Finite Group Databasebeta

← C2×#81753 #11664994

Subgroups — C2×#81753 #11664994

Order 768.

1970 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5731C211yesmaximal, characteristic, complemented
2C2×#5719C211yesmaximal, characteristic
3C2×C3×#492C211yesmaximal, characteristic, complemented
4C2×#5174C212yesmaximal, complemented
5C2×#5171C212yesmaximal, complemented
6C2×#5164C212yesmaximal, complemented
7C2×Dic3×SD16C212yesmaximal, complemented
8C2×#5082C211yesmaximal, characteristic, complemented
9C2×#5071C211yesmaximal, characteristic, complemented
10#81753C2116yesmaximal, direct product factor
11C2×#4689C211yesmaximal, characteristic, complemented
12C2×#4683C211yesmaximal, characteristic, complemented
13C2×#3174—31nomaximal, Sylow
14C22×C3×SD16C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127C2211yescharacteristic
18C2×#946C2212yes—
19C2×Q8×Dic3C2212yes—
20C2×#939—22no—
21C2×#938C2212yes—
22C2×#935—22no—
23C2×D4×Dic3C2212yes—
24C22×ΓL(1,𝔽25)C2211yescharacteristic
25C22×D3×C8C2211yescharacteristic
26#5731—24no—
27#5731C2218yescomplemented
28#5719—24no—
29#5719C2218yes—
30C2×#898—22no—
31C2×#889—22no—
32C2×#872C2211yescharacteristic
33C2×#870C2211yescharacteristic
34C2×#870C2211yescharacteristic
35C3×#492C2218yescomplemented
36C2×C3×C4×C8C2211yescharacteristic
37C2×#835C2212yes—
38C2×#835C2212yes—
39C2×#832C2212yes—
40C2×#832C2212yes—
41#5174C22116yescomplemented
42#5171C22116yescomplemented
43#5164C22116yescomplemented
44Dic3×SD16C22116yescomplemented
45C2×#818C2212yes—
46C2×#817C2212yes—
47C2×#815C2211yescharacteristic
48C2×C8×Dic3C2211yescharacteristic
49#5082C2218yescomplemented
50#5071C2218yescomplemented
51C2×#803C2211yescharacteristic
52#4689C2218yescomplemented
53#4683C2218yescomplemented
54C2×#541—31no—
55C2×#540—31no—
56#3174—316no—
57C2×#499—31no—
58C2×#492D311yescharacteristic, complemented
59C2×#491—31no—
60C2×#490—31no—
61C2×#468—32no—
62C2×#465—32no—
63C2×#440—32no—
64C2×C4×SD16—32no—
65C2×#405—31no—
66C22×C3×Q8C2312yes—
67C22×C3×D4—22no—
68C22×C3×D4C2312yes—
69C22×#67—22no—
70C23×Dic3—22no—
71C22×C4×D3C2311yescharacteristic
72C2×C3×SD16—28no—
73C2×C3×SD16—28no—
74C2×C3×SD16C2318yes—
75C2×C3×SD16C2318yes—
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8C2314yes—
80C3×#127—22no—
81C3×#127C2314yes—
82C2×C3×C4⋊C4—22no—
83C2×C3×C42C2311yescharacteristic
84C2×#217—22no—
85C2×#217—22no—
86#946—24no—
87#946C2318yes—
88Q8×Dic3—24no—
89Q8×Dic3C2318yes—
90#939—216no—
91#938—24no—
92#938C2318yes—
93#935—216no—
94D4×Dic3—24no—
95D4×Dic3C2318yes—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4D412yes—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2312yes—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4D412yes—
105C2×C4×Dic3—22no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×ΓL(1,𝔽25)—24no—
110C2×ΓL(1,𝔽25)C2314yes—
111C2×D3×C8—24no—
112C2×D3×C8C2314yes—
113#898—216no—
114#889—216no—
115#872D414yes—
116#872C2314yes—
117#870—22no—
118#870—22no—
119#870C2314yes—
120#870C2314yes—
121C3×C4×C8C2314yes—
122#835C2318yes—
123#835C2318yes—
124#832C2318yes—
125#832C2318yes—
126#818C2318yes—
127#817C2318yes—
128#815C2314yes—
129C8×Dic3C2314yes—
130#803C2314yes—
131C22×SD16C2×D312yes—
132C22×SD16C2×D312yes—
133C22×M16—31no—
134C23×C8—31no—
135#541—38no—
136#541—64no—
137#540—38no—
138#540—64no—
139C2×C4⋊Q8C2×D311yescharacteristic
140C2×#127C2×D311yescharacteristic
141C2×#126—62no—
142C2×#125—31no—
143C2×#125—31no—
144C2×#125—62no—
145C2×#123—62no—
146C2×C22⋊Q8—32no—
