Finite Group Databasebeta

← #10595957⋊C3 #13152362

Subgroups — #10595957⋊C3 #13152362

Order 1728.

1315 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638⋊C3C211yesmaximal, characteristic
2C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4C4×#638⋊C3C214yesmaximal, complemented
5#64090⋊C3C214yesmaximal, complemented
6C22⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
7#64086⋊C3C212yesmaximal, complemented
8#10595957C311yesmaximal, characteristic, complemented
9C3×#5486—31nomaximal
10C24×H(3,𝔽3)C2211yescharacteristic
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2212yescomplemented
13C2×#638⋊C3C418yescomplemented
14C2×#638⋊C3C2212yes—
15C2×#638⋊C3C2212yes—
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2212yescomplemented
18C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
19C22⋊C4×H(3,𝔽3)C2212yes—
20C22⋊C4×H(3,𝔽3)C2212yescomplemented
21#3757⋊C3C2212yes—
22#3757⋊C3C2212yescomplemented
23#3752⋊C3C2214yescomplemented
24#3750⋊C3C2214yescomplemented
25C4×(C3⋊Dic3)⋊C3C2214yescomplemented
26C22×#638C2×C311yescharacteristic
27C22×C3×#67—31no—
28C2×C32×C22⋊C4—21no—
29C2×C32×C22⋊C4D311yescharacteristic, complemented
30C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
31C2×#3757C2×C311yescharacteristic, complemented
32C4×#638C2×C314yescomplemented
33#64090C2×C314yescomplemented
34C22⋊C4×C3⋊D3C2×C312yescomplemented
35#64086C2×C312yescomplemented
36C2×C3×#217—31no—
37C3×C4×#67—34no—
38C3×#67⋊C4—34no—
39C3×D3×C22⋊C4—32no—
40C3×#877—32no—
41C23×H(3,𝔽3)—42no—
42C23×H(3,𝔽3)C2×C414yescomplemented
43C23×H(3,𝔽3)C2311yescharacteristic
44C23×H(3,𝔽3)C2311yescharacteristic
45C23×H(3,𝔽3)C2311yescharacteristic
46C22×(C3⋊D3)⋊C3—42no—
47C22×(C3⋊D3)⋊C3C2×C414yescomplemented
48C22×(C3⋊D3)⋊C3C2312yes—
49C2×C4×H(3,𝔽3)—42no—
50C2×C4×H(3,𝔽3)C2314yes—
51#638⋊C3C2×C418yescomplemented
52#638⋊C3C2×C418yescomplemented
53C2×(C3⋊Dic3)⋊C3—42no—
54C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
55C2×(C3⋊Dic3)⋊C3C2314yes—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C4×(C3⋊D3)⋊C3—44no—
58C3×#518—91no—
59#5486—31no—
60#5486—91no—
61C24×C32—21no—
62C24×C32C2×D311yescharacteristic
63C24×C32C22×C311yescharacteristic
64C23×C3⋊D3C22×C311yescharacteristic
65C23×C3×D3—31no—
66C22×C32×C4—22no—
67C22×C32×C4C2×D312yescomplemented
68C22×C32×C4C22×C312yescomplemented
69C2×#638C3×C418yescomplemented
70C2×#638C22×C312yes—
71C2×#638C22×C312yes—
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C312yescomplemented
74C2×C4×C3⋊D3C22×C314yescomplemented
75C2×C3×#67—38no—
76C2×C3×#67—32no—
