Finite Group Databasebeta

← #10596186⋊C3 #13153274

Subgroups — #10596186⋊C3 #13153274

Order 1728.

1277 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3830⋊C3C211yesmaximal, characteristic, complemented
2C2×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
4#64149⋊C3C211yesmaximal, characteristic, complemented
5#64148⋊C3C214yesmaximal, complemented
6#64146⋊C3C214yesmaximal, complemented
7#64145⋊C3C211yesmaximal, characteristic, complemented
8#64084⋊C3C211yesmaximal, characteristic, complemented
9#64081⋊C3C211yesmaximal, characteristic, complemented
10#10596186C311yesmaximal, characteristic, complemented
11C3×#5715—31nomaximal
12C23×(C3⋊D3)⋊C3C2212yes—
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×D4×H(3,𝔽3)C2214yescomplemented
16C2×#638⋊C3C2212yes—
17C2×#638⋊C3C2212yes—
18C2×#638⋊C3C2214yescomplemented
19C22×(C3⋊Dic3)⋊C3C2212yescomplemented
20#3830⋊C3—24no—
21D4×(C3⋊D3)⋊C3—24no—
22C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
23C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
24C2×#634⋊C3C2211yescharacteristic, complemented
25C42×H(3,𝔽3)C2211yescharacteristic, complemented
26#3757⋊C3C2214yescomplemented
27#3752⋊C3C2212yes—
28#3752⋊C3C2214yescomplemented
29#3750⋊C3C2212yescomplemented
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
31C2×#3830C2×C311yescharacteristic, complemented
32C2×D4×C3⋊D3C2×C311yescharacteristic, complemented
33C2×C3×#237—31no—
34C2×C3×D3×D4—31no—
35C32×#127—21no—
36C32×#127D311yescharacteristic, complemented
37C32×#127C2×C311yescharacteristic, complemented
38#64149C2×C311yescharacteristic, complemented
39#64148C2×C314yescomplemented
40#64146C2×C314yescomplemented
41#64145C2×C311yescharacteristic, complemented
42#64084C2×C311yescharacteristic, complemented
43#64081C2×C311yescharacteristic, complemented
44C3×#940—31no—
45C3×#939—34no—
46C3×#937—34no—
47C3×#936—31no—
48C3×#875—31no—
49C3×#872—31no—
50C23×H(3,𝔽3)C2312yes—
51C23×H(3,𝔽3)C2312yes—
52C22×(C3⋊D3)⋊C3—24no—
53C22×(C3⋊D3)⋊C3—44no—
54C22×(C3⋊D3)⋊C3C2312yes—
55C22×(C3⋊D3)⋊C3C2311yescharacteristic
56D4×H(3,𝔽3)—22no—
57D4×H(3,𝔽3)—22no—
58D4×H(3,𝔽3)—44no—
59C2×C4×H(3,𝔽3)C2312yes—
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61#638⋊C3—24no—
62#638⋊C3—24no—
63#638⋊C3—44no—
64C2×(C3⋊Dic3)⋊C3—24no—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C4⋊((C3⋊D3)⋊C3)—22no—
69C4×(C3⋊D3)⋊C3D414yescomplemented
70#634⋊C3—22no—
71C3×#535—91no—
72#5715—31no—
73#5715—91no—
74C23×C3⋊D3C22×C312yes—
75C23×C3×D3—32no—
76C2×C32×D4—21no—
77C2×C32×D4—21no—
78C2×C32×D4—24no—
79C2×C32×D4C2×D311yescharacteristic
80C2×C32×D4C2×D311yescharacteristic, complemented
81C2×C32×D4C2×D314yescomplemented
82C2×C32×D4C22×C311yescharacteristic
