Finite Group Databasebeta

← #10596139⋊C3 #13153118

Subgroups — #10596139⋊C3 #13153118

Order 1728.

1161 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C211yesmaximal, characteristic, complemented
2C2×#3830⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
4#64153⋊C3C211yesmaximal, characteristic, complemented
5#64149⋊C3C211yesmaximal, characteristic, complemented
6#64093⋊C3C214yesmaximal, complemented
7#64090⋊C3C214yesmaximal, complemented
8#64084⋊C3C211yesmaximal, characteristic, complemented
9C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
10#10596139C311yesmaximal, characteristic, complemented
11C3×#5668—31nomaximal
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C2×#638⋊C3C2214yes—
16C22×(C3⋊Dic3)⋊C3C2212yes—
17#88722—24no—
18#3830⋊C3—24no—
19C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
20C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
21C2×C4×(C3⋊D3)⋊C3C2212yes—
22C2×C4×(C3⋊D3)⋊C3C2212yes—
23C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
24C2×#634⋊C3C2211yescharacteristic
25C22⋊C4×H(3,𝔽3)C2214yes—
26C42×H(3,𝔽3)C2211yescharacteristic
27#3752⋊C3C2214yes—
28#3750⋊C3C2214yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2212yes—
31C2×#3832C2×C311yescharacteristic, complemented
32C2×#3830C2×C311yescharacteristic, complemented
33C2×C3×#239—31no—
34C2×C3×#237—31no—
35C32×#124—21no—
36C32×#124D311yescharacteristic, complemented
37C32×#124C2×C311yescharacteristic, complemented
38#64153C2×C311yescharacteristic, complemented
39#64149C2×C311yescharacteristic, complemented
40#64093C2×C314yescomplemented
41#64090C2×C314yescomplemented
42#64084C2×C311yescharacteristic, complemented
43C42×C3⋊D3C2×C311yescharacteristic, complemented
44C3×#944—31no—
45C3×#940—31no—
46C3×#884—34no—
47C3×#67⋊C4—34no—
48C3×#875—31no—
49C3×C42×D3—31no—
50C23×H(3,𝔽3)C2312yes—
51C22×(C3⋊D3)⋊C3C2312yes—
52C22×(C3⋊D3)⋊C3C2311yescharacteristic
53Q8×H(3,𝔽3)—22no—
54D4×H(3,𝔽3)—22no—
55C2×C4×H(3,𝔽3)C2312yes—
56C2×C4×H(3,𝔽3)C2312yes—
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58#638⋊C3—24no—
59#638⋊C3—28no—
60C2×(C3⋊Dic3)⋊C3—24no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C4⋊((C3⋊D3)⋊C3)—24no—
65C4⋊((C3⋊D3)⋊C3)—22no—
66C4×(C3⋊D3)⋊C3—24no—
67C4×(C3⋊D3)⋊C3—24no—
68C4×(C3⋊D3)⋊C3D414yescomplemented
69#634⋊C3—22no—
70C3×#532—91no—
71#5668—31no—
72#5668—91no—
73C2×C32×Q8—21no—
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C22×C311yescharacteristic
76C2×C32×D4—21no—
77C2×C32×D4C2×D311yescharacteristic
78C2×C32×D4C22×C311yescharacteristic
79C2×#638C22×C314yes—
80C2×#638C22×C312yes—
81C22×C3⋊Dic3C22×C312yes—
82#3832—24no—
83#3830—24no—
84C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
85C2×C4⋊(C3⋊D3)C22×C312yes—
86C2×C4×C3⋊D3C22×C312yes—
87C2×C4×C3⋊D3C22×C312yes—
88C2×C4×C3⋊D3C22×C311yescharacteristic
89C2×#634C22×C311yescharacteristic
90C2×C3×#67—34no—
91C2×C3×#67—32no—
92C22×C3×Dic3—32no—
93C3×#239—64no—
94C3×#237—64no—
95C2×C3×D12—32no—
96C2×C3×D12—31no—
97C2×C3×C4×D3—31no—
98C2×C3×C4×D3—32no—
