Finite Group Databasebeta

← #10596141⋊C3 #13153111

Subgroups — #10596141⋊C3 #13153111

Order 1728.

902 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic, complemented
3#64153⋊C3C211yesmaximal, characteristic, complemented
4#64148⋊C3C212yesmaximal, complemented
5#64094⋊C3C212yesmaximal, complemented
6#64093⋊C3C212yesmaximal, complemented
7#64090⋊C3C212yesmaximal, complemented
8#64086⋊C3C212yesmaximal, complemented
9#64081⋊C3C212yesmaximal, complemented
10#10596141C311yesmaximal, characteristic, complemented
11C3×#5670—31nomaximal
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C2×#638⋊C3C2212yes—
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20C22⋊C4×H(3,𝔽3)C2212yes—
21C22⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23#3757⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33C32×#124—21no—
34C32×#124D311yescharacteristic, complemented
35C32×#124C2×C311yescharacteristic, complemented
36#64156C2×C311yescharacteristic, complemented
37#64153C2×C311yescharacteristic, complemented
38#64148C2×C312yescomplemented
39#64094C2×C312yescomplemented
40#64093C2×C312yescomplemented
41#64090C2×C312yescomplemented
42#64086C2×C312yescomplemented
43#64081C2×C312yescomplemented
44C3×#947—31no—
45C3×#944—31no—
46C3×#939—32no—
47C3×#885—32no—
48C3×#884—32no—
49C3×#67⋊C4—32no—
50C3×#877—32no—
51C3×#872—32no—
52C23×H(3,𝔽3)C2312yes—
53C22×(C3⋊D3)⋊C3C2312yes—
54Q8×H(3,𝔽3)—41no—
55D4×H(3,𝔽3)—41no—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×C4×H(3,𝔽3)C2312yes—
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60#638⋊C3—24no—
61#638⋊C3—42no—
62C2×(C3⋊Dic3)⋊C3—24no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C4⋊((C3⋊D3)⋊C3)—41no—
67C4×(C3⋊D3)⋊C3—24no—
68#634⋊C3—41no—
69C3×#555—91no—
70#5670—31no—
71#5670—91no—
72C2×C32×Q8—21no—
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C22×C311yescharacteristic
75C2×C32×D4—21no—
76C2×C32×D4C2×D311yescharacteristic
77C2×C32×D4C22×C311yescharacteristic
78C2×#638C22×C312yes—
79C2×#638C22×C312yes—
80C22×C3⋊Dic3C22×C312yes—
81C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
82C2×C4×C3⋊D3C22×C312yes—
83C2×#634C22×C311yescharacteristic
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C22×C3×Dic3—32no—
87C2×C3×D12—31no—
88C2×C3×C4×D3—32no—
89C2×C3×Dic6—31no—
90C32×C22⋊C4—22no—
91C32×C22⋊C4—22no—
92C32×C22⋊C4C2×D312yes—
93C32×C22⋊C4C2×D312yes—
94C32×C22⋊C4C22×C312yes—
95C32×C22⋊C4C22×C312yes—
96C32×C42—21no—
97C32×C42C2×D311yescharacteristic
98C32×C42C22×C311yescharacteristic
99#3757C22×C312yes—
100#3752C22×C312yes—
101#3752C22×C312yes—
102#3752C22×C312yes—
103#3750C22×C312yes—
104#3750C22×C312yes—
105#3750C22×C312yes—
106C4×C3⋊Dic3C22×C312yes—
107C4×C3⋊Dic3C22×C311yescharacteristic
108C4×C3⋊Dic3C22×C311yescharacteristic
109C3×#217—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—32no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—31no—
119C22×H(3,𝔽3)—22no—
120C22×H(3,𝔽3)—22no—
121C22×H(3,𝔽3)—22no—
122C22×H(3,𝔽3)C2411yescharacteristic
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C2×(C3⋊D3)⋊C3—22no—
126C4×H(3,𝔽3)—22no—
127C4×H(3,𝔽3)—21no—
128C4×H(3,𝔽3)—21no—
129C4×H(3,𝔽3)—21no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3#4114yes—
133C3×C4⋊Q8—91no—
134C3×#124—31no—
135C3×#124—31no—
136C3×#124—61no—
137C3×#124—92no—
138C3×#124—91no—
139C3×#124C3×D311yescharacteristic
140C3×#121—92no—
141C3×#121—92no—
142C3×C4×D4—92no—
143C3×#117—92no—
144C3×#117—92no—
145#947—31no—
146#947—91no—
147#944—31no—
148#944—91no—
149#939—32no—
150#939—92no—
151#885—32no—
152#885—92no—
153#884—32no—
154#884—92no—
155#67⋊C4—32no—
156#67⋊C4—92no—
157#877—32no—
158#877—92no—
159#872—32no—
160#872—92no—
