Finite Group Databasebeta

← #10596135⋊C3 #13153041

Subgroups — #10596135⋊C3 #13153041

Order 1728.

904 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#123C211yesmaximal, characteristic, complemented
2#64149⋊C3C211yesmaximal, characteristic
3#64144⋊C3C211yesmaximal, characteristic, complemented
4C4×#638⋊C3C212yesmaximal, complemented
5#64103⋊C3C212yesmaximal, complemented
6#64096⋊C3C212yesmaximal, complemented
7#64094⋊C3C212yesmaximal, complemented
8#64094⋊C3C212yesmaximal, complemented
9#64090⋊C3C212yesmaximal, complemented
10#10596135C311yesmaximal, characteristic, complemented
11C3×#5664—31nomaximal
12C2×D4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C2×#638⋊C3C2212yes—
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2212yes—
20C4⋊C4×H(3,𝔽3)C2212yes—
21C22⋊C4×H(3,𝔽3)C2212yes—
22C22⋊C4×H(3,𝔽3)C2211yescharacteristic
23#3757⋊C3C2212yes—
24#3757⋊C3C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2212yes—
33C32×#123—21no—
34C32×#123D311yescharacteristic, complemented
35C32×#123C2×C311yescharacteristic, complemented
36#64149C2×C311yescharacteristic
37#64144C2×C311yescharacteristic, complemented
38C4×#638C2×C312yescomplemented
39#64103C2×C312yescomplemented
40#64096C2×C312yescomplemented
41#64094C2×C312yescomplemented
42#64094C2×C312yescomplemented
43#64090C2×C312yescomplemented
44C3×#940—31no—
45C3×#935—31no—
46C3×C4×#67—32no—
47C3×#894—32no—
48C3×#887—32no—
49C3×#885—32no—
50C3×#885—32no—
51C3×#67⋊C4—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×(C3⋊D3)⋊C3C2312yes—
55D4×H(3,𝔽3)—41no—
56C2×C4×H(3,𝔽3)—41no—
57C2×C4×H(3,𝔽3)C2312yes—
58C2×C4×H(3,𝔽3)C2312yes—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60#638⋊C3—24no—
61#638⋊C3—24no—
62C2×(C3⋊Dic3)⋊C3—41no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C4⋊((C3⋊D3)⋊C3)—41no—
67C4×(C3⋊D3)⋊C3—42no—
68#634⋊C3—24no—
69C3×#559—91no—
70#5664—31no—
71#5664—91no—
72C2×C32×D4—21no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C22×C311yescharacteristic
75C22×C32×C4—21no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C22×C311yescharacteristic
78C2×#638C22×C312yes—
79C2×#638C22×C312yes—
80C22×C3⋊Dic3C22×C311yescharacteristic
81C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
82C2×C4×C3⋊D3C22×C312yes—
83C2×#634C22×C312yes—
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C22×C3×Dic3—31no—
87C2×C3×D12—31no—
88C2×C3×C4×D3—32no—
89C2×C3×Dic6—32no—
90C32×C4⋊C4—22no—
91C32×C4⋊C4C2×D312yes—
92C32×C4⋊C4C22×C312yes—
93C32×C22⋊C4—21no—
94C32×C22⋊C4—22no—
95C32×C22⋊C4C2×D312yes—
96C32×C22⋊C4C2×D311yescharacteristic
97C32×C22⋊C4C22×C312yes—
98C32×C22⋊C4C22×C311yescharacteristic
99#3757C22×C311yescharacteristic
100#3757C22×C312yes—
101#3752C22×C312yes—
102#3752C22×C312yes—
103#3752C22×C312yes—
104#3750C22×C312yes—
105#3750C22×C312yes—
106C4×C3⋊Dic3C22×C312yes—
107C4×C3⋊Dic3C22×C312yes—
108C4×C3⋊Dic3C22×C311yescharacteristic
109C3×#217—31no—
110C3×#217—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—32no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—32no—
116C3×C4×Dic3—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—22no—
127C4×H(3,𝔽3)—22no—
128C4×H(3,𝔽3)—22no—
129C4×H(3,𝔽3)—21no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3#4114yes—
133C3×#127—91no—
134C3×#124—92no—
135C3×#124—92no—
136C3×#123—31no—
137C3×#123—31no—
138C3×#123—61no—
139C3×#123—92no—
140C3×#123—91no—
141C3×#123C3×D311yescharacteristic
142C3×C4×Q8—92no—
143C3×C4×D4—92no—
144C3×C4×D4—92no—
145#940—31no—
146#940—91no—
147#935—31no—
148#935—91no—
149C4×#67—32no—
150C4×#67—92no—
151#894—32no—
152#894—92no—
153#887—32no—
154#887—92no—
155#885—32no—
156#885—32no—
157#885—92no—
158#885—92no—
159#67⋊C4—32no—
160#67⋊C4—92no—
161C23×C32—21no—
162C23×C32—21no—