147C2×#121—32no—
148C2×#121—62no—
149C2×C4×Q8—32no—
150C2×C4×D4—32no—
151C2×#117—31no—
152#499—38no—
153#492C2×D318yescomplemented
154#491—38no—
155#490—38no—
156#468—316no—
157#465—316no—
158#440—316no—
159C4×SD16—316no—
160C2×#107—32no—
161C2×#106—32no—
162C2×#103—32no—
163C2×#103—32no—
164C2×#102—32no—
165C2×#102—32no—
166#405—38no—
167C2×#97—31no—
168C2×#97—31no—
169C2×C4×C8—31no—
170C2×C4×C8C2×D311yescharacteristic
171C24×C3—22no—
172C23×D3C2×D411yescharacteristic
173C2×C3×Q8—22no—
174C2×C3×Q8—24no—
175C2×C3×Q8—24no—
176C2×C3×Q8C2414yes—
177C2×C3×D4—22no—
178C2×C3×D4—28no—
179C2×C3×D4—24no—
180C2×C3×D4—24no—
181C2×C3×D4—84no—
182C2×C3×D4C2414yes—
183C22×C3×C4—22no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2411yescharacteristic
186C2×#67—28no—
187C2×#67—84no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—28no—
191C22×Dic3—84no—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×D411yescharacteristic
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3C2×D412yes—
197C2×C4×D3C2412yes—
198C3×SD16—216no—
199C3×SD16—216no—
200C2×C3×C8#4114yes—
201C2×C3×C8#4114yes—
202C2×C3×C8C2412yes—
203C2×C3×C8C2412yes—
204C3×C4⋊C4—28no—
205C3×C42C2×D412yes—
206C3×C42C2412yes—
207#217—28no—
208#217—28no—
209(C2×D3)⋊C4—28no—
210(C2×D3)⋊C4C2×D418yes—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—28no—
214C4⋊Dic3C2414yes—
215C4⋊Dic3C2414yes—
216Dic3⋊C4—28no—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219Dic3⋊C4C2×D418yes—
220C4×Dic3—28no—
221C4×Dic3C2×D412yes—
222C4×Dic3C2412yes—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8C2412yes—
226C2×C3⋊C8C2412yes—
227ΓL(1,𝔽25)—28no—
228D3×C8—28no—
229C22×Q8C22×D312yes—
230C22×D4—22no—
231C22×D4—62no—
232C22×D4C22×D312yes—
233C23×C4—31no—
234C23×C4—62no—
235C2×SD16—28no—
236C2×SD16—28no—
237C2×SD16C22×D318yes—
238C2×SD16C22×D318yes—
239C2×M16—34no—
240C2×M16—64no—
241C22×C8—31no—
242C22×C8—31no—
243C22×C8—34no—
244C22×C8—64no—
245C22×C8C22×D311yescharacteristic
246C22×C8C22×D311yescharacteristic
247C4⋊Q8—22no—
248C4⋊Q8C22×D314yes—
249#127—22no—
250#127C22×D314yes—
251#126—616no—
252#125—34no—
253#125—34no—
254#125—616no—
255#125—62no—
256#125—62no—
257#123—616no—
258C22⋊Q8—38no—
259C22⋊Q8—64no—
260#121—38no—
261#121—616no—
262#121—64no—
263C4×Q8—38no—
264C4×Q8—64no—
265C4×D4—38no—
266C4×D4—64no—
267#117—34no—
268#117—34no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—62no—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C42—31no—
282C2×C42—62no—
283C2×C42C22×D311yescharacteristic
284#107—38no—
285#106—38no—
286#103—38no—
287#103—38no—
288#102—38no—
289#102—38no—
290#97—34no—
291#97—34no—
292C4×C8—34no—
293C4×C8C22×D314yes—
294C23×C3—24no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3C22×D411yescharacteristic
299C22×D3—22no—
300C22×D3—24no—
301C22×D3C22×D412yes—
302C3×Q8—24no—
303C3×Q8—24no—
304C3×Q8—44no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—44no—
308C3×D4—88no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—44no—
312C2×C3×C4C22×D414yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4112yes—
316C2×C3×C4C2511yescharacteristic
317#67—88no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—44no—
323C2×Dic3—44no—
324C2×Dic3—88no—
325C2×Dic3C22×D414yes—
326C2×Dic3C22×D412yes—
327C4×D3—24no—
328C4×D3—22no—
329C4×D3—22no—
330C3×C8C2×#4114yes—
331C3×C8C2×#4114yes—
332C3⋊C8—22no—
333C3⋊C8—22no—
334C24—22no—
335C24—31no—
336C2×Q8—22no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8C23×D314yes—
