Finite Group Databasebeta

← C2×#64088⋊C3 #13152358

Subgroups — C2×#64088⋊C3 #13152358

Order 1728.

993 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×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3757⋊C3C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×#3750⋊C3C211yesmaximal, characteristic, complemented
6C2×#3750⋊C3C211yesmaximal, characteristic, complemented
7C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#64088⋊C3C218yesmaximal, direct product factor
9C2×#64088C311yesmaximal, characteristic, complemented
10C2×C3×#879—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22⋊C4×H(3,𝔽3)C2214yes—
19#3757⋊C3C2214yes—
20#3757⋊C3C2214yes—
21C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
22#3750⋊C3C2214yescomplemented
23#3750⋊C3C2214yescomplemented
24C4×(C3⋊Dic3)⋊C3C2214yescomplemented
25C2×C32×C22⋊C4—21no—
26C2×C32×C22⋊C4D311yescharacteristic
27C2×C32×C22⋊C4C2×C311yescharacteristic
28C2×#3757C2×C311yescharacteristic
29C2×#3757C2×C311yescharacteristic
30C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
31C2×#3750C2×C311yescharacteristic, complemented
32C2×#3750C2×C311yescharacteristic, complemented
33C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
34#64088C2×C318yescomplemented
35C2×C3×#217—31no—
36C2×C3×#217—31no—
37C2×C3×C4⋊Dic3—31no—
38C2×C3×Dic3⋊C4—31no—
39C2×C3×Dic3⋊C4—31no—
40C2×C3×C4×Dic3—31no—
41C3×#879—38no—
42C23×H(3,𝔽3)—22no—
43C23×H(3,𝔽3)—22no—
44C23×H(3,𝔽3)—22no—
45C23×H(3,𝔽3)C2312yes—
46C23×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)—22no—
48C2×C4×H(3,𝔽3)—22no—
49C2×C4×H(3,𝔽3)C2312yes—
50C2×C4×H(3,𝔽3)C2312yes—
51C2×(C3⋊Dic3)⋊C3—22no—
52C2×(C3⋊Dic3)⋊C3—22no—
53C2×(C3⋊Dic3)⋊C3—22no—
54C2×(C3⋊Dic3)⋊C3—22no—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C2×(C3⋊Dic3)⋊C3C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×C3×#126—91no—
60C2×#879—31no—
61C2×#879—91no—
62C24×C32—21no—
63C24×C32C2×D311yescharacteristic
64C24×C32C22×C311yescharacteristic
65C22×C32×C4—21no—
66C22×C32×C4—21no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C22×C3⋊Dic3C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C311yescharacteristic
74C22×C3⋊Dic3C22×C311yescharacteristic
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C32×C22⋊C4—24no—
80C32×C22⋊C4C2×D314yes—
81C32×C22⋊C4C22×C314yes—
82#3757C22×C314yes—
83#3757C22×C314yes—
84C4⋊(C3⋊Dic3)C22×C314yescomplemented
85#3750C22×C314yescomplemented
86#3750C22×C314yescomplemented
87C4×C3⋊Dic3C22×C314yescomplemented
88C3×#217—34no—
89C3×#217—34no—
90C3×C4⋊Dic3—34no—
91C3×Dic3⋊C4—34no—
92C3×Dic3⋊C4—34no—
93C3×C4×Dic3—34no—
94C22×H(3,𝔽3)—22no—
95C22×H(3,𝔽3)—22no—
96C22×H(3,𝔽3)—22no—
97C22×H(3,𝔽3)—44no—
98C22×H(3,𝔽3)#4112yes—
99C22×H(3,𝔽3)#4112yes—
100C22×H(3,𝔽3)#4112yes—
101C22×H(3,𝔽3)C2411yescharacteristic
102C4×H(3,𝔽3)—22no—
103C4×H(3,𝔽3)—22no—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—22no—
107(C3⋊Dic3)⋊C3—22no—
108C3×#126—98no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C4⋊C4—91no—
112C2×C3×C22⋊C4—31no—
113C2×C3×C22⋊C4—31no—
114C2×C3×C22⋊C4—61no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×C3×C22⋊C4C3×D311yescharacteristic
118C2×C3×C42—91no—
119C2×#217—31no—
120C2×#217—31no—
121C2×#217—91no—
122C2×#217—91no—
123C2×C4⋊Dic3—31no—
124C2×C4⋊Dic3—91no—
125C2×Dic3⋊C4—31no—
126C2×Dic3⋊C4—31no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4—91no—
129C2×C4×Dic3—31no—
130C2×C4×Dic3—91no—
131#879—38no—
132#879—98no—
133C23×C32—24no—
134C23×C32—22no—
135C23×C32—22no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—21no—
141C23×C32—22no—
142C23×C32—42no—
143C23×C32—42no—
