Finite Group Databasebeta

← #10596130⋊C3 #13153049

Subgroups — #10596130⋊C3 #13153049

Order 1728.

946 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#123C211yesmaximal, characteristic, complemented
2#64146⋊C3C211yesmaximal, characteristic
3#64145⋊C3C211yesmaximal, characteristic, complemented
4C4×#638⋊C3C212yesmaximal, complemented
5#64105⋊C3C212yesmaximal, complemented
6#64101⋊C3C212yesmaximal, complemented
7#64092⋊C3C212yesmaximal, complemented
8C22⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
9#64088⋊C3C212yesmaximal, complemented
10#10596130C311yesmaximal, characteristic, complemented
11C3×#5659—31nomaximal
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×#638⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic, complemented
19C4⋊C4×H(3,𝔽3)C2212yescomplemented
20C22⋊C4×H(3,𝔽3)C2212yes—
21C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
22#3757⋊C3C2212yes—
23#3757⋊C3C2211yescharacteristic
24#3757⋊C3C2212yescomplemented
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
29#3750⋊C3C2212yescomplemented
30#3750⋊C3C2212yescomplemented
31C4×(C3⋊Dic3)⋊C3C2212yescomplemented
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
33C32×#123—21no—
34C32×#123D311yescharacteristic, complemented
35C32×#123C2×C311yescharacteristic, complemented
36#64146C2×C311yescharacteristic
37#64145C2×C311yescharacteristic, complemented
38C4×#638C2×C312yescomplemented
39#64105C2×C312yescomplemented
40#64101C2×C312yescomplemented
41#64092C2×C312yescomplemented
42C22⋊C4×C3⋊D3C2×C312yescomplemented
43#64088C2×C312yescomplemented
44C3×#937—31no—
45C3×#936—31no—
46C3×C4×#67—32no—
47C3×#896—32no—
48C3×#892—32no—
49C3×#883—32no—
50C3×D3×C22⋊C4—32no—
51C3×#879—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×(C3⋊D3)⋊C3—24no—
55C22×(C3⋊D3)⋊C3—41no—
56C22×(C3⋊D3)⋊C3C2312yes—
57D4×H(3,𝔽3)—41no—
58C2×C4×H(3,𝔽3)—41no—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2312yes—
61C2×C4×H(3,𝔽3)C2312yes—
62#638⋊C3—24no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C4×(C3⋊D3)⋊C3—24no—
67C4×(C3⋊D3)⋊C3—42no—
68#634⋊C3—41no—
69C3×#551—91no—
70#5659—31no—
71#5659—91no—
72C23×C3⋊D3C22×C311yescharacteristic
73C23×C3×D3—31no—
74C2×C32×D4—21no—
75C2×C32×D4C2×D311yescharacteristic
76C2×C32×D4C22×C311yescharacteristic
77C22×C32×C4—21no—
78C22×C32×C4C2×D311yescharacteristic, complemented
79C22×C32×C4C22×C311yescharacteristic, complemented
80C2×#638C22×C312yes—
81C2×C4×C3⋊D3C22×C312yes—
82C2×C4×C3⋊D3C22×C312yes—
83C2×#634C22×C311yescharacteristic, complemented
84C2×C3×#67—32no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—32no—
87C2×C3×Dic6—31no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4C2×D312yescomplemented
90C32×C4⋊C4C22×C312yescomplemented
91C32×C22⋊C4—22no—
92C32×C22⋊C4—21no—
93C32×C22⋊C4C2×D312yes—
94C32×C22⋊C4C2×D311yescharacteristic, complemented
95C32×C22⋊C4C22×C312yes—
96C32×C22⋊C4C22×C311yescharacteristic, complemented
97#3757C22×C311yescharacteristic
98#3757C22×C312yes—
99#3757C22×C312yescomplemented
100#3752C22×C312yes—
101#3752C22×C312yes—
102#3752C22×C312yes—
103C4⋊(C3⋊Dic3)C22×C312yescomplemented
104#3750C22×C312yescomplemented
105#3750C22×C312yescomplemented
106C4×C3⋊Dic3C22×C312yescomplemented
107C4×C3⋊Dic3C22×C311yescharacteristic, complemented
108C3×#217—31no—