83C2×C32×D4C22×C311yescharacteristic, complemented
84C2×C32×D4C22×C314yescomplemented
85C2×#638C22×C312yes—
86C2×#638C22×C312yes—
87C2×#638C22×C314yescomplemented
88C22×C3⋊Dic3C22×C312yescomplemented
89#3830—24no—
90D4×C3⋊D3—24no—
91C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
92C2×C4×C3⋊D3C22×C311yescharacteristic
93C2×#634C22×C311yescharacteristic, complemented
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C2×C3×#67—34no—
97C22×C3×Dic3—32no—
98C3×#237—64no—
99C3×D3×D4—64no—
100C2×C3×D12—31no—
101C2×C3×C4×D3—31no—
102C2×C3×Dic6—31no—
103C32×C42—21no—
104C32×C42C2×D311yescharacteristic, complemented
105C32×C42C22×C311yescharacteristic, complemented
106#3757C22×C314yescomplemented
107#3752C22×C312yes—
108#3752C22×C314yescomplemented
109#3750C22×C312yescomplemented
110C4×C3⋊Dic3C22×C311yescharacteristic, complemented
111C3×#217—34no—
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)—22no—
118C22×H(3,𝔽3)—24no—
119C22×H(3,𝔽3)—24no—
120C22×H(3,𝔽3)C2411yescharacteristic
121C2×(C3⋊D3)⋊C3—22no—
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—24no—
124C2×(C3⋊D3)⋊C3—48no—
125C2×(C3⋊D3)⋊C3C2×D412yes—
126C4×H(3,𝔽3)—22no—
127C4×H(3,𝔽3)C2×D412yes—
128(C3⋊Dic3)⋊C3—22no—
129(C3⋊Dic3)⋊C3—22no—
130(C3⋊Dic3)⋊C3C2×D412yes—
131C2×C3×#41—91no—
132C22×C3×D4—91no—
133C2×#237—31no—
134C2×#237—91no—
135C2×D3×D4—31no—
136C2×D3×D4—91no—
137C3×#127—31no—
138C3×#127—31no—
139C3×#127—61no—
140C3×#127—91no—
141C3×#127C3×D311yescharacteristic
142C3×#124—91no—
143C3×#124—91no—
144C3×#121—94no—
145C3×#120—94no—
146C3×#117—91no—
147#940—31no—
148#940—91no—
149#939—34no—
150#939—94no—
151#937—34no—
152#937—94no—
153#936—31no—
154#936—91no—
155#875—31no—
156#875—91no—
157#872—31no—
158#872—91no—
159C23×C32—22no—
160C23×C32—22no—
161C23×C32C22×D312yes—
162C23×C32C22×D312yes—
163C23×C32C23×C312yes—
164C23×C32C23×C312yes—
165C22×C3⋊D3—24no—
166C22×C3⋊D3—44no—
167C22×C3⋊D3C23×C312yes—
168C22×C3⋊D3C23×C311yescharacteristic
169C22×C3×D3—31no—
170C22×C3×D3—32no—
171C22×C3×D3—64no—
172C22×C3×D3—124no—
173C32×D4—22no—
174C32×D4—22no—
175C32×D4—22no—
176C32×D4—22no—
177C32×D4—42no—
178C32×D4—42no—
179C32×D4—44no—
180C32×D4—44no—
181C32×D4—48no—
182C2×C32×C4—22no—
183C2×C32×C4—21no—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C23×C312yes—
187C2×C32×C4C23×C311yescharacteristic
188#638—24no—
189#638—24no—
190#638—44no—
191C2×C3⋊Dic3—24no—
192C2×C3⋊Dic3C23×C312yes—
193C2×C3⋊Dic3C23×C312yes—
194C2×C3⋊Dic3C23×C311yescharacteristic
195C4⋊(C3⋊D3)—22no—
196C4×C3⋊D3C3×D414yescomplemented
197#634—22no—
198C3×#67—64no—
199C3×#67—64no—
200C3×#67—124no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—64no—
205C3×D12—62no—
206C3×C4×D3—34no—
207C3×Dic6—62no—
208#535—91noSylow
