Finite Group Databasebeta

← #10595975⋊C3 #13152436

Subgroups — #10595975⋊C3 #13152436

Order 1728.

1121 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C4×#638⋊C3C214yesmaximal, complemented
4#64135⋊C3C211yesmaximal, characteristic, complemented
5#64102⋊C3C212yesmaximal, complemented
6#64101⋊C3C212yesmaximal, complemented
7C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
8#64096⋊C3C212yesmaximal, complemented
9#10595975C311yesmaximal, characteristic, complemented
10C3×#5504—31nomaximal
11C22×C4×H(3,𝔽3)C2212yes—
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#638⋊C3C2212yes—
14#3828⋊C3C418yescomplemented
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2214yes—
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2212yes—
21#3757⋊C3C2212yes—
22#3752⋊C3C2214yes—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2214yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2214yes—
28C2×#3828C2×C311yescharacteristic
29C2×C3×#235—31no—
30C2×C32×C4⋊C4—21no—
31C2×C32×C4⋊C4D311yescharacteristic, complemented
32C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
33C4×#638C2×C314yescomplemented
34#64135C2×C311yescharacteristic, complemented
35#64102C2×C312yescomplemented
36#64101C2×C312yescomplemented
37C4⋊C4×C3⋊D3C2×C312yescomplemented
38#64096C2×C312yescomplemented
39C3×C4×#67—34no—
40C3×#926—31no—
41C3×#893—32no—
42C3×#892—32no—
43C3×D3×C4⋊C4—32no—
44C3×#887—32no—
45C23×H(3,𝔽3)C2311yescharacteristic
46C22×(C3⋊D3)⋊C3C2312yes—
47C2×C4×H(3,𝔽3)—42no—
48C2×C4×H(3,𝔽3)C2×C414yescomplemented
49C2×C4×H(3,𝔽3)C2314yes—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52#638⋊C3C2×C418yescomplemented
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
57C4×(C3⋊D3)⋊C3—44no—
58C4×(C3⋊D3)⋊C3C2×C418yescomplemented
59#634⋊C3C2×C414yescomplemented
60C3×#520—91no—
61#5504—31no—
62#5504—91no—
63C22×C32×C4—21no—
64C22×C32×C4—22no—
65C22×C32×C4C2×D312yes—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C312yes—
68C22×C32×C4C22×C311yescharacteristic
69C2×#638C22×C312yes—
70#3828C3×C418yescomplemented
71C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
72C2×C4×C3⋊D3C22×C314yes—
73C2×C4×C3⋊D3C22×C312yes—
74C2×#634C22×C311yescharacteristic
75C2×C3×#67—32no—
76C3×#235—38no—
77C2×C3×D12—31no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—34no—
80C2×C3×Dic6—31no—
81C32×C4⋊C4—22no—
82C32×C4⋊C4—22no—
83C32×C4⋊C4C2×D312yes—
84C32×C4⋊C4C2×D312yes—
85C32×C4⋊C4C22×C312yes—
86C32×C4⋊C4C22×C312yes—
87#3757C22×C312yes—
88#3752C22×C314yes—
89C4⋊(C3⋊Dic3)C22×C311yescharacteristic
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic
91#3750C22×C314yes—
92C4×C3⋊Dic3C22×C314yes—
93C4×C3⋊Dic3C22×C312yes—
94C3×#217—32no—
95C3×(C2×D3)⋊C4—34no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—31no—
98C3×Dic3⋊C4—34no—
99C3×C4×Dic3—34no—
100C3×C4×Dic3—32no—
101C22×H(3,𝔽3)—21no—
102C22×H(3,𝔽3)—21no—
103C22×H(3,𝔽3)C22×C412yes—
