Finite Group Databasebeta

← C3×D4×GO+(4,𝔽2) #13181036

Subgroups — C3×D4×GO+(4,𝔽2) #13181036

Order 1728.

1436 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×GO+(4,𝔽2)C212yesmaximal, complemented
2C3×D32×D4C212yesmaximal, complemented
3C3×D4×#157C211yesmaximal, characteristic, complemented
4C3×#64242C212yesmaximal, complemented
5C3×#64241C214yesmaximal, complemented
6C3×#64231C211yesmaximal, characteristic, complemented
7C3×#64230C212yesmaximal, complemented
8C3×C4×GO+(4,𝔽2)C211yesmaximal, characteristic, complemented
9D4×GO+(4,𝔽2)C311yesmaximal, characteristic, direct product factor
10C22×C3×D32C2214yescomplemented
11C22×C3×#157C2212yescomplemented
12C2×C3×GO+(4,𝔽2)—28no—
13C2×C3×GO+(4,𝔽2)C2211yescharacteristic
14C2×C3×GO+(4,𝔽2)C2214yescomplemented
15C2×C3×GO+(4,𝔽2)C2212yescomplemented
16C2×C3×GO+(4,𝔽2)C2212yescomplemented
17C3×D4×C3⋊D3C2211yescharacteristic
18C32×D3×D4—22no—
19C3×#3811C2214yescomplemented
20C3×D3×#67—24no—
21C3×#3802C2212yescomplemented
22C3×D3×D12—22no—
23C3×C4×D32C2212yescomplemented
24C3×C22⋊#157C2212yescomplemented
25C3×C4⋊#157C2211yescharacteristic, complemented
26C3×C4×#157C2211yescharacteristic, complemented
27C3×#3773C2211yescharacteristic, complemented
28C3×D32⋊C4C2212yescomplemented
29C3×D32⋊C4C2214yescomplemented
30C22×GO+(4,𝔽2)C2×C312yescomplemented
31C2×C3×D3×D4—62no—
32D32×D4C2×C312yescomplemented
33D4×#157C2×C311yescharacteristic, complemented
34#64242C2×C312yescomplemented
35#64241C2×C314yescomplemented
36#64231C2×C311yescharacteristic, complemented
37#64230C2×C312yescomplemented
38C4×GO+(4,𝔽2)C2×C311yescharacteristic, complemented
39C22×C3×C3⋊D3C2312yes—
40C22×C32×D3—24no—
41C2×C3×D32—28no—
42C2×C3×D32—24no—
43C2×C3×D32—44no—
44C2×C3×D32C2314yes—
45C2×C3×D32C2312yes—
46C2×C3×#157—24no—
47C2×C3×#157C2312yes—
48C2×C3×#157C2311yescharacteristic
49C2×C3×#157C2311yescharacteristic
50C3×GO+(4,𝔽2)—28no—
51C3×GO+(4,𝔽2)—24no—
52C3×GO+(4,𝔽2)—42no—
53C3×GO+(4,𝔽2)D414yesdirect product factor
54C33×D4D411yescharacteristic, complemented
55C3×#638D412yescomplemented
56C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
57C3×C4×C3⋊D3C2311yescharacteristic
58C32×#67—24no—
59C32×D12—22no—
60C32×C4×D3—22no—
61C3×#626—24no—
62C3×#625—22no—
63C3×#624C2312yes—
64C3×D3×Dic3—22no—
65C3×D42—91nomaximal
66C23×C3×D3—64no—
67C22×D32C22×C314yescomplemented
68C22×#157C22×C312yescomplemented
69C2×GO+(4,𝔽2)—28no—
70C2×GO+(4,𝔽2)C22×C311yescharacteristic
71C2×GO+(4,𝔽2)C22×C314yescomplemented
72C2×GO+(4,𝔽2)C22×C312yescomplemented
73C2×GO+(4,𝔽2)C22×C312yescomplemented
74C2×C32×D4—62no—
75D4×C3⋊D3C22×C311yescharacteristic
76C2×C3×#67—64no—