209C2×H(3,𝔽3)—42no—
210C2×H(3,𝔽3)—42no—
211C2×H(3,𝔽3)C22×D411yescharacteristic
212C2×H(3,𝔽3)H(5,𝔽2)12yes—
213(C3⋊D3)⋊C3—22no—
214(C3⋊D3)⋊C3—42no—
215C24×C3—92no—
216C23×D3—32no—
217C23×D3—92no—
218C3×#41—184no—
219C2×C3×Q8—91no—
220C2×C3×D4—31no—
221C2×C3×D4—31no—
222C2×C3×D4—31no—
223C2×C3×D4—31no—
224C2×C3×D4—34no—
225C2×C3×D4—34no—
226C2×C3×D4—61no—
227C2×C3×D4—61no—
228C2×C3×D4—64no—
229C2×C3×D4—91no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—94no—
233C2×C3×D4—184no—
234C2×C3×D4C2×C3×D311yescharacteristic
235C2×C3×D4C2×C3×D311yescharacteristic
236C2×C3×D4C2×C3×D314yes—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C2×#67—32no—
240C2×#67—32no—
241C2×#67—34no—
242C2×#67—92no—
243C2×#67—92no—
244C2×#67—94no—
245C22×Dic3—32no—
246C22×Dic3—92no—
247#237—64no—
248#237—184no—
249D3×D4—64no—
250D3×D4—184no—
251C2×D12—31no—
252C2×D12—91no—
253C2×C4×D3—31no—
254C2×C4×D3—91no—
255C2×Dic6—31no—
256C2×Dic6—91no—
257C3×C4⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—94no—
260C3×C22⋊C4—94no—
261C3×C42—31no—
262C3×C42—31no—
263C3×C42—61no—
264C3×C42—91no—
265C3×C42C2×C3×D311yescharacteristic
266#217—34no—
267#217—94no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—34no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—94no—
272Dic3⋊C4—32no—
273Dic3⋊C4—92no—
274C4×Dic3—31no—
275C4×Dic3—91no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—24no—
279C22×C32—24no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—21no—
285C22×C32—42no—
286C22×C32—42no—
287C22×C32—44no—
288C22×C32—44no—
289C22×C32C23×D311yescharacteristic
290C22×C32C24×C311yescharacteristic
291C2×C3⋊D3—22no—
292C2×C3⋊D3—22no—
293C2×C3⋊D3—24no—
294C2×C3⋊D3—48no—
295C2×C3⋊D3C2×C3×D412yes—
296C2×C3×D3—32no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—64no—
300C2×C3×D3—128no—
301C32×C4—24no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4D3×D412yes—
306C32×C4C2×C3×D412yes—
307C3⋊Dic3—22no—
308C3⋊Dic3—22no—
309C3⋊Dic3C2×C3×D412yes—
310C3×Dic3—32no—
311C3×Dic3—62no—
312C3×Dic3—62no—
313C2×#41—91no—
314C22×D4—91no—
315#127—91no—
316#127(C3⋊D3)⋊C311yescharacteristic, complemented
317#124—91no—
318#124—91no—
319#121—94no—
320#120—94no—
321#117—91no—
322H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
323C23×C3—32no—
324C23×C3—32no—
325C23×C3—32no—
326C23×C3—32no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—91no—
330C23×C3—92no—
331C23×C3—184no—
332C23×C3—364no—
333C23×C3C22×C3×D312yes—
334C23×C3C22×C3×D312yes—
335C22×D3—31no—
336C22×D3—32no—
337C22×D3—64no—
338C22×D3—91no—
339C22×D3—92no—
340C22×D3—124no—
341C22×D3—184no—
342C22×D3—364no—
343C3×Q8—182no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—44no—
347C3×D4—62no—
