Finite Group Databasebeta

← C2×#85853 #11703445

Subgroups — C2×#85853 #11703445

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×C4×#41C211yesmaximal, characteristic, complemented
2C2×#5834C212yesmaximal, complemented
3#85853C2132yesmaximal, direct product factor
4C2×#5570C211yesmaximal, characteristic, complemented
5C2×#5577C211yesmaximal, characteristic, complemented
6C2×#5574C212yesmaximal, complemented
7C2×#5551C213yesmaximal, complemented
8C2×#5562C216yesmaximal, complemented
9C2×#5553C213yesmaximal, complemented
10C2×#5530C216yesmaximal, complemented
11C2×#5477C213yesmaximal, complemented
12C2×#5479C213yesmaximal, complemented
13C2×#3517—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C22×#239C2212yes—
16C22×#235C2216yes—
17C3×C4×#41C22116yescomplemented
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2213yes—
21C2×C3×C4×D4C2213yes—
22C2×C3×#117C2213yes—
23C22×C3×C42C2213yes—
24#5834C22132yescomplemented
25C2×Q8×Dic3C2212yes—
26C2×#940C2216yes—
27C2×#938C2216yes—
28C2×#930C2216yes—
29C2×#930C2213yes—
30C2×#930C2216yes—
31C2×#929C2216yes—
32C2×C4×#67C2216yes—
33C2×#924C2213yes—
34#5577C22116yescomplemented
35#5574C22132yescomplemented
36#5570C22116yescomplemented
37#5562C22196yescomplemented
38#5553C22148yescomplemented
39#5551C22148yescomplemented
40#5530C22196yescomplemented
41C2×#895C2216yes—
42C2×#893C2216yes—
43C2×D3×C4⋊C4C2216yes—
44C2×#890C2216yes—
45C2×#884C22112yes—
46C2×#883C22112yes—
47#5479C22148yescomplemented
48C2×#876C2216yes—
49#5477C22148yescomplemented
50C2×#875C2213yes—
51C2×#874C2213yes—
52C2×C4×D12C2216yes—
53C2×C4×D12C2213yes—
54C2×C4×D12C2216yes—
55C2×#872C2216yes—
56C2×#870C2211yescharacteristic
57C2×#869C2213yes—
58C2×C4×Dic6C2213yes—
59#3517—332no—
60C2×#557—31no—
61C2×#553—36no—
62C2×#550—32no—
63C2×#550—31no—
64C2×#550—33no—
65C2×#547—33no—
66C2×#540—36no—
67C2×#532—33no—
68C2×#529—33no—
69C2×#520—32no—
70C2×C4×#41D311yescharacteristic, complemented
71C2×C3×#41—28no—
72C2×C3×#41C2318yes—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4C2313yes—
75C23×C3×C4C2313yes—
76C23×C3×C4C2313yes—
77C23×C3×C4C2313yes—
78C22×#67C23112yes—
79C22×#67C2316yes—
80C2×#239—216no—
81C2×#239C23116yes—
82C2×#235—424no—
83C2×#235C23148yes—
84C22×D12C2313yes—
85C22×D12C2316yes—
86C22×D12C2316yes—
87C22×C4×D3C2316yes—
88C22×C4×D3C23112yes—
89C22×Dic6C2313yes—
90C3×C4×Q8C2318yes—
91C3×C4×Q8C2318yes—
92C3×C4×D4C23124yes—
93C3×C4×D4C23124yes—
94C3×#117C23124yes—
95C2×C3×C4⋊C4C2313yes—
96C2×C3×C4⋊C4C2313yes—
97C2×C3×C22⋊C4C2316yes—
98C2×C3×C42C2313yes—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2313yes—
101C2×C3×C42C2313yes—
102C2×C3×C42C23124yes—
103C2×#217C2316yes—
104Q8×Dic3C23116yes—
105#940C23148yes—
106#938C23148yes—
107#930C23148yes—
108#930C23148yes—
109#930C23124yes—
110#929C23148yes—
111C4×#67C23148yes—
112C2×(C2×D3)⋊C4C2316yes—
113C2×(C2×D3)⋊C4C2316yes—
114C2×(C2×D3)⋊C4C2316yes—