104C22×H(3,𝔽3)C2411yescharacteristic
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3C22×C414yes—
108C4×H(3,𝔽3)—24no—
109C4×H(3,𝔽3)C22×C412yes—
110C4×H(3,𝔽3)C22×C412yes—
111(C3⋊Dic3)⋊C3—22no—
112(C3⋊Dic3)⋊C3—22no—
113(C3⋊Dic3)⋊C3C22×C414yes—
114C2×C3×#41—91no—
115C2×#235—31no—
116C2×#235—91no—
117C3×C4×Q8—92no—
118C3×C4×D4—92no—
119C3×C4×D4—94no—
120C3×#117—92no—
121C3×#117—91no—
122C2×C3×C4⋊C4—31no—
123C2×C3×C4⋊C4—31no—
124C2×C3×C4⋊C4—61no—
125C2×C3×C4⋊C4—92no—
126C2×C3×C4⋊C4C3×D311yescharacteristic
127C4×#67—34no—
128C4×#67—94no—
129#926—31no—
130#926—91no—
131#893—32no—
132#893—92no—
133#892—32no—
134#892—92no—
135D3×C4⋊C4—32no—
136D3×C4⋊C4—92no—
137#887—32no—
138#887—92no—
139C23×C32—21no—
140C23×C32C22×D311yescharacteristic
141C23×C32C23×C311yescharacteristic
142C22×C3⋊D3C23×C312yes—
143C22×C3×D3—32no—
144C2×C32×C4—24no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—24no—
148C2×C32×C4—42no—
149C2×C32×C4—42no—
150C2×C32×C4—44no—
151C2×C32×C4C4×D314yescomplemented
152C2×C32×C4C2×C3×C414yescomplemented
153C2×C32×C4C22×D314yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C23×C314yes—
157C2×C32×C4C23×C311yescharacteristic
158C2×C32×C4C23×C311yescharacteristic
159#638C2×C3×C418yescomplemented
160C2×C3⋊Dic3C23×C312yes—
161C2×C3⋊Dic3C23×C312yes—
162C2×C3⋊Dic3C23×C312yes—
163C4⋊(C3⋊D3)C2×C3×C414yescomplemented
164C4×C3⋊D3—44no—
165C4×C3⋊D3C2×C3×C418yescomplemented
166#634C2×C3×C414yescomplemented
167C3×#67—38no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C3×D12—34no—
172C3×C4×D3—38no—
173C3×C4×D3—124no—
174C3×Dic6—34no—
175#520—91noSylow
176C2×H(3,𝔽3)—22no—
177C2×H(3,𝔽3)C23×C411yescharacteristic
178C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
179C2×H(3,𝔽3)2−1+411yescharacteristic
180(C3⋊D3)⋊C3—24no—
181C3×#41—98no—
182C2×C3×Q8—91no—
183C2×C3×D4—91no—
184C2×C3×D4—92no—
185C22×C3×C4—31no—
186C22×C3×C4—31no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—61no—
190C22×C3×C4—62no—
191C22×C3×C4—92no—
192C22×C3×C4—94no—
193C22×C3×C4C2×C3×D312yes—
194C22×C3×C4C2×C3×D311yescharacteristic
195C2×#67—32no—
196C2×#67—92no—
197#235—38no—
198#235—98no—
199C2×D12—31no—
200C2×D12—91no—
201C2×C4×D3—32no—
202C2×C4×D3—34no—
203C2×C4×D3—92no—
204C2×C4×D3—94no—
205C2×Dic6—31no—
206C2×Dic6—91no—
207C3×C4⋊C4—32no—
208C3×C4⋊C4—32no—
209C3×C4⋊C4—32no—
210C3×C4⋊C4—32no—
211C3×C4⋊C4—62no—
212C3×C4⋊C4—62no—
213C3×C4⋊C4—94no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4—91no—
216C3×C4⋊C4C2×C3×D312yes—
217C3×C4⋊C4C2×C3×D312yes—
218C3×C22⋊C4—94no—
219C3×C22⋊C4—92no—
220C3×C42—92no—
221C3×C42—94no—
222#217—32no—
223#217—92no—
224(C2×D3)⋊C4—34no—
225(C2×D3)⋊C4—94no—
226C4⋊Dic3—31no—
227C4⋊Dic3—31no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230Dic3⋊C4—34no—
231Dic3⋊C4—94no—
232C4×Dic3—34no—
233C4×Dic3—32no—
234C4×Dic3—94no—
235C4×Dic3—92no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32C2×C4×D312yes—