77C2×C3×#67—32no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—32no—
80C2×C3×C4×D3—34no—
81C32×C22⋊C4—22no—
82C32×C22⋊C4—22no—
83C32×C22⋊C4C2×D312yes—
84C32×C22⋊C4C2×D312yescomplemented
85C32×C22⋊C4C22×C312yes—
86C32×C22⋊C4C22×C312yescomplemented
87#3757C22×C312yes—
88#3757C22×C312yescomplemented
89#3752C22×C314yescomplemented
90#3750C22×C314yescomplemented
91C4×C3⋊Dic3C22×C314yescomplemented
92C3×#217—32no—
93C3×#217—32no—
94C3×(C2×D3)⋊C4—34no—
95C3×Dic3⋊C4—34no—
96C3×C4×Dic3—34no—
97C22×H(3,𝔽3)—22no—
98C22×H(3,𝔽3)—22no—
99C22×H(3,𝔽3)—22no—
100C22×H(3,𝔽3)—44no—
101C22×H(3,𝔽3)C22×C412yes—
102C22×H(3,𝔽3)C22×C412yes—
103C22×H(3,𝔽3)C22×C412yes—
104C22×H(3,𝔽3)C2411yescharacteristic
105C2×(C3⋊D3)⋊C3—24no—
106C2×(C3⋊D3)⋊C3—44no—
107C2×(C3⋊D3)⋊C3C22×C414yes—
108C4×H(3,𝔽3)—24no—
109(C3⋊Dic3)⋊C3—24no—
110(C3⋊Dic3)⋊C3C22×C414yes—
111C22×C3×D4—91no—
112C22×#67—31no—
113C22×#67—91no—
114C3×C4×D4—94no—
115C3×C4×D4—94no—
116C3×#117—92no—
117C2×C3×C22⋊C4—31no—
118C2×C3×C22⋊C4—31no—
119C2×C3×C22⋊C4—61no—
120C2×C3×C22⋊C4—92no—
121C2×C3×C22⋊C4—91no—
122C2×C3×C22⋊C4C3×D311yescharacteristic
123C2×#217—31no—
124C2×#217—91no—
125C4×#67—34no—
126C4×#67—94no—
127#67⋊C4—34no—
128#67⋊C4—94no—
129D3×C22⋊C4—32no—
130D3×C22⋊C4—92no—
131#877—32no—
132#877—92no—
133C23×C32—24no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—42no—
138C23×C32—42no—
139C23×C32—44no—
140C23×C32C4×D314yescomplemented
141C23×C32C2×C3×C414yescomplemented
142C23×C32C22×D311yescharacteristic
143C23×C32C22×D311yescharacteristic
144C23×C32C22×D311yescharacteristic
145C23×C32C23×C311yescharacteristic
146C23×C32C23×C311yescharacteristic
147C23×C32C23×C311yescharacteristic
148C22×C3⋊D3—42no—
149C22×C3⋊D3C2×C3×C414yescomplemented
150C22×C3⋊D3C23×C312yes—
151C22×C3×D3—32no—
152C22×C3×D3—34no—
153C22×C3×D3—122no—
154C2×C32×C4—24no—
155C2×C32×C4—42no—
156C2×C32×C4—42no—
157C2×C32×C4—44no—
158C2×C32×C4C22×D314yes—
159C2×C32×C4C23×C314yes—
160#638C2×C3×C418yescomplemented
161#638C2×C3×C418yescomplemented
162C2×C3⋊Dic3—42no—
163C2×C3⋊Dic3C2×C3×C414yescomplemented
164C2×C3⋊Dic3C23×C314yes—
165C2×C3⋊Dic3C23×C312yes—
166C4×C3⋊D3—44no—
167C3×#67—38no—
168C3×#67—38no—
169C2×C3×Dic3—34no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—34no—
172C2×C3×Dic3—122no—
173C3×C4×D3—124no—
174#518—91noSylow
175C2×H(3,𝔽3)—22no—
176C2×H(3,𝔽3)—22no—
177C2×H(3,𝔽3)—22no—
178C2×H(3,𝔽3)C23×C411yescharacteristic
179C2×H(3,𝔽3)H(5,𝔽2)12yes—
180(C3⋊D3)⋊C3—24no—
181C24×C3—31no—
182C24×C3—31no—
183C24×C3—61no—
184C24×C3—91no—
185C24×C3C2×C3×D311yescharacteristic
