×#596611yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×#550C2
socleC23×C3C25
hypercenterC2×#550D3
hypocenterC3C2×#3517
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×C3×#550C2semidirect
2. C2×#5834C2semidirect
3. C2×#5827C2semidirect
4. #86773C2direct (N × H)
5. C2×#5718C2semidirect
6. C2×#5716C2semidirect
7. C2×#5618C2semidirect
8. C2×#5619C2semidirect
9. C2×#5570C2semidirect
10. C2×C4×#239C2semidirect
11. C2×#5557C2semidirect
12. C2×#5555C2semidirect
17. C3×#550C22semidirect
22. #5834C22semidirect
23. #5827C22semidirect
39. #5718C22semidirect
40. #5716C22semidirect
41. #5619C22semidirect
42. #5618C22semidirect
43. C4×#239C22semidirect
44. #5570C22semidirect
45. #5557C22semidirect
46. #5555C22semidirect
64. C2×#550D3semidirect
149. #550C2×D3semidirect
190. C2×C3×Q8#41semidirect
328. C3×Q8C2×#41semidirect
366. C2×Q8#237semidirect
451. Q8C2×#237semidirect
482. C2×C3#3517semidirect
508. C3C2×#3517semidirect
511. C2#86773direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×#58342
C2×#58272
#867732
C2×#57182
C2×#57162
C2×#56182
C2×#56192
C2×#55702
C2×C4×#2392
C2×#55572
C2×#55552
C2×#35173