Finite Group Databasebeta

← #10596145⋊C3 #13153116

Subgroups — #10596145⋊C3 #13153116

Order 1728.

1245 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C211yesmaximal, characteristic, complemented
2C2×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
4#64156⋊C3C211yesmaximal, characteristic, complemented
5#64149⋊C3C211yesmaximal, characteristic, complemented
6#64093⋊C3C214yesmaximal, complemented
7#64091⋊C3C214yesmaximal, complemented
8#11721385C211yesmaximal, characteristic, complemented
9#64081⋊C3C211yesmaximal, characteristic, complemented
10#10596145C311yesmaximal, characteristic, complemented
11C3×#5674—31nomaximal
12C23×(C3⋊D3)⋊C3C2212yes—
13C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×D4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C2×#638⋊C3C2214yescomplemented
17#88722—24no—
18D4×(C3⋊D3)⋊C3—24no—
19C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
20C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
21C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
22C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
23C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
24C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
25C22⋊C4×H(3,𝔽3)C2214yescomplemented
26C42×H(3,𝔽3)C2211yescharacteristic, complemented
27#3752⋊C3C2212yes—
28#3752⋊C3C2214yescomplemented
29#3750⋊C3C2212yescomplemented
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
31C2×#3832C2×C311yescharacteristic, complemented
32C2×D4×C3⋊D3C2×C311yescharacteristic, complemented
33C2×C3×#239—31no—
34C2×C3×D3×D4—31no—
35C32×#124—21no—
36C32×#124D311yescharacteristic, complemented
37C32×#124C2×C311yescharacteristic, complemented
38#64156C2×C311yescharacteristic, complemented
39#64149C2×C311yescharacteristic, complemented
40#64093C2×C314yescomplemented
41#64091C2×C314yescomplemented
42#64083C2×C311yescharacteristic, complemented
43#64081C2×C311yescharacteristic, complemented
44C3×#947—31no—
45C3×#940—31no—
46C3×#884—34no—
47C3×#882—34no—
48C3×#874—31no—
49C3×#872—31no—
50C23×H(3,𝔽3)C2312yes—
51C22×(C3⋊D3)⋊C3—24no—
52C22×(C3⋊D3)⋊C3—44no—
53C22×(C3⋊D3)⋊C3C2312yes—
54C22×(C3⋊D3)⋊C3C2312yes—
55C22×(C3⋊D3)⋊C3C2311yescharacteristic
56Q8×H(3,𝔽3)—22no—
57D4×H(3,𝔽3)—22no—
58C2×C4×H(3,𝔽3)C2312yes—
59C2×C4×H(3,𝔽3)C2312yes—
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61#638⋊C3—24no—
62#638⋊C3—44no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
65C4⋊((C3⋊D3)⋊C3)—24no—
66C4⋊((C3⋊D3)⋊C3)—22no—
67C4⋊((C3⋊D3)⋊C3)—22no—
68C4⋊((C3⋊D3)⋊C3)—44no—
69C4×(C3⋊D3)⋊C3—24no—
70C4×(C3⋊D3)⋊C3D414yescomplemented
71C3×#535—91no—
72#5674—31no—
73#5674—91no—
74C23×C3⋊D3C22×C312yes—
75C23×C3×D3—32no—
76C2×C32×Q8—21no—
77C2×C32×Q8C2×D311yescharacteristic, complemented
78C2×C32×Q8C22×C311yescharacteristic, complemented