186C23×D3—31no—
187C23×D3—91no—
188C2×C3×D4—92no—
189C2×C3×D4—92no—
190C2×C3×D4—98no—
191C22×C3×C4—32no—
192C22×C3×C4—32no—
193C22×C3×C4—62no—
194C22×C3×C4—94no—
195C22×C3×C4—91no—
196C22×C3×C4—92no—
197C22×C3×C4C2×C3×D312yes—
198C2×#67—38no—
199C2×#67—32no—
200C2×#67—32no—
201C2×#67—92no—
202C2×#67—92no—
203C2×#67—98no—
204C22×Dic3—31no—
205C22×Dic3—32no—
206C22×Dic3—91no—
207C22×Dic3—92no—
208C2×C4×D3—34no—
209C2×C4×D3—94no—
210C3×C4⋊C4—94no—
211C3×C22⋊C4—32no—
212C3×C22⋊C4—32no—
213C3×C22⋊C4—32no—
214C3×C22⋊C4—32no—
215C3×C22⋊C4—62no—
216C3×C22⋊C4—62no—
217C3×C22⋊C4—94no—
218C3×C22⋊C4—92no—
219C3×C22⋊C4—92no—
220C3×C22⋊C4C2×C3×D312yes—
221C3×C22⋊C4C2×C3×D312yes—
222C3×C42—94no—
223#217—32no—
224#217—32no—
225#217—92no—
226#217—92no—
227(C2×D3)⋊C4—34no—
228(C2×D3)⋊C4—94no—
229Dic3⋊C4—34no—
230Dic3⋊C4—94no—
231C4×Dic3—34no—
232C4×Dic3—94no—
233C22×C32—24no—
234C22×C32—22no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—21no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—42no—
245C22×C32—42no—
246C22×C32—44no—
247C22×C32—44no—
248C22×C32—48no—
249C22×C32C2×C4×D312yes—
250C22×C32C2×C4×D312yes—
251C22×C32C2×C4×D312yes—
252C22×C32C22×C3×C412yes—
253C22×C32C22×C3×C412yes—
254C22×C32C22×C3×C412yes—
255C22×C32C23×D311yescharacteristic
256C22×C32C24×C311yescharacteristic
257C2×C3⋊D3—24no—
258C2×C3⋊D3—44no—
259C2×C3⋊D3C22×C3×C414yes—
260C2×C3×D3—34no—
261C2×C3×D3—64no—
262C2×C3×D3—124no—
263C32×C4—24no—
264C32×C4—24no—
265C32×C4—44no—
266C3⋊Dic3—24no—
267C3⋊Dic3C22×C3×C414yes—
268C3×Dic3—34no—
269C3×Dic3—64no—
270C22×D4—91no—
271C4×D4—94no—
272C4×D4—94no—
273#117—92no—
274C2×C22⋊C4—91no—
275C2×C22⋊C4—92no—
276C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
277H(3,𝔽3)#51811yescharacteristic, complemented, Sylow
278C23×C3—34no—
279C23×C3—34no—
280C23×C3—31no—
281C23×C3—31no—
282C23×C3—31no—
283C23×C3—31no—
284C23×C3—31no—
285C23×C3—31no—
286C23×C3—42no—
287C23×C3—64no—
288C23×C3—61no—
289C23×C3—61no—
290C23×C3—61no—
291C23×C3—92no—
292C23×C3—94no—
293C23×C3—122no—
294C23×C3—122no—
295C23×C3—124no—
296C23×C3—362no—
297C23×C3C3×C4×D314yes—
298C23×C3C22×C3×D311yescharacteristic
299C23×C3C22×C3×D311yescharacteristic
300C23×C3C22×C3×D311yescharacteristic
301C22×D3—32no—
302C22×D3—34no—
303C22×D3—92no—
304C22×D3—94no—
305C22×D3—122no—
306C22×D3—362no—
307C3×D4—98no—
308C3×D4—98no—
309C2×C3×C4—34no—
310C2×C3×C4—34no—
311C2×C3×C4—42no—
312C2×C3×C4—64no—
313C2×C3×C4—92no—
314C2×C3×C4—94no—
315C2×C3×C4—94no—
316C2×C3×C4—122no—
317C2×C3×C4—122no—
318C2×C3×C4—124no—
319C2×C3×C4—364no—
320C2×C3×C4—362no—