245C22×C32C22×C3×C412yes—
246C22×C32C23×D311yescharacteristic
247C22×C32C24×C311yescharacteristic
248C2×C3⋊D3—22no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3C22×C3×C414yes—
251C2×C3×D3—34no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C32×C4—24no—
255C32×C4—24no—
256C32×C4—22no—
257C32×C4—22no—
258C32×C4—44no—
259C32×C4C2×C4×D312yes—
260C32×C4C2×C4×D312yes—
261C32×C4C22×C3×C412yes—
262C32×C4C22×C3×C412yes—
263C3⋊Dic3—22no—
264C3⋊Dic3—22no—
265C3⋊Dic3C22×C3×C414yes—
266C3×Dic3—34no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C2×#41—91no—
270C4×Q8—92no—
271C4×D4—94no—
272C4×D4—92no—
273#117—91no—
274#117—92no—
275C2×C4⋊C4—92no—
276C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
277H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
278C23×C3—31no—
279C23×C3—31no—
280C23×C3—61no—
281C23×C3—92no—
282C23×C3C22×C3×D311yescharacteristic
283C22×D3—32no—
284C22×D3—92no—
285C3×Q8—94no—
286C3×D4—94no—
287C3×D4—98no—
288C2×C3×C4—34no—
289C2×C3×C4—34no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—34no—
295C2×C3×C4—34no—
296C2×C3×C4—42no—
297C2×C3×C4—64no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—64no—
301C2×C3×C4—98no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—122no—
306C2×C3×C4—122no—
307C2×C3×C4—124no—
308C2×C3×C4—364no—
309C2×C3×C4C3×C4×D314yes—
310C2×C3×C4C22×C3×D314yes—
311C2×C3×C4C22×C3×D311yescharacteristic
312C2×C3×C4C22×C3×D311yescharacteristic
313#67—38no—
314#67—98no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—92no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321D12—34no—
322D12—94no—
323C4×D3—38no—
324C4×D3—98no—
325C4×D3—124no—
326C4×D3—364no—
327Dic6—34no—
328Dic6—94no—
329C2×C32—24no—
330C2×C32—22no—
331C2×C32—22no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32C22×C4×D311yescharacteristic
336C2×C32#100411yescharacteristic
337C2×C32#100711yescharacteristic
338C2×C32C23×C3×C411yescharacteristic
339C2×C32C3×H(5,𝔽2)11yescharacteristic
340C2×C32C3×2−1+411yescharacteristic
341C3⋊D3—24no—
342C3×D3—64no—
343#41—98no—
344C2×Q8—91no—
345C2×D4—92no—
346C2×D4—91no—
347C22×C4—92no—
348C22×C4—94no—
349C22×C4C2×(C3⋊D3)⋊C312yes—
350C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
351C4⋊C4—91no—
352C4⋊C4—94no—
353C4⋊C4—91no—
354C4⋊C4C2×(C3⋊D3)⋊C312yes—
355C4⋊C4C2×(C3⋊D3)⋊C312yes—
356C22⋊C4—92no—
357C22⋊C4—94no—
358C42—94no—
359C42—92no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—32no—
365C22×C3—32no—
366C22×C3—61no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—62no—
371C22×C3—62no—
372C22×C3—61no—
373C22×C3—62no—
374C22×C3—94no—
375C22×C3—182no—
376C22×C3—182no—
377C22×C3C2×C3×C4×D312yes—
378C22×C3C23×C3×D311yescharacteristic
379C2×D3—34no—
380C2×D3—62no—
381C2×D3—62no—
382C2×D3—94no—
383C2×D3—182no—
384C2×D3—182no—
385C3×C4—24no—
386C3×C4—32no—
387C3×C4—32no—
388C3×C4—32no—
389C3×C4—32no—