115C2×(C2×D3)⋊C4C2316yes—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2316yes—
118C2×C4⋊Dic3C2316yes—
119C2×C4⋊Dic3C2311yescharacteristic
120#924C23124yes—
121C2×Dic3⋊C4C2316yes—
122C2×Dic3⋊C4C2312yes—
123C2×Dic3⋊C4C2312yes—
124C2×Dic3⋊C4C2316yes—
125C2×C4×Dic3C2316yes—
126#895C23148yes—
127#893C23148yes—
128D3×C4⋊C4C23148yes—
129#890C23148yes—
130#884C23196yes—
131#883C23196yes—
132#876C23148yes—
133#875C23124yes—
134#874C23124yes—
135C4×D12C23148yes—
136C4×D12C23148yes—
137C4×D12C23124yes—
138#872C23148yes—
139#870C2318yes—
140#869C23124yes—
141C4×Dic6C23124yes—
142C22×#41—32no—
143C22×#41—36no—
144C22×#41C2×D311yescharacteristic
145#557—316no—
146#553—396no—
147#550—332no—
148#550—316no—
149#550—348no—
150#547—348no—
151#540—396no—
152#532—348no—
153C2×C4⋊Q8—33no—
154C2×#127—36no—
155C2×#127—33no—
156#529—348no—
157C2×#126—36no—
158C2×#125—36no—
159C2×#125—31no—
160C2×#124—33no—
161C2×#123—36no—
162C2×#123—312no—
163C2×C22⋊Q8—33no—
164C2×#121—36no—
165C2×#121—33no—
166C2×#121—36no—
167C2×#121—36no—
168C2×#121—36no—
169C2×#121—312no—
170#520—332no—
171C4×#41C2×D3116yescomplemented
172C2×C4×Q8—32no—
173C2×C4×Q8—33no—
174C2×C4×Q8C2×D311yescharacteristic
175C2×C4×Q8C2×D311yescharacteristic
176C2×C4×D4—36no—
177C2×C4×D4—36no—
178C2×C4×D4—36no—
179C2×C4×D4—33no—
180C2×C4×D4—36no—
181C2×C4×D4C2×D313yes—
182C2×C4×D4C2×D313yes—
183C2×#117—36no—
184C2×#117C2×D313yes—
185C22×C4⋊C4—36no—
186C22×C42C2×D313yes—
187C24×C3C2413yes—
188C23×D3C2416yes—
189C3×#41—232no—
190C2×C3×Q8#4118yescomplemented
191C2×C3×Q8C2414yes—
192C2×C3×D4—212no—
193C2×C3×D4C24112yes—
194C22×C3×C4—212no—
195C22×C3×C4—212no—
196C22×C3×C4—46no—
197C22×C3×C4C2413yes—
198C22×C3×C4C2411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×C3×C4C2413yes—
201C22×C3×C4C24112yes—
202C22×C3×C4C24112yes—
203C22×C3×C4C24112yes—
204C22×C3×C4C2413yes—
205C22×C3×C4C2411yescharacteristic
206C2×#67—424no—
207C2×#67—412no—
208C2×#67C24148yes—
209C2×#67C24124yes—
210C22×Dic3C2412yes—
211C22×Dic3C2416yes—
212C22×Dic3C2412yes—
213#239—264no—
214#235—496no—
215C2×D12—224no—
216C2×D12—212no—
217C2×D12—412no—
218C2×D12C24112yes—
219C2×D12C24124yes—
220C2×D12C24124yes—
221C2×C4×D3—224no—
222C2×C4×D3—424no—
223C2×C4×D3C24124yes—
224C2×C4×D3C24148yes—
225C2×Dic6—212no—
226C2×Dic6C24112yes—
227C3×C4⋊C4C24112yes—
228C3×C4⋊C4C24112yes—
229C3×C22⋊C4C24124yes—
230C3×C42C24112yes—
231C3×C42C2414yes—
232C3×C42C24112yes—
233C3×C42C24112yes—
234#217C24124yes—
235(C2×D3)⋊C4C24124yes—
236(C2×D3)⋊C4C24124yes—
237(C2×D3)⋊C4C24124yes—
238(C2×D3)⋊C4C24124yes—
239C4⋊Dic3C2414yes—
240C4⋊Dic3C24124yes—
241C4⋊Dic3C24124yes—
242C4⋊Dic3C2414yes—
243Dic3⋊C4C24124yes—
244Dic3⋊C4C2418yes—
245Dic3⋊C4C2418yes—
246Dic3⋊C4C24124yes—
247C4×Dic3C24124yes—
248C2×#41—28no—
249C2×#41—316no—
250C2×#41—348no—
251C2×#41—616no—
252C2×#41—1224no—
253C2×#41C22×D318yes—
254C22×Q8—33no—
255C22×Q8C22×D311yescharacteristic
256C22×D4—36no—
257C22×D4—312no—
