Finite Group Databasebeta

← C2×#80852 #11665136

Subgroups — C2×#80852 #11665136

Order 768.

2018 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5600C212yesmaximal, complemented
2C2×C3×#469C211yesmaximal, characteristic
3C2×#5155C211yesmaximal, characteristic, complemented
4C2×#5152C211yesmaximal, characteristic, complemented
5C2×#5137C211yesmaximal, characteristic, complemented
6C2×Dic3×M16C211yesmaximal, characteristic, complemented
7C2×#5042C212yesmaximal, complemented
8#80852C2116yesmaximal, direct product factor
9C2×#4858C211yesmaximal, characteristic, complemented
10C2×#4853C211yesmaximal, characteristic, complemented
11C2×#4768C212yesmaximal, complemented
12C2×#4757C212yesmaximal, complemented
13C2×#3183—31nomaximal, Sylow
14C22×C3×D8C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#121C2212yes—
17C2×#936C2212yes—
18C2×#935—22no—
19C2×D4×Dic3—22no—
20C2×D4×Dic3C2212yes—
21C22×#214C2212yes—
22C2×#924C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C22×Dic12C2211yescharacteristic
26#5600—28no—
27#5600C22116yescomplemented
28C2×#890C2212yes—
29C2×#887C2212yes—
30C2×#879—22no—
31C2×#877—22no—
32C3×#469C2218yes—
33C2×C3×#106C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×#832C2212yes—
36#5155C2218yescomplemented
37#5152C2218yescomplemented
38#5137C2218yescomplemented
39Dic3×M16C2218yescomplemented
40C2×#817C2211yescharacteristic
41C2×#816C2212yes—
42C2×#815C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46#5042C22116yescomplemented
47C2×#810C2212yes—
48C2×#807C2212yes—
49#4858C2218yescomplemented
50#4853C2218yescomplemented
51#4768C22116yescomplemented
52#4757C22116yescomplemented
53C2×#529—32no—
54#3183—316no—
55C2×#499—31no—
56C2×#495—31no—
57C2×#488—32no—
58C2×#470—31no—
59C2×#469D311yescharacteristic
60C2×#468—32no—
61C2×#442—31no—
62C2×#441—31no—
63C2×#440—32no—
64C2×C4×M16—31no—
65C22×C3×D4—22no—
66C22×C3×D4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3—22no—
69C23×Dic3D412yes—
70C22×Dic6C2312yes—
71C2×C3×D8—24no—
72C2×C3×D8C2314yes—
73C2×C3×M16—24no—
74C2×C3×M16C2314yes—
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C3×#121—24no—
78C3×#121C2318yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×#217—22no—
82C2×#217—22no—
83C2×#217—22no—
84C2×#217—22no—
85#936—24no—
86#936C2318yes—
87#935—216no—
88D4×Dic3—216no—
89D4×Dic3—24no—
90D4×Dic3C2318yes—
91C2×#214—28no—
92C2×#214C2318yes—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3C2312yes—
95#924—24no—
96#924C2318yes—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3D412yes—
102C2×C4×Dic3D414yes—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2314yes—
106C2×#208—24no—
107C2×#208C2314yes—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×Dic12—24no—
111C2×Dic12C2314yes—
112#890—24no—
113#890C2318yes—
114#887—24no—
115#887C2318yes—
116#879—216no—
117#877—216no—
118C3×#106C2314yes—
119C3×#102C2318yes—
120#832C2318yes—
121#817C2314yes—
122#816C2318yes—
123#815C2314yes—
124#815C2314yes—
125C8×Dic3C2314yes—
126C8×Dic3C2314yes—
127#810C2318yes—
128#807C2318yes—
129C22×Q16—31no—
130C22×SD16—32no—
131C22×D8C2×D311yescharacteristic
132C22×M16—31no—
133C22×M16C2×D311yescharacteristic
134#529—316no—
135#529—68no—
136C2×#126—62no—
137C2×#125—32no—
138C2×#124—32no—
139C2×#123—62no—
140C2×C22⋊Q8—32no—
141C2×#121C2×D312yes—
142C2×C4×Q8—32no—