340C2×D4—22no—
341C2×D4—28no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—68no—
345C2×D4—84no—
346C2×D4—244no—
347C2×D4C23×D314yes—
348C22×C4—22no—
349C22×C4—32no—
350C22×C4—31no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—68no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—244no—
359C22×C4D3×D412yes—
360C22×C4C23×D311yescharacteristic
361SD16—216no—
362SD16—216no—
363M16—68no—
364C2×C8—32no—
365C2×C8—32no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—68no—
369C2×C8#23714yes—
370C2×C8#23714yes—
371C2×C8C23×D312yes—
372C2×C8C23×D312yes—
373C4⋊C4—28no—
374C4⋊C4—38no—
375C4⋊C4—34no—
376C4⋊C4—34no—
377C4⋊C4—68no—
378C4⋊C4—68no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—68no—
382C4⋊C4—68no—
383C22⋊C4—38no—
384C22⋊C4—68no—
385C22⋊C4—68no—
386C22⋊C4—68no—
387C42—32no—
388C42—32no—
389C42—68no—
390C42D3×D412yes—
391C42C23×D312yes—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—88no—
396C22×C3#53212yes—
397C22×C3#57714yes—
398C22×C3C23×D411yescharacteristic
399C2×D3—22no—
400C2×D3—24no—
401C2×D3—44no—
402C3×C4—24no—
403C3×C4—44no—
404C3×C4#53212yes—
405C3×C4C22×#4111yescharacteristic
406C3×C4C22×#4111yescharacteristic
407Dic3—22no—
408Dic3—24no—
409Dic3—44no—
410Dic3—44no—
411C23—24no—
412C23—32no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—62no—
417C23—64no—
418C23C2×D3×D411yescharacteristic
419Q8—24no—
420Q8—24no—
421Q8—44no—
422D4—24no—
423D4—24no—
424D4—44no—
425D4—88no—
426D4—248no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—34no—
430C2×C4—32no—
431C2×C4—44no—
432C2×C4—62no—
433C2×C4—64no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—64no—
439C2×C4—124no—
440C2×C4—124no—
441C2×C4—248no—
442C2×C4C2×D3×D414yes—
443C2×C4C2×D3×D411yescharacteristic
444C2×C4C2×#23712yes—
445C2×C4C2×#23712yes—
446C2×C4C24×D311yescharacteristic
447C8—62no—
448C8—62no—
449C8C2×#23714yes—
450C8C2×#23714yes—
451C2×C3—84no—
452C2×C3#317414yescomplemented
453C2×C3C2×#53211yescharacteristic
454C2×C3C2×#57712yes—
455D3—42no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—64no—
460C22—62no—
461C22—88no—
462C22—124no—
463C22#571612yes—
464C22#576214yes—
465C22C22×D3×D411yescharacteristic
466C4—24no—
467C4—44no—
468C4—64no—
469C4—62no—
470C4—124no—
471C4—124no—
472C4#571612yes—
473C4C22×#23711yescharacteristic
474C4C22×#23711yescharacteristic
475C3C2×#317411yescharacteristic, complemented, Sylow
476C2—84no—
477C2—122no—
478C2#8175314yesdirect product factor
479C2C2×#571611yescharacteristic
480C2C2×#576212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#492C2
socleC23×C3C22×D4
prime radicalC2×#5719C2
hypercenterC2×#492D3
hypocenterC3C2×#3174
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
3. C2×C3×#492C2semidirect
4. C2×#5174C2semidirect
5. C2×#5171C2semidirect
6. C2×#5164C2semidirect
7. C2×Dic3×SD16C2semidirect
8. C2×#5082C2semidirect
9. C2×#5071C2semidirect
10. #81753C2direct (N × H)
11. C2×#4689C2semidirect
12. C2×#4683C2semidirect
27. #5731C22semidirect
35. C3×#492C22semidirect
41. #5174C22semidirect
42. #5171C22semidirect
43. #5164C22semidirect
44. Dic3×SD16C22semidirect
49. #5082C22semidirect
50. #5071C22semidirect
52. #4689C22semidirect
53. #4683C22semidirect
58. C2×#492D3semidirect
153. #492C2×D3semidirect
452. C2×C3#3174semidirect
475. C3C2×#3174semidirect
478. C2#81753direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57312
C2×#57192
C2×C3×#4922
C2×#51742
C2×#51712
C2×#51642
C2×Dic3×SD162
C2×#50822
C2×#50712
#817532
C2×#46892
C2×#46832
C2×#31743