348C3×D4—62no—
349C3×D4—62no—
350C3×D4—62no—
351C3×D4—122no—
352C3×D4—122no—
353C3×D4—124no—
354C3×D4—124no—
355C3×D4—128no—
356C3×D4—182no—
357C3×D4—184no—
358C3×D4—184no—
359C3×D4—364no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—32no—
363C2×C3×C4—32no—
364C2×C3×C4—62no—
365C2×C3×C4—61no—
366C2×C3×C4—94no—
367C2×C3×C4—91no—
368C2×C3×C4—92no—
369C2×C3×C4—92no—
370C2×C3×C4—184no—
371C2×C3×C4C22×C3×D312yes—
372C2×C3×C4C22×C3×D311yescharacteristic
373#67—64no—
374#67—64no—
375#67—124no—
376#67—184no—
377#67—184no—
378#67—364no—
379C2×Dic3—31no—
380C2×Dic3—32no—
381C2×Dic3—32no—
382C2×Dic3—64no—
383C2×Dic3—91no—
384C2×Dic3—92no—
385C2×Dic3—92no—
386C2×Dic3—184no—
387D12—62no—
388D12—182no—
389C4×D3—34no—
390C4×D3—94no—
391Dic6—62no—
392Dic6—182no—
393C2×C32—21no—
394C2×C32—22no—
395C2×C32—42no—
396C2×C32—42no—
397C2×C32—42no—
398C2×C32—42no—
399C2×C32—82no—
400C2×C32—82no—
401C2×C32C2×D3×D411yescharacteristic
402C2×C32#100412yes—
403C2×C32C22×C3×D411yescharacteristic
404C2×C32C3×H(5,𝔽2)12yes—
405C3⋊D3—22no—
406C3⋊D3—42no—
407C3×D3—62no—
408C3×D3—122no—
409C24—92no—
410#41—184no—
411C2×Q8—91no—
412C2×D4—92no—
413C2×D4—92no—
414C2×D4—94no—
415C2×D4—91no—
416C2×D4—184no—
417C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
418C2×D4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
419C2×D4C2×(C3⋊D3)⋊C314yescomplemented
420C22×C4—91no—
421C22×C4—92no—
422C4⋊C4—92no—
423C22⋊C4—94no—
424C22⋊C4—92no—
425C22⋊C4—94no—
426C42—91no—
427C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—24no—
431C22×C3—24no—
432C22×C3—31no—
433C22×C3—31no—
434C22×C3—62no—
435C22×C3—62no—
436C22×C3—64no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3—64no—
440C22×C3—64no—
441C22×C3—64no—
442C22×C3—61no—
443C22×C3—92no—
444C22×C3—122no—
445C22×C3—122no—
446C22×C3—124no—
447C22×C3—124no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—184no—
451C22×C3—368no—
452C22×C3C23×C3×D311yescharacteristic
453C2×D3—32no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—64no—
457C2×D3—92no—
458C2×D3—128no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—184no—
462C2×D3—368no—
463C3×C4—22no—
464C3×C4—32no—
465C3×C4—32no—
466C3×C4—62no—
467C3×C4—62no—
468C3×C4—64no—
469C3×C4—62no—
470C3×C4—92no—
471C3×C4—182no—
472C3×C4—182no—
473C3×C4C3×D3×D412yes—
474Dic3—32no—
475Dic3—62no—
476Dic3—62no—
477Dic3—92no—
478Dic3—182no—
479Dic3—182no—
480C32—21no—
481C32#571511yescharacteristic, complemented
482C32C3×#53511yescharacteristic, complemented
483C23—91no—
484C23—92no—
485C23—184no—
486C23—364no—
487C23C22×(C3⋊D3)⋊C312yes—
488C23C22×(C3⋊D3)⋊C312yes—
489Q8—182no—
490D4—22no—
491D4—22no—
492D4—44no—
493D4—184no—
494D4—184no—
495D4—182no—
496D4—364no—
497C2×C4—91no—