143C2×C4×D4—32no—
144C2×C4×D4—62no—
145C2×#117—62no—
146C22×C42—31no—
147#499—38no—
148#495—38no—
149#488—316no—
150#470—38no—
151#469C2×D318yes—
152#468—316no—
153#442—38no—
154#441—38no—
155#440—316no—
156C2×#107—32no—
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×#103—32no—
160C2×#103—32no—
161C2×#102—32no—
162C2×#102C2×D312yes—
163C4×M16—38no—
164C2×#97—31no—
165C2×#97—31no—
166C2×C4×C8—31no—
167C2×C4×C8—31no—
168C24×C3—22no—
169C24×C3C2×D411yescharacteristic
170C2×C3×D4—28no—
171C2×C3×D4—22no—
172C2×C3×D4—24no—
173C2×C3×D4—24no—
174C2×C3×D4—48no—
175C2×C3×D4C2414yes—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2411yescharacteristic
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—28no—
186C22×Dic3—44no—
187C22×Dic3—48no—
188C22×Dic3C2×D412yes—
189C22×Dic3C2×D412yes—
190C22×Dic3C2×D418yes—
191C2×Dic6—24no—
192C2×Dic6—24no—
193C2×Dic6—22no—
194C2×Dic6C2414yes—
195C3×D8—28no—
196C3×M16—28no—
197C2×C3×C8—22no—
198C2×C3×C8#4114yes—
199C2×C3×C8C2412yes—
200C2×C3×C8C2412yes—
201C3×C4⋊C4—22no—
202C3×C4⋊C4C2414yes—
203C3×C22⋊C4—28no—
204#217—28no—
205#217—28no—
206#217—28no—
207#217—28no—
208#214—216no—
209C4⋊Dic3—22no—
210C4⋊Dic3—28no—
211C4⋊Dic3C2414yes—
212Dic3⋊C4—28no—
213Dic3⋊C4—28no—
214Dic3⋊C4—28no—
215C4×Dic3—28no—
216C4×Dic3C2×D418yes—
217C4×Dic3C2×D412yes—
218C4×Dic3C2×D412yes—
219C4×Dic3C2412yes—
220C4×Dic3C2412yes—
221#208—28no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8#4114yes—
224C2×C3⋊C8C2412yes—
225C2×C3⋊C8C2412yes—
226Dic12—28no—
227C22×Q8—32no—
228C22×D4—22no—
229C22×D4C22×D312yes—
230C23×C4—32no—
231C23×C4—62no—
232C23×C4C22×D311yescharacteristic
233C2×Q16—34no—
234C2×Q16—64no—
235C2×SD16—38no—
236C2×SD16—68no—
237C2×D8—24no—
238C2×D8C22×D314yes—
239C2×M16—24no—
240C2×M16—34no—
241C2×M16—64no—
242C2×M16C22×D314yes—
243C22×C8—31no—
244C22×C8—31no—
245C22×C8C22×D311yescharacteristic
246C22×C8C22×D311yescharacteristic
247#126—616no—
248#125—38no—
249#125—64no—
250#124—38no—
251#124—64no—
252#123—616no—
253C22⋊Q8—38no—
254C22⋊Q8—64no—
255#121—24no—
256#121C22×D318yes—
257C4×Q8—38no—
258C4×Q8—64no—
259C4×D4—38no—
260C4×D4—64no—
261C4×D4—616no—
262#117—616no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4C22×D312yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C42—32no—
275C2×C42—34no—
276C2×C42—31no—
277C2×C42—31no—
278C2×C42—34no—
279C2×C42—62no—
280#107—38no—
281#106—34no—
282#106C22×D314yes—
283#103—38no—
284#103—38no—
285#102—38no—
286#102C22×D318yes—
287#97—34no—
288#97—34no—
289C4×C8—34no—
290C4×C8—34no—
291C23×C3—22no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3C22×D412yes—
298C23×C3C22×D411yescharacteristic
299C3×D4—24no—
300C3×D4—24no—
301C3×D4—416no—
302C3×D4—44no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—44no—
309C2×C3×C4C22×D412yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4112yes—
312C2×C3×C4C2×#4112yes—
313C2×C3×C4C2511yescharacteristic
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—48no—
319C2×Dic3—416no—
320C2×Dic3—44no—
321C2×Dic3—44no—
322C2×Dic3C22×D414yes—
323C2×Dic3C22×D414yes—
324Dic6—24no—
325Dic6—24no—
326Dic6—44no—
327C3×C8—22no—
328C3×C8C2×#4114yes—
329C3⋊C8—22no—
330C3⋊C8C2×#4114yes—
331C24—22no—
332C24D3×D411yescharacteristic
333C2×Q8—34no—
334C2×Q8—62no—
335C2×Q8—64no—
336C2×Q8—64no—
337C2×D4—28no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—48no—