99C2×C3×C4×D3—32no—
100C2×C3×Dic6—31no—
101C32×C22⋊C4—24no—
102C32×C22⋊C4C2×D314yes—
103C32×C22⋊C4C22×C314yes—
104C32×C42—21no—
105C32×C42C2×D311yescharacteristic
106C32×C42C22×C311yescharacteristic
107#3752C22×C314yes—
108#3750C22×C314yes—
109C4×C3⋊Dic3C22×C312yes—
110C4×C3⋊Dic3C22×C311yescharacteristic
111C3×(C2×D3)⋊C4—34no—
112C3×Dic3⋊C4—34no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—31no—
115C22×H(3,𝔽3)—22no—
116C22×H(3,𝔽3)—24no—
117C22×H(3,𝔽3)C2411yescharacteristic
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—22no—
120C2×(C3⋊D3)⋊C3—24no—
121C2×(C3⋊D3)⋊C3C2×D412yes—
122C4×H(3,𝔽3)—22no—
123C4×H(3,𝔽3)—22no—
124C4×H(3,𝔽3)C2×D412yes—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3C2×D412yes—
127(C3⋊Dic3)⋊C3#4114yescomplemented
128C2×C3×#41—91no—
129C2×C3×#41—91no—
130C2×#239—31no—
131C2×#239—91no—
132C2×#237—31no—
133C2×#237—91no—
134C3×C4⋊Q8—91no—
135C3×#127—91no—
136C3×#124—31no—
137C3×#124—31no—
138C3×#124—61no—
139C3×#124—91no—
140C3×#124C3×D311yescharacteristic
141C3×#121—94no—
142C3×C4×D4—94no—
143C2×C3×C42—91no—
144#944—31no—
145#944—91no—
146#940—31no—
147#940—91no—
148#884—34no—
149#884—94no—
150#67⋊C4—34no—
151#67⋊C4—94no—
152#875—31no—
153#875—91no—
154C42×D3—31no—
155C42×D3—91no—
156C23×C32—22no—
157C23×C32C22×D312yes—
158C23×C32C23×C312yes—
159C22×C3⋊D3C23×C312yes—
160C22×C3⋊D3C23×C311yescharacteristic
161C22×C3×D3—31no—
162C22×C3×D3—32no—
163C32×Q8—22no—
164C32×Q8—22no—
165C32×Q8—42no—
166C32×D4—22no—
167C32×D4—22no—
168C32×D4—42no—
169C2×C32×C4—21no—
170C2×C32×C4—22no—
171C2×C32×C4—22no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C23×C312yes—
176C2×C32×C4C23×C312yes—
177C2×C32×C4C23×C311yescharacteristic
178#638—24no—
179#638—28no—
180C2×C3⋊Dic3—24no—
181C2×C3⋊Dic3C23×C312yes—
182C2×C3⋊Dic3C23×C311yescharacteristic
183C2×C3⋊Dic3C23×C312yes—
184C4⋊(C3⋊D3)—24no—
185C4⋊(C3⋊D3)—22no—
186C4×C3⋊D3—24no—
187C4×C3⋊D3—24no—
188C4×C3⋊D3C3×D414yescomplemented
189#634—22no—
190C3×#67—68no—
191C3×#67—64no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—64no—
196C3×D12—64no—
197C3×D12—62no—
198C3×C4×D3—34no—
199C3×C4×D3—64no—
200C3×C4×D3—64no—
201C3×Dic6—62no—
202#532—91noSylow
203C2×H(3,𝔽3)—42no—
204C2×H(3,𝔽3)C22×D411yescharacteristic
205C2×H(3,𝔽3)C2×#4112yes—
206(C3⋊D3)⋊C3—22no—
207(C3⋊D3)⋊C3—42no—
208C3×#41—184no—
209C3×#41—184no—
210C2×C3×Q8—31no—
211C2×C3×Q8—31no—
212C2×C3×Q8—61no—
213C2×C3×Q8—91no—
214C2×C3×Q8C2×C3×D311yescharacteristic
215C2×C3×D4—31no—
216C2×C3×D4—31no—
217C2×C3×D4—61no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C2×C3×D4—94no—
221C2×C3×D4—92no—
222C2×C3×D4C2×C3×D311yescharacteristic
223C22×C3×C4—91no—
224C22×C3×C4—92no—
225C22×C3×C4—92no—
226C22×C3×C4—92no—
227C2×#67—34no—
228C2×#67—32no—
229C2×#67—94no—
230C2×#67—92no—
231C22×Dic3—32no—
232C22×Dic3—92no—
233#239—64no—
234#239—184no—
235#237—64no—
236#237—184no—
237C2×D12—31no—
238C2×D12—32no—
239C2×D12—91no—
240C2×D12—92no—
241C2×C4×D3—31no—
242C2×C4×D3—32no—
243C2×C4×D3—32no—
244C2×C4×D3—91no—