109C3×#217—32no—
110C3×#217—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—32no—
114C3×C4⋊Dic3—32no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—31no—
119C22×H(3,𝔽3)—22no—
120C22×H(3,𝔽3)—22no—
121C22×H(3,𝔽3)—21no—
122C22×H(3,𝔽3)—21no—
123C22×H(3,𝔽3)C2411yescharacteristic
124C2×(C3⋊D3)⋊C3—22no—
125C2×(C3⋊D3)⋊C3—22no—
126C2×(C3⋊D3)⋊C3—44no—
127C2×(C3⋊D3)⋊C3#4114yes—
128C4×H(3,𝔽3)—22no—
129C4×H(3,𝔽3)—21no—
130C4×H(3,𝔽3)—22no—
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3—22no—
133(C3⋊Dic3)⋊C3—22no—
134C3×#126—92no—
135C3×#124—91no—
136C3×#123—31no—
137C3×#123—31no—
138C3×#123—61no—
139C3×#123—92no—
140C3×#123C3×D311yescharacteristic
141C3×C22⋊Q8—92no—
142C3×#120—91no—
143C3×C4×D4—92no—
144C3×#117—92no—
145C2×C3×C22⋊C4—92no—
146#937—31no—
147#937—91no—
148#936—31no—
149#936—91no—
150C4×#67—32no—
151C4×#67—92no—
152#896—32no—
153#896—92no—
154#892—32no—
155#892—92no—
156#883—32no—
157#883—92no—
158D3×C22⋊C4—32no—
159D3×C22⋊C4—92no—
160#879—32no—
161#879—92no—
162C23×C32—21no—
163C23×C32—21no—
164C23×C32C22×D311yescharacteristic
165C23×C32C22×D311yescharacteristic
166C23×C32C23×C311yescharacteristic
167C23×C32C23×C311yescharacteristic
168C22×C3⋊D3—24no—
169C22×C3⋊D3—41no—
170C22×C3⋊D3C23×C312yes—
171C22×C3×D3—32no—
172C22×C3×D3—64no—
173C22×C3×D3—121no—
174C32×D4—41no—
175C32×D4—41no—
176C32×D4—42no—
177C2×C32×C4—22no—
178C2×C32×C4—22no—
179C2×C32×C4—21no—
180C2×C32×C4—41no—
181C2×C32×C4—41no—
182C2×C32×C4—42no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D312yes—
186C2×C32×C4C23×C312yes—
187C2×C32×C4C23×C311yescharacteristic
188C2×C32×C4C23×C312yes—
189#638—24no—
190C2×C3⋊Dic3C23×C312yes—
191C2×C3⋊Dic3C23×C312yes—
192C2×C3⋊Dic3C23×C312yes—
193C4×C3⋊D3—24no—
194C4×C3⋊D3—42no—
195#634—41no—
196C3×#67—64no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C3×C4×D3—64no—
201C3×C4×D3—122no—
202C3×Dic6—121no—
203#551—91noSylow
204C2×H(3,𝔽3)—41no—
205C2×H(3,𝔽3)—41no—
206C2×H(3,𝔽3)C2×#4112yes—
207C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
208(C3⋊D3)⋊C3—24no—
209C24×C3—91no—
210C23×D3—31no—
211C23×D3—91no—
212C2×C3×Q8—91no—
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C2×C3×D4—61no—
216C2×C3×D4—92no—
217C2×C3×D4C2×C3×D311yescharacteristic
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—61no—
221C22×C3×C4—92no—
222C22×C3×C4—92no—
223C22×C3×C4C2×C3×D311yescharacteristic
224C2×#67—32no—
225C2×#67—92no—
226C2×C4×D3—32no—
227C2×C4×D3—32no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×Dic6—31no—
231C2×Dic6—91no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—62no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4C2×C3×D312yes—
239C3×C22⋊C4—32no—
240C3×C22⋊C4—32no—
241C3×C22⋊C4—31no—
242C3×C22⋊C4—31no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—61no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4—91no—
251C3×C22⋊C4C2×C3×D312yes—
252C3×C22⋊C4C2×C3×D311yescharacteristic
253C3×C42—91no—
254C3×C42—92no—
255#217—31no—
256#217—32no—
257#217—32no—
258#217—91no—
259#217—92no—
260#217—92no—
261(C2×D3)⋊C4—32no—
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—32no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267C4⋊Dic3—32no—
268C4⋊Dic3—92no—
269Dic3⋊C4—32no—
270Dic3⋊C4—32no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273C4×Dic3—32no—
