Finite Group Databasebeta

← C2×#86776 #11703389

Subgroups — C2×#86776 #11703389

Order 768.

7758 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×#5831C212yesmaximal, complemented
3C2×#5827C212yesmaximal, complemented
4#86776C2132yesmaximal, direct product factor
5C2×#5713C213yesmaximal, complemented
6C2×#5716C213yesmaximal, complemented
7C2×#5611C216yesmaximal, complemented
8C2×#5603C216yesmaximal, complemented
9C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
10C2×Q8×Dic6C211yesmaximal, characteristic, complemented
11C2×#5557C213yesmaximal, complemented
12C2×#5543C213yesmaximal, complemented
13C2×#3522—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C22×D3×Q8C2211yescharacteristic
16C22×#237C2216yes—
17C3×#550C22116yescomplemented
18C2×C3×#127C2213yes—
19C2×C3×#121C2216yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2213yes—
22#5831C22132yescomplemented
23#5827C22132yescomplemented
24C2×Q8×Dic3C2212yes—
25C2×Q8×Dic3C2211yescharacteristic
26C2×#938C2213yes—
27C2×#938C2216yes—
28C2×#936C2213yes—
29C2×#936C2216yes—
30C2×#936C2216yes—
31C2×#935C22112yes—
32C2×D4×Dic3C2216yes—
33C2×D4×Dic3C2216yes—
34C2×D4×Dic3C2216yes—
35C2×C4×#67C2216yes—
36C2×#926C2216yes—
37C22×C4⋊Dic3C2216yes—
38C2×#924C2216yes—
39#5716C22148yescomplemented
40#5713C22148yescomplemented
41#5611C22196yescomplemented
42#5603C22196yescomplemented
43C4×D3×Q8C22116yescomplemented
44Q8×Dic6C22116yescomplemented
45#5557C22148yescomplemented
46#5543C22148yescomplemented
47C2×#897C2211yescharacteristic
48C2×D3×C4⋊C4C2213yes—
49C2×#890C2213yes—
50C2×#888C2216yes—
51C2×#887C2213yes—
52C2×#883C2216yes—
53C2×#879C2216yes—
54C2×#878C22112yes—
55C2×C42×D3C2213yes—
56C2×#869C2213yes—
57C2×C4×Dic6C2213yes—
58C2×C4×Dic6C2213yes—
59#3522—332no—
60C2×#559—33no—
61C2×Q82—31no—
62C2×#554—33no—
63C2×#550D311yescharacteristic, complemented
64C2×#548—32no—
65C2×#547—36no—
66C2×#542—36no—
67C2×#532—33no—
68C2×#529—33no—
69C2×#520—32no—
70C22×C4×Q8—31no—
71C2×C3×#41—216no—
72C2×C3×#41C23116yes—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4C2316yes—
75C22×C3×D4C2313yes—
76C22×C3×D4C2316yes—
77C23×C3×C4C2316yes—
78C22×#67C2316yes—
79C23×Dic3C23112yes—
80C2×D3×Q8—28no—
81C2×D3×Q8C2318yes—
82C2×#237—424no—
83C2×#237C23148yes—
84C22×C4×D3C2313yes—
85C22×C4×D3C2313yes—
86C22×C4×D3C2311yescharacteristic
87C22×Dic6C2313yes—
88C22×Dic6C2311yescharacteristic
89C22×Dic6C2316yes—
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×#217C23112yes—
99C2×#217C2316yes—
100C2×#217C2316yes—
101C2×#217C2316yes—
102Q8×Dic3C23116yes—
103Q8×Dic3C2318yes—
104#938C23124yes—
105#938C23148yes—
106#936C23124yes—
107#936C23148yes—
108#936C23148yes—
109#935C23196yes—
110D4×Dic3C23148yes—
111D4×Dic3C23148yes—
112D4×Dic3C23148yes—
113C4×#67C23148yes—
114C2×(C2×D3)⋊C4C2312yes—