161C23×C32—22no—
162C23×C32C22×D312yes—
163C23×C32C23×C312yes—
164C22×C3⋊D3C23×C312yes—
165C22×C3×D3—32no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—42no—
169C32×D4—41no—
170C32×D4—41no—
171C32×D4—42no—
172C2×C32×C4—21no—
173C2×C32×C4—22no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C23×C311yescharacteristic
181C2×C32×C4C23×C312yes—
182C2×C32×C4C23×C311yescharacteristic
183C2×C32×C4C23×C311yescharacteristic
184#638—24no—
185#638—42no—
186C2×C3⋊Dic3—24no—
187C2×C3⋊Dic3C23×C312yes—
188C2×C3⋊Dic3C23×C312yes—
189C2×C3⋊Dic3C23×C312yes—
190C4⋊(C3⋊D3)—41no—
191C4×C3⋊D3—24no—
192#634—41no—
193C3×#67—64no—
194C3×#67—122no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—64no—
199C3×D12—121no—
200C3×C4×D3—64no—
201C3×Dic6—121no—
202#555—91noSylow
203C2×H(3,𝔽3)—42no—
204C2×H(3,𝔽3)C2×#4112yes—
205C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
206(C3⋊D3)⋊C3—42no—
207C2×C3×Q8—31no—
208C2×C3×Q8—31no—
209C2×C3×Q8—61no—
210C2×C3×Q8—91no—
211C2×C3×Q8C2×C3×D311yescharacteristic
212C2×C3×D4—31no—
213C2×C3×D4—31no—
214C2×C3×D4—61no—
215C2×C3×D4—91no—
216C2×C3×D4—92no—
217C2×C3×D4—92no—
218C2×C3×D4C2×C3×D311yescharacteristic
219C22×C3×C4—92no—
220C22×C3×C4—92no—
221C2×#67—32no—
222C2×#67—32no—
223C2×#67—92no—
224C2×#67—92no—
225C22×Dic3—32no—
226C22×Dic3—92no—
227C2×D12—31no—
228C2×D12—91no—
229C2×C4×D3—32no—
230C2×C4×D3—92no—
231C2×Dic6—31no—
232C2×Dic6—91no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C22⋊C4—32no—
237C3×C22⋊C4—32no—
238C3×C22⋊C4—32no—
239C3×C22⋊C4—32no—
240C3×C22⋊C4—62no—
241C3×C22⋊C4—62no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4C2×C3×D312yes—
247C3×C22⋊C4C2×C3×D312yes—
248C3×C42—31no—
249C3×C42—31no—
250C3×C42—61no—
251C3×C42—92no—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42C2×C3×D311yescharacteristic
255#217—32no—
256#217—92no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263Dic3⋊C4—32no—
264Dic3⋊C4—32no—
265Dic3⋊C4—32no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269C4×Dic3—32no—
270C4×Dic3—31no—
271C4×Dic3—31no—
272C4×Dic3—92no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C22×C32—22no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—21no—
282C22×C32—42no—
283C22×C32—42no—
284C22×C32—42no—
285C22×C32C23×D311yescharacteristic
286C22×C32C24×C311yescharacteristic
287C2×C3⋊D3—22no—
288C2×C3⋊D3—22no—
289C2×C3⋊D3—22no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—22no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—42no—
303C32×C4—41no—
304C32×C4—41no—
305C3⋊Dic3—22no—
306C3⋊Dic3—22no—
307C3⋊Dic3C3×#4114yes—
308C3×Dic3—34no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C4⋊Q8—91no—
312#124—91no—
313#124—92no—
314#124(C3⋊D3)⋊C311yescharacteristic, complemented
315#121—92no—
316#121—92no—
317C4×D4—92no—
318#117—92no—
319#117—92no—
320H(3,𝔽3)#55511yescharacteristic, complemented, Sylow
321C23×C3—32no—
322C23×C3—32no—
323C23×C3—62no—
324C23×C3—92no—
325C23×C3C22×C3×D312yes—
326C22×D3—32no—
327C22×D3—92no—
328C3×Q8—41no—
329C3×Q8—121no—
330C3×Q8—121no—
331C3×Q8—122no—
332C3×Q8—361no—
333C3×D4—41no—
334C3×D4—121no—
335C3×D4—121no—
336C3×D4—122no—
337C3×D4—184no—
338C3×D4—361no—
339C3×D4—362no—
340C2×C3×C4—32no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—32no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—61no—
349C2×C3×C4—62no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—92no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4—184no—
356C2×C3×C4—184no—
357C2×C3×C4C22×C3×D311yescharacteristic
358C2×C3×C4C22×C3×D312yes—
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×C3×C4C22×C3×D311yescharacteristic
361#67—64no—
362#67—122no—
363#67—184no—