258C22×D4—36no—
259C22×D4—33no—
260C22×D4—36no—
261C22×D4C22×D313yes—
262C23×C4—36no—
263C23×C4—312no—
264C23×C4C22×D313yes—
265C23×C4C22×D313yes—
266C23×C4C22×D313yes—
267C4⋊Q8—324no—
268#127—348no—
269#127—324no—
270#126—348no—
271#125—348no—
272#125—38no—
273#124—324no—
274#123—348no—
275#123—396no—
276C22⋊Q8—324no—
277#121—348no—
278#121—324no—
279#121—348no—
280#121—348no—
281#121—348no—
282#121—396no—
283C4×Q8—316no—
284C4×Q8—324no—
285C4×Q8C22×D318yes—
286C4×Q8C22×D318yes—
287C4×D4—348no—
288C4×D4—348no—
289C4×D4—348no—
290C4×D4—324no—
291C4×D4—348no—
292C4×D4C22×D3124yes—
293C4×D4C22×D3124yes—
294#117—348no—
295#117C22×D3124yes—
296C2×C4⋊C4—36no—
297C2×C4⋊C4—36no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—36no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—36no—
304C2×C4⋊C4—348no—
305C2×C4⋊C4C22×D313yes—
306C2×C4⋊C4C22×D313yes—
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—36no—
314C2×C42C22×D313yes—
315C2×C42C22×D313yes—
316C2×C42C22×D313yes—
317C2×C42C22×D3124yes—
318C2×C42C22×D311yescharacteristic
319C23×C3—26no—
320C23×C3—26no—
321C23×C3—26no—
322C23×C3C2516yes—
323C23×C3C2511yescharacteristic
324C22×D3—212no—
325C22×D3—212no—
326C22×D3—212no—
327C22×D3C25112yes—
328C3×Q8C2×#41116yescomplemented
329C3×D4—224no—
330C2×C3×C4—22no—
331C2×C3×C4—26no—
332C2×C3×C4—26no—
333C2×C3×C4—22no—
334C2×C3×C4—224no—
335C2×C3×C4—224no—
336C2×C3×C4—412no—
337C2×C3×C4C2×#41112yes—
338C2×C3×C4H(5,𝔽2)14yes—
339C2×C3×C4C2516yes—
340C2×C3×C4C2512yes—
341C2×C3×C4C2512yes—
342C2×C3×C4C2516yes—
343C2×C3×C4C2516yes—
344C2×C3×C4C2512yes—
345#67—448no—
346#67—424no—
347C2×Dic3—24no—
348C2×Dic3—212no—
349C2×Dic3—24no—
350C2×Dic3C2514yes—
351C2×Dic3C25112yes—
352C2×Dic3C2514yes—
353D12—248no—
354D12—224no—
355D12—424no—
356C4×D3—248no—
357C4×D3—448no—
358Dic6—224no—
359C24—36no—
360C24C23×D313yes—
361#41—232no—
362#41—664no—
363#41—1296no—
364C2×Q8—312no—
365C2×Q8—612no—
366C2×Q8#23518yescomplemented
367C2×Q8C23×D314yes—
368C2×D4—212no—
369C2×D4—324no—
370C2×D4—348no—
371C2×D4—324no—
372C2×D4—312no—
373C2×D4—324no—
374C2×D4—624no—
375C2×D4—612no—
376C2×D4—1212no—
377C2×D4—1224no—
378C2×D4—1212no—
379C2×D4C23×D3112yes—
380C22×C4—212no—
381C22×C4—212no—
382C22×C4—32no—
383C22×C4—36no—
384C22×C4—32no—
385C22×C4—324no—
386C22×C4—348no—
387C22×C4—46no—
388C22×C4—624no—
389C22×C4—1224no—
390C22×C4C23×D3112yes—
391C22×C4C23×D3112yes—
392C22×C4C23×D311yescharacteristic
393C22×C4C23×D311yescharacteristic
394C22×C4C23×D313yes—
395C22×C4C23×D313yes—
396C22×C4C23×D3112yes—
397C22×C4C23×D313yes—
398C22×C4C23×D311yescharacteristic
399C4⋊C4—324no—
400C4⋊C4—324no—
401C4⋊C4—38no—
402C4⋊C4—34no—
403C4⋊C4—38no—
404C4⋊C4—324no—
405C4⋊C4—34no—
406C4⋊C4—324no—
407C4⋊C4C23×D3112yes—
408C4⋊C4C23×D3112yes—
409C22⋊C4—324no—
410C22⋊C4—324no—
411C22⋊C4—324no—
412C22⋊C4—324no—
413C22⋊C4—324no—
414C22⋊C4C23×D3124yes—
415C42—324no—
416C42C23×D314yes—
417C42C23×D3112yes—