498C2×C4—94no—
499C2×C4—92no—
500C2×C4—92no—
501C2×C4—184no—
502C2×C4C22×(C3⋊D3)⋊C312yes—
503C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—32no—
507C2×C3—32no—
508C2×C3—42no—
509C2×C3—42no—
510C2×C3—61no—
511C2×C3—62no—
512C2×C3—122no—
513C2×C3—122no—
514C2×C3—122no—
515C2×C3—122no—
516C2×C3—182no—
517C2×C3—242no—
518C2×C3—242no—
519C2×C3—362no—
520C2×C3C2×C3×D3×D411yescharacteristic
521C2×C3C3×#100412yes—
522D3—62no—
523D3—122no—
524D3—182no—
525D3—362no—
526C22—22no—
527C22—22no—
528C22—24no—
529C22—24no—
530C22—92no—
531C22—182no—
532C22—182no—
533C22—184no—
534C22—368no—
535C22C23×(C3⋊D3)⋊C311yescharacteristic
536C4—22no—
537C4—92no—
538C4—182no—
539C4—182no—
540C4D4×(C3⋊D3)⋊C312yes—
541C3—31no—
542C3—31no—
543C3—61no—
544C3C3×#571511yescharacteristic
545C2—42no—
546C2—42no—
547C2—182no—
548C2—362no—
549C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
550C2#64361⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#127C2
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#3830⋊C3C2semidirect
2. C2×D4×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#127C2semidirect
4. #64149⋊C3C2semidirect
5. #64148⋊C3C2semidirect
6. #64146⋊C3C2semidirect
7. #64145⋊C3C2semidirect
8. #64084⋊C3C2semidirect
9. #64081⋊C3C2semidirect
10. #10596186C3semidirect
14. C2×D4×H(3,𝔽3)C22semidirect
15. C2×D4×H(3,𝔽3)C22semidirect
18. C2×#638⋊C3C22semidirect
19. C22×(C3⋊Dic3)⋊C3C22semidirect
24. C2×#634⋊C3C22semidirect
25. C42×H(3,𝔽3)C22semidirect
26. #3757⋊C3C22semidirect
28. #3752⋊C3C22semidirect
29. #3750⋊C3C22semidirect
30. C4×(C3⋊Dic3)⋊C3C22semidirect
31. C2×#3830C2×C3semidirect
32. C2×D4×C3⋊D3C2×C3semidirect
36. C32×#127D3semidirect
37. C32×#127C2×C3semidirect
38. #64149C2×C3semidirect
39. #64148C2×C3semidirect
40. #64146C2×C3semidirect
41. #64145C2×C3semidirect
42. #64084C2×C3semidirect
43. #64081C2×C3semidirect
69. C4×(C3⋊D3)⋊C3D4semidirect
80. C2×C32×D4C2×D3semidirect
81. C2×C32×D4C2×D3semidirect
83. C2×C32×D4C22×C3semidirect
84. C2×C32×D4C22×C3semidirect
87. C2×#638C22×C3semidirect
88. C22×C3⋊Dic3C22×C3semidirect
93. C2×#634C22×C3semidirect
104. C32×C42C2×D3semidirect
105. C32×C42C22×C3semidirect
106. #3757C22×C3semidirect
108. #3752C22×C3semidirect
109. #3750C22×C3semidirect
110. C4×C3⋊Dic3C22×C3semidirect
196. C4×C3⋊D3C3×D4semidirect
316. #127(C3⋊D3)⋊C3semidirect
322. H(3,𝔽3)#535semidirect
418. C2×D4C2×(C3⋊D3)⋊C3semidirect
419. C2×D4C2×(C3⋊D3)⋊C3semidirect
427. C42C2×(C3⋊D3)⋊C3semidirect
481. C32#5715semidirect
482. C32C3×#535semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
C2×D4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1272
#64149⋊C32
#64148⋊C32
#64146⋊C32
#64145⋊C32
#64084⋊C32
#64081⋊C32
#105961863
C3×#57153