115#926C23148yes—
116C2×C4⋊Dic3C2313yes—
117C2×C4⋊Dic3C2313yes—
118C2×C4⋊Dic3C2316yes—
119C2×C4⋊Dic3C2316yes—
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C23148yes—
122#924C23148yes—
123C2×Dic3⋊C4C2316yes—
124C2×Dic3⋊C4C2316yes—
125C2×Dic3⋊C4C2316yes—
126C2×C4×Dic3C2313yes—
127C2×C4×Dic3C2316yes—
128C2×C4×Dic3C2313yes—
129C2×C4×Dic3C2313yes—
130#897C2318yes—
131D3×C4⋊C4C23124yes—
132#890C23124yes—
133#888C23148yes—
134#887C23124yes—
135#883C23148yes—
136#879C23148yes—
137#878C23196yes—
138C42×D3C23124yes—
139#869C23124yes—
140C4×Dic6C23124yes—
141C4×Dic6C23124yes—
142C22×#41—36no—
143C22×#41C2×D312yes—
144C23×Q8—31no—
145#559—348no—
146Q82—316no—
147#554—348no—
148#550C2×D3116yescomplemented
149#548—332no—
150#547—396no—
151#542—396no—
152#532—348no—
153C2×C4⋊Q8—36no—
154C2×C4⋊Q8—33no—
155C2×#127C2×D313yes—
156#529—348no—
157C2×#126—36no—
158C2×#125—33no—
159C2×#124—33no—
160C2×#124—36no—
161C2×#124—36no—
162C2×#123—312no—
163C2×#123—36no—
164C2×C22⋊Q8—36no—
165C2×C22⋊Q8—31no—
166C2×C22⋊Q8—312no—
167C2×#121—33no—
168C2×#121—36no—
169C2×#121C2×D316yes—
170#520—332no—
171C2×C4×Q8—31no—
172C2×C4×Q8—32no—
173C2×C4×Q8—316no—
174C2×C4×Q8—33no—
175C2×C4×Q8—33no—
176C2×C4×Q8—33no—
177C2×C4×Q8C2×D311yescharacteristic
178C2×C4×D4—36no—
179C2×C4×D4—36no—
180C2×C4×D4—36no—
181C2×C4×D4—36no—
182C2×C4×D4C2×D313yes—
183C2×#117—36no—
184C22×C4⋊C4—36no—
185C22×C4⋊C4—33no—
186C22×C42—33no—
187C24×C3C2416yes—
188C23×D3C2411yescharacteristic
189C3×#41—264no—
190C2×C3×Q8#4118yescomplemented
191C2×C3×Q8C2414yes—
192C2×C3×D4—224no—
193C2×C3×D4—212no—
194C2×C3×D4—412no—
195C2×C3×D4C24112yes—
196C2×C3×D4C24124yes—
197C2×C3×D4C24124yes—
198C22×C3×C4—224no—
199C22×C3×C4C2413yes—
200C22×C3×C4C2412yes—
201C22×C3×C4C24124yes—
202C22×C3×C4C2413yes—
203C22×C3×C4C2411yescharacteristic
204C2×#67—224no—
205C2×#67C24124yes—
206C22×Dic3—424no—
207C22×Dic3C2411yescharacteristic
208C22×Dic3C2412yes—
209C22×Dic3C2413yes—
210C22×Dic3C2413yes—
211C22×Dic3C2416yes—
212C22×Dic3C24148yes—
213D3×Q8—232no—
214#237—496no—
215C2×C4×D3—212no—
216C2×C4×D3—212no—
217C2×C4×D3—24no—
218C2×C4×D3C2414yes—
219C2×C4×D3C24112yes—
220C2×C4×D3C24112yes—
221C2×Dic6—212no—
222C2×Dic6—24no—
223C2×Dic6—412no—
224C2×Dic6C2414yes—
225C2×Dic6C24112yes—
226C2×Dic6C24124yes—
227C3×C4⋊C4C24112yes—
228C3×C4⋊C4C2414yes—
229C3×C22⋊C4C24124yes—
230C3×C42C24112yes—
231#217C24148yes—
232#217C24124yes—
233#217C24124yes—
234#217C24124yes—
235(C2×D3)⋊C4C2418yes—
236C4⋊Dic3C24112yes—
237C4⋊Dic3C24112yes—
238C4⋊Dic3C24124yes—
239C4⋊Dic3C24124yes—
240C4⋊Dic3C2418yes—
241Dic3⋊C4C24124yes—
242Dic3⋊C4C24124yes—
243Dic3⋊C4C24124yes—
244C4×Dic3C24112yes—
245C4×Dic3C24124yes—
246C4×Dic3C24112yes—
247C4×Dic3C24112yes—
248C2×#41—216no—
249C2×#41—348no—
250C2×#41—1224no—
251C2×#41C22×D3116yes—
