Finite Group Databasebeta

← C2×#78922 #11664466

Subgroups — C2×#78922 #11664466

Order 768.

2246 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#936C211yesmaximal, characteristic
2C2×#5517C211yesmaximal, characteristic, complemented
3C2×C3×#418C211yesmaximal, characteristic
4C2×#5215C211yesmaximal, characteristic
5C2×#5120C211yesmaximal, characteristic, complemented
6C2×Dic3×M16C211yesmaximal, characteristic, complemented
7C2×#5004C211yesmaximal, characteristic, complemented
8#78922C2116yesmaximal, direct product factor
9C2×#4765C214yesmaximal, complemented
10C2×#4768C214yesmaximal, complemented
11C2×#3140—31nomaximal, Sylow
12C23×C3×D4C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#217—22no—
17C2×#936—22no—
18C2×#936—24no—
19C2×#936C2218yes—
20C2×#936C2212yes—
21C2×#936C2212yes—
22C2×#926C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×#208C2211yescharacteristic
25#5517—24no—
26#5517C2218yescomplemented
27C2×#890C2212yescomplemented
28C2×#888C2212yescomplemented
29C2×#878—22no—
30C2×C4×Dic6—22no—
31C3×#418C2218yes—
32C2×C3×#102C2214yes—
33#5215C2218yes—
34C2×#832C2214yes—
35#5120C2218yescomplemented
36Dic3×M16C2218yescomplemented
37C2×#816C2214yescomplemented
38C2×#815C2212yescomplemented
39C2×C8×Dic3C2212yescomplemented
40#5004C2218yescomplemented
41C2×#810C2214yescomplemented
42#4768C22132yescomplemented
43#4765C22132yescomplemented
44C2×#533—31no—
45C22×#124—31no—
46#3140—316no—
47C2×#489—34no—
48C2×#488—34no—
49C2×#419—31no—
50C2×#419—31no—
51C2×#418—31no—
52C2×#418D311yescharacteristic
53C2×C4×M16—31no—
54C25×C3—21no—
55C22×C3×D4—21no—
56C22×C3×D4—22no—
57C22×C3×D4—24no—
58C22×C3×D4—24no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2314yes—
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—21no—
63C23×Dic3D412yes—
64C22×Dic6—24no—
65C22×Dic6—24no—
66C22×Dic6—21no—
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6C2314yes—
70C22×Dic6C2312yes—
71C2×C3×M16—24no—
72C2×C3×M16C2314yes—
73C22×C3×C8C2312yes—
74C3×#117—22no—
75C3×#117C2314yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×#217—21no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—216no—
83#936—216no—
84#936—24no—
85#936—24no—
86#936C2318yes—
87#936C2318yes—
88#926—22no—
89#926C2314yes—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3D412yes—
96C2×C4×Dic3D414yes—
97C2×C4×Dic3C2311yescharacteristic
98C2×C4×Dic3C2311yescharacteristic
99C2×C4×Dic3C2314yes—
100C2×#208—24no—
101C2×#208C2314yes—
102C22×C3⋊C8C2312yes—
103#890—24no—
104#890C2318yescomplemented
105#888—24no—
106#888C2318yescomplemented
107#878—216no—
108C4×Dic6—216no—
109C3×#102C23116yes—
110#832C23116yes—
111#816C23116yescomplemented
112#815C2318yescomplemented
113C8×Dic3C2318yescomplemented
114#810C23116yescomplemented
115C23×Q8—31no—
116C23×D4C2×D311yescharacteristic
117C22×M16—31no—
118C22×M16C2×D311yescharacteristic
119#533—38no—
120#533—64no—
121C2×C4⋊Q8—32no—
122C2×#125—32no—
123C2×#124—32no—
124C2×#124—32no—
125C2×#124—38no—
126C2×#124—62no—
127C2×#124—64no—
128C2×C22⋊Q8—62no—
129C2×C4×Q8—62no—
130C2×#117—31no—
131C2×#117C2×D311yescharacteristic
132C22×C22⋊C4—62no—
133C22×C42—31no—
134#489—332no—
135#488—332no—
136#419—38no—
137#419—38no—
138#418—38no—
139#418C2×D318yes—
140C2×#103—34no—
141C2×#103—34no—
142C2×#102—34no—
143C2×#102C2×D314yes—
144C4×M16—38no—
145C2×#97—32no—
146C2×C4×C8—32no—
147C24×C3—24no—
148C24×C3—22no—
149C24×C3—48no—
150C24×C3—44no—