163C23×C32C22×D311yescharacteristic
164C23×C32C22×D311yescharacteristic
165C23×C32C23×C311yescharacteristic
166C23×C32C23×C311yescharacteristic
167C22×C3⋊D3C23×C312yes—
168C22×C3×D3—32no—
169C32×D4—41no—
170C32×D4—41no—
171C32×D4—42no—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4—21no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—42no—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D312yes—
181C2×C32×C4C23×C311yescharacteristic
182C2×C32×C4C23×C312yes—
183C2×C32×C4C23×C312yes—
184#638—24no—
185#638—24no—
186C2×C3⋊Dic3—41no—
187C2×C3⋊Dic3C23×C312yes—
188C2×C3⋊Dic3C23×C312yes—
189C2×C3⋊Dic3C23×C312yes—
190C4⋊(C3⋊D3)—41no—
191C4×C3⋊D3—42no—
192#634—24no—
193C3×#67—64no—
194C3×#67—64no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—121no—
199C3×D12—121no—
200C3×C4×D3—122no—
201C3×Dic6—64no—
202#559—91noSylow
203C2×H(3,𝔽3)—41no—
204C2×H(3,𝔽3)—41no—
205C2×H(3,𝔽3)C2×#4112yes—
206C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
207(C3⋊D3)⋊C3—42no—
208C2×C3×Q8—92no—
209C2×C3×D4—31no—
210C2×C3×D4—31no—
211C2×C3×D4—61no—
212C2×C3×D4—91no—
213C2×C3×D4—92no—
214C2×C3×D4—92no—
215C2×C3×D4C2×C3×D311yescharacteristic
216C22×C3×C4—31no—
217C22×C3×C4—31no—
218C22×C3×C4—61no—
219C22×C3×C4—92no—
220C22×C3×C4—91no—
221C22×C3×C4C2×C3×D311yescharacteristic
222C2×#67—32no—
223C2×#67—32no—
224C2×#67—92no—
225C2×#67—92no—
226C22×Dic3—31no—
227C22×Dic3—91no—
228C2×D12—31no—
229C2×D12—91no—
230C2×C4×D3—32no—
231C2×C4×D3—92no—
232C2×Dic6—32no—
233C2×Dic6—92no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—32no—
236C3×C4⋊C4—62no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4C2×C3×D312yes—
240C3×C22⋊C4—32no—
241C3×C22⋊C4—31no—
242C3×C22⋊C4—32no—
243C3×C22⋊C4—31no—
244C3×C22⋊C4—62no—
245C3×C22⋊C4—61no—
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—92no—
254C3×C42—92no—
255C3×C42—91no—
256#217—31no—
257#217—32no—
258#217—91no—
259#217—92no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—32no—
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266Dic3⋊C4—32no—
267Dic3⋊C4—32no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270C4×Dic3—32no—
271C4×Dic3—32no—
272C4×Dic3—31no—
273C4×Dic3—92no—
274C4×Dic3—92no—
275C4×Dic3—91no—
276C22×C32—22no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—42no—
287C22×C32—41no—
288C22×C32—42no—
289C22×C32C23×D311yescharacteristic
290C22×C32C24×C311yescharacteristic
291C2×C3⋊D3—22no—
292C2×C3⋊D3—22no—
293C2×C3⋊D3—22no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C32×C4—22no—
298C32×C4—21no—
299C32×C4—22no—
300C32×C4—22no—
301C32×C4—22no—
302C32×C4—21no—
303C32×C4—42no—
304C32×C4—42no—
305C32×C4—41no—
306C3⋊Dic3—22no—
307C3⋊Dic3—22no—
308C3⋊Dic3C3×#4114yes—
309C3×Dic3—34no—
310C3×Dic3—62no—
311C3×Dic3—62no—
312#127—91no—
313#124—92no—
314#124—92no—
315#123—91no—
316#123—92no—
317#123(C3⋊D3)⋊C311yescharacteristic, complemented
318C4×Q8—92no—
319C4×D4—92no—
320C4×D4—92no—
321H(3,𝔽3)#55911yescharacteristic, complemented, Sylow
322C23×C3—31no—
323C23×C3—31no—
324C23×C3—31no—
325C23×C3—31no—
326C23×C3—61no—
327C23×C3—61no—
328C23×C3—92no—
329C23×C3C22×C3×D311yescharacteristic
330C23×C3C22×C3×D311yescharacteristic
331C22×D3—32no—
332C22×D3—92no—
333C3×Q8—184no—
334C3×D4—41no—
335C3×D4—121no—
336C3×D4—121no—
337C3×D4—122no—
338C3×D4—184no—
339C3×D4—184no—
340C3×D4—361no—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—41no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—61no—
351C2×C3×C4—92no—
352C2×C3×C4—92no—
353C2×C3×C4—92no—
354C2×C3×C4—121no—
355C2×C3×C4—121no—
356C2×C3×C4—122no—
357C2×C3×C4—362no—
358C2×C3×C4—361no—
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×C3×C4C22×C3×D312yes—
361C2×C3×C4C22×C3×D312yes—
362#67—64no—
363#67—64no—
364#67—184no—