79C2×C32×D4—21no—
80C2×C32×D4C2×D311yescharacteristic
81C2×C32×D4C22×C311yescharacteristic
82C2×#638C22×C312yes—
83C2×#638C22×C314yescomplemented
84#3832—24no—
85D4×C3⋊D3—24no—
86C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
87C2×C4⋊(C3⋊D3)C22×C312yes—
88C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
89C2×C4⋊(C3⋊D3)C22×C314yescomplemented
90C2×C4×C3⋊D3C22×C311yescharacteristic
91C2×C4×C3⋊D3C22×C312yescomplemented
92C2×C3×#67—32no—
93C2×C3×#67—34no—
94C3×#239—64no—
95C3×D3×D4—64no—
96C2×C3×D12—32no—
97C2×C3×D12—31no—
98C2×C3×D12—31no—
99C2×C3×D12—34no—
100C2×C3×C4×D3—31no—
101C2×C3×C4×D3—32no—
102C32×C22⋊C4—24no—
103C32×C22⋊C4C2×D314yescomplemented
104C32×C22⋊C4C22×C314yescomplemented
105C32×C42—21no—
106C32×C42C2×D311yescharacteristic, complemented
107C32×C42C22×C311yescharacteristic, complemented
108#3752C22×C312yes—
109#3752C22×C314yescomplemented
110#3750C22×C312yescomplemented
111C4×C3⋊Dic3C22×C311yescharacteristic, complemented
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—34no—
114C3×Dic3⋊C4—32no—
115C3×C4×Dic3—31no—
116C22×H(3,𝔽3)—22no—
117C22×H(3,𝔽3)—24no—
118C22×H(3,𝔽3)C2411yescharacteristic
119C2×(C3⋊D3)⋊C3—22no—
120C2×(C3⋊D3)⋊C3—22no—
121C2×(C3⋊D3)⋊C3—22no—
122C2×(C3⋊D3)⋊C3—24no—
123C2×(C3⋊D3)⋊C3—24no—
124C2×(C3⋊D3)⋊C3—48no—
125C2×(C3⋊D3)⋊C3C2×D412yes—
126C4×H(3,𝔽3)—22no—
127C4×H(3,𝔽3)—22no—
128C4×H(3,𝔽3)C2×D412yes—
129(C3⋊Dic3)⋊C3—22no—
130(C3⋊Dic3)⋊C3C2×D412yes—
131C2×C3×#41—91no—
132C22×C3×D4—91no—
133C2×#239—31no—
134C2×#239—91no—
135C2×D3×D4—31no—
136C2×D3×D4—91no—
137C3×#127—91no—
138C3×#127—91no—
139C3×#124—31no—
140C3×#124—31no—
141C3×#124—61no—
142C3×#124—91no—
143C3×#124C3×D311yescharacteristic
144C3×#121—94no—
145C3×#120—94no—
146C3×#117—91no—
147#947—31no—
148#947—91no—
149#940—31no—
150#940—91no—
151#884—34no—
152#884—94no—
153#882—34no—
154#882—94no—
155#874—31no—
156#874—91no—
157#872—31no—
158#872—91no—
159C23×C32—22no—
160C23×C32C22×D312yes—
161C23×C32C23×C312yes—
162C22×C3⋊D3—24no—
163C22×C3⋊D3—44no—
164C22×C3⋊D3C23×C312yes—
165C22×C3⋊D3C23×C312yes—
166C22×C3⋊D3C23×C311yescharacteristic
167C22×C3×D3—31no—
168C22×C3×D3—32no—
169C22×C3×D3—32no—
170C22×C3×D3—64no—
171C22×C3×D3—124no—
172C32×Q8—22no—
173C32×Q8—22no—
174C32×Q8—42no—
175C32×D4—22no—
176C32×D4—22no—
177C32×D4—42no—
178C2×C32×C4—21no—
179C2×C32×C4—22no—
180C2×C32×C4—22no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C23×C312yes—
185C2×C32×C4C23×C312yes—
186C2×C32×C4C23×C311yescharacteristic
187#638—24no—
188#638—44no—
189C2×C3⋊Dic3C23×C312yes—
190C2×C3⋊Dic3C23×C311yescharacteristic
191C4⋊(C3⋊D3)—24no—
192C4⋊(C3⋊D3)—22no—
193C4⋊(C3⋊D3)—22no—
194C4⋊(C3⋊D3)—44no—
195C4×C3⋊D3—24no—
196C4×C3⋊D3C3×D414yescomplemented
197C3×#67—64no—