77C3×D3×D4—22no—
78C3×D3×D4—22no—
79C3×D3×D4—42no—
80C3×D3×D4—62no—
81C3×D3×D4—62no—
82C3×D3×D4—64no—
83C3×D3×D4—62no—
84C3×D3×D4—64no—
85C2×C3×D12—62no—
86C2×C3×C4×D3—62no—
87#3811C22×C314yescomplemented
88D3×#67—24no—
89#3802C22×C312yescomplemented
90D3×D12—22no—
91C4×D32C22×C312yescomplemented
92C22⋊#157C22×C312yescomplemented
93C4⋊#157C22×C311yescharacteristic, complemented
94C4×#157C22×C311yescharacteristic, complemented
95#3773C22×C311yescharacteristic, complemented
96D32⋊C4C22×C312yescomplemented
97D32⋊C4C22×C314yescomplemented
98C22×C33C2×D412yescomplemented
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3C2×D412yescomplemented
102C2×C3×C3⋊D3C2411yescharacteristic
103C2×C32×D3—22no—
104C2×C32×D3—24no—
105C2×C32×D3—48no—
106C3×D32—22no—
107C3×D32—24no—
108C3×D32—24no—
109C3×D32—48no—
110C3×D32C2×D414yescomplemented
111C3×#157—22no—
112C3×#157—21no—
113C3×#157C2×D412yescomplemented
114C33×C4C2×D411yescharacteristic, complemented
115C3×C3⋊Dic3C2×D411yescharacteristic, complemented
116C32×Dic3—22no—
117C22×C3×D4—92no—
118C22×C3×D4—92no—
119C2×D3×D4—62no—
120C3×#127—91no—
121C3×#121—92no—
122C3×#121—92no—
123C3×#120—94no—
124C3×C4×D4—91no—
125C3×C4×D4—91no—
126C23×C32—64no—
127C22×C3⋊D3C23×C312yes—
128C22×C3×D3—24no—
129C22×C3×D3—24no—
130C22×C3×D3—44no—
131C22×C3×D3—62no—
132C22×C3×D3—64no—
133C22×C3×D3—64no—
134C22×C3×D3—64no—
135C22×C3×D3—64no—
136C22×C3×D3—128no—
137C22×C3×D3—128no—
138C2×D32—24no—
139C2×D32—28no—
140C2×D32—44no—
141C2×D32C23×C314yes—
142C2×D32C23×C312yes—
143C2×#157—24no—
144C2×#157C23×C312yes—
145C2×#157C23×C311yescharacteristic
146C2×#157C23×C311yescharacteristic
147GO+(4,𝔽2)—28no—
148GO+(4,𝔽2)—24no—
149GO+(4,𝔽2)—42no—
150GO+(4,𝔽2)C3×D414yesdirect product factor
151C32×D4—22no—
152C32×D4—42no—
153C32×D4—62no—
154C32×D4—64no—
155C32×D4C3×D411yescharacteristic, complemented
156C2×C32×C4—62no—
157#638C3×D412yescomplemented
158C4⋊(C3⋊D3)C3×D411yescharacteristic, complemented
159C4×C3⋊D3C23×C311yescharacteristic
160C3×#67—24no—
161C3×#67—24no—
162C3×#67—44no—
163C3×#67—64no—
164C3×#67—64no—
165C3×#67—64no—
166C2×C3×Dic3—62no—
167C3×D12—22no—
168C3×D12—22no—
169C3×D12—42no—
170C3×D12—62no—
171C3×D12—64no—
172C3×C4×D3—22no—
173C3×C4×D3—22no—
174C3×C4×D3—42no—
175C3×C4×D3—62no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178#626—24no—
179#625—22no—
180#624C23×C312yes—
181D3×Dic3—22no—
182D42—91noSylow
183C2×C33—22no—
184C2×C33C22×D411yescharacteristic
185C3×C3⋊D3—22no—
186C3×C3⋊D3C22×D411yescharacteristic
187C3×C3⋊D3H(5,𝔽2)11yescharacteristic
188C32×D3—24no—
189C32×D3—44no—
190C24×C3—94no—
191C23×D3—64no—
192C2×C3×D4—62no—