390C3×C4—64no—
391C3×C4—64no—
392C3×C4—62no—
393C3×C4—62no—
394C3×C4—94no—
395C3×C4—124no—
396C3×C4—182no—
397C3×C4—182no—
398C3×C4C2×C3×C4×D312yes—
399C3×C4C2×C3×C4×D312yes—
400Dic3—34no—
401Dic3—62no—
402Dic3—62no—
403Dic3—94no—
404Dic3—182no—
405Dic3—182no—
406C32—21no—
407C32#550411yescharacteristic, complemented
408C32C3×#52011yescharacteristic, complemented
409C23—92no—
410C23C22×(C3⋊D3)⋊C311yescharacteristic
411Q8—94no—
412D4—98no—
413D4—94no—
414C2×C4—42no—
415C2×C4—92no—
416C2×C4—98no—
417C2×C4—92no—
418C2×C4—92no—
419C2×C4—364no—
420C2×C4C4×(C3⋊D3)⋊C314yescomplemented
421C2×C4C22×(C3⋊D3)⋊C314yes—
422C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
423C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
424C2×C3—22no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—62no—
432C2×C3—62no—
433C2×C3—64no—
434C2×C3—61no—
435C2×C3—61no—
436C2×C3—61no—
437C2×C3—184no—
438C2×C3C22×C3×C4×D311yescharacteristic
439C2×C3C3×#100411yescharacteristic
440C2×C3C3×#100711yescharacteristic
441D3—64no—
442D3—184no—
443C22—21no—
444C22—21no—
445C22—94no—
446C22—182no—
447C22—182no—
448C22C2×C4×(C3⋊D3)⋊C312yes—
449C22C23×(C3⋊D3)⋊C311yescharacteristic
450C4—24no—
451C4—94no—
452C4—182no—
453C4—182no—
454C4C2×C4×(C3⋊D3)⋊C312yes—
455C4C2×C4×(C3⋊D3)⋊C312yes—
456C3—31no—
457C3—31no—
458C3—61no—
459C3C3×#550411yescharacteristic
460C2—22no—
461C2—184no—
462C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
463C2#64361⋊C311yescharacteristic
464C2#1172262711yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
Frattini, socleC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC2×#3828⋊C3C2
cosocleC22×C32C24×C3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. C4×#638⋊C3C2semidirect
4. #64135⋊C3C2semidirect
5. #64102⋊C3C2semidirect
6. #64101⋊C3C2semidirect
7. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
8. #64096⋊C3C2semidirect
9. #10595975C3semidirect
14. #3828⋊C3C4semidirect
31. C2×C32×C4⋊C4D3semidirect
32. C2×C32×C4⋊C4C2×C3semidirect
33. C4×#638C2×C3semidirect
34. #64135C2×C3semidirect
35. #64102C2×C3semidirect
36. #64101C2×C3semidirect
37. C4⋊C4×C3⋊D3C2×C3semidirect
38. #64096C2×C3semidirect
48. C2×C4×H(3,𝔽3)C2×C4semidirect
52. #638⋊C3C2×C4semidirect
56. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
58. C4×(C3⋊D3)⋊C3C2×C4semidirect
59. #634⋊C3C2×C4semidirect
70. #3828C3×C4semidirect
151. C2×C32×C4C4×D3semidirect
152. C2×C32×C4C2×C3×C4semidirect
159. #638C2×C3×C4semidirect
163. C4⋊(C3⋊D3)C2×C3×C4semidirect
165. C4×C3⋊D3C2×C3×C4semidirect
166. #634C2×C3×C4semidirect
276. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
277. H(3,𝔽3)#520semidirect
407. C32#5504semidirect
408. C32C3×#520semidirect
420. C2×C4C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×C4⋊C4×H(3,𝔽3)2
C4×#638⋊C32
#64135⋊C32
#64102⋊C32
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64096⋊C32
#105959753
C3×#55043