198C3×#67—124no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C3×D12—64no—
202C3×D12—62no—
203C3×D12—62no—
204C3×D12—124no—
205C3×C4×D3—34no—
206C3×C4×D3—64no—
207#535—91noSylow
208C2×H(3,𝔽3)—42no—
209C2×H(3,𝔽3)C22×D411yescharacteristic
210C2×H(3,𝔽3)H(5,𝔽2)12yes—
211(C3⋊D3)⋊C3—22no—
212(C3⋊D3)⋊C3—42no—
213(C3⋊D3)⋊C3—42no—
214C24×C3—92no—
215C23×D3—32no—
216C23×D3—92no—
217C3×#41—184no—
218C2×C3×Q8—31no—
219C2×C3×Q8—31no—
220C2×C3×Q8—61no—
221C2×C3×Q8C2×C3×D311yescharacteristic
222C2×C3×D4—31no—
223C2×C3×D4—31no—
224C2×C3×D4—61no—
225C2×C3×D4—92no—
226C2×C3×D4—91no—
227C2×C3×D4—91no—
228C2×C3×D4—94no—
229C2×C3×D4—92no—
230C2×C3×D4—94no—
231C2×C3×D4—184no—
232C2×C3×D4C2×C3×D311yescharacteristic
233C22×C3×C4—91no—
234C22×C3×C4—92no—
235C2×#67—32no—
236C2×#67—34no—
237C2×#67—92no—
238C2×#67—94no—
239#239—64no—
240#239—184no—
241D3×D4—64no—
242D3×D4—184no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—32no—
246C2×D12—34no—
247C2×D12—91no—
248C2×D12—91no—
249C2×D12—92no—
250C2×D12—94no—
251C2×C4×D3—31no—
252C2×C4×D3—32no—
253C2×C4×D3—91no—
254C2×C4×D3—92no—
255C3×C4⋊C4—92no—
256C3×C22⋊C4—34no—
257C3×C22⋊C4—34no—
258C3×C22⋊C4—64no—
259C3×C22⋊C4—92no—
260C3×C22⋊C4—94no—
261C3×C22⋊C4C2×C3×D314yes—
262C3×C42—31no—
263C3×C42—31no—
264C3×C42—61no—
265C3×C42—91no—
266C3×C42C2×C3×D311yescharacteristic
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—34no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—94no—
271Dic3⋊C4—32no—
272Dic3⋊C4—92no—
273C4×Dic3—31no—
274C4×Dic3—91no—
275C22×C32—22no—
276C22×C32—24no—
277C22×C32—22no—
278C22×C32—24no—
279C22×C32—21no—
280C22×C32—42no—
281C22×C32—44no—
282C22×C32C23×D311yescharacteristic
283C22×C32C24×C311yescharacteristic
284C2×C3⋊D3—22no—
285C2×C3⋊D3—22no—
286C2×C3⋊D3—22no—
287C2×C3⋊D3—24no—
288C2×C3⋊D3—24no—
289C2×C3⋊D3—48no—
290C2×C3⋊D3C2×C3×D412yes—
291C2×C3×D3—32no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—64no—
296C2×C3×D3—64no—
297C2×C3×D3—128no—
298C32×C4—22no—
299C32×C4—22no—
300C32×C4—22no—
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—42no—
304C32×C4—42no—
305C32×C4D3×D412yes—
306C32×C4C2×C3×D412yes—
307C3⋊Dic3—22no—
308C3⋊Dic3C2×C3×D412yes—
309C3×Dic3—32no—
310C3×Dic3—62no—
311C2×#41—91no—
312C22×D4—91no—
313#127—91no—
314#127—91no—
315#124—91no—
316#124(C3⋊D3)⋊C311yescharacteristic, complemented
317#121—94no—
318#120—94no—
319#117—91no—
320H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
321C23×C3—32no—
322C23×C3—32no—
323C23×C3—62no—
324C23×C3—91no—
325C23×C3—92no—
326C23×C3—92no—
327C23×C3—184no—
328C23×C3—364no—
329C23×C3C22×C3×D312yes—
330C22×D3—31no—
331C22×D3—32no—
332C22×D3—32no—
333C22×D3—64no—
334C22×D3—91no—
335C22×D3—92no—
336C22×D3—92no—
337C22×D3—124no—
338C22×D3—184no—
339C22×D3—364no—
340C3×Q8—22no—
341C3×Q8—62no—