193C2×C3×D4—62no—
194C2×C3×D4—94no—
195C2×C3×D4—92no—
196C2×C3×D4—91no—
197C2×C3×D4—91no—
198C2×C3×D4—94no—
199C2×C3×D4—92no—
200C2×C3×D4—92no—
201C2×C3×D4—122no—
202C2×C3×D4—182no—
203C2×C3×D4—184no—
204C2×C3×D4—188no—
205C22×C3×C4—92no—
206C22×C3×C4—92no—
207C2×#67—64no—
208D3×D4—22no—
209D3×D4—62no—
210D3×D4—64no—
211D3×D4—62no—
212D3×D4—62no—
213D3×D4—64no—
214C2×D12—62no—
215C2×C4×D3—62no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—91no—
218C3×C22⋊C4—92no—
219C3×C22⋊C4—94no—
220C3×C22⋊C4—92no—
221C3×C42—91no—
222C22×C32—24no—
223C22×C32—44no—
224C22×C32—62no—
225C22×C32—64no—
226C22×C32—128no—
227C22×C32C2×C3×D412yescomplemented
228C2×C3⋊D3—22no—
229C2×C3⋊D3—22no—
230C2×C3⋊D3C2×C3×D412yescomplemented
231C2×C3⋊D3C24×C311yescharacteristic
232C2×C3×D3—22no—
233C2×C3×D3—22no—
234C2×C3×D3—24no—
235C2×C3×D3—24no—
236C2×C3×D3—42no—
237C2×C3×D3—44no—
238C2×C3×D3—48no—
239C2×C3×D3—48no—
240C2×C3×D3—64no—
241C2×C3×D3—62no—
242C2×C3×D3—64no—
243C2×C3×D3—64no—
244C2×C3×D3—88no—
245C2×C3×D3—124no—
246C2×C3×D3—124no—
247C2×C3×D3—124no—
248C2×C3×D3—124no—
249C2×C3×D3—128no—
250C2×C3×D3—128no—
251D32—22no—
252D32—24no—
253D32—24no—
254D32—48no—
255D32C2×C3×D414yescomplemented
256#157—22no—
257#157—21no—
258#157C2×C3×D412yescomplemented
259C32×C4—22no—
260C32×C4—42no—
261C32×C4—62no—
262C32×C4C2×C3×D411yescharacteristic, complemented
263C3⋊Dic3C2×C3×D411yescharacteristic, complemented
264C3×Dic3—22no—
265C3×Dic3—22no—
266C3×Dic3—42no—
267C3×Dic3—62no—
268C22×D4—92no—
269C22×D4—92no—
270#127—91no—
271#121—92no—
272#121—92no—
273#120—94no—
274C4×D4—91no—
275C4×D4—91no—
276C33D4211yescharacteristic, complemented, Sylow
277C23×C3—64no—
278C23×C3—64no—
279C23×C3—92no—
280C23×C3—94no—
281C23×C3—92no—
282C23×C3—124no—
283C23×C3—184no—
284C23×C3—188no—
285C23×C3—364no—
286C22×D3—24no—
287C22×D3—62no—
288C22×D3—64no—
289C22×D3—64no—
290C22×D3—64no—
291C22×D3—64no—
292C22×D3—128no—
293C22×D3—128no—
294C3×D4—22no—
295C3×D4—42no—
296C3×D4—62no—
297C3×D4—64no—
298C3×D4—64no—
299C3×D4—62no—
300C3×D4—94no—
301C3×D4—92no—
302C3×D4—91no—
303C3×D4—122no—
304C3×D4—124no—
305C3×D4—184no—
306C3×D4—182no—
307C3×D4—184no—
308C3×D4—188no—
309C3×D4—362no—
310C3×D4GO+(4,𝔽2)11yescharacteristic, direct product factor
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—92no—
314C2×C3×C4—91no—
315C2×C3×C4—91no—
316C2×C3×C4—91no—
317C2×C3×C4—92no—
318C2×C3×C4—122no—
319C2×C3×C4—182no—
320C2×C3×C4—184no—
321#67—24no—
322#67—64no—
323#67—64no—
324#67—64no—
325C2×Dic3—62no—
326D12—22no—
327D12—62no—
328D12—64no—
329C4×D3—22no—
330C4×D3—62no—
331C4×D3—62no—