245C2×C4×D3—92no—
246C2×C4×D3—92no—
247C2×Dic6—31no—
248C2×Dic6—91no—
249C3×C4⋊C4—94no—
250C3×C22⋊C4—34no—
251C3×C22⋊C4—34no—
252C3×C22⋊C4—64no—
253C3×C22⋊C4—94no—
254C3×C22⋊C4C2×C3×D314yes—
255C3×C42—31no—
256C3×C42—31no—
257C3×C42—61no—
258C3×C42—92no—
259C3×C42—91no—
260C3×C42C2×C3×D311yescharacteristic
261(C2×D3)⋊C4—34no—
262(C2×D3)⋊C4—94no—
263Dic3⋊C4—34no—
264Dic3⋊C4—94no—
265C4×Dic3—32no—
266C4×Dic3—31no—
267C4×Dic3—92no—
268C4×Dic3—91no—
269C22×C32—22no—
270C22×C32—24no—
271C22×C32—22no—
272C22×C32—24no—
273C22×C32—21no—
274C22×C32—42no—
275C22×C32—44no—
276C22×C32C23×D311yescharacteristic
277C22×C32C24×C311yescharacteristic
278C2×C3⋊D3—22no—
279C2×C3⋊D3—22no—
280C2×C3⋊D3—24no—
281C2×C3⋊D3C2×C3×D412yes—
282C2×C3×D3—32no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—64no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—42no—
292C32×C4—42no—
293C32×C4D3×D412yes—
294C32×C4C2×C3×D412yes—
295C3⋊Dic3—22no—
296C3⋊Dic3C2×C3×D412yes—
297C3⋊Dic3C3×#4114yescomplemented
298C3×Dic3—32no—
299C3×Dic3—34no—
300C3×Dic3—62no—
301C2×#41—91no—
302C2×#41—91no—
303C4⋊Q8—91no—
304#127—91no—
305#124—91no—
306#124(C3⋊D3)⋊C311yescharacteristic, complemented
307#121—94no—
308C4×D4—94no—
309C2×C42—91no—
310H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
311C23×C3—32no—
312C23×C3—32no—
313C23×C3—62no—
314C23×C3—91no—
315C23×C3—92no—
316C23×C3C22×C3×D312yes—
317C22×D3—31no—
318C22×D3—32no—
319C22×D3—91no—
320C22×D3—92no—
321C3×Q8—22no—
322C3×Q8—62no—
323C3×Q8—62no—
324C3×Q8—122no—
325C3×Q8—182no—
326C3×D4—22no—
327C3×D4—62no—
328C3×D4—62no—
329C3×D4—122no—
330C3×D4—182no—
331C3×D4—184no—
332C3×D4—188no—
333C3×D4—184no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—32no—
337C2×C3×C4—32no—
338C2×C3×C4—32no—
339C2×C3×C4—32no—
340C2×C3×C4—61no—
341C2×C3×C4—62no—
342C2×C3×C4—62no—
343C2×C3×C4—94no—
344C2×C3×C4—92no—
345C2×C3×C4—91no—
346C2×C3×C4—92no—
347C2×C3×C4—184no—
348C2×C3×C4—184no—
349C2×C3×C4—184no—
350C2×C3×C4C22×C3×D312yes—
351C2×C3×C4C22×C3×D312yes—
352C2×C3×C4C22×C3×D311yescharacteristic
353#67—68no—
354#67—64no—
355#67—188no—
356#67—184no—
357C2×Dic3—32no—
358C2×Dic3—31no—
359C2×Dic3—32no—
360C2×Dic3—64no—
361C2×Dic3—92no—
362C2×Dic3—91no—
363C2×Dic3—92no—
364C2×Dic3—184no—
365D12—64no—
366D12—62no—
367D12—182no—
368D12—184no—
369C4×D3—34no—
370C4×D3—64no—
371C4×D3—64no—
372C4×D3—94no—
373C4×D3—184no—
374C4×D3—184no—
375Dic6—62no—
376Dic6—182no—
377C2×C32—21no—
378C2×C32—22no—
379C2×C32—42no—
380C2×C32—42no—
381C2×C32—82no—
382C2×C32C2×D3×D411yescharacteristic
383C2×C32D3×#4112yes—
384C2×C32C22×C3×D411yescharacteristic
385C2×C32C2×C3×#4112yes—
386C3⋊D3—22no—
387C3⋊D3—42no—
388C3×D3—62no—
389C3×D3—122no—
390#41—184no—
391#41—184no—
392C2×Q8—91no—
393C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
394C2×D4—94no—
395C2×D4—92no—
396C2×D4—92no—
397C2×D4—91no—
398C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
399C22×C4—91no—
400C22×C4—92no—
401C22×C4—92no—
402C22×C4—92no—
403C4⋊C4—94no—
404C22⋊C4—94no—