151C24×C3C2×D411yescharacteristic
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C2×C3×D4—24no—
155C2×C3×D4—28no—
156C2×C3×D4—24no—
157C2×C3×D4—28no—
158C2×C3×D4—48no—
159C2×C3×D4—44no—
160C2×C3×D4C2414yes—
161C22×C3×C4—22no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4#4114yes—
165C22×C3×C4#4114yes—
166C22×C3×C4C2411yescharacteristic
167C22×C3×C4C2412yes—
168C22×Dic3—22no—
169C22×Dic3—24no—
170C22×Dic3—22no—
171C22×Dic3—44no—
172C22×Dic3—44no—
173C22×Dic3C2×D412yes—
174C22×Dic3C2×D412yes—
175C22×Dic3C2×D418yes—
176C2×Dic6—24no—
177C2×Dic6—28no—
178C2×Dic6—24no—
179C2×Dic6—28no—
180C2×Dic6—24no—
181C2×Dic6—22no—
182C2×Dic6—28no—
183C2×Dic6—48no—
184C2×Dic6—48no—
185C2×Dic6—44no—
186C2×Dic6C2414yes—
187C3×M16—28no—
188C2×C3×C8—24no—
189C2×C3×C8C2414yes—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2414yes—
192C3×C22⋊C4—24no—
193C3×C42—24no—
194#217—24no—
195#217—216no—
196#217—216no—
197C4⋊Dic3—22no—
198C4⋊Dic3—28no—
199C4⋊Dic3C2414yes—
200Dic3⋊C4—28no—
201Dic3⋊C4—28no—
202C4×Dic3—24no—
203C4×Dic3C2×D418yes—
204C4×Dic3C2×D412yes—
205C4×Dic3C2×D412yes—
206C4×Dic3C2412yes—
207C4×Dic3C2412yes—
208#208—28no—
209C2×C3⋊C8—24no—
210C2×C3⋊C8C2414yes—
211C25—21no—
212C22×Q8—32no—
213C22×Q8—34no—
214C22×Q8—62no—
215C22×Q8—62no—
216C22×Q8—61no—
217C22×Q8—64no—
218C22×Q8—64no—
219C22×D4—24no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—24no—
223C22×D4C22×D312yes—
224C22×D4C22×D314yes—
225C23×C4—32no—
226C23×C4—61no—
227C23×C4C22×D311yescharacteristic
228C2×M16—24no—
229C2×M16—34no—
230C2×M16—64no—
231C2×M16C22×D314yes—
232C22×C8—32no—
233C22×C8C22×D312yes—
234C4⋊Q8—38no—
235C4⋊Q8—64no—
236#125—38no—
237#125—64no—
238#124—38no—
239#124—38no—
240#124—616no—
241#124—64no—
242#124—64no—
243C22⋊Q8—616no—
244C4×Q8—616no—
245#117—22no—
246#117—34no—
247#117—62no—
248#117C22×D314yes—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4C22×D312yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—616no—
258C2×C22⋊C4—61no—
259C2×C42—21no—
260C2×C42—32no—
261C2×C42—34no—
262C2×C42—31no—
263C2×C42—31no—
264C2×C42—34no—
265C2×C42—61no—
266#103—316no—
267#103—316no—
268#102—316no—
269#102C22×D3116yes—
270#97—38no—
271C4×C8—38no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—416no—
275C23×C3—48no—
276C23×C3—44no—
277C23×C3—48no—
278C23×C3—816no—
279C23×C3#12414yes—
280C23×C3C22×D412yes—
281C23×C3C22×D411yescharacteristic
282C3×D4—24no—
283C3×D4—24no—
284C3×D4—48no—
285C3×D4—44no—
286C2×C3×C4—24no—
287C2×C3×C4—44no—
288C2×C3×C4#12418yes—
289C2×C3×C4#12414yes—
290C2×C3×C4C22×D412yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4112yes—
293C2×C3×C4C2×#4114yes—
294C2×C3×C4C2×#4112yes—
295C2×C3×C4C2×#4114yes—
296C2×C3×C4C2511yescharacteristic
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—48no—
302C2×Dic3—48no—
303C2×Dic3—44no—
304C2×Dic3—44no—
305C2×Dic3C22×D414yes—
306C2×Dic3C22×D414yes—
307Dic6—24no—
308Dic6—24no—
309Dic6—416no—
310Dic6—48no—
311Dic6—44no—
312C3×C8—24no—
313C3⋊C8—24no—
314C24—24no—
315C24—22no—
316C24—48no—
317C24—44no—
318C24D3×D411yescharacteristic
319C2×Q8—34no—
320C2×Q8—68no—
321C2×Q8—64no—
322C2×Q8—62no—
323C2×Q8—64no—
324C2×Q8—68no—
325C2×Q8—64no—
326C2×Q8—68no—
327C2×Q8—128no—
328C2×Q8—128no—
329C2×Q8—124no—
330C2×D4—24no—
331C2×D4—28no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—28no—
336C2×D4—48no—
337C2×D4—44no—
338C2×D4C23×D314yes—
339C22×C4—22no—
340C22×C4—32no—