321C2×C3×C4C22×C3×D314yes—
322#67—38no—
323#67—38no—
324#67—98no—
325#67—98no—
326C2×Dic3—32no—
327C2×Dic3—34no—
328C2×Dic3—34no—
329C2×Dic3—94no—
330C2×Dic3—92no—
331C2×Dic3—94no—
332C2×Dic3—122no—
333C2×Dic3—362no—
334C4×D3—124no—
335C4×D3—364no—
336C2×C32—24no—
337C2×C32—22no—
338C2×C32—22no—
339C2×C32—22no—
340C2×C32—22no—
341C2×C32—22no—
342C2×C32—22no—
343C2×C32—21no—
344C2×C32—22no—
345C2×C32—42no—
346C2×C32—42no—
347C2×C32C22×C4×D311yescharacteristic
348C2×C32#100412yes—
349C2×C32C23×C3×C411yescharacteristic
350C2×C32C3×H(5,𝔽2)12yes—
351C3⋊D3—24no—
352C3×D3—64no—
353C24—91no—
354C24C2×(C3⋊D3)⋊C311yescharacteristic
355C2×D4—98no—
356C2×D4—92no—
357C2×D4—92no—
358C22×C4—91no—
359C22×C4—92no—
360C22×C4—94no—
361C22×C4C2×(C3⋊D3)⋊C312yescomplemented
362C4⋊C4—94no—
363C22⋊C4—92no—
364C22⋊C4—92no—
365C22⋊C4—94no—
366C22⋊C4C2×(C3⋊D3)⋊C312yes—
367C22⋊C4C2×(C3⋊D3)⋊C312yescomplemented
368C42—94no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—32no—
375C22×C3—32no—
376C22×C3—32no—
377C22×C3—32no—
378C22×C3—32no—
379C22×C3—32no—
380C22×C3—44no—
381C22×C3—62no—
382C22×C3—62no—
383C22×C3—62no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—62no—
387C22×C3—64no—
388C22×C3—61no—
389C22×C3—62no—
390C22×C3—62no—
391C22×C3—62no—
392C22×C3—94no—
393C22×C3—122no—
394C22×C3—122no—
395C22×C3—124no—
396C22×C3—124no—
397C22×C3—128no—
398C22×C3—184no—
399C22×C3—364no—
400C22×C3C2×C3×C4×D312yes—
401C22×C3C2×C3×C4×D312yes—
402C22×C3C2×C3×C4×D312yes—
403C22×C3C23×C3×D311yescharacteristic
404C2×D3—34no—
405C2×D3—64no—
406C2×D3—94no—
407C2×D3—124no—
408C2×D3—184no—
409C2×D3—364no—
410C3×C4—24no—
411C3×C4—64no—
412C3×C4—64no—
413C3×C4—94no—
414C3×C4—124no—
415C3×C4—184no—
416Dic3—34no—
417Dic3—64no—
418Dic3—94no—
419Dic3—184no—
420C32—21no—
421C32#548611yescharacteristic, complemented
422C32C3×#51811yescharacteristic, complemented
423C23—42no—
424C23—92no—
425C23—94no—
426C23—362no—
427C23C4×(C3⋊D3)⋊C314yescomplemented
428C23C22×(C3⋊D3)⋊C311yescharacteristic
429C23C22×(C3⋊D3)⋊C311yescharacteristic
430C23C22×(C3⋊D3)⋊C311yescharacteristic
431D4—98no—
432D4—98no—
433C2×C4—42no—
434C2×C4—92no—
435C2×C4—94no—
436C2×C4—94no—
437C2×C4—364no—
438C2×C4—362no—
439C2×C4C22×(C3⋊D3)⋊C314yes—
440C2×C3—22no—
441C2×C3—22no—
442C2×C3—22no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—32no—
446C2×C3—32no—
447C2×C3—62no—
448C2×C3—62no—
449C2×C3—62no—
450C2×C3—62no—
451C2×C3—62no—
452C2×C3—62no—
453C2×C3—64no—
454C2×C3—61no—
455C2×C3—62no—
456C2×C3—122no—
457C2×C3—122no—
458C2×C3—184no—
459C2×C3C22×C3×C4×D311yescharacteristic