332C4×D3—62no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—42no—
336C2×C32—44no—
337C2×C32—64no—
338C2×C32—84no—
339C2×C32—124no—
340C2×C32C22×C3×D411yescharacteristic
341C3⋊D3—22no—
342C3⋊D3C22×C3×D411yescharacteristic
343C3⋊D3C3×H(5,𝔽2)11yescharacteristic
344C3×D3—24no—
345C3×D3—24no—
346C3×D3—44no—
347C3×D3—44no—
348C3×D3—44no—
349C3×D3—62no—
350C3×D3—62no—
351C3×D3—84no—
352C3×D3—124no—
353C24—94no—
354C2×D4—62no—
355C2×D4—94no—
356C2×D4—92no—
357C2×D4—91no—
358C2×D4—91no—
359C2×D4—94no—
360C2×D4—92no—
361C2×D4—92no—
362C2×D4—182no—
363C2×D4—184no—
364C2×D4—188no—
365C22×C4—92no—
366C22×C4—92no—
367C4⋊C4—91no—
368C4⋊C4—91no—
369C22⋊C4—92no—
370C22⋊C4—92no—
371C22⋊C4—94no—
372C42—91no—
373C22×C3—24no—
374C22×C3—44no—
375C22×C3—62no—
376C22×C3—64no—
377C22×C3—62no—
378C22×C3—64no—
379C22×C3—94no—
380C22×C3—91no—
381C22×C3—92no—
382C22×C3—122no—
383C22×C3—124no—
384C22×C3—128no—
385C22×C3—128no—
386C22×C3—182no—
387C22×C3—182no—
388C22×C3—182no—
389C22×C3—184no—
390C22×C3—184no—
391C22×C3—248no—
392C22×C3—368no—
393C22×C3C2×GO+(4,𝔽2)12yescomplemented
394C2×D3—22no—
395C2×D3—24no—
396C2×D3—48no—
397C2×D3—64no—
398C2×D3—62no—
399C2×D3—64no—
400C2×D3—64no—
401C2×D3—124no—
402C2×D3—124no—
403C2×D3—124no—
404C2×D3—124no—
405C2×D3—128no—
406C2×D3—128no—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4—62no—
410C3×C4—62no—
411C3×C4—92no—
412C3×C4—91no—
413C3×C4—122no—
414C3×C4—182no—
415C3×C4—181no—
416C3×C4C2×GO+(4,𝔽2)11yescharacteristic, complemented
417Dic3—22no—
418Dic3—62no—
419C32—22no—
420C32—42no—
421C32C3×D4211yescharacteristic, complemented
422C23—64no—
423C23—92no—
424C23—94no—
425C23—92no—
426C23—184no—
427C23—188no—
428C23—364no—
429D4—64no—
430D4—62no—
431D4—94no—
432D4—92no—
433D4—91no—
434D4—184no—
435D4—182no—
436D4—184no—
437D4—188no—
438D4—362no—
439D4C3×GO+(4,𝔽2)11yescharacteristic, direct product factor
440C2×C4—62no—
441C2×C4—92no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—92no—
446C2×C4—182no—
447C2×C4—184no—
448C2×C3—22no—
449C2×C3—22no—
450C2×C3—44no—
451C2×C3—42no—
452C2×C3—64no—
453C2×C3—64no—
454C2×C3—84no—
455C2×C3—91no—
456C2×C3—91no—
457C2×C3—124no—
458C2×C3—124no—
459C2×C3—124no—
460C2×C3—182no—
461C2×C3—244no—
462C2×C3C22×GO+(4,𝔽2)11yescharacteristic
463D3—24no—
464D3—44no—
465D3—62no—
466D3—62no—
467D3—124no—
468C22—62no—
469C22—64no—
470C22—94no—
471C22—91no—
472C22—92no—
473C22—128no—
474C22—182no—
475C22—182no—
476C22—182no—
477C22—184no—
478C22—184no—
479C22—368no—
480C22C2×C3×GO+(4,𝔽2)12yescomplemented
481C4—62no—
482C4—92no—
483C4—91no—
484C4—182no—
485C4—181no—
486C4C2×C3×GO+(4,𝔽2)11yescharacteristic, complemented