274C4×Dic3—31no—
275C4×Dic3—92no—
276C4×Dic3—91no—
277C22×C32—22no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—42no—
288C22×C32—41no—
289C22×C32—42no—
290C22×C32C23×D311yescharacteristic
291C22×C32C24×C311yescharacteristic
292C2×C3⋊D3—22no—
293C2×C3⋊D3—22no—
294C2×C3⋊D3—44no—
295C2×C3⋊D3C3×#4114yes—
296C2×C3×D3—34no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—124no—
300C32×C4—21no—
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—21no—
305C32×C4—22no—
306C32×C4—42no—
307C32×C4—42no—
308C32×C4—41no—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3—22no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—62no—
315#126—92no—
316#124—91no—
317#123—92no—
318#123(C3⋊D3)⋊C311yescharacteristic, complemented
319C22⋊Q8—92no—
320#120—91no—
321C4×D4—92no—
322#117—92no—
323C2×C22⋊C4—92no—
324H(3,𝔽3)#55111yescharacteristic, complemented, Sylow
325C23×C3—31no—
326C23×C3—31no—
327C23×C3—31no—
328C23×C3—31no—
329C23×C3—61no—
330C23×C3—61no—
331C23×C3—92no—
332C23×C3—184no—
333C23×C3—361no—
334C23×C3C22×C3×D311yescharacteristic
335C23×C3C22×C3×D311yescharacteristic
336C22×D3—32no—
337C22×D3—64no—
338C22×D3—92no—
339C22×D3—121no—
340C22×D3—184no—
341C22×D3—361no—
342C3×Q8—361no—
343C3×D4—41no—
344C3×D4—121no—
345C3×D4—121no—
346C3×D4—122no—
347C3×D4—184no—
348C2×C3×C4—32no—
349C2×C3×C4—31no—
350C2×C3×C4—32no—
351C2×C3×C4—32no—
352C2×C3×C4—31no—
353C2×C3×C4—32no—
354C2×C3×C4—41no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—61no—
358C2×C3×C4—92no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—121no—
362C2×C3×C4—121no—
363C2×C3×C4—122no—
364C2×C3×C4—184no—
365C2×C3×C4—362no—
366C2×C3×C4C22×C3×D312yes—
367C2×C3×C4C22×C3×D311yescharacteristic
368C2×C3×C4C22×C3×D312yes—
369#67—64no—
370#67—184no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—92no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C4×D3—64no—
378C4×D3—122no—
379C4×D3—184no—
380C4×D3—362no—
381Dic6—121no—
382Dic6—361no—
383C2×C32—21no—
384C2×C32—22no—
385C2×C32—42no—
386C2×C32—41no—
387C2×C32—41no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—81no—
391C2×C32#100411yescharacteristic
392C2×C32D3×#4112yes—
393C2×C32C2×C3×#4112yes—
394C2×C32C3×H(5,𝔽2)11yescharacteristic
395C3⋊D3—24no—
396C3×D3—64no—
397C24—91no—
398C2×Q8—91no—
399C2×D4—92no—
400C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
401C22×C4—92no—
402C22×C4—92no—
403C22×C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
404C4⋊C4—92no—
405C4⋊C4—92no—
406C4⋊C4—92no—
407C4⋊C4C2×(C3⋊D3)⋊C312yescomplemented
408C22⋊C4—92no—
409C22⋊C4—92no—
410C22⋊C4—91no—
411C22⋊C4—92no—
412C22⋊C4—92no—
413C22⋊C4—92no—
414C22⋊C4C2×(C3⋊D3)⋊C312yes—
415C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
416C42—91no—
417C42—92no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—31no—
423C22×C3—31no—
424C22×C3—62no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—62no—
433C22×C3—61no—
434C22×C3—94no—
435C22×C3—122no—
436C22×C3—121no—
437C22×C3—122no—
438C22×C3—182no—
439C22×C3—182no—
440C22×C3—364no—
441C22×C3C23×C3×D311yescharacteristic
442C2×D3—34no—
443C2×D3—62no—
444C2×D3—62no—
445C2×D3—94no—
446C2×D3—124no—
447C2×D3—182no—
448C2×D3—182no—
449C2×D3—364no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—61no—