342C3×Q8—62no—
343C3×Q8—122no—
344C3×D4—22no—
345C3×D4—62no—
346C3×D4—62no—
347C3×D4—122no—
348C3×D4—182no—
349C3×D4—182no—
350C3×D4—184no—
351C3×D4—184no—
352C3×D4—364no—
353C3×D4—364no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—32no—
359C2×C3×C4—32no—
360C2×C3×C4—61no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—94no—
364C2×C3×C4—91no—
365C2×C3×C4—92no—
366C2×C3×C4—184no—
367C2×C3×C4C22×C3×D312yes—
368C2×C3×C4C22×C3×D312yes—
369C2×C3×C4C22×C3×D311yescharacteristic
370#67—64no—
371#67—124no—
372#67—184no—
373#67—364no—
374C2×Dic3—31no—
375C2×Dic3—32no—
376C2×Dic3—91no—
377C2×Dic3—92no—
378D12—64no—
379D12—62no—
380D12—62no—
381D12—124no—
382D12—182no—
383D12—182no—
384D12—184no—
385D12—364no—
386C4×D3—34no—
387C4×D3—64no—
388C4×D3—94no—
389C4×D3—184no—
390C2×C32—21no—
391C2×C32—22no—
392C2×C32—42no—
393C2×C32—42no—
394C2×C32—82no—
395C2×C32C2×D3×D411yescharacteristic
396C2×C32#100912yes—
397C2×C32C22×C3×D411yescharacteristic
398C2×C32C3×H(5,𝔽2)12yes—
399C3⋊D3—22no—
400C3⋊D3—42no—
401C3⋊D3—42no—
402C3×D3—62no—
403C3×D3—122no—
404C3×D3—122no—
405C24—92no—
406#41—184no—
407C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
408C2×D4—92no—
409C2×D4—94no—
410C2×D4—92no—
411C2×D4—91no—
412C2×D4—91no—
413C2×D4—94no—
414C2×D4—184no—
415C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
416C22×C4—91no—
417C22×C4—92no—
418C4⋊C4—92no—
419C22⋊C4—92no—
420C22⋊C4—94no—
421C22⋊C4C2×(C3⋊D3)⋊C314yescomplemented
422C42—91no—
423C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
424C22×C3—22no—
425C22×C3—24no—
426C22×C3—31no—
427C22×C3—31no—
428C22×C3—62no—
429C22×C3—64no—
430C22×C3—62no—
431C22×C3—64no—
432C22×C3—61no—
433C22×C3—92no—
434C22×C3—122no—
435C22×C3—124no—
436C22×C3—182no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3—184no—
440C22×C3—184no—
441C22×C3—368no—
442C22×C3C23×C3×D311yescharacteristic
443C2×D3—32no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—64no—
448C2×D3—64no—
449C2×D3—92no—
450C2×D3—128no—
451C2×D3—182no—
452C2×D3—182no—
453C2×D3—182no—
454C2×D3—184no—
455C2×D3—184no—
456C2×D3—368no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—32no—
460C3×C4—32no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—62no—
464C3×C4—62no—
465C3×C4—62no—
466C3×C4—92no—
467C3×C4—122no—
468C3×C4—122no—
469C3×C4—182no—
470C3×C4C3×D3×D412yes—
471Dic3—32no—
472Dic3—62no—
473Dic3—92no—
474Dic3—182no—
475C32—21no—
476C32#567411yescharacteristic, complemented
477C32C3×#53511yescharacteristic, complemented
478C23—91no—
479C23—92no—
480C23—92no—
481C23—184no—
482C23—364no—
483C23C22×(C3⋊D3)⋊C312yes—
484Q8—22no—
485D4—22no—
486D4—184no—
487D4—184no—
488D4—182no—
489D4—182no—
490D4—364no—
491D4—364no—
492C2×C4—91no—
493C2×C4—94no—
494C2×C4—92no—
495C2×C4—184no—