487C3—22no—
488C3—42no—
489C3D4×GO+(4,𝔽2)11yescharacteristic, direct product factor
490C2—22no—
491C2—64no—
492C2—91no—
493C2—91no—
494C2—124no—
495C2—182no—
496C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC33×D4D4
socleC2×C33C22×D4
hypercenterC3×D4GO+(4,𝔽2)
hypocenterC32C3×D42
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×GO+(4,𝔽2)C2semidirect
2. C3×D32×D4C2semidirect
3. C3×D4×#157C2semidirect
4. C3×#64242C2semidirect
5. C3×#64241C2semidirect
6. C3×#64231C2semidirect
7. C3×#64230C2semidirect
8. C3×C4×GO+(4,𝔽2)C2semidirect
9. D4×GO+(4,𝔽2)C3direct (N × H)
10. C22×C3×D32C22semidirect
11. C22×C3×#157C22semidirect
14. C2×C3×GO+(4,𝔽2)C22semidirect
15. C2×C3×GO+(4,𝔽2)C22semidirect
16. C2×C3×GO+(4,𝔽2)C22semidirect
19. C3×#3811C22semidirect
21. C3×#3802C22semidirect
23. C3×C4×D32C22semidirect
24. C3×C22⋊#157C22semidirect
25. C3×C4⋊#157C22semidirect
26. C3×C4×#157C22semidirect
27. C3×#3773C22semidirect
28. C3×D32⋊C4C22semidirect
29. C3×D32⋊C4C22semidirect
30. C22×GO+(4,𝔽2)C2×C3semidirect
32. D32×D4C2×C3semidirect
33. D4×#157C2×C3semidirect
34. #64242C2×C3semidirect
35. #64241C2×C3semidirect
36. #64231C2×C3semidirect
37. #64230C2×C3semidirect
38. C4×GO+(4,𝔽2)C2×C3semidirect
53. C3×GO+(4,𝔽2)D4direct (N × H)
54. C33×D4D4semidirect
55. C3×#638D4semidirect
56. C3×C4⋊(C3⋊D3)D4semidirect
67. C22×D32C22×C3semidirect
68. C22×#157C22×C3semidirect
71. C2×GO+(4,𝔽2)C22×C3semidirect
72. C2×GO+(4,𝔽2)C22×C3semidirect
73. C2×GO+(4,𝔽2)C22×C3semidirect
87. #3811C22×C3semidirect
89. #3802C22×C3semidirect
91. C4×D32C22×C3semidirect
92. C22⋊#157C22×C3semidirect
93. C4⋊#157C22×C3semidirect
94. C4×#157C22×C3semidirect
95. #3773C22×C3semidirect
96. D32⋊C4C22×C3semidirect
97. D32⋊C4C22×C3semidirect
98. C22×C33C2×D4semidirect
101. C2×C3×C3⋊D3C2×D4semidirect
110. C3×D32C2×D4semidirect
113. C3×#157C2×D4semidirect
114. C33×C4C2×D4semidirect
115. C3×C3⋊Dic3C2×D4semidirect
150. GO+(4,𝔽2)C3×D4direct (N × H)
155. C32×D4C3×D4semidirect
157. #638C3×D4semidirect
158. C4⋊(C3⋊D3)C3×D4semidirect
227. C22×C32C2×C3×D4semidirect
230. C2×C3⋊D3C2×C3×D4semidirect
255. D32C2×C3×D4semidirect
258. #157C2×C3×D4semidirect
262. C32×C4C2×C3×D4semidirect
263. C3⋊Dic3C2×C3×D4semidirect
276. C33D42semidirect
310. C3×D4GO+(4,𝔽2)direct (N × H)
393. C22×C3C2×GO+(4,𝔽2)semidirect
416. C3×C4C2×GO+(4,𝔽2)semidirect
421. C32C3×D42semidirect
439. D4C3×GO+(4,𝔽2)direct (N × H)
480. C22C2×C3×GO+(4,𝔽2)semidirect
486. C4C2×C3×GO+(4,𝔽2)semidirect
489. C3D4×GO+(4,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C3×D32×D42
C3×D4×#1572
C3×#642422
C3×#642412
C3×#642312
C3×#642302
C3×C4×GO+(4,𝔽2)2
D4×GO+(4,𝔽2)3
C3×D429