365#67—184no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—92no—
370C2×Dic3—92no—
371C2×Dic3—92no—
372C2×Dic3—121no—
373C2×Dic3—361no—
374D12—121no—
375D12—361no—
376C4×D3—122no—
377C4×D3—362no—
378Dic6—64no—
379Dic6—184no—
380C2×C32—21no—
381C2×C32—22no—
382C2×C32—42no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—81no—
388C2×C32#100411yescharacteristic
389C2×C32D3×#4112yes—
390C2×C32C2×C3×#4112yes—
391C2×C32C3×H(5,𝔽2)11yescharacteristic
392C3⋊D3—42no—
393C3×D3—122no—
394C2×Q8—92no—
395C2×D4—92no—
396C2×D4—92no—
397C2×D4—91no—
398C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
399C22×C4—91no—
400C22×C4—92no—
401C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
402C4⋊C4—92no—
403C4⋊C4—92no—
404C4⋊C4C2×(C3⋊D3)⋊C312yes—
405C22⋊C4—92no—
406C22⋊C4—91no—
407C22⋊C4—92no—
408C22⋊C4—92no—
409C22⋊C4—92no—
410C22⋊C4C2×(C3⋊D3)⋊C312yes—
411C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
412C42—92no—
413C42—92no—
414C42—91no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—31no—
420C22×C3—31no—
421C22×C3—61no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—61no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—61no—
431C22×C3—122no—
432C22×C3—121no—
433C22×C3—122no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3—182no—
437C22×C3C23×C3×D311yescharacteristic
438C2×D3—62no—
439C2×D3—62no—
440C2×D3—62no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—182no—
444C3×C4—22no—
445C3×C4—21no—
446C3×C4—22no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—61no—
450C3×C4—62no—
451C3×C4—61no—
452C3×C4—62no—
453C3×C4—94no—
454C3×C4—122no—
455C3×C4—121no—
456C3×C4—122no—
457C3×C4—182no—
458C3×C4—182no—
459Dic3—34no—
460Dic3—62no—
461Dic3—62no—
462Dic3—94no—
463Dic3—182no—
464Dic3—182no—
465C32—21no—
466C32#566411yescharacteristic, complemented
467C32C3×#55911yescharacteristic, complemented
468C23—92no—
469C23C22×(C3⋊D3)⋊C311yescharacteristic
470C23C22×(C3⋊D3)⋊C311yescharacteristic
471Q8—184no—
472D4—41no—
473D4—184no—
474D4—184no—
475D4—361no—
476C2×C4—41no—
477C2×C4—92no—
478C2×C4—92no—
479C2×C4—92no—
480C2×C4—362no—
481C2×C4—361no—
482C2×C4C22×(C3⋊D3)⋊C312yes—
483C2×C4C22×(C3⋊D3)⋊C312yes—
484C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—32no—
488C2×C3—32no—
489C2×C3—41no—
490C2×C3—41no—
491C2×C3—61no—
492C2×C3—62no—
493C2×C3—121no—
494C2×C3—121no—
495C2×C3—121no—
496C2×C3—121no—
497C2×C3—122no—
498C2×C3—241no—
499C2×C3—362no—
500C2×C3C3×#100411yescharacteristic
501C2×C3C3×D3×#4112yes—
502D3—122no—
503D3—362no—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—182no—
509C22—182no—
510C22—182no—
511C22C23×(C3⋊D3)⋊C311yescharacteristic
512C4—22no—
513C4—21no—
514C4—22no—
515C4—94no—
516C4—182no—
517C4—182no—
518C3—31no—
519C3—31no—
520C3—61no—
521C3C3×#566411yescharacteristic
522C2—41no—
523C2—41no—
524C2—362no—
525C2#64361⋊C311yescharacteristic
526C2#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#64149⋊C3C2
hypercenter#123(C3⋊D3)⋊C3
hypocenterC32C3×#559
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. #64144⋊C3C2semidirect
4. C4×#638⋊C3C2semidirect
5. #64103⋊C3C2semidirect
6. #64096⋊C3C2semidirect
7. #64094⋊C3C2semidirect
8. #64094⋊C3C2semidirect
9. #64090⋊C3C2semidirect
10. #10596135C3semidirect
34. C32×#123D3semidirect
35. C32×#123C2×C3semidirect
37. #64144C2×C3semidirect
38. C4×#638C2×C3semidirect
39. #64103C2×C3semidirect
40. #64096C2×C3semidirect
41. #64094C2×C3semidirect
42. #64094C2×C3semidirect
43. #64090C2×C3semidirect
317. #123(C3⋊D3)⋊C3semidirect
321. H(3,𝔽3)#559semidirect
466. C32#5664semidirect
467. C32C3×#559semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1232
#64149⋊C32
#64144⋊C32
C4×#638⋊C32
#64103⋊C32
#64096⋊C32
#64094⋊C32
#64090⋊C32
#105961353
C3×#56643