496C2×C4C22×(C3⋊D3)⋊C312yes—
497C2×C4C22×(C3⋊D3)⋊C312yes—
498C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—32no—
502C2×C3—32no—
503C2×C3—42no—
504C2×C3—61no—
505C2×C3—62no—
506C2×C3—122no—
507C2×C3—122no—
508C2×C3—182no—
509C2×C3—242no—
510C2×C3—362no—
511C2×C3—362no—
512C2×C3C2×C3×D3×D411yescharacteristic
513C2×C3C3×#100912yes—
514D3—62no—
515D3—122no—
516D3—122no—
517D3—182no—
518D3—362no—
519D3—362no—
520C22—22no—
521C22—24no—
522C22—92no—
523C22—182no—
524C22—182no—
525C22—182no—
526C22—184no—
527C22—184no—
528C22—368no—
529C22C23×(C3⋊D3)⋊C311yescharacteristic
530C4—22no—
531C4—22no—
532C4—92no—
533C4—182no—
534C4D4×(C3⋊D3)⋊C312yes—
535C3—31no—
536C3—31no—
537C3—61no—
538C3C3×#567411yescharacteristic
539C2—42no—
540C2—182no—
541C2—362no—
542C2—362no—
543C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
544C2#1172263812yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#124C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#88722C2semidirect
2. C2×D4×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#124C2semidirect
4. #64156⋊C3C2semidirect
5. #64149⋊C3C2semidirect
6. #64093⋊C3C2semidirect
7. #64091⋊C3C2semidirect
8. #11721385C2semidirect
9. #64081⋊C3C2semidirect
10. #10596145C3semidirect
13. C2×Q8×H(3,𝔽3)C22semidirect
16. C2×#638⋊C3C22semidirect
21. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
22. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
24. C2×C4×(C3⋊D3)⋊C3C22semidirect
25. C22⋊C4×H(3,𝔽3)C22semidirect
26. C42×H(3,𝔽3)C22semidirect
28. #3752⋊C3C22semidirect
29. #3750⋊C3C22semidirect
30. C4×(C3⋊Dic3)⋊C3C22semidirect
31. C2×#3832C2×C3semidirect
32. C2×D4×C3⋊D3C2×C3semidirect
36. C32×#124D3semidirect
37. C32×#124C2×C3semidirect
38. #64156C2×C3semidirect
39. #64149C2×C3semidirect
40. #64093C2×C3semidirect
41. #64091C2×C3semidirect
42. #64083C2×C3semidirect
43. #64081C2×C3semidirect
70. C4×(C3⋊D3)⋊C3D4semidirect
77. C2×C32×Q8C2×D3semidirect
78. C2×C32×Q8C22×C3semidirect
83. C2×#638C22×C3semidirect
88. C2×C4⋊(C3⋊D3)C22×C3semidirect
89. C2×C4⋊(C3⋊D3)C22×C3semidirect
91. C2×C4×C3⋊D3C22×C3semidirect
103. C32×C22⋊C4C2×D3semidirect
104. C32×C22⋊C4C22×C3semidirect
106. C32×C42C2×D3semidirect
107. C32×C42C22×C3semidirect
109. #3752C22×C3semidirect
110. #3750C22×C3semidirect
111. C4×C3⋊Dic3C22×C3semidirect
196. C4×C3⋊D3C3×D4semidirect
316. #124(C3⋊D3)⋊C3semidirect
320. H(3,𝔽3)#535semidirect
407. C2×Q8C2×(C3⋊D3)⋊C3semidirect
421. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
423. C42C2×(C3⋊D3)⋊C3semidirect
476. C32#5674semidirect
477. C32C3×#535semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×D4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1242
#64156⋊C32
#64149⋊C32
#64093⋊C32
#64091⋊C32
#117213852
#64081⋊C32
#105961453
C3×#56743