454C3×C4—62no—
455C3×C4—62no—
456C3×C4—62no—
457C3×C4—62no—
458C3×C4—61no—
459C3×C4—122no—
460C3×C4—121no—
461C3×C4—122no—
462C3×C4—182no—
463C3×C4—182no—
464C3×C4—182no—
465Dic3—62no—
466Dic3—62no—
467Dic3—62no—
468Dic3—182no—
469Dic3—182no—
470Dic3—182no—
471C32—21no—
472C32#565911yescharacteristic, complemented
473C32C3×#55111yescharacteristic, complemented
474C23—92no—
475C23—184no—
476C23—361no—
477C23C22×(C3⋊D3)⋊C311yescharacteristic
478C23C22×(C3⋊D3)⋊C311yescharacteristic
479Q8—361no—
480D4—41no—
481D4—184no—
482C2×C4—41no—
483C2×C4—92no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—184no—
487C2×C4—362no—
488C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
489C2×C4C22×(C3⋊D3)⋊C312yes—
490C2×C4C22×(C3⋊D3)⋊C312yes—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—32no—
494C2×C3—32no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—61no—
498C2×C3—62no—
499C2×C3—121no—
500C2×C3—121no—
501C2×C3—121no—
502C2×C3—121no—
503C2×C3—122no—
504C2×C3—184no—
505C2×C3—241no—
506C2×C3C3×#100411yescharacteristic
507C2×C3C3×D3×#4112yes—
508D3—64no—
509D3—184no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—94no—
515C22—182no—
516C22—182no—
517C22—364no—
518C22C23×(C3⋊D3)⋊C311yescharacteristic
519C4—22no—
520C4—21no—
521C4—22no—
522C4—182no—
523C4—182no—
524C4—182no—
525C3—31no—
526C3—31no—
527C3—61no—
528C3C3×#565911yescharacteristic
529C2—41no—
530C2—41no—
531C2—184no—
532C2#64361⋊C311yescharacteristic
533C2#41×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#123C2
prime radical#64146⋊C3C2
hypercenter#123(C3⋊D3)⋊C3
hypocenterC32C3×#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#123C2semidirect
3. #64145⋊C3C2semidirect
4. C4×#638⋊C3C2semidirect
5. #64105⋊C3C2semidirect
6. #64101⋊C3C2semidirect
7. #64092⋊C3C2semidirect
8. C22⋊C4×(C3⋊D3)⋊C3C2semidirect
9. #64088⋊C3C2semidirect
10. #10596130C3semidirect
14. C22×C4×H(3,𝔽3)C22semidirect
18. C2×#634⋊C3C22semidirect
19. C4⋊C4×H(3,𝔽3)C22semidirect
21. C22⋊C4×H(3,𝔽3)C22semidirect
24. #3757⋊C3C22semidirect
28. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
29. #3750⋊C3C22semidirect
30. #3750⋊C3C22semidirect
31. C4×(C3⋊Dic3)⋊C3C22semidirect
32. C4×(C3⋊Dic3)⋊C3C22semidirect
34. C32×#123D3semidirect
35. C32×#123C2×C3semidirect
37. #64145C2×C3semidirect
38. C4×#638C2×C3semidirect
39. #64105C2×C3semidirect
40. #64101C2×C3semidirect
41. #64092C2×C3semidirect
42. C22⋊C4×C3⋊D3C2×C3semidirect
43. #64088C2×C3semidirect
78. C22×C32×C4C2×D3semidirect
79. C22×C32×C4C22×C3semidirect
83. C2×#634C22×C3semidirect
89. C32×C4⋊C4C2×D3semidirect
90. C32×C4⋊C4C22×C3semidirect
94. C32×C22⋊C4C2×D3semidirect
96. C32×C22⋊C4C22×C3semidirect
99. #3757C22×C3semidirect
103. C4⋊(C3⋊Dic3)C22×C3semidirect
104. #3750C22×C3semidirect
105. #3750C22×C3semidirect
106. C4×C3⋊Dic3C22×C3semidirect
107. C4×C3⋊Dic3C22×C3semidirect
318. #123(C3⋊D3)⋊C3semidirect
324. H(3,𝔽3)#551semidirect
403. C22×C4C2×(C3⋊D3)⋊C3semidirect
407. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
415. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
472. C32#5659semidirect
473. C32C3×#551semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1232
#64146⋊C32
#64145⋊C32
C4×#638⋊C32
#64105⋊C32
#64101⋊C32
#64092⋊C32
C22⋊C4×(C3⋊D3)⋊C32
#64088⋊C32
#105961303
C3×#56593