252C22×Q8—38no—
253C22×Q8—31no—
254C22×Q8—33no—
255C22×Q8—36no—
256C22×Q8—68no—
257C22×Q8C22×D311yescharacteristic
258C22×D4—36no—
259C22×D4C22×D313yes—
260C22×D4C22×D316yes—
261C22×D4C22×D316yes—
262C23×C4—312no—
263C23×C4—33no—
264C23×C4—33no—
265C23×C4—31no—
266C23×C4C22×D316yes—
267C4⋊Q8—348no—
268C4⋊Q8—324no—
269#127C22×D3124yes—
270#126—348no—
271#125—324no—
272#124—324no—
273#124—348no—
274#124—348no—
275#123—396no—
276#123—348no—
277C22⋊Q8—348no—
278C22⋊Q8—38no—
279C22⋊Q8—396no—
280#121—324no—
281#121—348no—
282#121C22×D3148yes—
283C4×Q8—38no—
284C4×Q8—316no—
285C4×Q8—324no—
286C4×Q8—324no—
287C4×Q8—324no—
288C4×Q8C22×D318yes—
289C4×D4—348no—
290C4×D4—348no—
291C4×D4—348no—
292C4×D4—348no—
293C4×D4C22×D3124yes—
294#117—348no—
295C2×C4⋊C4—33no—
296C2×C4⋊C4—36no—
297C2×C4⋊C4—36no—
298C2×C4⋊C4—36no—
299C2×C4⋊C4—348no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—33no—
302C2×C4⋊C4—36no—
303C2×C4⋊C4—36no—
304C2×C4⋊C4—324no—
305C2×C4⋊C4C22×D313yes—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—36no—
308C2×C22⋊C4—36no—
309C2×C22⋊C4—312no—
310C2×C22⋊C4—36no—
311C2×C22⋊C4—32no—
312C2×C22⋊C4C22×D316yes—
313C2×C42—33no—
314C2×C42—36no—
315C2×C42—33no—
316C2×C42—33no—
317C2×C42—324no—
318C2×C42C22×D313yes—
319C23×C3—212no—
320C23×C3—212no—
321C23×C3—212no—
322C23×C3C25112yes—
323C23×C3C2511yescharacteristic
324C22×D3—22no—
325C22×D3—22no—
326C22×D3H(5,𝔽2)14yes—
327C22×D3C2512yes—
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×C42−1+414yes—
337C2×C3×C4C2516yes—
338C2×C3×C4C2514yes—
339C2×C3×C4C2516yes—
340C2×C3×C4C2512yes—
341#67—248no—
342C2×Dic3—26no—
343C2×Dic3—212no—
344C2×Dic3—26no—
345C2×Dic3—24no—
346C2×Dic3—22no—
347C2×Dic3—448no—
348C2×Dic3C2512yes—
349C2×Dic3C2514yes—
350C2×Dic3C2516yes—
351C2×Dic3C2516yes—
352C2×Dic3C25112yes—
353C4×D3—224no—
354C4×D3—224no—
355C4×D3—28no—
356Dic6—224no—
357Dic6—28no—
358Dic6—424no—
359C24—31no—
360C24C23×D316yes—
361#41—264no—
362#41—1296no—
363C2×Q8—34no—
364C2×Q8—312no—
365C2×Q8—324no—
366C2×Q8—632no—
367C2×Q8—612no—
368C2×Q8—64no—
369C2×Q8—1212no—
370C2×Q8#23718yescomplemented
371C2×Q8C23×D314yes—
372C2×D4—224no—
373C2×D4—212no—
374C2×D4—324no—
375C2×D4—412no—
376C2×D4—624no—
377C2×D4C23×D3112yes—
378C2×D4C23×D3124yes—
379C2×D4C23×D3124yes—
380C22×C4—224no—
381C22×C4—31no—
382C22×C4—32no—
383C22×C4—33no—
384C22×C4—33no—
385C22×C4—36no—
386C22×C4—348no—
387C22×C4—34no—
388C22×C4—312no—
389C22×C4—312no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—64no—
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—324no—
403C4⋊C4—38no—
404C4⋊C4—312no—
405C4⋊C4—324no—
406C4⋊C4—324no—
407C4⋊C4C23×D3112yes—
408C4⋊C4C23×D314yes—
409C22⋊C4—324no—
410C22⋊C4—324no—
411C22⋊C4—348no—
412C22⋊C4—324no—
413C22⋊C4—38no—
414C22⋊C4C23×D3124yes—
415C42—312no—