418C42C23×D3112yes—
419C42C23×D3112yes—
420C22×C3—26no—
421C22×C3—26no—
422C22×C3—26no—
423C22×C3—412no—
424C22×C3C22×#4112yes—
425C22×C3C2×H(5,𝔽2)12yes—
426C22×C3#58512yes—
427C22×C3C2611yescharacteristic
428C2×D3—212no—
429C2×D3—212no—
430C2×D3—212no—
431C2×D3—424no—
432C3×C4—22no—
433C3×C4—26no—
434C3×C4—26no—
435C3×C4—22no—
436C3×C4C22×#41112yes—
437C3×C4C2×H(5,𝔽2)14yes—
438Dic3—24no—
439Dic3—212no—
440Dic3—24no—
441C23—26no—
442C23—26no—
443C23—26no—
444C23—312no—
445C23—612no—
446C23—612no—
447C23—612no—
448C23C24×D316yes—
449C23C24×D311yescharacteristic
450Q8—624no—
451Q8C2×#235116yescomplemented
452D4—224no—
453D4—648no—
454D4—624no—
455D4—1224no—
456D4—1248no—
457D4—1224no—
458C2×C4—22no—
459C2×C4—26no—
460C2×C4—22no—
461C2×C4—26no—
462C2×C4—224no—
463C2×C4—224no—
464C2×C4—34no—
465C2×C4—312no—
466C2×C4—34no—
467C2×C4—412no—
468C2×C4—64no—
469C2×C4—612no—
470C2×C4—64no—
471C2×C4—648no—
472C2×C4—1248no—
473C2×C4C2×#235112yes—
474C2×C4#100914yes—
475C2×C4C24×D312yes—
476C2×C4C24×D312yes—
477C2×C4C24×D316yes—
478C2×C4C24×D316yes—
479C2×C4C24×D316yes—
480C2×C4C24×D312yes—
481C2×C3—46no—
482C2×C3#351714yescomplemented
483C2×C3C23×#4111yescharacteristic
484C2×C3C22×H(5,𝔽2)11yescharacteristic
485C2×C3C2×#58511yescharacteristic
486D3—412no—
487C22—26no—
488C22—26no—
489C22—26no—
490C22—412no—
491C22—612no—
492C22—612no—
493C22—612no—
494C22—1224no—
495C22C22×#23512yes—
496C22C2×#100912yes—
497C22#596412yes—
498C22C25×D311yescharacteristic
499C4—22no—
500C4—26no—
501C4—22no—
502C4—26no—
503C4—64no—
504C4—612no—
505C4—64no—
506C4C22×#235112yes—
507C4C2×#100914yes—
508C3C2×#351711yescharacteristic, complemented, Sylow
509C2—46no—
510C2—1212no—
511C2#8585314yesdirect product factor
512C2C23×#23511yescharacteristic
513C2C22×#100911yescharacteristic
514C2C2×#596411yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×C4×#41C2
socleC23×C3C25
hypercenterC2×C4×#41D3
hypocenterC3C2×#3517
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×#41C2semidirect
2. C2×#5834C2semidirect
3. #85853C2direct (N × H)
4. C2×#5570C2semidirect
5. C2×#5577C2semidirect
6. C2×#5574C2semidirect
7. C2×#5551C2semidirect
8. C2×#5562C2semidirect
9. C2×#5553C2semidirect
10. C2×#5530C2semidirect
11. C2×#5477C2semidirect
12. C2×#5479C2semidirect
17. C3×C4×#41C22semidirect
24. #5834C22semidirect
34. #5577C22semidirect
35. #5574C22semidirect
36. #5570C22semidirect
37. #5562C22semidirect
38. #5553C22semidirect
39. #5551C22semidirect
40. #5530C22semidirect
47. #5479C22semidirect
49. #5477C22semidirect
70. C2×C4×#41D3semidirect
171. C4×#41C2×D3semidirect
190. C2×C3×Q8#41semidirect
328. C3×Q8C2×#41semidirect
366. C2×Q8#235semidirect
451. Q8C2×#235semidirect
482. C2×C3#3517semidirect
508. C3C2×#3517semidirect
511. C2#85853direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C2×#58342
#858532
C2×#55702
C2×#55772
C2×#55742
C2×#55512
C2×#55622
C2×#55532
C2×#55302
C2×#54772
C2×#54792
C2×#35173