342C2×D4C23×D314yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—32no—
347C22×C4—32no—
348C22×C4—38no—
349C22×C4—68no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—124no—
353C22×C4—128no—
354C22×C4D3×D412yes—
355C22×C4D3×D411yescharacteristic
356C22×C4C23×D312yes—
357C22×C4C23×D311yescharacteristic
358Q16—68no—
359SD16—616no—
360D8—28no—
361M16—28no—
362M16—68no—
363C2×C8—22no—
364C2×C8—34no—
365C2×C8—32no—
366C2×C8—32no—
367C2×C8—62no—
368C2×C8#23714yes—
369C2×C8C23×D312yes—
370C2×C8C23×D312yes—
371C4⋊C4—22no—
372C4⋊C4—34no—
373C4⋊C4—68no—
374C4⋊C4—68no—
375C4⋊C4—68no—
376C4⋊C4—62no—
377C4⋊C4—68no—
378C4⋊C4C23×D314yes—
379C22⋊C4—28no—
380C22⋊C4—68no—
381C22⋊C4—68no—
382C22⋊C4—68no—
383C22⋊C4—68no—
384C42—38no—
385C42—32no—
386C42—32no—
387C42—32no—
388C42—32no—
389C42—68no—
390C22×C3—22no—
391C22×C3—24no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3—88no—
397C22×C3#53212yes—
398C22×C3#57514yes—
399C22×C3C23×D411yescharacteristic
400C3×C4—22no—
401C3×C4—44no—
402C3×C4#53212yes—
403C3×C4C22×#4111yescharacteristic
404C3×C4C22×#4111yescharacteristic
405Dic3—24no—
406Dic3—24no—
407Dic3—44no—
408Dic3—44no—
409C23—24no—
410C23—22no—
411C23—24no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23C2×D3×D412yes—
416C23C2×D3×D411yescharacteristic
417Q8—64no—
418Q8—64no—
419Q8—124no—
420D4—24no—
421D4—24no—
422D4—416no—
423D4—44no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—34no—
430C2×C4—34no—
431C2×C4—44no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—128no—
437C2×C4—1216no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4C2×D3×D412yes—
441C2×C4C2×D3×D411yescharacteristic
442C2×C4C2×#23712yes—
443C2×C4D3×#4112yes—
444C2×C4C24×D311yescharacteristic
445C8—22no—
446C8—34no—
447C8—62no—
448C8C2×#23714yes—
449C2×C3—42no—
450C2×C3—84no—
451C2×C3#318314yescomplemented
452C2×C3C2×#53211yescharacteristic
453C2×C3C2×#57512yes—
454C22—24no—
455C22—22no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—88no—
461C22#559612yes—
462C22#577314yes—
463C22C22×D3×D411yescharacteristic
464C4—22no—
465C4—44no—
466C4—64no—
467C4—64no—
468C4—124no—
469C4—124no—
470C4#559612yes—
471C4C22×#23711yescharacteristic
472C4C2×D3×#4111yescharacteristic
473C3C2×#318311yescharacteristic, complemented, Sylow
474C2—42no—
475C2—84no—
476C2#8085214yesdirect product factor
477C2C2×#559611yescharacteristic
478C2C2×#577312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC2×C3×#469C2
socleC23×C3C22×D4
hypercenterC2×#469D3
hypocenterC3C2×#3183
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. C2×#5600C2semidirect
3. C2×#5155C2semidirect
4. C2×#5152C2semidirect
5. C2×#5137C2semidirect
6. C2×Dic3×M16C2semidirect
7. C2×#5042C2semidirect
8. #80852C2direct (N × H)
9. C2×#4858C2semidirect
10. C2×#4853C2semidirect
11. C2×#4768C2semidirect
12. C2×#4757C2semidirect
27. #5600C22semidirect
36. #5155C22semidirect
37. #5152C22semidirect
38. #5137C22semidirect
39. Dic3×M16C22semidirect
46. #5042C22semidirect
49. #4858C22semidirect
50. #4853C22semidirect
51. #4768C22semidirect
52. #4757C22semidirect
451. C2×C3#3183semidirect
473. C3C2×#3183semidirect
476. C2#80852direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56002
C2×C3×#4692
C2×#51552
C2×#51522
C2×#51372
C2×Dic3×M162
C2×#50422
#808522
C2×#48582
C2×#48532
C2×#47682
C2×#47572
C2×#31833