460C2×C3C3×#100412yes—
461D3—64no—
462D3—184no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—44no—
467C22—94no—
468C22—184no—
469C22—364no—
470C22C2×C4×(C3⋊D3)⋊C312yes—
471C22C2×C4×(C3⋊D3)⋊C312yes—
472C22C2×C4×(C3⋊D3)⋊C312yes—
473C22C23×(C3⋊D3)⋊C311yescharacteristic
474C4—24no—
475C4—94no—
476C4—184no—
477C3—31no—
478C3—31no—
479C3—61no—
480C3C3×#548611yescharacteristic
481C2—22no—
482C2—22no—
483C2—22no—
484C2—184no—
485C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
486C2#64361⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
Frattini, socleC22×C3C23×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC22×#638⋊C3C2
cosocleC22×C32C24×C3
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#518
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×H(3,𝔽3)C2semidirect
3. C2×#3757⋊C3C2semidirect
4. C4×#638⋊C3C2semidirect
5. #64090⋊C3C2semidirect
6. C22⋊C4×(C3⋊D3)⋊C3C2semidirect
7. #64086⋊C3C2semidirect
8. #10595957C3semidirect
12. C22×C4×H(3,𝔽3)C22semidirect
13. C2×#638⋊C3C4semidirect
17. C22×(C3⋊Dic3)⋊C3C22semidirect
18. C2×C4×(C3⋊D3)⋊C3C22semidirect
20. C22⋊C4×H(3,𝔽3)C22semidirect
22. #3757⋊C3C22semidirect
23. #3752⋊C3C22semidirect
24. #3750⋊C3C22semidirect
25. C4×(C3⋊Dic3)⋊C3C22semidirect
29. C2×C32×C22⋊C4D3semidirect
30. C2×C32×C22⋊C4C2×C3semidirect
31. C2×#3757C2×C3semidirect
32. C4×#638C2×C3semidirect
33. #64090C2×C3semidirect
34. C22⋊C4×C3⋊D3C2×C3semidirect
35. #64086C2×C3semidirect
42. C23×H(3,𝔽3)C2×C4semidirect
47. C22×(C3⋊D3)⋊C3C2×C4semidirect
51. #638⋊C3C2×C4semidirect
52. #638⋊C3C2×C4semidirect
54. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
67. C22×C32×C4C2×D3semidirect
68. C22×C32×C4C22×C3semidirect
69. C2×#638C3×C4semidirect
73. C22×C3⋊Dic3C22×C3semidirect
74. C2×C4×C3⋊D3C22×C3semidirect
84. C32×C22⋊C4C2×D3semidirect
86. C32×C22⋊C4C22×C3semidirect
88. #3757C22×C3semidirect
89. #3752C22×C3semidirect
90. #3750C22×C3semidirect
91. C4×C3⋊Dic3C22×C3semidirect
140. C23×C32C4×D3semidirect
141. C23×C32C2×C3×C4semidirect
149. C22×C3⋊D3C2×C3×C4semidirect
160. #638C2×C3×C4semidirect
161. #638C2×C3×C4semidirect
163. C2×C3⋊Dic3C2×C3×C4semidirect
276. C2×C22⋊C4(C3⋊D3)⋊C3semidirect
277. H(3,𝔽3)#518semidirect
361. C22×C4C2×(C3⋊D3)⋊C3semidirect
367. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
421. C32#5486semidirect
422. C32C3×#518semidirect
427. C23C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#638⋊C32
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C4×#638⋊C32
#64090⋊C32
C22⋊C4×(C3⋊D3)⋊C32
#64086⋊C32
#105959573
C3×#54863