144C23×C32C22×D312yes—
145C23×C32C22×D311yescharacteristic
146C23×C32C23×C312yes—
147C23×C32C23×C311yescharacteristic
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—42no—
155C2×C32×C4—42no—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C23×C312yes—
159C2×C32×C4C23×C312yes—
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3—22no—
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3—22no—
164C2×C3⋊Dic3C23×C312yes—
165C2×C3⋊Dic3C23×C312yes—
166C2×C3⋊Dic3C23×C312yes—
167C2×C3⋊Dic3C23×C312yes—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×#126—91noSylow
177C2×H(3,𝔽3)—42no—
178C2×H(3,𝔽3)#12614yescomplemented
179C2×H(3,𝔽3)C2×#4111yescharacteristic
180C2×H(3,𝔽3)C2×#4111yescharacteristic
181C2×H(3,𝔽3)C2×#4111yescharacteristic
182C24×C3—31no—
183C24×C3—31no—
184C24×C3—61no—
185C24×C3C2×C3×D311yescharacteristic
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—61no—
191C22×C3×C4—61no—
192C22×C3×C4—91no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C22×C3×C4—91no—
196C22×C3×C4C2×C3×D311yescharacteristic
197C22×C3×C4C2×C3×D311yescharacteristic
198C22×Dic3—31no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—91no—
203C22×Dic3—91no—
204C22×Dic3—91no—
205C22×Dic3—91no—
206C3×C4⋊C4—94no—
207C3×C4⋊C4—94no—
208C3×C4⋊C4—94no—
209C3×C22⋊C4—34no—
210C3×C22⋊C4—34no—
211C3×C22⋊C4—64no—
212C3×C22⋊C4—94no—
213C3×C22⋊C4—94no—
214C3×C22⋊C4C2×C3×D314yes—
215C3×C42—94no—
216#217—34no—
217#217—34no—
218#217—94no—
219#217—94no—
220C4⋊Dic3—34no—
221C4⋊Dic3—94no—
222Dic3⋊C4—34no—
223Dic3⋊C4—34no—
224Dic3⋊C4—94no—
225Dic3⋊C4—94no—
226C4×Dic3—34no—
227C4×Dic3—94no—
228C22×C32—24no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—21no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—42no—
240C22×C32—42no—
241C22×C32—48no—
242C22×C32—44no—
243C22×C32—44no—
244C22×C32#23512yes—
245C22×C32#23712yes—
246C22×C32#23712yes—
247C22×C32C3×#4112yes—
248C22×C32C3×#4112yes—
249C22×C32C3×#4112yes—
250C22×C32C23×D311yescharacteristic
251C22×C32C24×C311yescharacteristic
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—42no—
257C32×C4—42no—
258C3⋊Dic3—22no—
259C3⋊Dic3—22no—
260C3⋊Dic3—22no—
261C3⋊Dic3—22no—
262C3×Dic3—62no—
263C3×Dic3—62no—
264C3×Dic3—62no—
265C3×Dic3—62no—
266#126—98no—
267C2×C4⋊C4—91no—
268C2×C4⋊C4—91no—
269C2×C4⋊C4—91no—
270C2×C22⋊C4—91no—
271C2×C22⋊C4—91no—
272C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
273C2×C42—91no—
274H(3,𝔽3)C2×#12611yescharacteristic, complemented, Sylow
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—32no—
279C23×C3—32no—
280C23×C3—31no—
281C23×C3—31no—
282C23×C3—62no—
283C23×C3—62no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—62no—
288C23×C3—64no—
289C23×C3—62no—
290C23×C3—61no—
291C23×C3—122no—
292C23×C3—122no—
293C23×C3C22×C3×D312yes—
294C23×C3C22×C3×D311yescharacteristic
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—32no—
300C2×C3×C4—32no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—92no—
308C2×C3×C4—92no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—122no—
312C2×C3×C4—122no—
313C2×C3×C4—182no—
314C2×C3×C4—182no—
315C2×C3×C4—182no—
316C2×C3×C4—182no—
317C2×C3×C4C22×C3×D312yes—
318C2×C3×C4C22×C3×D312yes—
319C2×Dic3—32no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—62no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—92no—
330C2×Dic3—92no—
331C2×Dic3—182no—
332C2×Dic3—182no—
333C2×Dic3—182no—
334C2×Dic3—182no—
335C2×C32—24no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—44no—
340C2×C32—42no—
341C2×C32—42no—
342C2×C32#87914yescomplemented
343C2×C32C3×#12614yescomplemented
344C2×C32C2×#23511yescharacteristic