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×C3C2×2−1+412yes—
427C22×C3C2611yescharacteristic
428C2×D3—22no—
429C2×D3—22no—
430C2×D3—28no—
431C2×D3C2×H(5,𝔽2)14yes—
432C3×C4—24no—
433C3×C4—26no—
434C3×C4C22×#41112yes—
435C3×C4C2×2−1+414yes—
436Dic3—26no—
437Dic3—212no—
438Dic3—26no—
439Dic3—24no—
440Dic3—22no—
441C23—212no—
442C23—212no—
443C23—212no—
444C23—34no—
445C23—32no—
446C23—62no—
447C23—62no—
448C23C24×D3112yes—
449C23C24×D311yescharacteristic
450Q8—624no—
451Q8—68no—
452Q8—1224no—
453Q8C2×#237116yescomplemented
454D4—248no—
455D4—224no—
456D4—424no—
457D4—648no—
458C2×C4—24no—
459C2×C4—26no—
460C2×C4—248no—
461C2×C4—32no—
462C2×C4—34no—
463C2×C4—36no—
464C2×C4—36no—
465C2×C4—312no—
466C2×C4—624no—
467C2×C4—66no—
468C2×C4—62no—
469C2×C4—64no—
470C2×C4—66no—
471C2×C4—612no—
472C2×C4—624no—
473C2×C4—68no—
474C2×C4—1248no—
475C2×C4C2×#237112yes—
476C2×C4#101014yes—
477C2×C4C24×D316yes—
478C2×C4C24×D314yes—
479C2×C4C24×D316yes—
480C2×C4C24×D312yes—
481C2×C3—412no—
482C2×C3#352214yescomplemented
483C2×C3C23×#4111yescharacteristic
484C2×C3C22×H(5,𝔽2)11yescharacteristic
485C2×C3C22×2−1+411yescharacteristic
486D3—24no—
487C22—212no—
488C22—212no—
489C22—212no—
490C22—34no—
491C22—424no—
492C22—62no—
493C22—62no—
494C22—68no—
495C22C22×#23712yes—
496C22C2×#101012yes—
497C22D3×H(5,𝔽2)12yes—
498C22C25×D311yescharacteristic
499C4—24no—
500C4—26no—
501C4—66no—
502C4—62no—
503C4—64no—
504C4—66no—
505C4—612no—
506C4C22×#237112yes—
507C4C2×#101014yes—
508C3C2×#352211yescharacteristic, complemented, Sylow
509C2—412no—
510C2—64no—
511C2#8677614yesdirect product factor
512C2C23×#23711yescharacteristic
513C2C22×#101011yescharacteristic
514C2C2×D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×#550C2
socleC23×C3C25
hypercenterC2×#550D3
hypocenterC3C2×#3522
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×#5831C2semidirect
3. C2×#5827C2semidirect
4. #86776C2direct (N × H)
5. C2×#5713C2semidirect
6. C2×#5716C2semidirect
7. C2×#5611C2semidirect
8. C2×#5603C2semidirect
9. C2×C4×D3×Q8C2semidirect
10. C2×Q8×Dic6C2semidirect
11. C2×#5557C2semidirect
12. C2×#5543C2semidirect
17. C3×#550C22semidirect
22. #5831C22semidirect
23. #5827C22semidirect
39. #5716C22semidirect
40. #5713C22semidirect
41. #5611C22semidirect
42. #5603C22semidirect
43. C4×D3×Q8C22semidirect
44. Q8×Dic6C22semidirect
45. #5557C22semidirect
46. #5543C22semidirect
63. C2×#550D3semidirect
148. #550C2×D3semidirect
190. C2×C3×Q8#41semidirect
328. C3×Q8C2×#41semidirect
370. C2×Q8#237semidirect
453. Q8C2×#237semidirect
482. C2×C3#3522semidirect
508. C3C2×#3522semidirect
511. C2#86776direct (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×#58312
C2×#58272
#867762
C2×#57132
C2×#57162
C2×#56112
C2×#56032
C2×C4×D3×Q82
C2×Q8×Dic62
C2×#55572
C2×#55432
C2×#35223