405C22⋊C4C2×(C3⋊D3)⋊C314yes—
406C42—92no—
407C42—91no—
408C42C2×(C3⋊D3)⋊C311yescharacteristic
409C22×C3—22no—
410C22×C3—24no—
411C22×C3—31no—
412C22×C3—31no—
413C22×C3—62no—
414C22×C3—64no—
415C22×C3—62no—
416C22×C3—64no—
417C22×C3—61no—
418C22×C3—92no—
419C22×C3—122no—
420C22×C3—124no—
421C22×C3—182no—
422C22×C3—182no—
423C22×C3—184no—
424C22×C3C23×C3×D311yescharacteristic
425C2×D3—32no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—64no—
429C2×D3—92no—
430C2×D3—182no—
431C2×D3—182no—
432C2×D3—184no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—32no—
436C3×C4—32no—
437C3×C4—62no—
438C3×C4—62no—
439C3×C4—62no—
440C3×C4—62no—
441C3×C4—62no—
442C3×C4—92no—
443C3×C4—94no—
444C3×C4—122no—
445C3×C4—122no—
446C3×C4—182no—
447C3×C4C3×D3×D412yes—
448Dic3—32no—
449Dic3—34no—
450Dic3—62no—
451Dic3—92no—
452Dic3—94no—
453Dic3—182no—
454C32—21no—
455C32#566811yescharacteristic, complemented
456C32C3×#53211yescharacteristic, complemented
457C23—91no—
458C23—92no—
459C23C22×(C3⋊D3)⋊C312yes—
460Q8—22no—
461Q8—182no—
462D4—22no—
463D4—184no—
464D4—188no—
465D4—184no—
466D4—182no—
467C2×C4—92no—
468C2×C4—91no—
469C2×C4—94no—
470C2×C4—92no—
471C2×C4—184no—
472C2×C4—184no—
473C2×C4—184no—
474C2×C4C22×(C3⋊D3)⋊C312yes—
475C2×C4C22×(C3⋊D3)⋊C312yes—
476C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
477C2×C3—31no—
478C2×C3—31no—
479C2×C3—32no—
480C2×C3—32no—
481C2×C3—42no—
482C2×C3—61no—
483C2×C3—62no—
484C2×C3—122no—
485C2×C3—122no—
486C2×C3—182no—
487C2×C3—242no—
488C2×C3—362no—
489C2×C3C2×C3×D3×D411yescharacteristic
490C2×C3C3×D3×#4112yes—
491D3—62no—
492D3—122no—
493D3—182no—
494D3—362no—
495C22—22no—
496C22—24no—
497C22—92no—
498C22—182no—
499C22—182no—
500C22—184no—
501C22C23×(C3⋊D3)⋊C311yescharacteristic
502C4—22no—
503C4—22no—
504C4—92no—
505C4—94no—
506C4—182no—
507C4D4×(C3⋊D3)⋊C312yes—
508C3—31no—
509C3—31no—
510C3—61no—
511C3C3×#566811yescharacteristic
512C2—42no—
513C2—182no—
514C2—362no—
515C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
516C2#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×#532
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×#3830⋊C3C2semidirect
3. H(3,𝔽3)×#124C2semidirect
4. #64153⋊C3C2semidirect
5. #64149⋊C3C2semidirect
6. #64093⋊C3C2semidirect
7. #64090⋊C3C2semidirect
8. #64084⋊C3C2semidirect
9. C42×(C3⋊D3)⋊C3C2semidirect
10. #10596139C3semidirect
31. C2×#3832C2×C3semidirect
32. C2×#3830C2×C3semidirect
36. C32×#124D3semidirect
37. C32×#124C2×C3semidirect
38. #64153C2×C3semidirect
39. #64149C2×C3semidirect
40. #64093C2×C3semidirect
41. #64090C2×C3semidirect
42. #64084C2×C3semidirect
43. C42×C3⋊D3C2×C3semidirect
68. C4×(C3⋊D3)⋊C3D4semidirect
127. (C3⋊Dic3)⋊C3#41semidirect
188. C4×C3⋊D3C3×D4semidirect
297. C3⋊Dic3C3×#41semidirect
306. #124(C3⋊D3)⋊C3semidirect
310. H(3,𝔽3)#532semidirect
455. C32#5668semidirect
456. C32C3×#532semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×#3830⋊C32
H(3,𝔽3)×#1242
#64153⋊C32
#64149⋊C32
#64093⋊C32
#64090⋊C32
#64084⋊C32
C42×(C3⋊D3)⋊C32
#105961393
C3×#56683