345C2×C32C2×#23711yescharacteristic
346C2×C32C2×#23711yescharacteristic
347C2×C32C2×C3×#4111yescharacteristic
348C2×C32C2×C3×#4111yescharacteristic
349C2×C32C2×C3×#4111yescharacteristic
350C24C2×(C3⋊D3)⋊C311yescharacteristic
351C22×C4—91no—
352C22×C4—91no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
356C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
357C4⋊C4—94no—
358C4⋊C4—94no—
359C4⋊C4—94no—
360C22⋊C4—94no—
361C22⋊C4—94no—
362C22⋊C4C2×(C3⋊D3)⋊C314yes—
363C42—94no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—32no—
370C22×C3—32no—
371C22×C3—32no—
372C22×C3—32no—
373C22×C3—32no—
374C22×C3—32no—
375C22×C3—44no—
376C22×C3—62no—
377C22×C3—62no—
378C22×C3—62no—
379C22×C3—62no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3—64no—
383C22×C3—61no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—62no—
387C22×C3—122no—
388C22×C3—122no—
389C22×C3—124no—
390C22×C3—124no—
391C22×C3—128no—
392C22×C3C3×#23512yes—
393C22×C3C3×#23712yes—
394C22×C3C3×#23712yes—
395C22×C3C23×C3×D311yescharacteristic
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—62no—
399C3×C4—62no—
400C3×C4—62no—
401C3×C4—62no—
402C3×C4—122no—
403C3×C4—122no—
404C3×C4—182no—
405C3×C4—182no—
406C3×C4—182no—
407C3×C4—182no—
408Dic3—62no—
409Dic3—62no—
410Dic3—62no—
411Dic3—62no—
412Dic3—182no—
413Dic3—182no—
414Dic3—182no—
415Dic3—182no—
416C32—21no—
417C32C2×#87911yescharacteristic, complemented
418C32C2×C3×#12611yescharacteristic, complemented
419C23—22no—
420C23—22no—
421C23—22no—
422C23C22×(C3⋊D3)⋊C312yes—
423C23C22×(C3⋊D3)⋊C311yescharacteristic
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—92no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—92no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—182no—
433C2×C4—182no—
434C2×C4C22×(C3⋊D3)⋊C312yes—
435C2×C4C22×(C3⋊D3)⋊C312yes—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—34no—
443C2×C3—34no—
444C2×C3—42no—
445C2×C3—61no—
446C2×C3—61no—
447C2×C3—61no—
448C2×C3—64no—
449C2×C3—122no—
450C2×C3—122no—
451C2×C3—124no—
452C2×C3C3×#87914yes—
453C2×C3C2×C3×#23511yescharacteristic
454C2×C3C2×C3×#23711yescharacteristic
455C2×C3C2×C3×#23711yescharacteristic
456C22—22no—
457C22—22no—
458C22—22no—
459C22—44no—
460C22#3828⋊C312yes—
461C22#3830⋊C312yes—
462C22#3830⋊C312yes—
463C22C23×(C3⋊D3)⋊C311yescharacteristic
464C4—22no—
465C4—22no—
466C4—182no—
467C4—182no—
468C4—182no—
469C4—182no—
470C3—31no—
471C3—31no—
472C3—61no—
473C3C2×C3×#87911yescharacteristic
474C2—42no—
475C2#64088⋊C314yesdirect product factor
476C2C2×#3828⋊C311yescharacteristic
477C2C2×#3830⋊C311yescharacteristic
478C2C2×#3830⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#126
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#3750⋊C3C2semidirect
6. C2×#3750⋊C3C2semidirect
7. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
8. #64088⋊C3C2direct (N × H)
9. C2×#64088C3semidirect
21. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
22. #3750⋊C3C22semidirect
23. #3750⋊C3C22semidirect
24. C4×(C3⋊Dic3)⋊C3C22semidirect
30. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
31. C2×#3750C2×C3semidirect
32. C2×#3750C2×C3semidirect
33. C2×C4×C3⋊Dic3C2×C3semidirect
34. #64088C2×C3semidirect
84. C4⋊(C3⋊Dic3)C22×C3semidirect
85. #3750C22×C3semidirect
86. #3750C22×C3semidirect
87. C4×C3⋊Dic3C22×C3semidirect
178. C2×H(3,𝔽3)#126semidirect
274. H(3,𝔽3)C2×#126semidirect
342. C2×C32#879semidirect
343. C2×C32C3×#126semidirect
417. C32C2×#879semidirect
418. C32C2×C3×#126semidirect
475. C2#64088⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#64088⋊C32
C2×#640883
C2×C3×#8793