341C22×C4—32no—
342C22×C4—38no—
343C22×C4—64no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—124no—
347C22×C4—124no—
348C22×C4#23514yes—
349C22×C4D3×D412yes—
350C22×C4D3×D411yescharacteristic
351C22×C4#23714yes—
352C22×C4C23×D312yes—
353C22×C4C23×D311yescharacteristic
354M16—28no—
355M16—68no—
356C2×C8—24no—
357C2×C8—34no—
358C2×C8—64no—
359C2×C8C23×D314yes—
360C4⋊C4—22no—
361C4⋊C4—34no—
362C4⋊C4—68no—
363C4⋊C4—68no—
364C4⋊C4—68no—
365C4⋊C4—62no—
366C4⋊C4C23×D314yes—
367C22⋊C4—24no—
368C22⋊C4—616no—
369C22⋊C4—616no—
370C22⋊C4—64no—
371C42—24no—
372C42—38no—
373C42—32no—
374C42—32no—
375C42—32no—
376C42—32no—
377C42—64no—
378C22×C3—28no—
379C22×C3—24no—
380C22×C3—44no—
381C22×C3—48no—
382C22×C3—816no—
383C22×C3—88no—
384C22×C3C2×#12414yes—
385C22×C3C2×#12412yes—
386C22×C3#57514yes—
387C22×C3C23×D411yescharacteristic
388C3×C4—44no—
389C3×C4C2×#12414yes—
390C3×C4C2×#12412yes—
391C3×C4C22×#4111yescharacteristic
392C3×C4C22×#4111yescharacteristic
393Dic3—24no—
394Dic3—24no—
395Dic3—44no—
396Dic3—44no—
397C23—24no—
398C23—24no—
399C23—416no—
400C23—48no—
401C23—44no—
402C23—48no—
403C23—816no—
404C23#88514yes—
405C23C2×D3×D412yes—
406C23C2×D3×D411yescharacteristic
407Q8—64no—
408Q8—64no—
409Q8—1216no—
410Q8—124no—
411Q8—128no—
412D4—24no—
413D4—24no—
414D4—44no—
415D4—48no—
416C2×C4—24no—
417C2×C4—34no—
418C2×C4—34no—
419C2×C4—44no—
420C2×C4—64no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—64no—
424C2×C4—128no—
425C2×C4—128no—
426C2×C4—124no—
427C2×C4—124no—
428C2×C4#88518yes—
429C2×C4#88514yes—
430C2×C4C2×#23512yes—
431C2×C4C2×#23514yes—
432C2×C4C2×D3×D412yes—
433C2×C4C2×D3×D411yescharacteristic
434C2×C4C2×#23712yes—
435C2×C4C2×#23714yes—
436C2×C4C24×D311yescharacteristic
437C8—24no—
438C8—64no—
439C2×C3—24no—
440C2×C3—84no—
441C2×C3#314014yescomplemented
442C2×C3C22×#12411yescharacteristic
443C2×C3C2×#57512yes—
444C22—28no—
445C22—24no—
446C22—48no—
447C22—44no—
448C22—816no—
449C22—88no—
450C22C2×#88514yes—
451C22C2×#88512yes—
452C22#577314yes—
453C22C22×D3×D411yescharacteristic
454C4—44no—
455C4—64no—
456C4—64no—
457C4—124no—
458C4—124no—
459C4C2×#88512yes—
460C4C2×#88514yes—
461C4C22×#23511yescharacteristic
462C4C22×#23711yescharacteristic
463C3C2×#314011yescharacteristic, complemented, Sylow
464C2—24no—
465C2—84no—
466C2#7892214yesdirect product factor
467C2C22×#88511yescharacteristic
468C2C2×#577312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#418C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenterC2×#418D3
hypocenterC3C2×#3140
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#5517C2semidirect
5. C2×#5120C2semidirect
6. C2×Dic3×M16C2semidirect
7. C2×#5004C2semidirect
8. #78922C2direct (N × H)
9. C2×#4765C2semidirect
10. C2×#4768C2semidirect
26. #5517C22semidirect
27. C2×#890C22semidirect
28. C2×#888C22semidirect
35. #5120C22semidirect
36. Dic3×M16C22semidirect
37. C2×#816C22semidirect
38. C2×#815C22semidirect
39. C2×C8×Dic3C22semidirect
40. #5004C22semidirect
41. C2×#810C22semidirect
42. #4768C22semidirect
43. #4765C22semidirect
104. #890C23semidirect
106. #888C23semidirect
111. #816C23semidirect
112. #815C23semidirect
113. C8×Dic3C23semidirect
114. #810C23semidirect
441. C2×C3#3140semidirect
463. C3C2×#3140semidirect
466. C2#78922direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9362
C2×#55172
C2×C3×#4182
C2×#52152
C2×#51202
C2×Dic3×M162
C2×#50042
#789222
C2×#47652
C2×#47682
C2×#31403