364#67—362no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—64no—
369C2×Dic3—92no—
370C2×Dic3—92no—
371C2×Dic3—92no—
372C2×Dic3—184no—
373D12—121no—
374D12—361no—
375C4×D3—64no—
376C4×D3—184no—
377Dic6—121no—
378Dic6—361no—
379C2×C32—21no—
380C2×C32—22no—
381C2×C32—42no—
382C2×C32—42no—
383C2×C32—82no—
384C2×C32#100411yescharacteristic
385C2×C32D3×#4112yes—
386C2×C32C2×C3×#4112yes—
387C2×C32C3×H(5,𝔽2)11yescharacteristic
388C3⋊D3—42no—
389C3×D3—122no—
390C2×Q8—91no—
391C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
392C2×D4—92no—
393C2×D4—92no—
394C2×D4—91no—
395C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
396C22×C4—92no—
397C22×C4—92no—
398C4⋊C4—92no—
399C4⋊C4—92no—
400C4⋊C4—92no—
401C22⋊C4—92no—
402C22⋊C4—92no—
403C22⋊C4—92no—
404C22⋊C4—92no—
405C22⋊C4C2×(C3⋊D3)⋊C312yes—
406C22⋊C4C2×(C3⋊D3)⋊C312yes—
407C42—91no—
408C42—92no—
409C42—91no—
410C42C2×(C3⋊D3)⋊C311yescharacteristic
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—31no—
415C22×C3—31no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—62no—
420C22×C3—62no—
421C22×C3—62no—
422C22×C3—61no—
423C22×C3—122no—
424C22×C3—122no—
425C22×C3—122no—
426C22×C3—182no—
427C22×C3—182no—
428C22×C3—182no—
429C22×C3C23×C3×D311yescharacteristic
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—182no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—22no—
440C3×C4—61no—
441C3×C4—62no—
442C3×C4—61no—
443C3×C4—62no—
444C3×C4—61no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—62no—
449C3×C4—94no—
450C3×C4—121no—
451C3×C4—121no—
452C3×C4—122no—
453C3×C4—182no—
454C3×C4—182no—
455Dic3—34no—
456Dic3—62no—
457Dic3—62no—
458Dic3—94no—
459Dic3—182no—
460Dic3—182no—
461C32—21no—
462C32#567011yescharacteristic, complemented
463C32C3×#55511yescharacteristic, complemented
464C23—92no—
465C23C22×(C3⋊D3)⋊C312yes—
466Q8—41no—
467Q8—361no—
468D4—41no—
469D4—184no—
470D4—361no—
471D4—362no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—184no—
476C2×C4—184no—
477C2×C4C22×(C3⋊D3)⋊C312yes—
478C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
479C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
480C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
481C2×C3—31no—
482C2×C3—31no—
483C2×C3—32no—
484C2×C3—32no—
485C2×C3—42no—
486C2×C3—61no—
487C2×C3—62no—
488C2×C3—122no—
489C2×C3—122no—
490C2×C3—242no—
491C2×C3—362no—
492C2×C3C3×#100411yescharacteristic
493C2×C3C3×D3×#4112yes—
494D3—122no—
495D3—362no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—182no—
500C22—182no—
501C22—182no—
502C22C23×(C3⋊D3)⋊C311yescharacteristic
503C4—21no—
504C4—21no—
505C4—21no—
506C4—22no—
507C4—94no—
508C4—182no—
509C4—182no—
510C3—31no—
511C3—31no—
512C3—61no—
513C3C3×#567011yescharacteristic
514C2—42no—
515C2—362no—
516C2#64361⋊C311yescharacteristic
517C2#41×(C3⋊D3)⋊C312yes—
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×#555
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. #64156⋊C3C2semidirect
3. #64153⋊C3C2semidirect
4. #64148⋊C3C2semidirect
5. #64094⋊C3C2semidirect
6. #64093⋊C3C2semidirect
7. #64090⋊C3C2semidirect
8. #64086⋊C3C2semidirect
9. #64081⋊C3C2semidirect
10. #10596141C3semidirect
34. C32×#124D3semidirect
35. C32×#124C2×C3semidirect
36. #64156C2×C3semidirect
37. #64153C2×C3semidirect
38. #64148C2×C3semidirect
39. #64094C2×C3semidirect
40. #64093C2×C3semidirect
41. #64090C2×C3semidirect
42. #64086C2×C3semidirect
43. #64081C2×C3semidirect
314. #124(C3⋊D3)⋊C3semidirect
320. H(3,𝔽3)#555semidirect
462. C32#5670semidirect
463. C32C3×#555semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
#64156⋊C32
#64153⋊C32
#64148⋊C32
#64094⋊C32
#64093⋊C32
#64090⋊C32
#64086⋊C32
#64081⋊C32
#105961413
C3×#56703